3.6.56 \(\int \frac {1}{(d+e x)^{3/2} (a+c x^2)^2} \, dx\)

Optimal. Leaf size=845 \[ \frac {e \left (c d^2-5 a e^2\right )}{2 a \left (c d^2+a e^2\right )^2 \sqrt {d+e x}}+\frac {\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt {c} e^2 d+\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{4 \sqrt {2} a \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt {c} e^2 d+\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{4 \sqrt {2} a \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt {c} e^2 d-\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c} (d+e x)-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c d^2+a e^2}\right )}{8 \sqrt {2} a \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt {c} e^2 d-\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c} (d+e x)+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c d^2+a e^2}\right )}{8 \sqrt {2} a \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) \sqrt {d+e x} \left (c x^2+a\right )} \]

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Rubi [A]  time = 3.34, antiderivative size = 845, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {741, 829, 827, 1169, 634, 618, 206, 628} \begin {gather*} \frac {e \left (c d^2-5 a e^2\right )}{2 a \left (c d^2+a e^2\right )^2 \sqrt {d+e x}}+\frac {\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt {c} e^2 d+\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{4 \sqrt {2} a \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt {c} e^2 d+\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{4 \sqrt {2} a \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt {c} e^2 d-\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c} (d+e x)-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c d^2+a e^2}\right )}{8 \sqrt {2} a \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt {c} e^2 d-\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c} (d+e x)+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c d^2+a e^2}\right )}{8 \sqrt {2} a \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) \sqrt {d+e x} \left (c x^2+a\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d + e*x)^(3/2)*(a + c*x^2)^2),x]

[Out]

(e*(c*d^2 - 5*a*e^2))/(2*a*(c*d^2 + a*e^2)^2*Sqrt[d + e*x]) + (a*e + c*d*x)/(2*a*(c*d^2 + a*e^2)*Sqrt[d + e*x]
*(a + c*x^2)) + (c^(1/4)*e*(c^(3/2)*d^3 + 13*a*Sqrt[c]*d*e^2 + (c*d^2 - 5*a*e^2)*Sqrt[c*d^2 + a*e^2])*ArcTanh[
(Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]] - Sqrt[2]*c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]]
)/(4*Sqrt[2]*a*(c*d^2 + a*e^2)^(5/2)*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (c^(1/4)*e*(c^(3/2)*d^3 + 13*a*S
qrt[c]*d*e^2 + (c*d^2 - 5*a*e^2)*Sqrt[c*d^2 + a*e^2])*ArcTanh[(Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]] + Sqrt[2]
*c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]])/(4*Sqrt[2]*a*(c*d^2 + a*e^2)^(5/2)*Sqrt[Sqrt[c
]*d - Sqrt[c*d^2 + a*e^2]]) - (c^(1/4)*e*(c^(3/2)*d^3 + 13*a*Sqrt[c]*d*e^2 - (c*d^2 - 5*a*e^2)*Sqrt[c*d^2 + a*
e^2])*Log[Sqrt[c*d^2 + a*e^2] - Sqrt[2]*c^(1/4)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]*Sqrt[d + e*x] + Sqrt[c]*
(d + e*x)])/(8*Sqrt[2]*a*(c*d^2 + a*e^2)^(5/2)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]) + (c^(1/4)*e*(c^(3/2)*d^
3 + 13*a*Sqrt[c]*d*e^2 - (c*d^2 - 5*a*e^2)*Sqrt[c*d^2 + a*e^2])*Log[Sqrt[c*d^2 + a*e^2] + Sqrt[2]*c^(1/4)*Sqrt
[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]*Sqrt[d + e*x] + Sqrt[c]*(d + e*x)])/(8*Sqrt[2]*a*(c*d^2 + a*e^2)^(5/2)*Sqrt[
Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]])

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 741

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(m + 1)*(a*e + c*d*x)*(
a + c*x^2)^(p + 1))/(2*a*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[1/(2*a*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*
Simp[c*d^2*(2*p + 3) + a*e^2*(m + 2*p + 3) + c*e*d*(m + 2*p + 4)*x, x]*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a
, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]

Rule 827

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2, Subst[Int[(e*f
- d*g + g*x^2)/(c*d^2 + a*e^2 - 2*c*d*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0]

Rule 829

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Simp[((e*f - d*g)*(d
+ e*x)^(m + 1))/((m + 1)*(c*d^2 + a*e^2)), x] + Dist[1/(c*d^2 + a*e^2), Int[((d + e*x)^(m + 1)*Simp[c*d*f + a*
e*g - c*(e*f - d*g)*x, x])/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2, 0] &&
FractionQ[m] && LtQ[m, -1]

Rule 1169

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r =
Rt[2*q - b/c, 2]}, Dist[1/(2*c*q*r), Int[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Dist[1/(2*c*q*r), Int[(
d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]

Rubi steps

\begin {align*} \int \frac {1}{(d+e x)^{3/2} \left (a+c x^2\right )^2} \, dx &=\frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) \sqrt {d+e x} \left (a+c x^2\right )}-\frac {\int \frac {\frac {1}{2} \left (-2 c d^2-5 a e^2\right )-\frac {3}{2} c d e x}{(d+e x)^{3/2} \left (a+c x^2\right )} \, dx}{2 a \left (c d^2+a e^2\right )}\\ &=\frac {e \left (c d^2-5 a e^2\right )}{2 a \left (c d^2+a e^2\right )^2 \sqrt {d+e x}}+\frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) \sqrt {d+e x} \left (a+c x^2\right )}-\frac {\int \frac {-c d \left (c d^2+4 a e^2\right )-\frac {1}{2} c e \left (c d^2-5 a e^2\right ) x}{\sqrt {d+e x} \left (a+c x^2\right )} \, dx}{2 a \left (c d^2+a e^2\right )^2}\\ &=\frac {e \left (c d^2-5 a e^2\right )}{2 a \left (c d^2+a e^2\right )^2 \sqrt {d+e x}}+\frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) \sqrt {d+e x} \left (a+c x^2\right )}-\frac {\operatorname {Subst}\left (\int \frac {\frac {1}{2} c d e \left (c d^2-5 a e^2\right )-c d e \left (c d^2+4 a e^2\right )-\frac {1}{2} c e \left (c d^2-5 a e^2\right ) x^2}{c d^2+a e^2-2 c d x^2+c x^4} \, dx,x,\sqrt {d+e x}\right )}{a \left (c d^2+a e^2\right )^2}\\ &=\frac {e \left (c d^2-5 a e^2\right )}{2 a \left (c d^2+a e^2\right )^2 \sqrt {d+e x}}+\frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) \sqrt {d+e x} \left (a+c x^2\right )}-\frac {\operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (\frac {1}{2} c d e \left (c d^2-5 a e^2\right )-c d e \left (c d^2+4 a e^2\right )\right )}{\sqrt [4]{c}}-\left (\frac {1}{2} c d e \left (c d^2-5 a e^2\right )+\frac {1}{2} \sqrt {c} e \left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}-c d e \left (c d^2+4 a e^2\right )\right ) x}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{2 \sqrt {2} a \sqrt [4]{c} \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}-\frac {\operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \left (\frac {1}{2} c d e \left (c d^2-5 a e^2\right )-c d e \left (c d^2+4 a e^2\right )\right )}{\sqrt [4]{c}}+\left (\frac {1}{2} c d e \left (c d^2-5 a e^2\right )+\frac {1}{2} \sqrt {c} e \left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}-c d e \left (c d^2+4 a e^2\right )\right ) x}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{2 \sqrt {2} a \sqrt [4]{c} \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\\ &=\frac {e \left (c d^2-5 a e^2\right )}{2 a \left (c d^2+a e^2\right )^2 \sqrt {d+e x}}+\frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) \sqrt {d+e x} \left (a+c x^2\right )}-\frac {\left (\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt {c} d e^2-\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right )\right ) \operatorname {Subst}\left (\int \frac {-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 x}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{8 \sqrt {2} a \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\left (\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt {c} d e^2-\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right )\right ) \operatorname {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 x}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{8 \sqrt {2} a \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\left (e \left (c^{3/2} d^3+13 a \sqrt {c} d e^2+\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{8 a \left (c d^2+a e^2\right )^{5/2}}+\frac {\left (e \left (c^{3/2} d^3+13 a \sqrt {c} d e^2+\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{8 a \left (c d^2+a e^2\right )^{5/2}}\\ &=\frac {e \left (c d^2-5 a e^2\right )}{2 a \left (c d^2+a e^2\right )^2 \sqrt {d+e x}}+\frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) \sqrt {d+e x} \left (a+c x^2\right )}-\frac {\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt {c} d e^2-\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{8 \sqrt {2} a \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt {c} d e^2-\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{8 \sqrt {2} a \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}-\frac {\left (e \left (c^{3/2} d^3+13 a \sqrt {c} d e^2+\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{2 \left (d-\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}\right )-x^2} \, dx,x,-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 \sqrt {d+e x}\right )}{4 a \left (c d^2+a e^2\right )^{5/2}}-\frac {\left (e \left (c^{3/2} d^3+13 a \sqrt {c} d e^2+\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{2 \left (d-\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}\right )-x^2} \, dx,x,\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 \sqrt {d+e x}\right )}{4 a \left (c d^2+a e^2\right )^{5/2}}\\ &=\frac {e \left (c d^2-5 a e^2\right )}{2 a \left (c d^2+a e^2\right )^2 \sqrt {d+e x}}+\frac {a e+c d x}{2 a \left (c d^2+a e^2\right ) \sqrt {d+e x} \left (a+c x^2\right )}+\frac {\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt {c} d e^2+\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}-\sqrt {2} \sqrt {d+e x}\right )}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{4 \sqrt {2} a \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt {c} d e^2+\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+\sqrt {2} \sqrt {d+e x}\right )}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{4 \sqrt {2} a \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt {c} d e^2-\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{8 \sqrt {2} a \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\sqrt [4]{c} e \left (c^{3/2} d^3+13 a \sqrt {c} d e^2-\left (c d^2-5 a e^2\right ) \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{8 \sqrt {2} a \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\\ \end {align*}

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Mathematica [C]  time = 0.50, size = 346, normalized size = 0.41 \begin {gather*} \frac {\frac {3 c^{3/4} d \left (\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {-a} e}}\right )}{\sqrt {\sqrt {c} d-\sqrt {-a} e}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {-a} e+\sqrt {c} d}}\right )}{\sqrt {\sqrt {-a} e+\sqrt {c} d}}\right )}{2 \sqrt {-a}}-\frac {\left (5 a e^2-c d^2\right ) \left (\left (a e-\sqrt {-a} \sqrt {c} d\right ) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )+\left (\sqrt {-a} \sqrt {c} d+a e\right ) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )\right )}{2 a \sqrt {d+e x} \left (a e^2+c d^2\right )}+\frac {a e+c d x}{\left (a+c x^2\right ) \sqrt {d+e x}}}{2 a \left (a e^2+c d^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d + e*x)^(3/2)*(a + c*x^2)^2),x]

[Out]

((a*e + c*d*x)/(Sqrt[d + e*x]*(a + c*x^2)) + (3*c^(3/4)*d*(ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sq
rt[-a]*e]]/Sqrt[Sqrt[c]*d - Sqrt[-a]*e] - ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[-a]*e]]/Sqrt[S
qrt[c]*d + Sqrt[-a]*e]))/(2*Sqrt[-a]) - ((-(c*d^2) + 5*a*e^2)*((-(Sqrt[-a]*Sqrt[c]*d) + a*e)*Hypergeometric2F1
[-1/2, 1, 1/2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - Sqrt[-a]*e)] + (Sqrt[-a]*Sqrt[c]*d + a*e)*Hypergeometric2F1[-1
/2, 1, 1/2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]))/(2*a*(c*d^2 + a*e^2)*Sqrt[d + e*x]))/(2*a*(c*d^2 +
a*e^2))

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IntegrateAlgebraic [C]  time = 1.37, size = 409, normalized size = 0.48 \begin {gather*} \frac {i \left (2 c d+5 i \sqrt {a} \sqrt {c} e\right ) \tan ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {-c d-i \sqrt {a} \sqrt {c} e}}{\sqrt {c} d+i \sqrt {a} e}\right )}{4 a^{3/2} \left (\sqrt {c} d+i \sqrt {a} e\right )^2 \sqrt {-i \sqrt {c} \left (\sqrt {a} e-i \sqrt {c} d\right )}}-\frac {i \left (2 c d-5 i \sqrt {a} \sqrt {c} e\right ) \tan ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {-c d+i \sqrt {a} \sqrt {c} e}}{\sqrt {c} d-i \sqrt {a} e}\right )}{4 a^{3/2} \left (\sqrt {c} d-i \sqrt {a} e\right )^2 \sqrt {i \sqrt {c} \left (\sqrt {a} e+i \sqrt {c} d\right )}}-\frac {e \left (4 a^2 e^4+4 a c d^2 e^2+5 a c e^2 (d+e x)^2-11 a c d e^2 (d+e x)+c^2 d^3 (d+e x)-c^2 d^2 (d+e x)^2\right )}{2 a \sqrt {d+e x} \left (a e^2+c d^2\right )^2 \left (a e^2+c d^2-2 c d (d+e x)+c (d+e x)^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/((d + e*x)^(3/2)*(a + c*x^2)^2),x]

[Out]

-1/2*(e*(4*a*c*d^2*e^2 + 4*a^2*e^4 + c^2*d^3*(d + e*x) - 11*a*c*d*e^2*(d + e*x) - c^2*d^2*(d + e*x)^2 + 5*a*c*
e^2*(d + e*x)^2))/(a*(c*d^2 + a*e^2)^2*Sqrt[d + e*x]*(c*d^2 + a*e^2 - 2*c*d*(d + e*x) + c*(d + e*x)^2)) + ((I/
4)*(2*c*d + (5*I)*Sqrt[a]*Sqrt[c]*e)*ArcTan[(Sqrt[-(c*d) - I*Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d + I*
Sqrt[a]*e)])/(a^(3/2)*(Sqrt[c]*d + I*Sqrt[a]*e)^2*Sqrt[(-I)*Sqrt[c]*((-I)*Sqrt[c]*d + Sqrt[a]*e)]) - ((I/4)*(2
*c*d - (5*I)*Sqrt[a]*Sqrt[c]*e)*ArcTan[(Sqrt[-(c*d) + I*Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d - I*Sqrt[
a]*e)])/(a^(3/2)*(Sqrt[c]*d - I*Sqrt[a]*e)^2*Sqrt[I*Sqrt[c]*(I*Sqrt[c]*d + Sqrt[a]*e)])

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fricas [B]  time = 1.05, size = 5698, normalized size = 6.74

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(3/2)/(c*x^2+a)^2,x, algorithm="fricas")

[Out]

-1/8*((a^2*c^2*d^5 + 2*a^3*c*d^3*e^2 + a^4*d*e^4 + (a*c^3*d^4*e + 2*a^2*c^2*d^2*e^3 + a^3*c*e^5)*x^3 + (a*c^3*
d^5 + 2*a^2*c^2*d^3*e^2 + a^3*c*d*e^4)*x^2 + (a^2*c^2*d^4*e + 2*a^3*c*d^2*e^3 + a^4*e^5)*x)*sqrt(-(4*c^4*d^7 +
35*a*c^3*d^5*e^2 + 70*a^2*c^2*d^3*e^4 - 105*a^3*c*d*e^6 + (a^3*c^5*d^10 + 5*a^4*c^4*d^8*e^2 + 10*a^5*c^3*d^6*
e^4 + 10*a^6*c^2*d^4*e^6 + 5*a^7*c*d^2*e^8 + a^8*e^10)*sqrt(-(1225*c^5*d^8*e^6 + 10780*a*c^4*d^6*e^8 + 21966*a
^2*c^3*d^4*e^10 - 7700*a^3*c^2*d^2*e^12 + 625*a^4*c*e^14)/(a^3*c^10*d^20 + 10*a^4*c^9*d^18*e^2 + 45*a^5*c^8*d^
16*e^4 + 120*a^6*c^7*d^14*e^6 + 210*a^7*c^6*d^12*e^8 + 252*a^8*c^5*d^10*e^10 + 210*a^9*c^4*d^8*e^12 + 120*a^10
*c^3*d^6*e^14 + 45*a^11*c^2*d^4*e^16 + 10*a^12*c*d^2*e^18 + a^13*e^20)))/(a^3*c^5*d^10 + 5*a^4*c^4*d^8*e^2 + 1
0*a^5*c^3*d^6*e^4 + 10*a^6*c^2*d^4*e^6 + 5*a^7*c*d^2*e^8 + a^8*e^10))*log(-(140*c^4*d^6*e^3 + 1491*a*c^3*d^4*e
^5 + 3750*a^2*c^2*d^2*e^7 - 625*a^3*c*e^9)*sqrt(e*x + d) + (35*a^2*c^4*d^7*e^4 + 609*a^3*c^3*d^5*e^6 + 1977*a^
4*c^2*d^3*e^8 - 325*a^5*c*d*e^10 + (2*a^3*c^7*d^14 + 19*a^4*c^6*d^12*e^2 + 60*a^5*c^5*d^10*e^4 + 85*a^6*c^4*d^
8*e^6 + 50*a^7*c^3*d^6*e^8 - 3*a^8*c^2*d^4*e^10 - 16*a^9*c*d^2*e^12 - 5*a^10*e^14)*sqrt(-(1225*c^5*d^8*e^6 + 1
0780*a*c^4*d^6*e^8 + 21966*a^2*c^3*d^4*e^10 - 7700*a^3*c^2*d^2*e^12 + 625*a^4*c*e^14)/(a^3*c^10*d^20 + 10*a^4*
c^9*d^18*e^2 + 45*a^5*c^8*d^16*e^4 + 120*a^6*c^7*d^14*e^6 + 210*a^7*c^6*d^12*e^8 + 252*a^8*c^5*d^10*e^10 + 210
*a^9*c^4*d^8*e^12 + 120*a^10*c^3*d^6*e^14 + 45*a^11*c^2*d^4*e^16 + 10*a^12*c*d^2*e^18 + a^13*e^20)))*sqrt(-(4*
c^4*d^7 + 35*a*c^3*d^5*e^2 + 70*a^2*c^2*d^3*e^4 - 105*a^3*c*d*e^6 + (a^3*c^5*d^10 + 5*a^4*c^4*d^8*e^2 + 10*a^5
*c^3*d^6*e^4 + 10*a^6*c^2*d^4*e^6 + 5*a^7*c*d^2*e^8 + a^8*e^10)*sqrt(-(1225*c^5*d^8*e^6 + 10780*a*c^4*d^6*e^8
+ 21966*a^2*c^3*d^4*e^10 - 7700*a^3*c^2*d^2*e^12 + 625*a^4*c*e^14)/(a^3*c^10*d^20 + 10*a^4*c^9*d^18*e^2 + 45*a
^5*c^8*d^16*e^4 + 120*a^6*c^7*d^14*e^6 + 210*a^7*c^6*d^12*e^8 + 252*a^8*c^5*d^10*e^10 + 210*a^9*c^4*d^8*e^12 +
120*a^10*c^3*d^6*e^14 + 45*a^11*c^2*d^4*e^16 + 10*a^12*c*d^2*e^18 + a^13*e^20)))/(a^3*c^5*d^10 + 5*a^4*c^4*d^
8*e^2 + 10*a^5*c^3*d^6*e^4 + 10*a^6*c^2*d^4*e^6 + 5*a^7*c*d^2*e^8 + a^8*e^10))) - (a^2*c^2*d^5 + 2*a^3*c*d^3*e
^2 + a^4*d*e^4 + (a*c^3*d^4*e + 2*a^2*c^2*d^2*e^3 + a^3*c*e^5)*x^3 + (a*c^3*d^5 + 2*a^2*c^2*d^3*e^2 + a^3*c*d*
e^4)*x^2 + (a^2*c^2*d^4*e + 2*a^3*c*d^2*e^3 + a^4*e^5)*x)*sqrt(-(4*c^4*d^7 + 35*a*c^3*d^5*e^2 + 70*a^2*c^2*d^3
*e^4 - 105*a^3*c*d*e^6 + (a^3*c^5*d^10 + 5*a^4*c^4*d^8*e^2 + 10*a^5*c^3*d^6*e^4 + 10*a^6*c^2*d^4*e^6 + 5*a^7*c
*d^2*e^8 + a^8*e^10)*sqrt(-(1225*c^5*d^8*e^6 + 10780*a*c^4*d^6*e^8 + 21966*a^2*c^3*d^4*e^10 - 7700*a^3*c^2*d^2
*e^12 + 625*a^4*c*e^14)/(a^3*c^10*d^20 + 10*a^4*c^9*d^18*e^2 + 45*a^5*c^8*d^16*e^4 + 120*a^6*c^7*d^14*e^6 + 21
0*a^7*c^6*d^12*e^8 + 252*a^8*c^5*d^10*e^10 + 210*a^9*c^4*d^8*e^12 + 120*a^10*c^3*d^6*e^14 + 45*a^11*c^2*d^4*e^
16 + 10*a^12*c*d^2*e^18 + a^13*e^20)))/(a^3*c^5*d^10 + 5*a^4*c^4*d^8*e^2 + 10*a^5*c^3*d^6*e^4 + 10*a^6*c^2*d^4
*e^6 + 5*a^7*c*d^2*e^8 + a^8*e^10))*log(-(140*c^4*d^6*e^3 + 1491*a*c^3*d^4*e^5 + 3750*a^2*c^2*d^2*e^7 - 625*a^
3*c*e^9)*sqrt(e*x + d) - (35*a^2*c^4*d^7*e^4 + 609*a^3*c^3*d^5*e^6 + 1977*a^4*c^2*d^3*e^8 - 325*a^5*c*d*e^10 +
(2*a^3*c^7*d^14 + 19*a^4*c^6*d^12*e^2 + 60*a^5*c^5*d^10*e^4 + 85*a^6*c^4*d^8*e^6 + 50*a^7*c^3*d^6*e^8 - 3*a^8
*c^2*d^4*e^10 - 16*a^9*c*d^2*e^12 - 5*a^10*e^14)*sqrt(-(1225*c^5*d^8*e^6 + 10780*a*c^4*d^6*e^8 + 21966*a^2*c^3
*d^4*e^10 - 7700*a^3*c^2*d^2*e^12 + 625*a^4*c*e^14)/(a^3*c^10*d^20 + 10*a^4*c^9*d^18*e^2 + 45*a^5*c^8*d^16*e^4
+ 120*a^6*c^7*d^14*e^6 + 210*a^7*c^6*d^12*e^8 + 252*a^8*c^5*d^10*e^10 + 210*a^9*c^4*d^8*e^12 + 120*a^10*c^3*d
^6*e^14 + 45*a^11*c^2*d^4*e^16 + 10*a^12*c*d^2*e^18 + a^13*e^20)))*sqrt(-(4*c^4*d^7 + 35*a*c^3*d^5*e^2 + 70*a^
2*c^2*d^3*e^4 - 105*a^3*c*d*e^6 + (a^3*c^5*d^10 + 5*a^4*c^4*d^8*e^2 + 10*a^5*c^3*d^6*e^4 + 10*a^6*c^2*d^4*e^6
+ 5*a^7*c*d^2*e^8 + a^8*e^10)*sqrt(-(1225*c^5*d^8*e^6 + 10780*a*c^4*d^6*e^8 + 21966*a^2*c^3*d^4*e^10 - 7700*a^
3*c^2*d^2*e^12 + 625*a^4*c*e^14)/(a^3*c^10*d^20 + 10*a^4*c^9*d^18*e^2 + 45*a^5*c^8*d^16*e^4 + 120*a^6*c^7*d^14
*e^6 + 210*a^7*c^6*d^12*e^8 + 252*a^8*c^5*d^10*e^10 + 210*a^9*c^4*d^8*e^12 + 120*a^10*c^3*d^6*e^14 + 45*a^11*c
^2*d^4*e^16 + 10*a^12*c*d^2*e^18 + a^13*e^20)))/(a^3*c^5*d^10 + 5*a^4*c^4*d^8*e^2 + 10*a^5*c^3*d^6*e^4 + 10*a^
6*c^2*d^4*e^6 + 5*a^7*c*d^2*e^8 + a^8*e^10))) + (a^2*c^2*d^5 + 2*a^3*c*d^3*e^2 + a^4*d*e^4 + (a*c^3*d^4*e + 2*
a^2*c^2*d^2*e^3 + a^3*c*e^5)*x^3 + (a*c^3*d^5 + 2*a^2*c^2*d^3*e^2 + a^3*c*d*e^4)*x^2 + (a^2*c^2*d^4*e + 2*a^3*
c*d^2*e^3 + a^4*e^5)*x)*sqrt(-(4*c^4*d^7 + 35*a*c^3*d^5*e^2 + 70*a^2*c^2*d^3*e^4 - 105*a^3*c*d*e^6 - (a^3*c^5*
d^10 + 5*a^4*c^4*d^8*e^2 + 10*a^5*c^3*d^6*e^4 + 10*a^6*c^2*d^4*e^6 + 5*a^7*c*d^2*e^8 + a^8*e^10)*sqrt(-(1225*c
^5*d^8*e^6 + 10780*a*c^4*d^6*e^8 + 21966*a^2*c^3*d^4*e^10 - 7700*a^3*c^2*d^2*e^12 + 625*a^4*c*e^14)/(a^3*c^10*
d^20 + 10*a^4*c^9*d^18*e^2 + 45*a^5*c^8*d^16*e^4 + 120*a^6*c^7*d^14*e^6 + 210*a^7*c^6*d^12*e^8 + 252*a^8*c^5*d
^10*e^10 + 210*a^9*c^4*d^8*e^12 + 120*a^10*c^3*d^6*e^14 + 45*a^11*c^2*d^4*e^16 + 10*a^12*c*d^2*e^18 + a^13*e^2
0)))/(a^3*c^5*d^10 + 5*a^4*c^4*d^8*e^2 + 10*a^5*c^3*d^6*e^4 + 10*a^6*c^2*d^4*e^6 + 5*a^7*c*d^2*e^8 + a^8*e^10)
)*log(-(140*c^4*d^6*e^3 + 1491*a*c^3*d^4*e^5 + 3750*a^2*c^2*d^2*e^7 - 625*a^3*c*e^9)*sqrt(e*x + d) + (35*a^2*c
^4*d^7*e^4 + 609*a^3*c^3*d^5*e^6 + 1977*a^4*c^2*d^3*e^8 - 325*a^5*c*d*e^10 - (2*a^3*c^7*d^14 + 19*a^4*c^6*d^12
*e^2 + 60*a^5*c^5*d^10*e^4 + 85*a^6*c^4*d^8*e^6 + 50*a^7*c^3*d^6*e^8 - 3*a^8*c^2*d^4*e^10 - 16*a^9*c*d^2*e^12
- 5*a^10*e^14)*sqrt(-(1225*c^5*d^8*e^6 + 10780*a*c^4*d^6*e^8 + 21966*a^2*c^3*d^4*e^10 - 7700*a^3*c^2*d^2*e^12
+ 625*a^4*c*e^14)/(a^3*c^10*d^20 + 10*a^4*c^9*d^18*e^2 + 45*a^5*c^8*d^16*e^4 + 120*a^6*c^7*d^14*e^6 + 210*a^7*
c^6*d^12*e^8 + 252*a^8*c^5*d^10*e^10 + 210*a^9*c^4*d^8*e^12 + 120*a^10*c^3*d^6*e^14 + 45*a^11*c^2*d^4*e^16 + 1
0*a^12*c*d^2*e^18 + a^13*e^20)))*sqrt(-(4*c^4*d^7 + 35*a*c^3*d^5*e^2 + 70*a^2*c^2*d^3*e^4 - 105*a^3*c*d*e^6 -
(a^3*c^5*d^10 + 5*a^4*c^4*d^8*e^2 + 10*a^5*c^3*d^6*e^4 + 10*a^6*c^2*d^4*e^6 + 5*a^7*c*d^2*e^8 + a^8*e^10)*sqrt
(-(1225*c^5*d^8*e^6 + 10780*a*c^4*d^6*e^8 + 21966*a^2*c^3*d^4*e^10 - 7700*a^3*c^2*d^2*e^12 + 625*a^4*c*e^14)/(
a^3*c^10*d^20 + 10*a^4*c^9*d^18*e^2 + 45*a^5*c^8*d^16*e^4 + 120*a^6*c^7*d^14*e^6 + 210*a^7*c^6*d^12*e^8 + 252*
a^8*c^5*d^10*e^10 + 210*a^9*c^4*d^8*e^12 + 120*a^10*c^3*d^6*e^14 + 45*a^11*c^2*d^4*e^16 + 10*a^12*c*d^2*e^18 +
a^13*e^20)))/(a^3*c^5*d^10 + 5*a^4*c^4*d^8*e^2 + 10*a^5*c^3*d^6*e^4 + 10*a^6*c^2*d^4*e^6 + 5*a^7*c*d^2*e^8 +
a^8*e^10))) - (a^2*c^2*d^5 + 2*a^3*c*d^3*e^2 + a^4*d*e^4 + (a*c^3*d^4*e + 2*a^2*c^2*d^2*e^3 + a^3*c*e^5)*x^3 +
(a*c^3*d^5 + 2*a^2*c^2*d^3*e^2 + a^3*c*d*e^4)*x^2 + (a^2*c^2*d^4*e + 2*a^3*c*d^2*e^3 + a^4*e^5)*x)*sqrt(-(4*c
^4*d^7 + 35*a*c^3*d^5*e^2 + 70*a^2*c^2*d^3*e^4 - 105*a^3*c*d*e^6 - (a^3*c^5*d^10 + 5*a^4*c^4*d^8*e^2 + 10*a^5*
c^3*d^6*e^4 + 10*a^6*c^2*d^4*e^6 + 5*a^7*c*d^2*e^8 + a^8*e^10)*sqrt(-(1225*c^5*d^8*e^6 + 10780*a*c^4*d^6*e^8 +
21966*a^2*c^3*d^4*e^10 - 7700*a^3*c^2*d^2*e^12 + 625*a^4*c*e^14)/(a^3*c^10*d^20 + 10*a^4*c^9*d^18*e^2 + 45*a^
5*c^8*d^16*e^4 + 120*a^6*c^7*d^14*e^6 + 210*a^7*c^6*d^12*e^8 + 252*a^8*c^5*d^10*e^10 + 210*a^9*c^4*d^8*e^12 +
120*a^10*c^3*d^6*e^14 + 45*a^11*c^2*d^4*e^16 + 10*a^12*c*d^2*e^18 + a^13*e^20)))/(a^3*c^5*d^10 + 5*a^4*c^4*d^8
*e^2 + 10*a^5*c^3*d^6*e^4 + 10*a^6*c^2*d^4*e^6 + 5*a^7*c*d^2*e^8 + a^8*e^10))*log(-(140*c^4*d^6*e^3 + 1491*a*c
^3*d^4*e^5 + 3750*a^2*c^2*d^2*e^7 - 625*a^3*c*e^9)*sqrt(e*x + d) - (35*a^2*c^4*d^7*e^4 + 609*a^3*c^3*d^5*e^6 +
1977*a^4*c^2*d^3*e^8 - 325*a^5*c*d*e^10 - (2*a^3*c^7*d^14 + 19*a^4*c^6*d^12*e^2 + 60*a^5*c^5*d^10*e^4 + 85*a^
6*c^4*d^8*e^6 + 50*a^7*c^3*d^6*e^8 - 3*a^8*c^2*d^4*e^10 - 16*a^9*c*d^2*e^12 - 5*a^10*e^14)*sqrt(-(1225*c^5*d^8
*e^6 + 10780*a*c^4*d^6*e^8 + 21966*a^2*c^3*d^4*e^10 - 7700*a^3*c^2*d^2*e^12 + 625*a^4*c*e^14)/(a^3*c^10*d^20 +
10*a^4*c^9*d^18*e^2 + 45*a^5*c^8*d^16*e^4 + 120*a^6*c^7*d^14*e^6 + 210*a^7*c^6*d^12*e^8 + 252*a^8*c^5*d^10*e^
10 + 210*a^9*c^4*d^8*e^12 + 120*a^10*c^3*d^6*e^14 + 45*a^11*c^2*d^4*e^16 + 10*a^12*c*d^2*e^18 + a^13*e^20)))*s
qrt(-(4*c^4*d^7 + 35*a*c^3*d^5*e^2 + 70*a^2*c^2*d^3*e^4 - 105*a^3*c*d*e^6 - (a^3*c^5*d^10 + 5*a^4*c^4*d^8*e^2
+ 10*a^5*c^3*d^6*e^4 + 10*a^6*c^2*d^4*e^6 + 5*a^7*c*d^2*e^8 + a^8*e^10)*sqrt(-(1225*c^5*d^8*e^6 + 10780*a*c^4*
d^6*e^8 + 21966*a^2*c^3*d^4*e^10 - 7700*a^3*c^2*d^2*e^12 + 625*a^4*c*e^14)/(a^3*c^10*d^20 + 10*a^4*c^9*d^18*e^
2 + 45*a^5*c^8*d^16*e^4 + 120*a^6*c^7*d^14*e^6 + 210*a^7*c^6*d^12*e^8 + 252*a^8*c^5*d^10*e^10 + 210*a^9*c^4*d^
8*e^12 + 120*a^10*c^3*d^6*e^14 + 45*a^11*c^2*d^4*e^16 + 10*a^12*c*d^2*e^18 + a^13*e^20)))/(a^3*c^5*d^10 + 5*a^
4*c^4*d^8*e^2 + 10*a^5*c^3*d^6*e^4 + 10*a^6*c^2*d^4*e^6 + 5*a^7*c*d^2*e^8 + a^8*e^10))) - 4*(2*a*c*d^2*e - 4*a
^2*e^3 + (c^2*d^2*e - 5*a*c*e^3)*x^2 + (c^2*d^3 + a*c*d*e^2)*x)*sqrt(e*x + d))/(a^2*c^2*d^5 + 2*a^3*c*d^3*e^2
+ a^4*d*e^4 + (a*c^3*d^4*e + 2*a^2*c^2*d^2*e^3 + a^3*c*e^5)*x^3 + (a*c^3*d^5 + 2*a^2*c^2*d^3*e^2 + a^3*c*d*e^4
)*x^2 + (a^2*c^2*d^4*e + 2*a^3*c*d^2*e^3 + a^4*e^5)*x)

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giac [A]  time = 0.96, size = 1350, normalized size = 1.60

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(3/2)/(c*x^2+a)^2,x, algorithm="giac")

[Out]

1/4*((a*c^2*d^4*e + 2*a^2*c*d^2*e^3 + a^3*e^5)^2*(c*d^2*e - 5*a*e^3)*abs(c) - (sqrt(-a*c)*c^3*d^7*e + 15*sqrt(
-a*c)*a*c^2*d^5*e^3 + 27*sqrt(-a*c)*a^2*c*d^3*e^5 + 13*sqrt(-a*c)*a^3*d*e^7)*abs(a*c^2*d^4*e + 2*a^2*c*d^2*e^3
+ a^3*e^5)*abs(c) + 2*(a*c^6*d^12*e + 8*a^2*c^5*d^10*e^3 + 22*a^3*c^4*d^8*e^5 + 28*a^4*c^3*d^6*e^7 + 17*a^5*c
^2*d^4*e^9 + 4*a^6*c*d^2*e^11)*abs(c))*arctan(sqrt(x*e + d)/sqrt(-(a*c^3*d^5 + 2*a^2*c^2*d^3*e^2 + a^3*c*d*e^4
+ sqrt((a*c^3*d^5 + 2*a^2*c^2*d^3*e^2 + a^3*c*d*e^4)^2 - (a*c^3*d^6 + 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 + a
^4*e^6)*(a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)))/(a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)))/((a^2*c^4*
d^8*e + sqrt(-a*c)*a*c^4*d^9 + 4*sqrt(-a*c)*a^2*c^3*d^7*e^2 + 4*a^3*c^3*d^6*e^3 + 6*sqrt(-a*c)*a^3*c^2*d^5*e^4
+ 6*a^4*c^2*d^4*e^5 + 4*sqrt(-a*c)*a^4*c*d^3*e^6 + 4*a^5*c*d^2*e^7 + sqrt(-a*c)*a^5*d*e^8 + a^6*e^9)*sqrt(-c^
2*d - sqrt(-a*c)*c*e)*abs(a*c^2*d^4*e + 2*a^2*c*d^2*e^3 + a^3*e^5)) + 1/4*((a*c^2*d^4*e + 2*a^2*c*d^2*e^3 + a^
3*e^5)^2*(c*d^2*e - 5*a*e^3)*abs(c) + (sqrt(-a*c)*c^3*d^7*e + 15*sqrt(-a*c)*a*c^2*d^5*e^3 + 27*sqrt(-a*c)*a^2*
c*d^3*e^5 + 13*sqrt(-a*c)*a^3*d*e^7)*abs(a*c^2*d^4*e + 2*a^2*c*d^2*e^3 + a^3*e^5)*abs(c) + 2*(a*c^6*d^12*e + 8
*a^2*c^5*d^10*e^3 + 22*a^3*c^4*d^8*e^5 + 28*a^4*c^3*d^6*e^7 + 17*a^5*c^2*d^4*e^9 + 4*a^6*c*d^2*e^11)*abs(c))*a
rctan(sqrt(x*e + d)/sqrt(-(a*c^3*d^5 + 2*a^2*c^2*d^3*e^2 + a^3*c*d*e^4 - sqrt((a*c^3*d^5 + 2*a^2*c^2*d^3*e^2 +
a^3*c*d*e^4)^2 - (a*c^3*d^6 + 3*a^2*c^2*d^4*e^2 + 3*a^3*c*d^2*e^4 + a^4*e^6)*(a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 +
a^3*c*e^4)))/(a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)))/((a^2*c^4*d^8*e - sqrt(-a*c)*a*c^4*d^9 - 4*sqrt(-a
*c)*a^2*c^3*d^7*e^2 + 4*a^3*c^3*d^6*e^3 - 6*sqrt(-a*c)*a^3*c^2*d^5*e^4 + 6*a^4*c^2*d^4*e^5 - 4*sqrt(-a*c)*a^4*
c*d^3*e^6 + 4*a^5*c*d^2*e^7 - sqrt(-a*c)*a^5*d*e^8 + a^6*e^9)*sqrt(-c^2*d + sqrt(-a*c)*c*e)*abs(a*c^2*d^4*e +
2*a^2*c*d^2*e^3 + a^3*e^5)) + 1/2*((x*e + d)^2*c^2*d^2*e - (x*e + d)*c^2*d^3*e - 5*(x*e + d)^2*a*c*e^3 + 11*(x
*e + d)*a*c*d*e^3 - 4*a*c*d^2*e^3 - 4*a^2*e^5)/((a*c^2*d^4 + 2*a^2*c*d^2*e^2 + a^3*e^4)*((x*e + d)^(5/2)*c - 2
*(x*e + d)^(3/2)*c*d + sqrt(x*e + d)*c*d^2 + sqrt(x*e + d)*a*e^2))

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maple [B]  time = 0.24, size = 8744, normalized size = 10.35 \begin {gather*} \text {output too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^(3/2)/(c*x^2+a)^2,x)

[Out]

result too large to display

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (c x^{2} + a\right )}^{2} {\left (e x + d\right )}^{\frac {3}{2}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^(3/2)/(c*x^2+a)^2,x, algorithm="maxima")

[Out]

integrate(1/((c*x^2 + a)^2*(e*x + d)^(3/2)), x)

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mupad [B]  time = 3.21, size = 8777, normalized size = 10.39

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + c*x^2)^2*(d + e*x)^(3/2)),x)

[Out]

atan((((-(4*a^3*c^4*d^7 - 25*a^2*e^7*(-a^9*c)^(1/2) + 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4 + 35*c^2*d^4*e^3
*(-a^9*c)^(1/2) - 105*a^6*c*d*e^6 + 154*a*c*d^2*e^5*(-a^9*c)^(1/2))/(64*(a^11*e^10 + a^6*c^5*d^10 + 5*a^10*c*d
^2*e^8 + 5*a^7*c^4*d^8*e^2 + 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2)*((d + e*x)^(1/2)*(-(4*a^3*c^4*d^
7 - 25*a^2*e^7*(-a^9*c)^(1/2) + 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4 + 35*c^2*d^4*e^3*(-a^9*c)^(1/2) - 105*
a^6*c*d*e^6 + 154*a*c*d^2*e^5*(-a^9*c)^(1/2))/(64*(a^11*e^10 + a^6*c^5*d^10 + 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8
*e^2 + 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2)*(2048*a^16*c^4*d*e^22 + 2048*a^6*c^14*d^21*e^2 + 20480
*a^7*c^13*d^19*e^4 + 92160*a^8*c^12*d^17*e^6 + 245760*a^9*c^11*d^15*e^8 + 430080*a^10*c^10*d^13*e^10 + 516096*
a^11*c^9*d^11*e^12 + 430080*a^12*c^8*d^9*e^14 + 245760*a^13*c^7*d^7*e^16 + 92160*a^14*c^6*d^5*e^18 + 20480*a^1
5*c^5*d^3*e^20) + 3328*a^14*c^4*d*e^21 + 256*a^5*c^13*d^19*e^3 + 5376*a^6*c^12*d^17*e^5 + 33792*a^7*c^11*d^15*
e^7 + 107520*a^8*c^10*d^13*e^9 + 204288*a^9*c^9*d^11*e^11 + 247296*a^10*c^8*d^9*e^13 + 193536*a^11*c^7*d^7*e^1
5 + 95232*a^12*c^6*d^5*e^17 + 26880*a^13*c^5*d^3*e^19) + (d + e*x)^(1/2)*(128*a^3*c^13*d^18*e^2 - 800*a^12*c^4
*e^20 + 1760*a^4*c^12*d^16*e^4 + 10240*a^5*c^11*d^14*e^6 + 30848*a^6*c^10*d^12*e^8 + 52480*a^7*c^9*d^10*e^10 +
51008*a^8*c^8*d^8*e^12 + 25600*a^9*c^7*d^6*e^14 + 3200*a^10*c^6*d^4*e^16 - 2432*a^11*c^5*d^2*e^18))*(-(4*a^3*
c^4*d^7 - 25*a^2*e^7*(-a^9*c)^(1/2) + 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4 + 35*c^2*d^4*e^3*(-a^9*c)^(1/2)
- 105*a^6*c*d*e^6 + 154*a*c*d^2*e^5*(-a^9*c)^(1/2))/(64*(a^11*e^10 + a^6*c^5*d^10 + 5*a^10*c*d^2*e^8 + 5*a^7*c
^4*d^8*e^2 + 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2)*1i - ((-(4*a^3*c^4*d^7 - 25*a^2*e^7*(-a^9*c)^(1/
2) + 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4 + 35*c^2*d^4*e^3*(-a^9*c)^(1/2) - 105*a^6*c*d*e^6 + 154*a*c*d^2*e
^5*(-a^9*c)^(1/2))/(64*(a^11*e^10 + a^6*c^5*d^10 + 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2 + 10*a^8*c^3*d^6*e^4 +
10*a^9*c^2*d^4*e^6)))^(1/2)*(3328*a^14*c^4*d*e^21 - (d + e*x)^(1/2)*(-(4*a^3*c^4*d^7 - 25*a^2*e^7*(-a^9*c)^(1
/2) + 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4 + 35*c^2*d^4*e^3*(-a^9*c)^(1/2) - 105*a^6*c*d*e^6 + 154*a*c*d^2*
e^5*(-a^9*c)^(1/2))/(64*(a^11*e^10 + a^6*c^5*d^10 + 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2 + 10*a^8*c^3*d^6*e^4
+ 10*a^9*c^2*d^4*e^6)))^(1/2)*(2048*a^16*c^4*d*e^22 + 2048*a^6*c^14*d^21*e^2 + 20480*a^7*c^13*d^19*e^4 + 92160
*a^8*c^12*d^17*e^6 + 245760*a^9*c^11*d^15*e^8 + 430080*a^10*c^10*d^13*e^10 + 516096*a^11*c^9*d^11*e^12 + 43008
0*a^12*c^8*d^9*e^14 + 245760*a^13*c^7*d^7*e^16 + 92160*a^14*c^6*d^5*e^18 + 20480*a^15*c^5*d^3*e^20) + 256*a^5*
c^13*d^19*e^3 + 5376*a^6*c^12*d^17*e^5 + 33792*a^7*c^11*d^15*e^7 + 107520*a^8*c^10*d^13*e^9 + 204288*a^9*c^9*d
^11*e^11 + 247296*a^10*c^8*d^9*e^13 + 193536*a^11*c^7*d^7*e^15 + 95232*a^12*c^6*d^5*e^17 + 26880*a^13*c^5*d^3*
e^19) - (d + e*x)^(1/2)*(128*a^3*c^13*d^18*e^2 - 800*a^12*c^4*e^20 + 1760*a^4*c^12*d^16*e^4 + 10240*a^5*c^11*d
^14*e^6 + 30848*a^6*c^10*d^12*e^8 + 52480*a^7*c^9*d^10*e^10 + 51008*a^8*c^8*d^8*e^12 + 25600*a^9*c^7*d^6*e^14
+ 3200*a^10*c^6*d^4*e^16 - 2432*a^11*c^5*d^2*e^18))*(-(4*a^3*c^4*d^7 - 25*a^2*e^7*(-a^9*c)^(1/2) + 35*a^4*c^3*
d^5*e^2 + 70*a^5*c^2*d^3*e^4 + 35*c^2*d^4*e^3*(-a^9*c)^(1/2) - 105*a^6*c*d*e^6 + 154*a*c*d^2*e^5*(-a^9*c)^(1/2
))/(64*(a^11*e^10 + a^6*c^5*d^10 + 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2 + 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*
e^6)))^(1/2)*1i)/(1000*a^10*c^4*e^19 - ((-(4*a^3*c^4*d^7 - 25*a^2*e^7*(-a^9*c)^(1/2) + 35*a^4*c^3*d^5*e^2 + 70
*a^5*c^2*d^3*e^4 + 35*c^2*d^4*e^3*(-a^9*c)^(1/2) - 105*a^6*c*d*e^6 + 154*a*c*d^2*e^5*(-a^9*c)^(1/2))/(64*(a^11
*e^10 + a^6*c^5*d^10 + 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2 + 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2)
*(3328*a^14*c^4*d*e^21 - (d + e*x)^(1/2)*(-(4*a^3*c^4*d^7 - 25*a^2*e^7*(-a^9*c)^(1/2) + 35*a^4*c^3*d^5*e^2 + 7
0*a^5*c^2*d^3*e^4 + 35*c^2*d^4*e^3*(-a^9*c)^(1/2) - 105*a^6*c*d*e^6 + 154*a*c*d^2*e^5*(-a^9*c)^(1/2))/(64*(a^1
1*e^10 + a^6*c^5*d^10 + 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2 + 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2
)*(2048*a^16*c^4*d*e^22 + 2048*a^6*c^14*d^21*e^2 + 20480*a^7*c^13*d^19*e^4 + 92160*a^8*c^12*d^17*e^6 + 245760*
a^9*c^11*d^15*e^8 + 430080*a^10*c^10*d^13*e^10 + 516096*a^11*c^9*d^11*e^12 + 430080*a^12*c^8*d^9*e^14 + 245760
*a^13*c^7*d^7*e^16 + 92160*a^14*c^6*d^5*e^18 + 20480*a^15*c^5*d^3*e^20) + 256*a^5*c^13*d^19*e^3 + 5376*a^6*c^1
2*d^17*e^5 + 33792*a^7*c^11*d^15*e^7 + 107520*a^8*c^10*d^13*e^9 + 204288*a^9*c^9*d^11*e^11 + 247296*a^10*c^8*d
^9*e^13 + 193536*a^11*c^7*d^7*e^15 + 95232*a^12*c^6*d^5*e^17 + 26880*a^13*c^5*d^3*e^19) - (d + e*x)^(1/2)*(128
*a^3*c^13*d^18*e^2 - 800*a^12*c^4*e^20 + 1760*a^4*c^12*d^16*e^4 + 10240*a^5*c^11*d^14*e^6 + 30848*a^6*c^10*d^1
2*e^8 + 52480*a^7*c^9*d^10*e^10 + 51008*a^8*c^8*d^8*e^12 + 25600*a^9*c^7*d^6*e^14 + 3200*a^10*c^6*d^4*e^16 - 2
432*a^11*c^5*d^2*e^18))*(-(4*a^3*c^4*d^7 - 25*a^2*e^7*(-a^9*c)^(1/2) + 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4
+ 35*c^2*d^4*e^3*(-a^9*c)^(1/2) - 105*a^6*c*d*e^6 + 154*a*c*d^2*e^5*(-a^9*c)^(1/2))/(64*(a^11*e^10 + a^6*c^5*
d^10 + 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2 + 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2) - ((-(4*a^3*c^4
*d^7 - 25*a^2*e^7*(-a^9*c)^(1/2) + 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4 + 35*c^2*d^4*e^3*(-a^9*c)^(1/2) - 1
05*a^6*c*d*e^6 + 154*a*c*d^2*e^5*(-a^9*c)^(1/2))/(64*(a^11*e^10 + a^6*c^5*d^10 + 5*a^10*c*d^2*e^8 + 5*a^7*c^4*
d^8*e^2 + 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2)*((d + e*x)^(1/2)*(-(4*a^3*c^4*d^7 - 25*a^2*e^7*(-a^
9*c)^(1/2) + 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4 + 35*c^2*d^4*e^3*(-a^9*c)^(1/2) - 105*a^6*c*d*e^6 + 154*a
*c*d^2*e^5*(-a^9*c)^(1/2))/(64*(a^11*e^10 + a^6*c^5*d^10 + 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2 + 10*a^8*c^3*d
^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2)*(2048*a^16*c^4*d*e^22 + 2048*a^6*c^14*d^21*e^2 + 20480*a^7*c^13*d^19*e^4
+ 92160*a^8*c^12*d^17*e^6 + 245760*a^9*c^11*d^15*e^8 + 430080*a^10*c^10*d^13*e^10 + 516096*a^11*c^9*d^11*e^12
+ 430080*a^12*c^8*d^9*e^14 + 245760*a^13*c^7*d^7*e^16 + 92160*a^14*c^6*d^5*e^18 + 20480*a^15*c^5*d^3*e^20) + 3
328*a^14*c^4*d*e^21 + 256*a^5*c^13*d^19*e^3 + 5376*a^6*c^12*d^17*e^5 + 33792*a^7*c^11*d^15*e^7 + 107520*a^8*c^
10*d^13*e^9 + 204288*a^9*c^9*d^11*e^11 + 247296*a^10*c^8*d^9*e^13 + 193536*a^11*c^7*d^7*e^15 + 95232*a^12*c^6*
d^5*e^17 + 26880*a^13*c^5*d^3*e^19) + (d + e*x)^(1/2)*(128*a^3*c^13*d^18*e^2 - 800*a^12*c^4*e^20 + 1760*a^4*c^
12*d^16*e^4 + 10240*a^5*c^11*d^14*e^6 + 30848*a^6*c^10*d^12*e^8 + 52480*a^7*c^9*d^10*e^10 + 51008*a^8*c^8*d^8*
e^12 + 25600*a^9*c^7*d^6*e^14 + 3200*a^10*c^6*d^4*e^16 - 2432*a^11*c^5*d^2*e^18))*(-(4*a^3*c^4*d^7 - 25*a^2*e^
7*(-a^9*c)^(1/2) + 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4 + 35*c^2*d^4*e^3*(-a^9*c)^(1/2) - 105*a^6*c*d*e^6 +
154*a*c*d^2*e^5*(-a^9*c)^(1/2))/(64*(a^11*e^10 + a^6*c^5*d^10 + 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2 + 10*a^8
*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2) - 32*a^2*c^12*d^16*e^3 - 232*a^3*c^11*d^14*e^5 + 280*a^4*c^10*d^12*
e^7 + 4760*a^5*c^9*d^10*e^9 + 13720*a^6*c^8*d^8*e^11 + 19208*a^7*c^7*d^6*e^13 + 14728*a^8*c^6*d^4*e^15 + 5960*
a^9*c^5*d^2*e^17))*(-(4*a^3*c^4*d^7 - 25*a^2*e^7*(-a^9*c)^(1/2) + 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4 + 35
*c^2*d^4*e^3*(-a^9*c)^(1/2) - 105*a^6*c*d*e^6 + 154*a*c*d^2*e^5*(-a^9*c)^(1/2))/(64*(a^11*e^10 + a^6*c^5*d^10
+ 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2 + 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2)*2i - ((2*e^3)/(a*e^2
+ c*d^2) + (c*e*(5*a*e^2 - c*d^2)*(d + e*x)^2)/(2*a*(a*e^2 + c*d^2)^2) - (c*d*e*(11*a*e^2 - c*d^2)*(d + e*x))
/(2*a*(a*e^2 + c*d^2)^2))/(c*(d + e*x)^(5/2) + (a*e^2 + c*d^2)*(d + e*x)^(1/2) - 2*c*d*(d + e*x)^(3/2)) + atan
((((-(4*a^3*c^4*d^7 + 25*a^2*e^7*(-a^9*c)^(1/2) + 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4 - 35*c^2*d^4*e^3*(-a
^9*c)^(1/2) - 105*a^6*c*d*e^6 - 154*a*c*d^2*e^5*(-a^9*c)^(1/2))/(64*(a^11*e^10 + a^6*c^5*d^10 + 5*a^10*c*d^2*e
^8 + 5*a^7*c^4*d^8*e^2 + 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2)*((d + e*x)^(1/2)*(-(4*a^3*c^4*d^7 +
25*a^2*e^7*(-a^9*c)^(1/2) + 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4 - 35*c^2*d^4*e^3*(-a^9*c)^(1/2) - 105*a^6*
c*d*e^6 - 154*a*c*d^2*e^5*(-a^9*c)^(1/2))/(64*(a^11*e^10 + a^6*c^5*d^10 + 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2
+ 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2)*(2048*a^16*c^4*d*e^22 + 2048*a^6*c^14*d^21*e^2 + 20480*a^7
*c^13*d^19*e^4 + 92160*a^8*c^12*d^17*e^6 + 245760*a^9*c^11*d^15*e^8 + 430080*a^10*c^10*d^13*e^10 + 516096*a^11
*c^9*d^11*e^12 + 430080*a^12*c^8*d^9*e^14 + 245760*a^13*c^7*d^7*e^16 + 92160*a^14*c^6*d^5*e^18 + 20480*a^15*c^
5*d^3*e^20) + 3328*a^14*c^4*d*e^21 + 256*a^5*c^13*d^19*e^3 + 5376*a^6*c^12*d^17*e^5 + 33792*a^7*c^11*d^15*e^7
+ 107520*a^8*c^10*d^13*e^9 + 204288*a^9*c^9*d^11*e^11 + 247296*a^10*c^8*d^9*e^13 + 193536*a^11*c^7*d^7*e^15 +
95232*a^12*c^6*d^5*e^17 + 26880*a^13*c^5*d^3*e^19) + (d + e*x)^(1/2)*(128*a^3*c^13*d^18*e^2 - 800*a^12*c^4*e^2
0 + 1760*a^4*c^12*d^16*e^4 + 10240*a^5*c^11*d^14*e^6 + 30848*a^6*c^10*d^12*e^8 + 52480*a^7*c^9*d^10*e^10 + 510
08*a^8*c^8*d^8*e^12 + 25600*a^9*c^7*d^6*e^14 + 3200*a^10*c^6*d^4*e^16 - 2432*a^11*c^5*d^2*e^18))*(-(4*a^3*c^4*
d^7 + 25*a^2*e^7*(-a^9*c)^(1/2) + 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4 - 35*c^2*d^4*e^3*(-a^9*c)^(1/2) - 10
5*a^6*c*d*e^6 - 154*a*c*d^2*e^5*(-a^9*c)^(1/2))/(64*(a^11*e^10 + a^6*c^5*d^10 + 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d
^8*e^2 + 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2)*1i - ((-(4*a^3*c^4*d^7 + 25*a^2*e^7*(-a^9*c)^(1/2) +
35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4 - 35*c^2*d^4*e^3*(-a^9*c)^(1/2) - 105*a^6*c*d*e^6 - 154*a*c*d^2*e^5*(
-a^9*c)^(1/2))/(64*(a^11*e^10 + a^6*c^5*d^10 + 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2 + 10*a^8*c^3*d^6*e^4 + 10*
a^9*c^2*d^4*e^6)))^(1/2)*(3328*a^14*c^4*d*e^21 - (d + e*x)^(1/2)*(-(4*a^3*c^4*d^7 + 25*a^2*e^7*(-a^9*c)^(1/2)
+ 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4 - 35*c^2*d^4*e^3*(-a^9*c)^(1/2) - 105*a^6*c*d*e^6 - 154*a*c*d^2*e^5*
(-a^9*c)^(1/2))/(64*(a^11*e^10 + a^6*c^5*d^10 + 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2 + 10*a^8*c^3*d^6*e^4 + 10
*a^9*c^2*d^4*e^6)))^(1/2)*(2048*a^16*c^4*d*e^22 + 2048*a^6*c^14*d^21*e^2 + 20480*a^7*c^13*d^19*e^4 + 92160*a^8
*c^12*d^17*e^6 + 245760*a^9*c^11*d^15*e^8 + 430080*a^10*c^10*d^13*e^10 + 516096*a^11*c^9*d^11*e^12 + 430080*a^
12*c^8*d^9*e^14 + 245760*a^13*c^7*d^7*e^16 + 92160*a^14*c^6*d^5*e^18 + 20480*a^15*c^5*d^3*e^20) + 256*a^5*c^13
*d^19*e^3 + 5376*a^6*c^12*d^17*e^5 + 33792*a^7*c^11*d^15*e^7 + 107520*a^8*c^10*d^13*e^9 + 204288*a^9*c^9*d^11*
e^11 + 247296*a^10*c^8*d^9*e^13 + 193536*a^11*c^7*d^7*e^15 + 95232*a^12*c^6*d^5*e^17 + 26880*a^13*c^5*d^3*e^19
) - (d + e*x)^(1/2)*(128*a^3*c^13*d^18*e^2 - 800*a^12*c^4*e^20 + 1760*a^4*c^12*d^16*e^4 + 10240*a^5*c^11*d^14*
e^6 + 30848*a^6*c^10*d^12*e^8 + 52480*a^7*c^9*d^10*e^10 + 51008*a^8*c^8*d^8*e^12 + 25600*a^9*c^7*d^6*e^14 + 32
00*a^10*c^6*d^4*e^16 - 2432*a^11*c^5*d^2*e^18))*(-(4*a^3*c^4*d^7 + 25*a^2*e^7*(-a^9*c)^(1/2) + 35*a^4*c^3*d^5*
e^2 + 70*a^5*c^2*d^3*e^4 - 35*c^2*d^4*e^3*(-a^9*c)^(1/2) - 105*a^6*c*d*e^6 - 154*a*c*d^2*e^5*(-a^9*c)^(1/2))/(
64*(a^11*e^10 + a^6*c^5*d^10 + 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2 + 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)
))^(1/2)*1i)/(1000*a^10*c^4*e^19 - ((-(4*a^3*c^4*d^7 + 25*a^2*e^7*(-a^9*c)^(1/2) + 35*a^4*c^3*d^5*e^2 + 70*a^5
*c^2*d^3*e^4 - 35*c^2*d^4*e^3*(-a^9*c)^(1/2) - 105*a^6*c*d*e^6 - 154*a*c*d^2*e^5*(-a^9*c)^(1/2))/(64*(a^11*e^1
0 + a^6*c^5*d^10 + 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2 + 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2)*(33
28*a^14*c^4*d*e^21 - (d + e*x)^(1/2)*(-(4*a^3*c^4*d^7 + 25*a^2*e^7*(-a^9*c)^(1/2) + 35*a^4*c^3*d^5*e^2 + 70*a^
5*c^2*d^3*e^4 - 35*c^2*d^4*e^3*(-a^9*c)^(1/2) - 105*a^6*c*d*e^6 - 154*a*c*d^2*e^5*(-a^9*c)^(1/2))/(64*(a^11*e^
10 + a^6*c^5*d^10 + 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2 + 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2)*(2
048*a^16*c^4*d*e^22 + 2048*a^6*c^14*d^21*e^2 + 20480*a^7*c^13*d^19*e^4 + 92160*a^8*c^12*d^17*e^6 + 245760*a^9*
c^11*d^15*e^8 + 430080*a^10*c^10*d^13*e^10 + 516096*a^11*c^9*d^11*e^12 + 430080*a^12*c^8*d^9*e^14 + 245760*a^1
3*c^7*d^7*e^16 + 92160*a^14*c^6*d^5*e^18 + 20480*a^15*c^5*d^3*e^20) + 256*a^5*c^13*d^19*e^3 + 5376*a^6*c^12*d^
17*e^5 + 33792*a^7*c^11*d^15*e^7 + 107520*a^8*c^10*d^13*e^9 + 204288*a^9*c^9*d^11*e^11 + 247296*a^10*c^8*d^9*e
^13 + 193536*a^11*c^7*d^7*e^15 + 95232*a^12*c^6*d^5*e^17 + 26880*a^13*c^5*d^3*e^19) - (d + e*x)^(1/2)*(128*a^3
*c^13*d^18*e^2 - 800*a^12*c^4*e^20 + 1760*a^4*c^12*d^16*e^4 + 10240*a^5*c^11*d^14*e^6 + 30848*a^6*c^10*d^12*e^
8 + 52480*a^7*c^9*d^10*e^10 + 51008*a^8*c^8*d^8*e^12 + 25600*a^9*c^7*d^6*e^14 + 3200*a^10*c^6*d^4*e^16 - 2432*
a^11*c^5*d^2*e^18))*(-(4*a^3*c^4*d^7 + 25*a^2*e^7*(-a^9*c)^(1/2) + 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4 - 3
5*c^2*d^4*e^3*(-a^9*c)^(1/2) - 105*a^6*c*d*e^6 - 154*a*c*d^2*e^5*(-a^9*c)^(1/2))/(64*(a^11*e^10 + a^6*c^5*d^10
+ 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2 + 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2) - ((-(4*a^3*c^4*d^7
+ 25*a^2*e^7*(-a^9*c)^(1/2) + 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4 - 35*c^2*d^4*e^3*(-a^9*c)^(1/2) - 105*a
^6*c*d*e^6 - 154*a*c*d^2*e^5*(-a^9*c)^(1/2))/(64*(a^11*e^10 + a^6*c^5*d^10 + 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*
e^2 + 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2)*((d + e*x)^(1/2)*(-(4*a^3*c^4*d^7 + 25*a^2*e^7*(-a^9*c)
^(1/2) + 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4 - 35*c^2*d^4*e^3*(-a^9*c)^(1/2) - 105*a^6*c*d*e^6 - 154*a*c*d
^2*e^5*(-a^9*c)^(1/2))/(64*(a^11*e^10 + a^6*c^5*d^10 + 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2 + 10*a^8*c^3*d^6*e
^4 + 10*a^9*c^2*d^4*e^6)))^(1/2)*(2048*a^16*c^4*d*e^22 + 2048*a^6*c^14*d^21*e^2 + 20480*a^7*c^13*d^19*e^4 + 92
160*a^8*c^12*d^17*e^6 + 245760*a^9*c^11*d^15*e^8 + 430080*a^10*c^10*d^13*e^10 + 516096*a^11*c^9*d^11*e^12 + 43
0080*a^12*c^8*d^9*e^14 + 245760*a^13*c^7*d^7*e^16 + 92160*a^14*c^6*d^5*e^18 + 20480*a^15*c^5*d^3*e^20) + 3328*
a^14*c^4*d*e^21 + 256*a^5*c^13*d^19*e^3 + 5376*a^6*c^12*d^17*e^5 + 33792*a^7*c^11*d^15*e^7 + 107520*a^8*c^10*d
^13*e^9 + 204288*a^9*c^9*d^11*e^11 + 247296*a^10*c^8*d^9*e^13 + 193536*a^11*c^7*d^7*e^15 + 95232*a^12*c^6*d^5*
e^17 + 26880*a^13*c^5*d^3*e^19) + (d + e*x)^(1/2)*(128*a^3*c^13*d^18*e^2 - 800*a^12*c^4*e^20 + 1760*a^4*c^12*d
^16*e^4 + 10240*a^5*c^11*d^14*e^6 + 30848*a^6*c^10*d^12*e^8 + 52480*a^7*c^9*d^10*e^10 + 51008*a^8*c^8*d^8*e^12
+ 25600*a^9*c^7*d^6*e^14 + 3200*a^10*c^6*d^4*e^16 - 2432*a^11*c^5*d^2*e^18))*(-(4*a^3*c^4*d^7 + 25*a^2*e^7*(-
a^9*c)^(1/2) + 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4 - 35*c^2*d^4*e^3*(-a^9*c)^(1/2) - 105*a^6*c*d*e^6 - 154
*a*c*d^2*e^5*(-a^9*c)^(1/2))/(64*(a^11*e^10 + a^6*c^5*d^10 + 5*a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2 + 10*a^8*c^3
*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2) - 32*a^2*c^12*d^16*e^3 - 232*a^3*c^11*d^14*e^5 + 280*a^4*c^10*d^12*e^7
+ 4760*a^5*c^9*d^10*e^9 + 13720*a^6*c^8*d^8*e^11 + 19208*a^7*c^7*d^6*e^13 + 14728*a^8*c^6*d^4*e^15 + 5960*a^9*
c^5*d^2*e^17))*(-(4*a^3*c^4*d^7 + 25*a^2*e^7*(-a^9*c)^(1/2) + 35*a^4*c^3*d^5*e^2 + 70*a^5*c^2*d^3*e^4 - 35*c^2
*d^4*e^3*(-a^9*c)^(1/2) - 105*a^6*c*d*e^6 - 154*a*c*d^2*e^5*(-a^9*c)^(1/2))/(64*(a^11*e^10 + a^6*c^5*d^10 + 5*
a^10*c*d^2*e^8 + 5*a^7*c^4*d^8*e^2 + 10*a^8*c^3*d^6*e^4 + 10*a^9*c^2*d^4*e^6)))^(1/2)*2i

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**(3/2)/(c*x**2+a)**2,x)

[Out]

Timed out

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