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

Optimal. Leaf size=726 \[ -\frac {e \left (-\sqrt {c} d \sqrt {a e^2+c d^2}+a e^2+c d^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {a e^2+c d^2}+\sqrt {c} (d+e x)\right )}{8 \sqrt {2} a c^{5/4} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}+\frac {e \left (-\sqrt {c} d \sqrt {a e^2+c d^2}+a e^2+c d^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt {d+e x} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {a e^2+c d^2}+\sqrt {c} (d+e x)\right )}{8 \sqrt {2} a c^{5/4} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}+\frac {e \left (\sqrt {c} d \sqrt {a e^2+c d^2}+a e^2+c d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}\right )}{4 \sqrt {2} a c^{5/4} \sqrt {a e^2+c d^2} \sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}-\frac {e \left (\sqrt {c} d \sqrt {a e^2+c d^2}+a e^2+c d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}\right )}{4 \sqrt {2} a c^{5/4} \sqrt {a e^2+c d^2} \sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}-\frac {\sqrt {d+e x} (a e-c d x)}{2 a c \left (a+c x^2\right )} \]

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Rubi [A]  time = 1.30, antiderivative size = 726, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {739, 827, 1169, 634, 618, 206, 628} \begin {gather*} -\frac {e \left (-\sqrt {c} d \sqrt {a e^2+c d^2}+a e^2+c d^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {a e^2+c d^2}+\sqrt {c} (d+e x)\right )}{8 \sqrt {2} a c^{5/4} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}+\frac {e \left (-\sqrt {c} d \sqrt {a e^2+c d^2}+a e^2+c d^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt {d+e x} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {a e^2+c d^2}+\sqrt {c} (d+e x)\right )}{8 \sqrt {2} a c^{5/4} \sqrt {a e^2+c d^2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}+\frac {e \left (\sqrt {c} d \sqrt {a e^2+c d^2}+a e^2+c d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}\right )}{4 \sqrt {2} a c^{5/4} \sqrt {a e^2+c d^2} \sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}-\frac {e \left (\sqrt {c} d \sqrt {a e^2+c d^2}+a e^2+c d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}\right )}{4 \sqrt {2} a c^{5/4} \sqrt {a e^2+c d^2} \sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}-\frac {\sqrt {d+e x} (a e-c d x)}{2 a c \left (a+c x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^(3/2)/(a + c*x^2)^2,x]

[Out]

-((a*e - c*d*x)*Sqrt[d + e*x])/(2*a*c*(a + c*x^2)) + (e*(c*d^2 + a*e^2 + Sqrt[c]*d*Sqrt[c*d^2 + a*e^2])*ArcTan
h[(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^(5/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (e*(c*d^2 + a*e^2 + Sqrt
[c]*d*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])/Sqr
t[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]])/(4*Sqrt[2]*a*c^(5/4)*Sqrt[c*d^2 + a*e^2]*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*
e^2]]) - (e*(c*d^2 + a*e^2 - Sqrt[c]*d*Sqrt[c*d^2 + a*e^2])*Log[Sqrt[c*d^2 + a*e^2] - Sqrt[2]*c^(1/4)*Sqrt[Sqr
t[c]*d + Sqrt[c*d^2 + a*e^2]]*Sqrt[d + e*x] + Sqrt[c]*(d + e*x)])/(8*Sqrt[2]*a*c^(5/4)*Sqrt[c*d^2 + a*e^2]*Sqr
t[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]) + (e*(c*d^2 + a*e^2 - Sqrt[c]*d*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^
(5/4)*Sqrt[c*d^2 + a*e^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 739

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*c*(p + 1)), x] + Dist[1/((p + 1)*(-2*a*c)), Int[(d + e*x)^(m - 2)*Simp[a*e^2*(m - 1) -
c*d^2*(2*p + 3) - d*c*e*(m + 2*p + 2)*x, x]*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^
2 + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 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 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 {(d+e x)^{3/2}}{\left (a+c x^2\right )^2} \, dx &=-\frac {(a e-c d x) \sqrt {d+e x}}{2 a c \left (a+c x^2\right )}+\frac {\int \frac {\frac {1}{2} \left (2 c d^2+a e^2\right )+\frac {1}{2} c d e x}{\sqrt {d+e x} \left (a+c x^2\right )} \, dx}{2 a c}\\ &=-\frac {(a e-c d x) \sqrt {d+e x}}{2 a c \left (a+c x^2\right )}+\frac {\operatorname {Subst}\left (\int \frac {-\frac {1}{2} c d^2 e+\frac {1}{2} e \left (2 c d^2+a e^2\right )+\frac {1}{2} c d e x^2}{c d^2+a e^2-2 c d x^2+c x^4} \, dx,x,\sqrt {d+e x}\right )}{a c}\\ &=-\frac {(a e-c d x) \sqrt {d+e x}}{2 a c \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^2 e+\frac {1}{2} e \left (2 c d^2+a e^2\right )\right )}{\sqrt [4]{c}}-\left (-\frac {1}{2} c d^2 e-\frac {1}{2} \sqrt {c} d e \sqrt {c d^2+a e^2}+\frac {1}{2} e \left (2 c d^2+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 c^{5/4} \sqrt {c d^2+a e^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^2 e+\frac {1}{2} e \left (2 c d^2+a e^2\right )\right )}{\sqrt [4]{c}}+\left (-\frac {1}{2} c d^2 e-\frac {1}{2} \sqrt {c} d e \sqrt {c d^2+a e^2}+\frac {1}{2} e \left (2 c d^2+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 c^{5/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\\ &=-\frac {(a e-c d x) \sqrt {d+e x}}{2 a c \left (a+c x^2\right )}-\frac {\left (e \left (c d^2+a e^2-\sqrt {c} d \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 c^{5/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\left (e \left (c d^2+a e^2-\sqrt {c} d \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 c^{5/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\left (e \left (c d^2+a e^2+\sqrt {c} d \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 c^{3/2} \sqrt {c d^2+a e^2}}+\frac {\left (e \left (c d^2+a e^2+\sqrt {c} d \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 c^{3/2} \sqrt {c d^2+a e^2}}\\ &=-\frac {(a e-c d x) \sqrt {d+e x}}{2 a c \left (a+c x^2\right )}-\frac {e \left (c d^2+a e^2-\sqrt {c} d \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 c^{5/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {e \left (c d^2+a e^2-\sqrt {c} d \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 c^{5/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}-\frac {\left (e \left (c d^2+a e^2+\sqrt {c} d \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 c^{3/2} \sqrt {c d^2+a e^2}}-\frac {\left (e \left (c d^2+a e^2+\sqrt {c} d \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 c^{3/2} \sqrt {c d^2+a e^2}}\\ &=-\frac {(a e-c d x) \sqrt {d+e x}}{2 a c \left (a+c x^2\right )}+\frac {e \left (c d^2+a e^2+\sqrt {c} d \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 c^{5/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \left (c d^2+a e^2+\sqrt {c} d \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 c^{5/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {e \left (c d^2+a e^2-\sqrt {c} d \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 c^{5/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {e \left (c d^2+a e^2-\sqrt {c} d \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 c^{5/4} \sqrt {c d^2+a e^2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\\ \end {align*}

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Mathematica [A]  time = 0.55, size = 208, normalized size = 0.29 \begin {gather*} \frac {\frac {2 a \sqrt [4]{c} \sqrt {d+e x} (c d x-a e)}{a+c x^2}-\sqrt {\sqrt {c} d-\sqrt {-a} e} \left (2 \sqrt {-a} \sqrt {c} d-a e\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {-a} e}}\right )+\sqrt {\sqrt {-a} e+\sqrt {c} d} \left (2 \sqrt {-a} \sqrt {c} d+a e\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {-a} e+\sqrt {c} d}}\right )}{4 a^2 c^{5/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((2*a*c^(1/4)*(-(a*e) + c*d*x)*Sqrt[d + e*x])/(a + c*x^2) - Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*(2*Sqrt[-a]*Sqrt[c]*d
- a*e)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[-a]*e]] + Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*(2*Sqrt[-
a]*Sqrt[c]*d + a*e)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[-a]*e]])/(4*a^2*c^(5/4))

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IntegrateAlgebraic [C]  time = 1.31, size = 314, normalized size = 0.43 \begin {gather*} -\frac {i \left (2 \sqrt {c} d-i \sqrt {a} e\right ) \sqrt {-i \sqrt {c} \left (\sqrt {a} e-i \sqrt {c} d\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} c^{3/2}}+\frac {i \left (2 \sqrt {c} d+i \sqrt {a} e\right ) \sqrt {i \sqrt {c} \left (\sqrt {a} e+i \sqrt {c} d\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} c^{3/2}}-\frac {e \sqrt {d+e x} \left (a e^2+c d^2-c d (d+e x)\right )}{2 a c \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[(d + e*x)^(3/2)/(a + c*x^2)^2,x]

[Out]

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

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fricas [A]  time = 0.44, size = 679, normalized size = 0.94 \begin {gather*} \frac {{\left (a c^{2} x^{2} + a^{2} c\right )} \sqrt {-\frac {a^{3} c^{2} \sqrt {-\frac {e^{6}}{a^{3} c^{5}}} + 4 \, c d^{3} + 3 \, a d e^{2}}{a^{3} c^{2}}} \log \left ({\left (4 \, c d^{2} e^{3} + a e^{5}\right )} \sqrt {e x + d} + {\left (2 \, a^{3} c^{4} d \sqrt {-\frac {e^{6}}{a^{3} c^{5}}} + a^{2} c e^{4}\right )} \sqrt {-\frac {a^{3} c^{2} \sqrt {-\frac {e^{6}}{a^{3} c^{5}}} + 4 \, c d^{3} + 3 \, a d e^{2}}{a^{3} c^{2}}}\right ) - {\left (a c^{2} x^{2} + a^{2} c\right )} \sqrt {-\frac {a^{3} c^{2} \sqrt {-\frac {e^{6}}{a^{3} c^{5}}} + 4 \, c d^{3} + 3 \, a d e^{2}}{a^{3} c^{2}}} \log \left ({\left (4 \, c d^{2} e^{3} + a e^{5}\right )} \sqrt {e x + d} - {\left (2 \, a^{3} c^{4} d \sqrt {-\frac {e^{6}}{a^{3} c^{5}}} + a^{2} c e^{4}\right )} \sqrt {-\frac {a^{3} c^{2} \sqrt {-\frac {e^{6}}{a^{3} c^{5}}} + 4 \, c d^{3} + 3 \, a d e^{2}}{a^{3} c^{2}}}\right ) - {\left (a c^{2} x^{2} + a^{2} c\right )} \sqrt {\frac {a^{3} c^{2} \sqrt {-\frac {e^{6}}{a^{3} c^{5}}} - 4 \, c d^{3} - 3 \, a d e^{2}}{a^{3} c^{2}}} \log \left ({\left (4 \, c d^{2} e^{3} + a e^{5}\right )} \sqrt {e x + d} + {\left (2 \, a^{3} c^{4} d \sqrt {-\frac {e^{6}}{a^{3} c^{5}}} - a^{2} c e^{4}\right )} \sqrt {\frac {a^{3} c^{2} \sqrt {-\frac {e^{6}}{a^{3} c^{5}}} - 4 \, c d^{3} - 3 \, a d e^{2}}{a^{3} c^{2}}}\right ) + {\left (a c^{2} x^{2} + a^{2} c\right )} \sqrt {\frac {a^{3} c^{2} \sqrt {-\frac {e^{6}}{a^{3} c^{5}}} - 4 \, c d^{3} - 3 \, a d e^{2}}{a^{3} c^{2}}} \log \left ({\left (4 \, c d^{2} e^{3} + a e^{5}\right )} \sqrt {e x + d} - {\left (2 \, a^{3} c^{4} d \sqrt {-\frac {e^{6}}{a^{3} c^{5}}} - a^{2} c e^{4}\right )} \sqrt {\frac {a^{3} c^{2} \sqrt {-\frac {e^{6}}{a^{3} c^{5}}} - 4 \, c d^{3} - 3 \, a d e^{2}}{a^{3} c^{2}}}\right ) + 4 \, {\left (c d x - a e\right )} \sqrt {e x + d}}{8 \, {\left (a c^{2} x^{2} + a^{2} c\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*((a*c^2*x^2 + a^2*c)*sqrt(-(a^3*c^2*sqrt(-e^6/(a^3*c^5)) + 4*c*d^3 + 3*a*d*e^2)/(a^3*c^2))*log((4*c*d^2*e^
3 + a*e^5)*sqrt(e*x + d) + (2*a^3*c^4*d*sqrt(-e^6/(a^3*c^5)) + a^2*c*e^4)*sqrt(-(a^3*c^2*sqrt(-e^6/(a^3*c^5))
+ 4*c*d^3 + 3*a*d*e^2)/(a^3*c^2))) - (a*c^2*x^2 + a^2*c)*sqrt(-(a^3*c^2*sqrt(-e^6/(a^3*c^5)) + 4*c*d^3 + 3*a*d
*e^2)/(a^3*c^2))*log((4*c*d^2*e^3 + a*e^5)*sqrt(e*x + d) - (2*a^3*c^4*d*sqrt(-e^6/(a^3*c^5)) + a^2*c*e^4)*sqrt
(-(a^3*c^2*sqrt(-e^6/(a^3*c^5)) + 4*c*d^3 + 3*a*d*e^2)/(a^3*c^2))) - (a*c^2*x^2 + a^2*c)*sqrt((a^3*c^2*sqrt(-e
^6/(a^3*c^5)) - 4*c*d^3 - 3*a*d*e^2)/(a^3*c^2))*log((4*c*d^2*e^3 + a*e^5)*sqrt(e*x + d) + (2*a^3*c^4*d*sqrt(-e
^6/(a^3*c^5)) - a^2*c*e^4)*sqrt((a^3*c^2*sqrt(-e^6/(a^3*c^5)) - 4*c*d^3 - 3*a*d*e^2)/(a^3*c^2))) + (a*c^2*x^2
+ a^2*c)*sqrt((a^3*c^2*sqrt(-e^6/(a^3*c^5)) - 4*c*d^3 - 3*a*d*e^2)/(a^3*c^2))*log((4*c*d^2*e^3 + a*e^5)*sqrt(e
*x + d) - (2*a^3*c^4*d*sqrt(-e^6/(a^3*c^5)) - a^2*c*e^4)*sqrt((a^3*c^2*sqrt(-e^6/(a^3*c^5)) - 4*c*d^3 - 3*a*d*
e^2)/(a^3*c^2))) + 4*(c*d*x - a*e)*sqrt(e*x + d))/(a*c^2*x^2 + a^2*c)

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giac [A]  time = 0.56, size = 411, normalized size = 0.57 \begin {gather*} -\frac {{\left (2 \, a c^{3} d^{3} + 2 \, a^{2} c^{2} d e^{2} - {\left (\sqrt {-a c} c d^{2} e + \sqrt {-a c} a e^{3}\right )} {\left | a \right |} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {a c^{2} d + \sqrt {a^{2} c^{4} d^{2} - {\left (a c^{2} d^{2} + a^{2} c e^{2}\right )} a c^{2}}}{a c^{2}}}}\right )}{4 \, {\left (a^{2} c^{2} e - \sqrt {-a c} a c^{2} d\right )} \sqrt {-c^{2} d + \sqrt {-a c} c e} {\left | a \right |}} - \frac {{\left (2 \, a c^{3} d^{3} + 2 \, a^{2} c^{2} d e^{2} + {\left (\sqrt {-a c} c d^{2} e + \sqrt {-a c} a e^{3}\right )} {\left | a \right |} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {a c^{2} d - \sqrt {a^{2} c^{4} d^{2} - {\left (a c^{2} d^{2} + a^{2} c e^{2}\right )} a c^{2}}}{a c^{2}}}}\right )}{4 \, {\left (a^{2} c^{2} e + \sqrt {-a c} a c^{2} d\right )} \sqrt {-c^{2} d - \sqrt {-a c} c e} {\left | a \right |}} + \frac {{\left (x e + d\right )}^{\frac {3}{2}} c d e - \sqrt {x e + d} c d^{2} e - \sqrt {x e + d} a e^{3}}{2 \, {\left ({\left (x e + d\right )}^{2} c - 2 \, {\left (x e + d\right )} c d + c d^{2} + a e^{2}\right )} a c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(2*a*c^3*d^3 + 2*a^2*c^2*d*e^2 - (sqrt(-a*c)*c*d^2*e + sqrt(-a*c)*a*e^3)*abs(a)*abs(c))*arctan(sqrt(x*e +
d)/sqrt(-(a*c^2*d + sqrt(a^2*c^4*d^2 - (a*c^2*d^2 + a^2*c*e^2)*a*c^2))/(a*c^2)))/((a^2*c^2*e - sqrt(-a*c)*a*c
^2*d)*sqrt(-c^2*d + sqrt(-a*c)*c*e)*abs(a)) - 1/4*(2*a*c^3*d^3 + 2*a^2*c^2*d*e^2 + (sqrt(-a*c)*c*d^2*e + sqrt(
-a*c)*a*e^3)*abs(a)*abs(c))*arctan(sqrt(x*e + d)/sqrt(-(a*c^2*d - sqrt(a^2*c^4*d^2 - (a*c^2*d^2 + a^2*c*e^2)*a
*c^2))/(a*c^2)))/((a^2*c^2*e + sqrt(-a*c)*a*c^2*d)*sqrt(-c^2*d - sqrt(-a*c)*c*e)*abs(a)) + 1/2*((x*e + d)^(3/2
)*c*d*e - sqrt(x*e + d)*c*d^2*e - sqrt(x*e + d)*a*e^3)/(((x*e + d)^2*c - 2*(x*e + d)*c*d + c*d^2 + a*e^2)*a*c)

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maple [B]  time = 0.19, size = 2851, normalized size = 3.93 \begin {gather*} \text {output too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*e/(c*e^2*x^2+a*e^2)*d/a*(e*x+d)^(3/2)-1/2*e^3/(c*e^2*x^2+a*e^2)/c*(e*x+d)^(1/2)-1/2*e/(c*e^2*x^2+a*e^2)/a*
(e*x+d)^(1/2)*d^2+1/16/e/a^2/c*ln((e*x+d)*c^(1/2)-(e*x+d)^(1/2)*(2*c*d+2*((a*e^2+c*d^2)*c)^(1/2))^(1/2)+(a*e^2
+c*d^2)^(1/2))*(2*c*d+2*(a*c*e^2+c^2*d^2)^(1/2))^(1/2)*(a*e^2+c*d^2)^(1/2)*d-1/16/e/a^2/c^(1/2)*ln((e*x+d)*c^(
1/2)-(e*x+d)^(1/2)*(2*c*d+2*((a*e^2+c*d^2)*c)^(1/2))^(1/2)+(a*e^2+c*d^2)^(1/2))*(2*c*d+2*(a*c*e^2+c^2*d^2)^(1/
2))^(1/2)*d^2-1/16/e/a^2/c^2*ln((e*x+d)*c^(1/2)-(e*x+d)^(1/2)*(2*c*d+2*((a*e^2+c*d^2)*c)^(1/2))^(1/2)+(a*e^2+c
*d^2)^(1/2))*(2*c*d+2*(a*c*e^2+c^2*d^2)^(1/2))^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)*(a*e^2+c*d^2)^(1/2)+1/16/e/a^2/c^
(3/2)*ln((e*x+d)*c^(1/2)-(e*x+d)^(1/2)*(2*c*d+2*((a*e^2+c*d^2)*c)^(1/2))^(1/2)+(a*e^2+c*d^2)^(1/2))*(2*c*d+2*(
a*c*e^2+c^2*d^2)^(1/2))^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)*d+1/2*e/a/c/(-2*c*d+4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*((a*
e^2+c*d^2)*c)^(1/2))^(1/2)*arctan((2*(e*x+d)^(1/2)*c^(1/2)-(2*c*d+2*((a*e^2+c*d^2)*c)^(1/2))^(1/2))/(-2*c*d+4*
(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*((a*e^2+c*d^2)*c)^(1/2))^(1/2))*(a*e^2+c*d^2)^(1/2)+1/8/e/a^2/c/(-2*c*d+4*(a*e^2
+c*d^2)^(1/2)*c^(1/2)-2*((a*e^2+c*d^2)*c)^(1/2))^(1/2)*arctan((2*(e*x+d)^(1/2)*c^(1/2)-(2*c*d+2*((a*e^2+c*d^2)
*c)^(1/2))^(1/2))/(-2*c*d+4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*((a*e^2+c*d^2)*c)^(1/2))^(1/2))*(2*c*d+2*((a*e^2+c*d
^2)*c)^(1/2))^(1/2)*(2*c*d+2*(a*c*e^2+c^2*d^2)^(1/2))^(1/2)*(a*e^2+c*d^2)^(1/2)*d-1/8/e/a^2/c^(1/2)/(-2*c*d+4*
(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*((a*e^2+c*d^2)*c)^(1/2))^(1/2)*arctan((2*(e*x+d)^(1/2)*c^(1/2)-(2*c*d+2*((a*e^2+
c*d^2)*c)^(1/2))^(1/2))/(-2*c*d+4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*((a*e^2+c*d^2)*c)^(1/2))^(1/2))*(2*c*d+2*((a*e
^2+c*d^2)*c)^(1/2))^(1/2)*(2*c*d+2*(a*c*e^2+c^2*d^2)^(1/2))^(1/2)*d^2-1/8/e/a^2/c^2/(-2*c*d+4*(a*e^2+c*d^2)^(1
/2)*c^(1/2)-2*((a*e^2+c*d^2)*c)^(1/2))^(1/2)*arctan((2*(e*x+d)^(1/2)*c^(1/2)-(2*c*d+2*((a*e^2+c*d^2)*c)^(1/2))
^(1/2))/(-2*c*d+4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*((a*e^2+c*d^2)*c)^(1/2))^(1/2))*(2*c*d+2*((a*e^2+c*d^2)*c)^(1/
2))^(1/2)*(2*c*d+2*(a*c*e^2+c^2*d^2)^(1/2))^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)*(a*e^2+c*d^2)^(1/2)+1/8/e/a^2/c^(3/2
)/(-2*c*d+4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*((a*e^2+c*d^2)*c)^(1/2))^(1/2)*arctan((2*(e*x+d)^(1/2)*c^(1/2)-(2*c*
d+2*((a*e^2+c*d^2)*c)^(1/2))^(1/2))/(-2*c*d+4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*((a*e^2+c*d^2)*c)^(1/2))^(1/2))*(2
*c*d+2*((a*e^2+c*d^2)*c)^(1/2))^(1/2)*(2*c*d+2*(a*c*e^2+c^2*d^2)^(1/2))^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)*d-1/16/e
/a^2/c*ln((e*x+d)*c^(1/2)+(e*x+d)^(1/2)*(2*c*d+2*((a*e^2+c*d^2)*c)^(1/2))^(1/2)+(a*e^2+c*d^2)^(1/2))*(2*c*d+2*
(a*c*e^2+c^2*d^2)^(1/2))^(1/2)*(a*e^2+c*d^2)^(1/2)*d+1/16/e/a^2/c^(1/2)*ln((e*x+d)*c^(1/2)+(e*x+d)^(1/2)*(2*c*
d+2*((a*e^2+c*d^2)*c)^(1/2))^(1/2)+(a*e^2+c*d^2)^(1/2))*(2*c*d+2*(a*c*e^2+c^2*d^2)^(1/2))^(1/2)*d^2+1/16/e/a^2
/c^2*ln((e*x+d)*c^(1/2)+(e*x+d)^(1/2)*(2*c*d+2*((a*e^2+c*d^2)*c)^(1/2))^(1/2)+(a*e^2+c*d^2)^(1/2))*(2*c*d+2*(a
*c*e^2+c^2*d^2)^(1/2))^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)*(a*e^2+c*d^2)^(1/2)-1/16/e/a^2/c^(3/2)*ln((e*x+d)*c^(1/2)
+(e*x+d)^(1/2)*(2*c*d+2*((a*e^2+c*d^2)*c)^(1/2))^(1/2)+(a*e^2+c*d^2)^(1/2))*(2*c*d+2*(a*c*e^2+c^2*d^2)^(1/2))^
(1/2)*(a*c*e^2+c^2*d^2)^(1/2)*d+1/2*e/a/c/(-2*c*d+4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*((a*e^2+c*d^2)*c)^(1/2))^(1/
2)*arctan((2*(e*x+d)^(1/2)*c^(1/2)+(2*c*d+2*((a*e^2+c*d^2)*c)^(1/2))^(1/2))/(-2*c*d+4*(a*e^2+c*d^2)^(1/2)*c^(1
/2)-2*((a*e^2+c*d^2)*c)^(1/2))^(1/2))*(a*e^2+c*d^2)^(1/2)+1/8/e/a^2/c/(-2*c*d+4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*
((a*e^2+c*d^2)*c)^(1/2))^(1/2)*arctan((2*(e*x+d)^(1/2)*c^(1/2)+(2*c*d+2*((a*e^2+c*d^2)*c)^(1/2))^(1/2))/(-2*c*
d+4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*((a*e^2+c*d^2)*c)^(1/2))^(1/2))*(2*c*d+2*((a*e^2+c*d^2)*c)^(1/2))^(1/2)*(2*c
*d+2*(a*c*e^2+c^2*d^2)^(1/2))^(1/2)*(a*e^2+c*d^2)^(1/2)*d-1/8/e/a^2/c^(1/2)/(-2*c*d+4*(a*e^2+c*d^2)^(1/2)*c^(1
/2)-2*((a*e^2+c*d^2)*c)^(1/2))^(1/2)*arctan((2*(e*x+d)^(1/2)*c^(1/2)+(2*c*d+2*((a*e^2+c*d^2)*c)^(1/2))^(1/2))/
(-2*c*d+4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*((a*e^2+c*d^2)*c)^(1/2))^(1/2))*(2*c*d+2*((a*e^2+c*d^2)*c)^(1/2))^(1/2
)*(2*c*d+2*(a*c*e^2+c^2*d^2)^(1/2))^(1/2)*d^2-1/8/e/a^2/c^2/(-2*c*d+4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*((a*e^2+c*
d^2)*c)^(1/2))^(1/2)*arctan((2*(e*x+d)^(1/2)*c^(1/2)+(2*c*d+2*((a*e^2+c*d^2)*c)^(1/2))^(1/2))/(-2*c*d+4*(a*e^2
+c*d^2)^(1/2)*c^(1/2)-2*((a*e^2+c*d^2)*c)^(1/2))^(1/2))*(2*c*d+2*((a*e^2+c*d^2)*c)^(1/2))^(1/2)*(2*c*d+2*(a*c*
e^2+c^2*d^2)^(1/2))^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)*(a*e^2+c*d^2)^(1/2)+1/8/e/a^2/c^(3/2)/(-2*c*d+4*(a*e^2+c*d^2
)^(1/2)*c^(1/2)-2*((a*e^2+c*d^2)*c)^(1/2))^(1/2)*arctan((2*(e*x+d)^(1/2)*c^(1/2)+(2*c*d+2*((a*e^2+c*d^2)*c)^(1
/2))^(1/2))/(-2*c*d+4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*((a*e^2+c*d^2)*c)^(1/2))^(1/2))*(2*c*d+2*((a*e^2+c*d^2)*c)
^(1/2))^(1/2)*(2*c*d+2*(a*c*e^2+c^2*d^2)^(1/2))^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)*d

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.44, size = 717, normalized size = 0.99 \begin {gather*} -\frac {\frac {\left (c\,d^2\,e+a\,e^3\right )\,\sqrt {d+e\,x}}{2\,a\,c}-\frac {d\,e\,{\left (d+e\,x\right )}^{3/2}}{2\,a}}{c\,{\left (d+e\,x\right )}^2+a\,e^2+c\,d^2-2\,c\,d\,\left (d+e\,x\right )}-2\,\mathrm {atanh}\left (\frac {2\,c\,e^6\,\sqrt {d+e\,x}\,\sqrt {-\frac {d^3}{16\,a^3\,c}-\frac {3\,d\,e^2}{64\,a^2\,c^2}-\frac {e^3\,\sqrt {-a^9\,c^5}}{64\,a^6\,c^5}}}{\frac {d\,e^7}{2\,a}+\frac {c\,d^3\,e^5}{2\,a^2}+\frac {e^8\,\sqrt {-a^9\,c^5}}{4\,a^5\,c^3}+\frac {d^2\,e^6\,\sqrt {-a^9\,c^5}}{4\,a^6\,c^2}}-\frac {2\,d\,e^5\,\sqrt {-a^9\,c^5}\,\sqrt {d+e\,x}\,\sqrt {-\frac {d^3}{16\,a^3\,c}-\frac {3\,d\,e^2}{64\,a^2\,c^2}-\frac {e^3\,\sqrt {-a^9\,c^5}}{64\,a^6\,c^5}}}{\frac {e^8\,\sqrt {-a^9\,c^5}}{4\,c^2}+\frac {a^3\,c^2\,d^3\,e^5}{2}+\frac {a^4\,c\,d\,e^7}{2}+\frac {d^2\,e^6\,\sqrt {-a^9\,c^5}}{4\,a\,c}}\right )\,\sqrt {-\frac {e^3\,\sqrt {-a^9\,c^5}+4\,a^3\,c^4\,d^3+3\,a^4\,c^3\,d\,e^2}{64\,a^6\,c^5}}-2\,\mathrm {atanh}\left (\frac {2\,c\,e^6\,\sqrt {d+e\,x}\,\sqrt {\frac {e^3\,\sqrt {-a^9\,c^5}}{64\,a^6\,c^5}-\frac {3\,d\,e^2}{64\,a^2\,c^2}-\frac {d^3}{16\,a^3\,c}}}{\frac {d\,e^7}{2\,a}+\frac {c\,d^3\,e^5}{2\,a^2}-\frac {e^8\,\sqrt {-a^9\,c^5}}{4\,a^5\,c^3}-\frac {d^2\,e^6\,\sqrt {-a^9\,c^5}}{4\,a^6\,c^2}}-\frac {2\,d\,e^5\,\sqrt {-a^9\,c^5}\,\sqrt {d+e\,x}\,\sqrt {\frac {e^3\,\sqrt {-a^9\,c^5}}{64\,a^6\,c^5}-\frac {3\,d\,e^2}{64\,a^2\,c^2}-\frac {d^3}{16\,a^3\,c}}}{\frac {e^8\,\sqrt {-a^9\,c^5}}{4\,c^2}-\frac {a^3\,c^2\,d^3\,e^5}{2}-\frac {a^4\,c\,d\,e^7}{2}+\frac {d^2\,e^6\,\sqrt {-a^9\,c^5}}{4\,a\,c}}\right )\,\sqrt {-\frac {4\,a^3\,c^4\,d^3-e^3\,\sqrt {-a^9\,c^5}+3\,a^4\,c^3\,d\,e^2}{64\,a^6\,c^5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^(3/2)/(a + c*x^2)^2,x)

[Out]

- (((a*e^3 + c*d^2*e)*(d + e*x)^(1/2))/(2*a*c) - (d*e*(d + e*x)^(3/2))/(2*a))/(c*(d + e*x)^2 + a*e^2 + c*d^2 -
2*c*d*(d + e*x)) - 2*atanh((2*c*e^6*(d + e*x)^(1/2)*(- d^3/(16*a^3*c) - (3*d*e^2)/(64*a^2*c^2) - (e^3*(-a^9*c
^5)^(1/2))/(64*a^6*c^5))^(1/2))/((d*e^7)/(2*a) + (c*d^3*e^5)/(2*a^2) + (e^8*(-a^9*c^5)^(1/2))/(4*a^5*c^3) + (d
^2*e^6*(-a^9*c^5)^(1/2))/(4*a^6*c^2)) - (2*d*e^5*(-a^9*c^5)^(1/2)*(d + e*x)^(1/2)*(- d^3/(16*a^3*c) - (3*d*e^2
)/(64*a^2*c^2) - (e^3*(-a^9*c^5)^(1/2))/(64*a^6*c^5))^(1/2))/((e^8*(-a^9*c^5)^(1/2))/(4*c^2) + (a^3*c^2*d^3*e^
5)/2 + (a^4*c*d*e^7)/2 + (d^2*e^6*(-a^9*c^5)^(1/2))/(4*a*c)))*(-(e^3*(-a^9*c^5)^(1/2) + 4*a^3*c^4*d^3 + 3*a^4*
c^3*d*e^2)/(64*a^6*c^5))^(1/2) - 2*atanh((2*c*e^6*(d + e*x)^(1/2)*((e^3*(-a^9*c^5)^(1/2))/(64*a^6*c^5) - (3*d*
e^2)/(64*a^2*c^2) - d^3/(16*a^3*c))^(1/2))/((d*e^7)/(2*a) + (c*d^3*e^5)/(2*a^2) - (e^8*(-a^9*c^5)^(1/2))/(4*a^
5*c^3) - (d^2*e^6*(-a^9*c^5)^(1/2))/(4*a^6*c^2)) - (2*d*e^5*(-a^9*c^5)^(1/2)*(d + e*x)^(1/2)*((e^3*(-a^9*c^5)^
(1/2))/(64*a^6*c^5) - (3*d*e^2)/(64*a^2*c^2) - d^3/(16*a^3*c))^(1/2))/((e^8*(-a^9*c^5)^(1/2))/(4*c^2) - (a^3*c
^2*d^3*e^5)/2 - (a^4*c*d*e^7)/2 + (d^2*e^6*(-a^9*c^5)^(1/2))/(4*a*c)))*(-(4*a^3*c^4*d^3 - e^3*(-a^9*c^5)^(1/2)
+ 3*a^4*c^3*d*e^2)/(64*a^6*c^5))^(1/2)

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

[Out]

Timed out

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