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

Optimal. Leaf size=663 \[ -\frac {2 e}{\sqrt {d+e x} \left (a e^2+c d^2\right )}-\frac {\sqrt [4]{c} e \left (\sqrt {a e^2+c d^2}+2 \sqrt {c} d\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {a e^2+c d^2}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} \left (a e^2+c d^2\right )^{3/2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}+\frac {\sqrt [4]{c} e \left (\sqrt {a e^2+c d^2}+2 \sqrt {c} d\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt {d+e x} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {a e^2+c d^2}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} \left (a e^2+c d^2\right )^{3/2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}+\frac {\sqrt [4]{c} e \left (2 \sqrt {c} d-\sqrt {a e^2+c d^2}\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}\right )}{\sqrt {2} \left (a e^2+c d^2\right )^{3/2} \sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}-\frac {\sqrt [4]{c} e \left (2 \sqrt {c} d-\sqrt {a e^2+c d^2}\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}\right )}{\sqrt {2} \left (a e^2+c d^2\right )^{3/2} \sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.97, antiderivative size = 663, 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 = {710, 827, 1169, 634, 618, 206, 628} \[ -\frac {2 e}{\sqrt {d+e x} \left (a e^2+c d^2\right )}-\frac {\sqrt [4]{c} e \left (\sqrt {a e^2+c d^2}+2 \sqrt {c} d\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {a e^2+c d^2}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} \left (a e^2+c d^2\right )^{3/2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}+\frac {\sqrt [4]{c} e \left (\sqrt {a e^2+c d^2}+2 \sqrt {c} d\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt {d+e x} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {a e^2+c d^2}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} \left (a e^2+c d^2\right )^{3/2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}+\frac {\sqrt [4]{c} e \left (2 \sqrt {c} d-\sqrt {a e^2+c d^2}\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}\right )}{\sqrt {2} \left (a e^2+c d^2\right )^{3/2} \sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}-\frac {\sqrt [4]{c} e \left (2 \sqrt {c} d-\sqrt {a e^2+c d^2}\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}\right )}{\sqrt {2} \left (a e^2+c d^2\right )^{3/2} \sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*e)/((c*d^2 + a*e^2)*Sqrt[d + e*x]) + (c^(1/4)*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])/Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]])/(Sqrt[2]*(c*
d^2 + a*e^2)^(3/2)*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (c^(1/4)*e*(2*Sqrt[c]*d - Sqrt[c*d^2 + a*e^2])*Arc
Tanh[(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]]])/(Sqrt[2]*(c*d^2 + a*e^2)^(3/2)*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (c^(1/4)*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)])/(2*Sqrt[2]*(c*d^2 + a*e^2)^(3/2)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]) + (c^(1/4)*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)])/(2*Sqrt[2]*(c*d^2 + a*e^2)^(3/2)*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 710

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

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

________________________________________________________________________________________

Mathematica [C]  time = 0.13, size = 132, normalized size = 0.20 \[ \frac {\frac {\, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {-a} e}\right )}{\sqrt {-a} \sqrt {c} d-a e}-\frac {\, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-\sqrt {-a} e}\right )}{\sqrt {-a} \sqrt {c} d+a e}}{\sqrt {d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(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)]/(Sqrt[-a]*Sqrt[c]*d - a*e))/Sqr
t[d + e*x]

________________________________________________________________________________________

fricas [B]  time = 1.27, size = 2863, normalized size = 4.32 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*((c*d^3 + a*d*e^2 + (c*d^2*e + a*e^3)*x)*sqrt(-(c^2*d^3 - 3*a*c*d*e^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)*sqrt(-(9*c^3*d^4*e^2 - 6*a*c^2*d^2*e^4 + a^2*c*e^6)/(a*c^6*d^12 + 6*a^2*c^5*d^10*e^
2 + 15*a^3*c^4*d^8*e^4 + 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 + 6*a^6*c*d^2*e^10 + a^7*e^12)))/(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))*log(-(3*c^2*d^2*e - a*c*e^3)*sqrt(e*x + d) + (6*a*c^2*d^3*e^2
- 2*a^2*c*d*e^4 + (a*c^4*d^8 + 2*a^2*c^3*d^6*e^2 - 2*a^4*c*d^2*e^6 - a^5*e^8)*sqrt(-(9*c^3*d^4*e^2 - 6*a*c^2*
d^2*e^4 + a^2*c*e^6)/(a*c^6*d^12 + 6*a^2*c^5*d^10*e^2 + 15*a^3*c^4*d^8*e^4 + 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d
^4*e^8 + 6*a^6*c*d^2*e^10 + a^7*e^12)))*sqrt(-(c^2*d^3 - 3*a*c*d*e^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)*sqrt(-(9*c^3*d^4*e^2 - 6*a*c^2*d^2*e^4 + a^2*c*e^6)/(a*c^6*d^12 + 6*a^2*c^5*d^10*e^2 + 15
*a^3*c^4*d^8*e^4 + 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 + 6*a^6*c*d^2*e^10 + a^7*e^12)))/(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))) - (c*d^3 + a*d*e^2 + (c*d^2*e + a*e^3)*x)*sqrt(-(c^2*d^3 - 3*a*c*d
*e^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)*sqrt(-(9*c^3*d^4*e^2 - 6*a*c^2*d^2*e^4 + a^
2*c*e^6)/(a*c^6*d^12 + 6*a^2*c^5*d^10*e^2 + 15*a^3*c^4*d^8*e^4 + 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 + 6*a
^6*c*d^2*e^10 + a^7*e^12)))/(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))*log(-(3*c^2*d^2*e - a
*c*e^3)*sqrt(e*x + d) - (6*a*c^2*d^3*e^2 - 2*a^2*c*d*e^4 + (a*c^4*d^8 + 2*a^2*c^3*d^6*e^2 - 2*a^4*c*d^2*e^6 -
a^5*e^8)*sqrt(-(9*c^3*d^4*e^2 - 6*a*c^2*d^2*e^4 + a^2*c*e^6)/(a*c^6*d^12 + 6*a^2*c^5*d^10*e^2 + 15*a^3*c^4*d^8
*e^4 + 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 + 6*a^6*c*d^2*e^10 + a^7*e^12)))*sqrt(-(c^2*d^3 - 3*a*c*d*e^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)*sqrt(-(9*c^3*d^4*e^2 - 6*a*c^2*d^2*e^4 + a^2*c*e^
6)/(a*c^6*d^12 + 6*a^2*c^5*d^10*e^2 + 15*a^3*c^4*d^8*e^4 + 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 + 6*a^6*c*d
^2*e^10 + a^7*e^12)))/(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))) + (c*d^3 + a*d*e^2 + (c*d^
2*e + a*e^3)*x)*sqrt(-(c^2*d^3 - 3*a*c*d*e^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)*sqr
t(-(9*c^3*d^4*e^2 - 6*a*c^2*d^2*e^4 + a^2*c*e^6)/(a*c^6*d^12 + 6*a^2*c^5*d^10*e^2 + 15*a^3*c^4*d^8*e^4 + 20*a^
4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 + 6*a^6*c*d^2*e^10 + a^7*e^12)))/(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))*log(-(3*c^2*d^2*e - a*c*e^3)*sqrt(e*x + d) + (6*a*c^2*d^3*e^2 - 2*a^2*c*d*e^4 - (a*c^4*d^8
+ 2*a^2*c^3*d^6*e^2 - 2*a^4*c*d^2*e^6 - a^5*e^8)*sqrt(-(9*c^3*d^4*e^2 - 6*a*c^2*d^2*e^4 + a^2*c*e^6)/(a*c^6*d^
12 + 6*a^2*c^5*d^10*e^2 + 15*a^3*c^4*d^8*e^4 + 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 + 6*a^6*c*d^2*e^10 + a^
7*e^12)))*sqrt(-(c^2*d^3 - 3*a*c*d*e^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)*sqrt(-(9*
c^3*d^4*e^2 - 6*a*c^2*d^2*e^4 + a^2*c*e^6)/(a*c^6*d^12 + 6*a^2*c^5*d^10*e^2 + 15*a^3*c^4*d^8*e^4 + 20*a^4*c^3*
d^6*e^6 + 15*a^5*c^2*d^4*e^8 + 6*a^6*c*d^2*e^10 + a^7*e^12)))/(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))) - (c*d^3 + a*d*e^2 + (c*d^2*e + a*e^3)*x)*sqrt(-(c^2*d^3 - 3*a*c*d*e^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)*sqrt(-(9*c^3*d^4*e^2 - 6*a*c^2*d^2*e^4 + a^2*c*e^6)/(a*c^6*d^12 + 6*a^2*c
^5*d^10*e^2 + 15*a^3*c^4*d^8*e^4 + 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 + 6*a^6*c*d^2*e^10 + a^7*e^12)))/(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))*log(-(3*c^2*d^2*e - a*c*e^3)*sqrt(e*x + d) - (6*a*c
^2*d^3*e^2 - 2*a^2*c*d*e^4 - (a*c^4*d^8 + 2*a^2*c^3*d^6*e^2 - 2*a^4*c*d^2*e^6 - a^5*e^8)*sqrt(-(9*c^3*d^4*e^2
- 6*a*c^2*d^2*e^4 + a^2*c*e^6)/(a*c^6*d^12 + 6*a^2*c^5*d^10*e^2 + 15*a^3*c^4*d^8*e^4 + 20*a^4*c^3*d^6*e^6 + 15
*a^5*c^2*d^4*e^8 + 6*a^6*c*d^2*e^10 + a^7*e^12)))*sqrt(-(c^2*d^3 - 3*a*c*d*e^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)*sqrt(-(9*c^3*d^4*e^2 - 6*a*c^2*d^2*e^4 + a^2*c*e^6)/(a*c^6*d^12 + 6*a^2*c^5*d^1
0*e^2 + 15*a^3*c^4*d^8*e^4 + 20*a^4*c^3*d^6*e^6 + 15*a^5*c^2*d^4*e^8 + 6*a^6*c*d^2*e^10 + a^7*e^12)))/(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))) + 4*sqrt(e*x + d)*e)/(c*d^3 + a*d*e^2 + (c*d^2*e + a*e^3
)*x)

________________________________________________________________________________________

giac [A]  time = 0.40, size = 646, normalized size = 0.97 \[ \frac {{\left ({\left (c d^{2} e + a e^{3}\right )}^{2} a {\left | c \right |} e + 2 \, {\left (\sqrt {-a c} c d^{3} e + \sqrt {-a c} a d e^{3}\right )} {\left | -c d^{2} e - a e^{3} \right |} {\left | c \right |} - {\left (c^{3} d^{6} e + 2 \, a c^{2} d^{4} e^{3} + a^{2} c d^{2} e^{5}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {c^{2} d^{3} + a c d e^{2} + \sqrt {{\left (c^{2} d^{3} + a c d e^{2}\right )}^{2} - {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} {\left (c^{2} d^{2} + a c e^{2}\right )}}}{c^{2} d^{2} + a c e^{2}}}}\right )}{{\left (a c^{2} d^{4} e - \sqrt {-a c} c^{2} d^{5} - 2 \, \sqrt {-a c} a c d^{3} e^{2} + 2 \, a^{2} c d^{2} e^{3} - \sqrt {-a c} a^{2} d e^{4} + a^{3} e^{5}\right )} \sqrt {-c^{2} d + \sqrt {-a c} c e} {\left | -c d^{2} e - a e^{3} \right |}} + \frac {{\left ({\left (c d^{2} e + a e^{3}\right )}^{2} a {\left | c \right |} e - 2 \, {\left (\sqrt {-a c} c d^{3} e + \sqrt {-a c} a d e^{3}\right )} {\left | -c d^{2} e - a e^{3} \right |} {\left | c \right |} - {\left (c^{3} d^{6} e + 2 \, a c^{2} d^{4} e^{3} + a^{2} c d^{2} e^{5}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {c^{2} d^{3} + a c d e^{2} - \sqrt {{\left (c^{2} d^{3} + a c d e^{2}\right )}^{2} - {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} {\left (c^{2} d^{2} + a c e^{2}\right )}}}{c^{2} d^{2} + a c e^{2}}}}\right )}{{\left (a c^{2} d^{4} e + \sqrt {-a c} c^{2} d^{5} + 2 \, \sqrt {-a c} a c d^{3} e^{2} + 2 \, a^{2} c d^{2} e^{3} + \sqrt {-a c} a^{2} d e^{4} + a^{3} e^{5}\right )} \sqrt {-c^{2} d - \sqrt {-a c} c e} {\left | -c d^{2} e - a e^{3} \right |}} - \frac {2 \, e}{{\left (c d^{2} + a e^{2}\right )} \sqrt {x e + d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((c*d^2*e + a*e^3)^2*a*abs(c)*e + 2*(sqrt(-a*c)*c*d^3*e + sqrt(-a*c)*a*d*e^3)*abs(-c*d^2*e - a*e^3)*abs(c) - (
c^3*d^6*e + 2*a*c^2*d^4*e^3 + a^2*c*d^2*e^5)*abs(c))*arctan(sqrt(x*e + d)/sqrt(-(c^2*d^3 + a*c*d*e^2 + sqrt((c
^2*d^3 + a*c*d*e^2)^2 - (c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4)*(c^2*d^2 + a*c*e^2)))/(c^2*d^2 + a*c*e^2)))/((a*c^
2*d^4*e - sqrt(-a*c)*c^2*d^5 - 2*sqrt(-a*c)*a*c*d^3*e^2 + 2*a^2*c*d^2*e^3 - sqrt(-a*c)*a^2*d*e^4 + a^3*e^5)*sq
rt(-c^2*d + sqrt(-a*c)*c*e)*abs(-c*d^2*e - a*e^3)) + ((c*d^2*e + a*e^3)^2*a*abs(c)*e - 2*(sqrt(-a*c)*c*d^3*e +
sqrt(-a*c)*a*d*e^3)*abs(-c*d^2*e - a*e^3)*abs(c) - (c^3*d^6*e + 2*a*c^2*d^4*e^3 + a^2*c*d^2*e^5)*abs(c))*arct
an(sqrt(x*e + d)/sqrt(-(c^2*d^3 + a*c*d*e^2 - sqrt((c^2*d^3 + a*c*d*e^2)^2 - (c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^
4)*(c^2*d^2 + a*c*e^2)))/(c^2*d^2 + a*c*e^2)))/((a*c^2*d^4*e + sqrt(-a*c)*c^2*d^5 + 2*sqrt(-a*c)*a*c*d^3*e^2 +
2*a^2*c*d^2*e^3 + sqrt(-a*c)*a^2*d*e^4 + a^3*e^5)*sqrt(-c^2*d - sqrt(-a*c)*c*e)*abs(-c*d^2*e - a*e^3)) - 2*e/
((c*d^2 + a*e^2)*sqrt(x*e + d))

________________________________________________________________________________________

maple [B]  time = 0.21, size = 5659, normalized size = 8.54 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 1.30, size = 4471, normalized size = 6.74 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- atan((((d + e*x)^(1/2)*(16*a^4*c^4*e^10 - 16*c^8*d^8*e^2 - 32*a*c^7*d^6*e^4 + 32*a^3*c^5*d^2*e^8) + (-(a*c^2
*d^3 + a*e^3*(-a^3*c)^(1/2) - 3*a^2*c*d*e^2 - 3*c*d^2*e*(-a^3*c)^(1/2))/(4*(a^5*e^6 + a^2*c^3*d^6 + 3*a^4*c*d^
2*e^4 + 3*a^3*c^2*d^4*e^2)))^(1/2)*(64*a*c^8*d^9*e^3 - (d + e*x)^(1/2)*(-(a*c^2*d^3 + a*e^3*(-a^3*c)^(1/2) - 3
*a^2*c*d*e^2 - 3*c*d^2*e*(-a^3*c)^(1/2))/(4*(a^5*e^6 + a^2*c^3*d^6 + 3*a^4*c*d^2*e^4 + 3*a^3*c^2*d^4*e^2)))^(1
/2)*(64*a*c^9*d^11*e^2 + 64*a^6*c^4*d*e^12 + 320*a^2*c^8*d^9*e^4 + 640*a^3*c^7*d^7*e^6 + 640*a^4*c^6*d^5*e^8 +
320*a^5*c^5*d^3*e^10) + 64*a^5*c^4*d*e^11 + 256*a^2*c^7*d^7*e^5 + 384*a^3*c^6*d^5*e^7 + 256*a^4*c^5*d^3*e^9))
*(-(a*c^2*d^3 + a*e^3*(-a^3*c)^(1/2) - 3*a^2*c*d*e^2 - 3*c*d^2*e*(-a^3*c)^(1/2))/(4*(a^5*e^6 + a^2*c^3*d^6 + 3
*a^4*c*d^2*e^4 + 3*a^3*c^2*d^4*e^2)))^(1/2)*1i + ((d + e*x)^(1/2)*(16*a^4*c^4*e^10 - 16*c^8*d^8*e^2 - 32*a*c^7
*d^6*e^4 + 32*a^3*c^5*d^2*e^8) - (-(a*c^2*d^3 + a*e^3*(-a^3*c)^(1/2) - 3*a^2*c*d*e^2 - 3*c*d^2*e*(-a^3*c)^(1/2
))/(4*(a^5*e^6 + a^2*c^3*d^6 + 3*a^4*c*d^2*e^4 + 3*a^3*c^2*d^4*e^2)))^(1/2)*((d + e*x)^(1/2)*(-(a*c^2*d^3 + a*
e^3*(-a^3*c)^(1/2) - 3*a^2*c*d*e^2 - 3*c*d^2*e*(-a^3*c)^(1/2))/(4*(a^5*e^6 + a^2*c^3*d^6 + 3*a^4*c*d^2*e^4 + 3
*a^3*c^2*d^4*e^2)))^(1/2)*(64*a*c^9*d^11*e^2 + 64*a^6*c^4*d*e^12 + 320*a^2*c^8*d^9*e^4 + 640*a^3*c^7*d^7*e^6 +
640*a^4*c^6*d^5*e^8 + 320*a^5*c^5*d^3*e^10) + 64*a*c^8*d^9*e^3 + 64*a^5*c^4*d*e^11 + 256*a^2*c^7*d^7*e^5 + 38
4*a^3*c^6*d^5*e^7 + 256*a^4*c^5*d^3*e^9))*(-(a*c^2*d^3 + a*e^3*(-a^3*c)^(1/2) - 3*a^2*c*d*e^2 - 3*c*d^2*e*(-a^
3*c)^(1/2))/(4*(a^5*e^6 + a^2*c^3*d^6 + 3*a^4*c*d^2*e^4 + 3*a^3*c^2*d^4*e^2)))^(1/2)*1i)/(((d + e*x)^(1/2)*(16
*a^4*c^4*e^10 - 16*c^8*d^8*e^2 - 32*a*c^7*d^6*e^4 + 32*a^3*c^5*d^2*e^8) - (-(a*c^2*d^3 + a*e^3*(-a^3*c)^(1/2)
- 3*a^2*c*d*e^2 - 3*c*d^2*e*(-a^3*c)^(1/2))/(4*(a^5*e^6 + a^2*c^3*d^6 + 3*a^4*c*d^2*e^4 + 3*a^3*c^2*d^4*e^2)))
^(1/2)*((d + e*x)^(1/2)*(-(a*c^2*d^3 + a*e^3*(-a^3*c)^(1/2) - 3*a^2*c*d*e^2 - 3*c*d^2*e*(-a^3*c)^(1/2))/(4*(a^
5*e^6 + a^2*c^3*d^6 + 3*a^4*c*d^2*e^4 + 3*a^3*c^2*d^4*e^2)))^(1/2)*(64*a*c^9*d^11*e^2 + 64*a^6*c^4*d*e^12 + 32
0*a^2*c^8*d^9*e^4 + 640*a^3*c^7*d^7*e^6 + 640*a^4*c^6*d^5*e^8 + 320*a^5*c^5*d^3*e^10) + 64*a*c^8*d^9*e^3 + 64*
a^5*c^4*d*e^11 + 256*a^2*c^7*d^7*e^5 + 384*a^3*c^6*d^5*e^7 + 256*a^4*c^5*d^3*e^9))*(-(a*c^2*d^3 + a*e^3*(-a^3*
c)^(1/2) - 3*a^2*c*d*e^2 - 3*c*d^2*e*(-a^3*c)^(1/2))/(4*(a^5*e^6 + a^2*c^3*d^6 + 3*a^4*c*d^2*e^4 + 3*a^3*c^2*d
^4*e^2)))^(1/2) - ((d + e*x)^(1/2)*(16*a^4*c^4*e^10 - 16*c^8*d^8*e^2 - 32*a*c^7*d^6*e^4 + 32*a^3*c^5*d^2*e^8)
+ (-(a*c^2*d^3 + a*e^3*(-a^3*c)^(1/2) - 3*a^2*c*d*e^2 - 3*c*d^2*e*(-a^3*c)^(1/2))/(4*(a^5*e^6 + a^2*c^3*d^6 +
3*a^4*c*d^2*e^4 + 3*a^3*c^2*d^4*e^2)))^(1/2)*(64*a*c^8*d^9*e^3 - (d + e*x)^(1/2)*(-(a*c^2*d^3 + a*e^3*(-a^3*c)
^(1/2) - 3*a^2*c*d*e^2 - 3*c*d^2*e*(-a^3*c)^(1/2))/(4*(a^5*e^6 + a^2*c^3*d^6 + 3*a^4*c*d^2*e^4 + 3*a^3*c^2*d^4
*e^2)))^(1/2)*(64*a*c^9*d^11*e^2 + 64*a^6*c^4*d*e^12 + 320*a^2*c^8*d^9*e^4 + 640*a^3*c^7*d^7*e^6 + 640*a^4*c^6
*d^5*e^8 + 320*a^5*c^5*d^3*e^10) + 64*a^5*c^4*d*e^11 + 256*a^2*c^7*d^7*e^5 + 384*a^3*c^6*d^5*e^7 + 256*a^4*c^5
*d^3*e^9))*(-(a*c^2*d^3 + a*e^3*(-a^3*c)^(1/2) - 3*a^2*c*d*e^2 - 3*c*d^2*e*(-a^3*c)^(1/2))/(4*(a^5*e^6 + a^2*c
^3*d^6 + 3*a^4*c*d^2*e^4 + 3*a^3*c^2*d^4*e^2)))^(1/2) + 16*a^3*c^4*e^9 + 16*c^7*d^6*e^3 + 48*a*c^6*d^4*e^5 + 4
8*a^2*c^5*d^2*e^7))*(-(a*c^2*d^3 + a*e^3*(-a^3*c)^(1/2) - 3*a^2*c*d*e^2 - 3*c*d^2*e*(-a^3*c)^(1/2))/(4*(a^5*e^
6 + a^2*c^3*d^6 + 3*a^4*c*d^2*e^4 + 3*a^3*c^2*d^4*e^2)))^(1/2)*2i - atan((((d + e*x)^(1/2)*(16*a^4*c^4*e^10 -
16*c^8*d^8*e^2 - 32*a*c^7*d^6*e^4 + 32*a^3*c^5*d^2*e^8) + (-(a*c^2*d^3 - a*e^3*(-a^3*c)^(1/2) - 3*a^2*c*d*e^2
+ 3*c*d^2*e*(-a^3*c)^(1/2))/(4*(a^5*e^6 + a^2*c^3*d^6 + 3*a^4*c*d^2*e^4 + 3*a^3*c^2*d^4*e^2)))^(1/2)*(64*a*c^8
*d^9*e^3 - (d + e*x)^(1/2)*(-(a*c^2*d^3 - a*e^3*(-a^3*c)^(1/2) - 3*a^2*c*d*e^2 + 3*c*d^2*e*(-a^3*c)^(1/2))/(4*
(a^5*e^6 + a^2*c^3*d^6 + 3*a^4*c*d^2*e^4 + 3*a^3*c^2*d^4*e^2)))^(1/2)*(64*a*c^9*d^11*e^2 + 64*a^6*c^4*d*e^12 +
320*a^2*c^8*d^9*e^4 + 640*a^3*c^7*d^7*e^6 + 640*a^4*c^6*d^5*e^8 + 320*a^5*c^5*d^3*e^10) + 64*a^5*c^4*d*e^11 +
256*a^2*c^7*d^7*e^5 + 384*a^3*c^6*d^5*e^7 + 256*a^4*c^5*d^3*e^9))*(-(a*c^2*d^3 - a*e^3*(-a^3*c)^(1/2) - 3*a^2
*c*d*e^2 + 3*c*d^2*e*(-a^3*c)^(1/2))/(4*(a^5*e^6 + a^2*c^3*d^6 + 3*a^4*c*d^2*e^4 + 3*a^3*c^2*d^4*e^2)))^(1/2)*
1i + ((d + e*x)^(1/2)*(16*a^4*c^4*e^10 - 16*c^8*d^8*e^2 - 32*a*c^7*d^6*e^4 + 32*a^3*c^5*d^2*e^8) - (-(a*c^2*d^
3 - a*e^3*(-a^3*c)^(1/2) - 3*a^2*c*d*e^2 + 3*c*d^2*e*(-a^3*c)^(1/2))/(4*(a^5*e^6 + a^2*c^3*d^6 + 3*a^4*c*d^2*e
^4 + 3*a^3*c^2*d^4*e^2)))^(1/2)*((d + e*x)^(1/2)*(-(a*c^2*d^3 - a*e^3*(-a^3*c)^(1/2) - 3*a^2*c*d*e^2 + 3*c*d^2
*e*(-a^3*c)^(1/2))/(4*(a^5*e^6 + a^2*c^3*d^6 + 3*a^4*c*d^2*e^4 + 3*a^3*c^2*d^4*e^2)))^(1/2)*(64*a*c^9*d^11*e^2
+ 64*a^6*c^4*d*e^12 + 320*a^2*c^8*d^9*e^4 + 640*a^3*c^7*d^7*e^6 + 640*a^4*c^6*d^5*e^8 + 320*a^5*c^5*d^3*e^10)
+ 64*a*c^8*d^9*e^3 + 64*a^5*c^4*d*e^11 + 256*a^2*c^7*d^7*e^5 + 384*a^3*c^6*d^5*e^7 + 256*a^4*c^5*d^3*e^9))*(-
(a*c^2*d^3 - a*e^3*(-a^3*c)^(1/2) - 3*a^2*c*d*e^2 + 3*c*d^2*e*(-a^3*c)^(1/2))/(4*(a^5*e^6 + a^2*c^3*d^6 + 3*a^
4*c*d^2*e^4 + 3*a^3*c^2*d^4*e^2)))^(1/2)*1i)/(((d + e*x)^(1/2)*(16*a^4*c^4*e^10 - 16*c^8*d^8*e^2 - 32*a*c^7*d^
6*e^4 + 32*a^3*c^5*d^2*e^8) - (-(a*c^2*d^3 - a*e^3*(-a^3*c)^(1/2) - 3*a^2*c*d*e^2 + 3*c*d^2*e*(-a^3*c)^(1/2))/
(4*(a^5*e^6 + a^2*c^3*d^6 + 3*a^4*c*d^2*e^4 + 3*a^3*c^2*d^4*e^2)))^(1/2)*((d + e*x)^(1/2)*(-(a*c^2*d^3 - a*e^3
*(-a^3*c)^(1/2) - 3*a^2*c*d*e^2 + 3*c*d^2*e*(-a^3*c)^(1/2))/(4*(a^5*e^6 + a^2*c^3*d^6 + 3*a^4*c*d^2*e^4 + 3*a^
3*c^2*d^4*e^2)))^(1/2)*(64*a*c^9*d^11*e^2 + 64*a^6*c^4*d*e^12 + 320*a^2*c^8*d^9*e^4 + 640*a^3*c^7*d^7*e^6 + 64
0*a^4*c^6*d^5*e^8 + 320*a^5*c^5*d^3*e^10) + 64*a*c^8*d^9*e^3 + 64*a^5*c^4*d*e^11 + 256*a^2*c^7*d^7*e^5 + 384*a
^3*c^6*d^5*e^7 + 256*a^4*c^5*d^3*e^9))*(-(a*c^2*d^3 - a*e^3*(-a^3*c)^(1/2) - 3*a^2*c*d*e^2 + 3*c*d^2*e*(-a^3*c
)^(1/2))/(4*(a^5*e^6 + a^2*c^3*d^6 + 3*a^4*c*d^2*e^4 + 3*a^3*c^2*d^4*e^2)))^(1/2) - ((d + e*x)^(1/2)*(16*a^4*c
^4*e^10 - 16*c^8*d^8*e^2 - 32*a*c^7*d^6*e^4 + 32*a^3*c^5*d^2*e^8) + (-(a*c^2*d^3 - a*e^3*(-a^3*c)^(1/2) - 3*a^
2*c*d*e^2 + 3*c*d^2*e*(-a^3*c)^(1/2))/(4*(a^5*e^6 + a^2*c^3*d^6 + 3*a^4*c*d^2*e^4 + 3*a^3*c^2*d^4*e^2)))^(1/2)
*(64*a*c^8*d^9*e^3 - (d + e*x)^(1/2)*(-(a*c^2*d^3 - a*e^3*(-a^3*c)^(1/2) - 3*a^2*c*d*e^2 + 3*c*d^2*e*(-a^3*c)^
(1/2))/(4*(a^5*e^6 + a^2*c^3*d^6 + 3*a^4*c*d^2*e^4 + 3*a^3*c^2*d^4*e^2)))^(1/2)*(64*a*c^9*d^11*e^2 + 64*a^6*c^
4*d*e^12 + 320*a^2*c^8*d^9*e^4 + 640*a^3*c^7*d^7*e^6 + 640*a^4*c^6*d^5*e^8 + 320*a^5*c^5*d^3*e^10) + 64*a^5*c^
4*d*e^11 + 256*a^2*c^7*d^7*e^5 + 384*a^3*c^6*d^5*e^7 + 256*a^4*c^5*d^3*e^9))*(-(a*c^2*d^3 - a*e^3*(-a^3*c)^(1/
2) - 3*a^2*c*d*e^2 + 3*c*d^2*e*(-a^3*c)^(1/2))/(4*(a^5*e^6 + a^2*c^3*d^6 + 3*a^4*c*d^2*e^4 + 3*a^3*c^2*d^4*e^2
)))^(1/2) + 16*a^3*c^4*e^9 + 16*c^7*d^6*e^3 + 48*a*c^6*d^4*e^5 + 48*a^2*c^5*d^2*e^7))*(-(a*c^2*d^3 - a*e^3*(-a
^3*c)^(1/2) - 3*a^2*c*d*e^2 + 3*c*d^2*e*(-a^3*c)^(1/2))/(4*(a^5*e^6 + a^2*c^3*d^6 + 3*a^4*c*d^2*e^4 + 3*a^3*c^
2*d^4*e^2)))^(1/2)*2i - (2*e)/((a*e^2 + c*d^2)*(d + e*x)^(1/2))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________