3.687 \(\int \frac{1}{(d+e x)^{3/2} \left (a+c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=532 \[ \frac{e \sqrt{a+c x^2} \left (-21 a^2 e^4+15 a c d^2 e^2+4 c^2 d^4\right )}{6 a^2 \sqrt{d+e x} \left (a e^2+c d^2\right )^3}+\frac{\sqrt{c} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (-21 a^2 e^4+15 a c d^2 e^2+4 c^2 d^4\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{6 (-a)^{3/2} \sqrt{a+c x^2} \left (a e^2+c d^2\right )^3 \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}-\frac{a e \left (c d^2-7 a e^2\right )-4 c d x \left (3 a e^2+c d^2\right )}{6 a^2 \sqrt{a+c x^2} \sqrt{d+e x} \left (a e^2+c d^2\right )^2}+\frac{a e+c d x}{3 a \left (a+c x^2\right )^{3/2} \sqrt{d+e x} \left (a e^2+c d^2\right )}-\frac{2 \sqrt{c} d \sqrt{\frac{c x^2}{a}+1} \left (3 a e^2+c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{3 (-a)^{3/2} \sqrt{a+c x^2} \sqrt{d+e x} \left (a e^2+c d^2\right )^2} \]

[Out]

(a*e + c*d*x)/(3*a*(c*d^2 + a*e^2)*Sqrt[d + e*x]*(a + c*x^2)^(3/2)) - (a*e*(c*d^
2 - 7*a*e^2) - 4*c*d*(c*d^2 + 3*a*e^2)*x)/(6*a^2*(c*d^2 + a*e^2)^2*Sqrt[d + e*x]
*Sqrt[a + c*x^2]) + (e*(4*c^2*d^4 + 15*a*c*d^2*e^2 - 21*a^2*e^4)*Sqrt[a + c*x^2]
)/(6*a^2*(c*d^2 + a*e^2)^3*Sqrt[d + e*x]) + (Sqrt[c]*(4*c^2*d^4 + 15*a*c*d^2*e^2
- 21*a^2*e^4)*Sqrt[d + e*x]*Sqrt[1 + (c*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt
[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(6*(-a)^(3/2)*(
c*d^2 + a*e^2)^3*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x
^2]) - (2*Sqrt[c]*d*(c*d^2 + 3*a*e^2)*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt
[-a]*e)]*Sqrt[1 + (c*x^2)/a]*EllipticF[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqr
t[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(3*(-a)^(3/2)*(c*d^2 + a*e^2)^2*Sqr
t[d + e*x]*Sqrt[a + c*x^2])

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Rubi [A]  time = 1.53672, antiderivative size = 532, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{e \sqrt{a+c x^2} \left (-21 a^2 e^4+15 a c d^2 e^2+4 c^2 d^4\right )}{6 a^2 \sqrt{d+e x} \left (a e^2+c d^2\right )^3}+\frac{\sqrt{c} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (-21 a^2 e^4+15 a c d^2 e^2+4 c^2 d^4\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{6 (-a)^{3/2} \sqrt{a+c x^2} \left (a e^2+c d^2\right )^3 \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}-\frac{a e \left (c d^2-7 a e^2\right )-4 c d x \left (3 a e^2+c d^2\right )}{6 a^2 \sqrt{a+c x^2} \sqrt{d+e x} \left (a e^2+c d^2\right )^2}+\frac{a e+c d x}{3 a \left (a+c x^2\right )^{3/2} \sqrt{d+e x} \left (a e^2+c d^2\right )}-\frac{2 \sqrt{c} d \sqrt{\frac{c x^2}{a}+1} \left (3 a e^2+c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{3 (-a)^{3/2} \sqrt{a+c x^2} \sqrt{d+e x} \left (a e^2+c d^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^(3/2)*(a + c*x^2)^(5/2)),x]

[Out]

(a*e + c*d*x)/(3*a*(c*d^2 + a*e^2)*Sqrt[d + e*x]*(a + c*x^2)^(3/2)) - (a*e*(c*d^
2 - 7*a*e^2) - 4*c*d*(c*d^2 + 3*a*e^2)*x)/(6*a^2*(c*d^2 + a*e^2)^2*Sqrt[d + e*x]
*Sqrt[a + c*x^2]) + (e*(4*c^2*d^4 + 15*a*c*d^2*e^2 - 21*a^2*e^4)*Sqrt[a + c*x^2]
)/(6*a^2*(c*d^2 + a*e^2)^3*Sqrt[d + e*x]) + (Sqrt[c]*(4*c^2*d^4 + 15*a*c*d^2*e^2
- 21*a^2*e^4)*Sqrt[d + e*x]*Sqrt[1 + (c*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt
[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(6*(-a)^(3/2)*(
c*d^2 + a*e^2)^3*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]*e)]*Sqrt[a + c*x
^2]) - (2*Sqrt[c]*d*(c*d^2 + 3*a*e^2)*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt
[-a]*e)]*Sqrt[1 + (c*x^2)/a]*EllipticF[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqr
t[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(3*(-a)^(3/2)*(c*d^2 + a*e^2)^2*Sqr
t[d + e*x]*Sqrt[a + c*x^2])

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**(3/2)/(c*x**2+a)**(5/2),x)

[Out]

Timed out

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Mathematica [C]  time = 8.06839, size = 669, normalized size = 1.26 \[ \frac{21 a^3 e^5+c (d+e x) \left (3 a^2 e^3 (7 d-3 e x)+a c d^2 e (d+15 e x)+4 c^2 d^4 x\right )-\frac{i c (d+e x)^{3/2} \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left (-21 a^2 e^4+15 a c d^2 e^2+4 c^2 d^4\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )}{e}+3 a^2 c e^3 \left (7 e^2 x^2-5 d^2\right )-12 a^2 e^5 \left (a+c x^2\right )+\frac{\sqrt{a} \sqrt{c} (d+e x)^{3/2} \left (33 i a^{3/2} \sqrt{c} d e^3-21 a^2 e^4+i \sqrt{a} c^{3/2} d^3 e+15 a c d^2 e^2+4 c^2 d^4\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )}{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}-a c^2 d^2 e \left (4 d^2+15 e^2 x^2\right )+\frac{2 a c (d+e x) \left (a e^2+c d^2\right ) \left (a e (2 d-e x)+c d^2 x\right )}{a+c x^2}-4 c^3 d^4 e x^2}{6 a^2 \sqrt{a+c x^2} \sqrt{d+e x} \left (a e^2+c d^2\right )^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^(3/2)*(a + c*x^2)^(5/2)),x]

[Out]

(21*a^3*e^5 - 4*c^3*d^4*e*x^2 - 12*a^2*e^5*(a + c*x^2) + 3*a^2*c*e^3*(-5*d^2 + 7
*e^2*x^2) - a*c^2*d^2*e*(4*d^2 + 15*e^2*x^2) + (2*a*c*(c*d^2 + a*e^2)*(d + e*x)*
(c*d^2*x + a*e*(2*d - e*x)))/(a + c*x^2) + c*(d + e*x)*(4*c^2*d^4*x + 3*a^2*e^3*
(7*d - 3*e*x) + a*c*d^2*e*(d + 15*e*x)) - (I*c*Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]*
(4*c^2*d^4 + 15*a*c*d^2*e^2 - 21*a^2*e^4)*Sqrt[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d
+ e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d + e*x)^(3/2)*Ellipti
cE[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqr
t[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)])/e + (Sqrt[a]*Sqrt[c]*(4*c^2*d^4 + I*Sqrt[a]*
c^(3/2)*d^3*e + 15*a*c*d^2*e^2 + (33*I)*a^(3/2)*Sqrt[c]*d*e^3 - 21*a^2*e^4)*Sqrt
[(e*((I*Sqrt[a])/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((I*Sqrt[a]*e)/Sqrt[c] - e*x)/(
d + e*x))]*(d + e*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-d - (I*Sqrt[a]*e)/Sqrt[c]]/
Sqrt[d + e*x]], (Sqrt[c]*d - I*Sqrt[a]*e)/(Sqrt[c]*d + I*Sqrt[a]*e)])/Sqrt[-d -
(I*Sqrt[a]*e)/Sqrt[c]])/(6*a^2*(c*d^2 + a*e^2)^3*Sqrt[d + e*x]*Sqrt[a + c*x^2])

_______________________________________________________________________________________

Maple [B]  time = 0.123, size = 3322, normalized size = 6.2 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^(3/2)/(c*x^2+a)^(5/2),x)

[Out]

1/6*(25*a^3*c*d^2*e^4+5*a^2*c^2*d^4*e^2+18*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e
-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*x^2*a^2*c^2*d^2
*e^4*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2
)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)-3*EllipticF((-
(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d
))^(1/2))*x^2*a*c^3*d^4*e^2*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c
)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d
))^(1/2)-4*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c
*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*x^2*c^3*d^5*e*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))
^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/
((-a*c)^(1/2)*e-c*d))^(1/2)*(-a*c)^(1/2)-6*EllipticE((-(e*x+d)*c/((-a*c)^(1/2)*e
-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*x^2*a^2*c^2*d^2
*e^4*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2
)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)+19*EllipticE((
-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*
d))^(1/2))*x^2*a*c^3*d^4*e^2*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*
c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*
d))^(1/2)-16*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e
-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*a^2*c*d^3*e^3*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d
))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*
e/((-a*c)^(1/2)*e-c*d))^(1/2)*(-a*c)^(1/2)-4*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)
*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*a*c^2*d^5*e*(
-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*
d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*(-a*c)^(1/2)-12*a^4*
e^6+21*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/
((-a*c)^(1/2)*e+c*d))^(1/2))*a^4*e^6*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-
c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1
/2)*e-c*d))^(1/2)+19*EllipticE((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)
^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*a^2*c^2*d^4*e^2*(-(e*x+d)*c/((-a*c)^(
1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*
c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)-12*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e
-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*x^2*a^2*c*d*e^5
*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+
c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)*(-a*c)^(1/2)-16*El
lipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(
1/2)*e+c*d))^(1/2))*x^2*a*c^2*d^3*e^3*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((
-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(
1/2)*e-c*d))^(1/2)*(-a*c)^(1/2)-21*EllipticE((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(
1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*a^4*e^6*(-(e*x+d)*c/((-
a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*
x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)+4*EllipticE((-(e*x+d)*c/((-a*c)^(1
/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*x^2*c^4*d^
6*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e
+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)+4*EllipticE((-(e*
x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^
(1/2))*a*c^3*d^6*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/
((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)-12
*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c
)^(1/2)*e+c*d))^(1/2))*a^3*d*e^5*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+
(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*
e-c*d))^(1/2)*(-a*c)^(1/2)-3*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(
-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*a^2*c^2*d^4*e^2*(-(e*x+d)*c/(
(-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((
c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)-6*EllipticE((-(e*x+d)*c/((-a*c)^
(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*a^3*c*d^
2*e^4*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/
2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)+21*EllipticF(
(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c
*d))^(1/2))*x^2*a^3*c*e^6*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^
(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))
^(1/2)-21*EllipticE((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),(-((-a*c)^(1/2)*e-c*
d)/((-a*c)^(1/2)*e+c*d))^(1/2))*x^2*a^3*c*e^6*(-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^
(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c*x+(-a*c)^(1/2))*e/(
(-a*c)^(1/2)*e-c*d))^(1/2)+18*EllipticF((-(e*x+d)*c/((-a*c)^(1/2)*e-c*d))^(1/2),
(-((-a*c)^(1/2)*e-c*d)/((-a*c)^(1/2)*e+c*d))^(1/2))*a^3*c*d^2*e^4*(-(e*x+d)*c/((
-a*c)^(1/2)*e-c*d))^(1/2)*((-c*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e+c*d))^(1/2)*((c
*x+(-a*c)^(1/2))*e/((-a*c)^(1/2)*e-c*d))^(1/2)+12*x^3*a^2*c^2*d*e^5+4*x^4*c^4*d^
4*e^2-21*x^4*a^2*c^2*e^6-35*x^2*a^3*c*e^6+4*x^3*c^4*d^5*e+16*x^3*a*c^3*d^3*e^3+1
4*x*a^3*c*d*e^5+20*x*a^2*c^2*d^3*e^3+6*x*a*c^3*d^5*e+36*x^2*a^2*c^2*d^2*e^4+7*x^
2*a*c^3*d^4*e^2+15*x^4*a*c^3*d^2*e^4)/(e*x+d)^(1/2)/(a*e^2+c*d^2)^3/a^2/(c*x^2+a
)^(3/2)/e

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2} + a\right )}^{\frac{5}{2}}{\left (e x + d\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + a)^(5/2)*(e*x + d)^(3/2)),x, algorithm="maxima")

[Out]

integrate(1/((c*x^2 + a)^(5/2)*(e*x + d)^(3/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (c^{2} e x^{5} + c^{2} d x^{4} + 2 \, a c e x^{3} + 2 \, a c d x^{2} + a^{2} e x + a^{2} d\right )} \sqrt{c x^{2} + a} \sqrt{e x + d}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + a)^(5/2)*(e*x + d)^(3/2)),x, algorithm="fricas")

[Out]

integral(1/((c^2*e*x^5 + c^2*d*x^4 + 2*a*c*e*x^3 + 2*a*c*d*x^2 + a^2*e*x + a^2*d
)*sqrt(c*x^2 + a)*sqrt(e*x + d)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + c x^{2}\right )^{\frac{5}{2}} \left (d + e x\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)**(3/2)/(c*x**2+a)**(5/2),x)

[Out]

Integral(1/((a + c*x**2)**(5/2)*(d + e*x)**(3/2)), x)

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + a)^(5/2)*(e*x + d)^(3/2)),x, algorithm="giac")

[Out]

Exception raised: RuntimeError