3.657 \(\int \frac{\left (a+c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=410 \[ -\frac{8 \sqrt{-a} c^{3/2} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (3 a e^2+4 c d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{5 e^4 \sqrt{a+c x^2} \left (a e^2+c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{32 \sqrt{-a} c^{3/2} d \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{5 e^4 \sqrt{a+c x^2} \sqrt{d+e x}}-\frac{4 c \sqrt{a+c x^2} \left (e x \left (3 a e^2+5 c d^2\right )+2 d \left (a e^2+2 c d^2\right )\right )}{5 e^3 (d+e x)^{3/2} \left (a e^2+c d^2\right )}-\frac{2 \left (a+c x^2\right )^{3/2}}{5 e (d+e x)^{5/2}} \]

[Out]

(-4*c*(2*d*(2*c*d^2 + a*e^2) + e*(5*c*d^2 + 3*a*e^2)*x)*Sqrt[a + c*x^2])/(5*e^3*
(c*d^2 + a*e^2)*(d + e*x)^(3/2)) - (2*(a + c*x^2)^(3/2))/(5*e*(d + e*x)^(5/2)) -
(8*Sqrt[-a]*c^(3/2)*(4*c*d^2 + 3*a*e^2)*Sqrt[d + e*x]*Sqrt[1 + (c*x^2)/a]*Ellip
ticE[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*
d - a*e)])/(5*e^4*(c*d^2 + a*e^2)*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]
*e)]*Sqrt[a + c*x^2]) + (32*Sqrt[-a]*c^(3/2)*d*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]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(5*e^4*Sqrt[d + e*x]*Sqrt[
a + c*x^2])

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Rubi [A]  time = 1.01069, antiderivative size = 410, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ -\frac{8 \sqrt{-a} c^{3/2} \sqrt{\frac{c x^2}{a}+1} \sqrt{d+e x} \left (3 a e^2+4 c d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{5 e^4 \sqrt{a+c x^2} \left (a e^2+c d^2\right ) \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}}}+\frac{32 \sqrt{-a} c^{3/2} d \sqrt{\frac{c x^2}{a}+1} \sqrt{\frac{\sqrt{c} (d+e x)}{\sqrt{-a} e+\sqrt{c} d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a e}{\sqrt{-a} \sqrt{c} d-a e}\right )}{5 e^4 \sqrt{a+c x^2} \sqrt{d+e x}}-\frac{4 c \sqrt{a+c x^2} \left (e x \left (3 a e^2+5 c d^2\right )+2 d \left (a e^2+2 c d^2\right )\right )}{5 e^3 (d+e x)^{3/2} \left (a e^2+c d^2\right )}-\frac{2 \left (a+c x^2\right )^{3/2}}{5 e (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + c*x^2)^(3/2)/(d + e*x)^(7/2),x]

[Out]

(-4*c*(2*d*(2*c*d^2 + a*e^2) + e*(5*c*d^2 + 3*a*e^2)*x)*Sqrt[a + c*x^2])/(5*e^3*
(c*d^2 + a*e^2)*(d + e*x)^(3/2)) - (2*(a + c*x^2)^(3/2))/(5*e*(d + e*x)^(5/2)) -
(8*Sqrt[-a]*c^(3/2)*(4*c*d^2 + 3*a*e^2)*Sqrt[d + e*x]*Sqrt[1 + (c*x^2)/a]*Ellip
ticE[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*
d - a*e)])/(5*e^4*(c*d^2 + a*e^2)*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[-a]
*e)]*Sqrt[a + c*x^2]) + (32*Sqrt[-a]*c^(3/2)*d*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]]/Sqrt[2]], (-2*a*e)/(Sqrt[-a]*Sqrt[c]*d - a*e)])/(5*e^4*Sqrt[d + e*x]*Sqrt[
a + c*x^2])

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Rubi in Sympy [A]  time = 144.497, size = 386, normalized size = 0.94 \[ \frac{32 c^{\frac{3}{2}} d \sqrt{- a} \sqrt{\frac{\sqrt{c} \sqrt{- a} \left (- d - e x\right )}{a e - \sqrt{c} d \sqrt{- a}}} \sqrt{1 + \frac{c x^{2}}{a}} F\left (\operatorname{asin}{\left (\sqrt{- \frac{\sqrt{c} x}{2 \sqrt{- a}} + \frac{1}{2}} \right )}\middle | \frac{2 a e}{a e - \sqrt{c} d \sqrt{- a}}\right )}{5 e^{4} \sqrt{a + c x^{2}} \sqrt{d + e x}} - \frac{8 c^{\frac{3}{2}} \sqrt{- a} \sqrt{1 + \frac{c x^{2}}{a}} \sqrt{d + e x} \left (3 a e^{2} + 4 c d^{2}\right ) E\left (\operatorname{asin}{\left (\sqrt{- \frac{\sqrt{c} x}{2 \sqrt{- a}} + \frac{1}{2}} \right )}\middle | \frac{2 a e}{a e - \sqrt{c} d \sqrt{- a}}\right )}{5 e^{4} \sqrt{\frac{\sqrt{c} \sqrt{- a} \left (- d - e x\right )}{a e - \sqrt{c} d \sqrt{- a}}} \sqrt{a + c x^{2}} \left (a e^{2} + c d^{2}\right )} - \frac{8 c \sqrt{a + c x^{2}} \left (d \left (a e^{2} + 2 c d^{2}\right ) + \frac{e x \left (3 a e^{2} + 5 c d^{2}\right )}{2}\right )}{5 e^{3} \left (d + e x\right )^{\frac{3}{2}} \left (a e^{2} + c d^{2}\right )} - \frac{2 \left (a + c x^{2}\right )^{\frac{3}{2}}}{5 e \left (d + e x\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+a)**(3/2)/(e*x+d)**(7/2),x)

[Out]

32*c**(3/2)*d*sqrt(-a)*sqrt(sqrt(c)*sqrt(-a)*(-d - e*x)/(a*e - sqrt(c)*d*sqrt(-a
)))*sqrt(1 + c*x**2/a)*elliptic_f(asin(sqrt(-sqrt(c)*x/(2*sqrt(-a)) + 1/2)), 2*a
*e/(a*e - sqrt(c)*d*sqrt(-a)))/(5*e**4*sqrt(a + c*x**2)*sqrt(d + e*x)) - 8*c**(3
/2)*sqrt(-a)*sqrt(1 + c*x**2/a)*sqrt(d + e*x)*(3*a*e**2 + 4*c*d**2)*elliptic_e(a
sin(sqrt(-sqrt(c)*x/(2*sqrt(-a)) + 1/2)), 2*a*e/(a*e - sqrt(c)*d*sqrt(-a)))/(5*e
**4*sqrt(sqrt(c)*sqrt(-a)*(-d - e*x)/(a*e - sqrt(c)*d*sqrt(-a)))*sqrt(a + c*x**2
)*(a*e**2 + c*d**2)) - 8*c*sqrt(a + c*x**2)*(d*(a*e**2 + 2*c*d**2) + e*x*(3*a*e*
*2 + 5*c*d**2)/2)/(5*e**3*(d + e*x)**(3/2)*(a*e**2 + c*d**2)) - 2*(a + c*x**2)**
(3/2)/(5*e*(d + e*x)**(5/2))

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Mathematica [C]  time = 5.13107, size = 602, normalized size = 1.47 \[ \frac{2 \left (-e^2 \left (a+c x^2\right ) \left (-4 c d (d+e x) \left (a e^2+c d^2\right )+c (d+e x)^2 \left (7 a e^2+11 c d^2\right )+\left (a e^2+c d^2\right )^2\right )+\frac{4 c (d+e x)^2 \left (\sqrt{c} (d+e x)^{3/2} \left (3 a^{3/2} e^3+4 \sqrt{a} c d^2 e-3 i a \sqrt{c} d e^2-4 i c^{3/2} d^3\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )+e^2 \sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}} \left (3 a^2 e^2+a c \left (4 d^2+3 e^2 x^2\right )+4 c^2 d^2 x^2\right )-\sqrt{a} \sqrt{c} e (d+e x)^{3/2} \left (i \sqrt{a} \sqrt{c} d e+3 a e^2+4 c d^2\right ) \sqrt{\frac{e \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{d+e x}} \sqrt{-\frac{-e x+\frac{i \sqrt{a} e}{\sqrt{c}}}{d+e x}} F\left (i \sinh ^{-1}\left (\frac{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}{\sqrt{d+e x}}\right )|\frac{\sqrt{c} d-i \sqrt{a} e}{\sqrt{c} d+i \sqrt{a} e}\right )\right )}{\sqrt{-d-\frac{i \sqrt{a} e}{\sqrt{c}}}}\right )}{5 e^5 \sqrt{a+c x^2} (d+e x)^{5/2} \left (a e^2+c d^2\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + c*x^2)^(3/2)/(d + e*x)^(7/2),x]

[Out]

(2*(-(e^2*(a + c*x^2)*((c*d^2 + a*e^2)^2 - 4*c*d*(c*d^2 + a*e^2)*(d + e*x) + c*(
11*c*d^2 + 7*a*e^2)*(d + e*x)^2)) + (4*c*(d + e*x)^2*(e^2*Sqrt[-d - (I*Sqrt[a]*e
)/Sqrt[c]]*(3*a^2*e^2 + 4*c^2*d^2*x^2 + a*c*(4*d^2 + 3*e^2*x^2)) + Sqrt[c]*((-4*
I)*c^(3/2)*d^3 + 4*Sqrt[a]*c*d^2*e - (3*I)*a*Sqrt[c]*d*e^2 + 3*a^(3/2)*e^3)*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)*EllipticE[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[a]*S
qrt[c]*e*(4*c*d^2 + I*Sqrt[a]*Sqrt[c]*d*e + 3*a*e^2)*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]]
))/(5*e^5*(c*d^2 + a*e^2)*(d + e*x)^(5/2)*Sqrt[a + c*x^2])

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Maple [B]  time = 0.046, size = 3410, normalized size = 8.3 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+a)^(3/2)/(e*x+d)^(7/2),x)

[Out]

-2/5*(24*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*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)+56*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*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)-24*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*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)-24*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*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)-32*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*c^2*d^4*e^2*(-a*c)^(1/2)*(-(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*c*d^3*e^3*
(-a*c)^(1/2)*(-(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)+a^3*e^
6+28*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^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)-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*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)-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))*x^2*c
^2*d^3*e^3*(-a*c)^(1/2)*(-(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)+10*x^3*a*c^2*d*e^5-12*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*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)+5*a^2*c*d^2*e^4+8*c^2*d^4*a*e^2+28*E
llipticE((-(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^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))*x^2*a*c*d*e^5*(-a*c)^(1/2)*(-(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)-32*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*a*c*
d^2*e^4*(-a*c)^(1/2)*(-(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*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))*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)+32*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*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)-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*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*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*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)+16*Ell
ipticE((-(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^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))*c^2*d^5*e*(-a*c)^(1/2)*(-(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))*a^2*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*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*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)+7*x^4*a*c^2*e^6+11*x^4*c^3*d^2*e^4+18*x^3*c^3*d^3*e^3+8*x^2*a^2*c*e^6+8*x^2
*c^3*d^4*e^2+16*x^2*a*c^2*d^2*e^4+10*x*a^2*c*d*e^5+18*x*a*c^2*d^3*e^3)/(c*x^2+a)
^(1/2)/(a*e^2+c*d^2)/(e*x+d)^(5/2)/e^5

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + a\right )}^{\frac{3}{2}}}{{\left (e x + d\right )}^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(3/2)/(e*x + d)^(7/2),x, algorithm="maxima")

[Out]

integrate((c*x^2 + a)^(3/2)/(e*x + d)^(7/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(3/2)/(e*x + d)^(7/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + c x^{2}\right )^{\frac{3}{2}}}{\left (d + e x\right )^{\frac{7}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+a)**(3/2)/(e*x+d)**(7/2),x)

[Out]

Integral((a + c*x**2)**(3/2)/(d + e*x)**(7/2), 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((c*x^2 + a)^(3/2)/(e*x + d)^(7/2),x, algorithm="giac")

[Out]

Exception raised: RuntimeError