3.1263 \(\int \frac{(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=516 \[ -\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} \left (24 A c e (2 c d-b e)-B \left (15 b^2 e^2-104 b c d e+128 c^2 d^2\right )\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{15 \sqrt{c} e^5 \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{2 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (3 A e \left (b^2 e^2-16 b c d e+16 c^2 d^2\right )-B d \left (43 b^2 e^2-168 b c d e+128 c^2 d^2\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{15 d e^5 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)}-\frac{2 \sqrt{b x+c x^2} \left (d \left (3 A c e (8 c d-7 b e)-B \left (15 b^2 e^2-76 b c d e+64 c^2 d^2\right )\right )-c e x (B d (16 c d-13 b e)-3 A e (2 c d-b e))\right )}{15 d e^4 \sqrt{d+e x} (c d-b e)}-\frac{2 \left (b x+c x^2\right )^{3/2} \left (d^2 (-3 A c e-5 b B e+8 B c d)+e x (B d (11 c d-8 b e)-3 A e (2 c d-b e))\right )}{15 d e^2 (d+e x)^{5/2} (c d-b e)} \]

[Out]

(-2*(d*(3*A*c*e*(8*c*d - 7*b*e) - B*(64*c^2*d^2 - 76*b*c*d*e + 15*b^2*e^2)) - c*
e*(B*d*(16*c*d - 13*b*e) - 3*A*e*(2*c*d - b*e))*x)*Sqrt[b*x + c*x^2])/(15*d*e^4*
(c*d - b*e)*Sqrt[d + e*x]) - (2*(d^2*(8*B*c*d - 5*b*B*e - 3*A*c*e) + e*(B*d*(11*
c*d - 8*b*e) - 3*A*e*(2*c*d - b*e))*x)*(b*x + c*x^2)^(3/2))/(15*d*e^2*(c*d - b*e
)*(d + e*x)^(5/2)) + (2*Sqrt[-b]*Sqrt[c]*(3*A*e*(16*c^2*d^2 - 16*b*c*d*e + b^2*e
^2) - B*d*(128*c^2*d^2 - 168*b*c*d*e + 43*b^2*e^2))*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sq
rt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(15*d*e^
5*(c*d - b*e)*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (2*Sqrt[-b]*(24*A*c*e*(2*c*
d - b*e) - B*(128*c^2*d^2 - 104*b*c*d*e + 15*b^2*e^2))*Sqrt[x]*Sqrt[1 + (c*x)/b]
*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(
15*Sqrt[c]*e^5*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

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Rubi [A]  time = 1.72529, antiderivative size = 516, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ -\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} \left (24 A c e (2 c d-b e)-B \left (15 b^2 e^2-104 b c d e+128 c^2 d^2\right )\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{15 \sqrt{c} e^5 \sqrt{b x+c x^2} \sqrt{d+e x}}+\frac{2 \sqrt{-b} \sqrt{c} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (3 A e \left (b^2 e^2-16 b c d e+16 c^2 d^2\right )-B d \left (43 b^2 e^2-168 b c d e+128 c^2 d^2\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{15 d e^5 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1} (c d-b e)}-\frac{2 \sqrt{b x+c x^2} \left (d \left (3 A c e (8 c d-7 b e)-B \left (15 b^2 e^2-76 b c d e+64 c^2 d^2\right )\right )-c e x (B d (16 c d-13 b e)-3 A e (2 c d-b e))\right )}{15 d e^4 \sqrt{d+e x} (c d-b e)}-\frac{2 \left (b x+c x^2\right )^{3/2} \left (d^2 (-3 A c e-5 b B e+8 B c d)+e x (B d (11 c d-8 b e)-3 A e (2 c d-b e))\right )}{15 d e^2 (d+e x)^{5/2} (c d-b e)} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(d*(3*A*c*e*(8*c*d - 7*b*e) - B*(64*c^2*d^2 - 76*b*c*d*e + 15*b^2*e^2)) - c*
e*(B*d*(16*c*d - 13*b*e) - 3*A*e*(2*c*d - b*e))*x)*Sqrt[b*x + c*x^2])/(15*d*e^4*
(c*d - b*e)*Sqrt[d + e*x]) - (2*(d^2*(8*B*c*d - 5*b*B*e - 3*A*c*e) + e*(B*d*(11*
c*d - 8*b*e) - 3*A*e*(2*c*d - b*e))*x)*(b*x + c*x^2)^(3/2))/(15*d*e^2*(c*d - b*e
)*(d + e*x)^(5/2)) + (2*Sqrt[-b]*Sqrt[c]*(3*A*e*(16*c^2*d^2 - 16*b*c*d*e + b^2*e
^2) - B*d*(128*c^2*d^2 - 168*b*c*d*e + 43*b^2*e^2))*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sq
rt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(15*d*e^
5*(c*d - b*e)*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (2*Sqrt[-b]*(24*A*c*e*(2*c*
d - b*e) - B*(128*c^2*d^2 - 104*b*c*d*e + 15*b^2*e^2))*Sqrt[x]*Sqrt[1 + (c*x)/b]
*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(
15*Sqrt[c]*e^5*Sqrt[d + e*x]*Sqrt[b*x + 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((B*x+A)*(c*x**2+b*x)**(3/2)/(e*x+d)**(7/2),x)

[Out]

Timed out

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Mathematica [C]  time = 5.62918, size = 530, normalized size = 1.03 \[ \frac{2 (x (b+c x))^{3/2} \left (e x \sqrt{\frac{b}{c}} (b+c x) \left ((d+e x)^2 \left (B d \left (23 b^2 e^2-93 b c d e+73 c^2 d^2\right )-3 A e \left (b^2 e^2-11 b c d e+11 c^2 d^2\right )\right )+3 d^2 (B d-A e) (c d-b e)^2-d (d+e x) (c d-b e) (6 A e (b e-2 c d)+B d (17 c d-11 b e))+5 B c d (d+e x)^3 (c d-b e)\right )+(d+e x)^2 \left (i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (3 A e \left (b^2 e^2-16 b c d e+16 c^2 d^2\right )+B d \left (-43 b^2 e^2+168 b c d e-128 c^2 d^2\right )\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+\sqrt{\frac{b}{c}} (b+c x) (d+e x) \left (3 A e \left (b^2 e^2-16 b c d e+16 c^2 d^2\right )+B d \left (-43 b^2 e^2+168 b c d e-128 c^2 d^2\right )\right )+i b e x^{3/2} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} (c d-b e) (3 A e (b e-8 c d)+4 B d (16 c d-7 b e)) F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )\right )\right )}{15 d e^5 x^2 \sqrt{\frac{b}{c}} (b+c x)^2 (d+e x)^{5/2} (c d-b e)} \]

Antiderivative was successfully verified.

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

[Out]

(2*(x*(b + c*x))^(3/2)*(Sqrt[b/c]*e*x*(b + c*x)*(3*d^2*(B*d - A*e)*(c*d - b*e)^2
- d*(c*d - b*e)*(B*d*(17*c*d - 11*b*e) + 6*A*e*(-2*c*d + b*e))*(d + e*x) + (-3*
A*e*(11*c^2*d^2 - 11*b*c*d*e + b^2*e^2) + B*d*(73*c^2*d^2 - 93*b*c*d*e + 23*b^2*
e^2))*(d + e*x)^2 + 5*B*c*d*(c*d - b*e)*(d + e*x)^3) + (d + e*x)^2*(Sqrt[b/c]*(B
*d*(-128*c^2*d^2 + 168*b*c*d*e - 43*b^2*e^2) + 3*A*e*(16*c^2*d^2 - 16*b*c*d*e +
b^2*e^2))*(b + c*x)*(d + e*x) + I*b*e*(B*d*(-128*c^2*d^2 + 168*b*c*d*e - 43*b^2*
e^2) + 3*A*e*(16*c^2*d^2 - 16*b*c*d*e + b^2*e^2))*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(
e*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] + I*b*e*(c*d
- b*e)*(4*B*d*(16*c*d - 7*b*e) + 3*A*e*(-8*c*d + b*e))*Sqrt[1 + b/(c*x)]*Sqrt[1
+ d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/(15*S
qrt[b/c]*d*e^5*(c*d - b*e)*x^2*(b + c*x)^2*(d + e*x)^(5/2))

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Maple [B]  time = 0.06, size = 4120, normalized size = 8. \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(x*(c*x+b))^(1/2)*(-43*B*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))
*x^2*b^4*d*e^5*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+6*A
*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^4*d*e^5*((c*x+b)/b)^(1/2
)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-86*B*EllipticE(((c*x+b)/b)^(1/2),(
b*e/(b*e-c*d))^(1/2))*x*b^4*d^2*e^4*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/
2)*(-c*x/b)^(1/2)+24*A*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c*
d^3*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-72*A*Ellip
ticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^2*d^4*e^2*((c*x+b)/b)^(1/2)*
(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+48*A*EllipticF(((c*x+b)/b)^(1/2),(b*
e/(b*e-c*d))^(1/2))*b*c^3*d^5*e*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(
-c*x/b)^(1/2)-51*A*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c*d^3*
e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+96*A*EllipticE
(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^2*d^4*e^2*((c*x+b)/b)^(1/2)*(-(e
*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-48*A*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b
*e-c*d))^(1/2))*b*c^3*d^5*e*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x
/b)^(1/2)-119*B*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c*d^4*e^2
*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+232*B*EllipticF((
(c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^2*d^5*e*((c*x+b)/b)^(1/2)*(-(e*x+d
)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+211*B*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-
c*d))^(1/2))*b^3*c*d^4*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/
b)^(1/2)-296*B*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^2*d^5*e*
((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-33*A*x^4*b*c^3*d*e
^5+5*B*x^5*b*c^3*d*e^5+15*B*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b
^4*d^3*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-64*B*x^
2*c^4*d^5*e+3*A*x^4*b^2*c^2*e^6+33*A*x^4*c^4*d^2*e^4-88*B*x^4*c^4*d^3*e^3+3*A*x^
3*b^3*c*e^6+54*A*x^3*c^4*d^3*e^3-144*B*x^3*c^4*d^4*e^2+24*A*x^2*c^4*d^4*e^2-5*B*
x^5*c^4*d^2*e^4+128*B*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b*c
^3*d^4*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+48*A*El
lipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^3*c*d^2*e^4*((c*x+b)/b)^(1/
2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-144*A*EllipticF(((c*x+b)/b)^(1/2)
,(b*e/(b*e-c*d))^(1/2))*x*b^2*c^2*d^3*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d
))^(1/2)*(-c*x/b)^(1/2)+96*A*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*
x*b*c^3*d^4*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-10
2*A*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^3*c*d^2*e^4*((c*x+b)/
b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+192*A*EllipticE(((c*x+b)/b)
^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^2*c^2*d^3*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b
*e-c*d))^(1/2)*(-c*x/b)^(1/2)+464*B*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^
(1/2))*x*b^2*c^2*d^4*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)
^(1/2)-96*A*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b*c^3*d^4*e^2*(
(c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-256*B*EllipticF(((c
*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b*c^3*d^5*e*((c*x+b)/b)^(1/2)*(-(e*x+d)*
c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-238*B*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*
d))^(1/2))*x*b^3*c*d^3*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/
b)^(1/2)+24*A*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b^3*c*d*e^5
*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-72*A*EllipticF(((
c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b^2*c^2*d^2*e^4*((c*x+b)/b)^(1/2)*(-(
e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+48*A*EllipticF(((c*x+b)/b)^(1/2),(b*e/(
b*e-c*d))^(1/2))*x^2*b*c^3*d^3*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2
)*(-c*x/b)^(1/2)-51*A*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b^3
*c*d*e^5*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+96*A*Elli
pticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b^2*c^2*d^2*e^4*((c*x+b)/b)^(
1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-48*A*EllipticE(((c*x+b)/b)^(1/2
),(b*e/(b*e-c*d))^(1/2))*x^2*b*c^3*d^3*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*
d))^(1/2)*(-c*x/b)^(1/2)-119*B*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2)
)*x^2*b^3*c*d^2*e^4*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2
)+232*B*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b^2*c^2*d^3*e^3*(
(c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+30*B*EllipticF(((c*
x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^4*d^2*e^4*((c*x+b)/b)^(1/2)*(-(e*x+d)*c
/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+15*B*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d)
)^(1/2))*x^2*b^4*d*e^5*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(
1/2)-64*B*x*b*c^3*d^5*e-18*B*x^4*b^2*c^2*d*e^5+103*B*x^4*b*c^3*d^2*e^4-33*A*x^3*
b^2*c^2*d*e^5-15*A*x^3*b*c^3*d^2*e^4-23*B*x^3*b^3*c*d*e^5+73*B*x^3*b^2*c^2*d^2*e
^4+85*B*x^3*b*c^3*d^3*e^3-48*A*x^2*b^2*c^2*d^2*e^4+33*A*x^2*b*c^3*d^3*e^3+158*B*
x^2*b^2*c^2*d^3*e^3-68*B*x^2*b*c^3*d^4*e^2-21*A*x*b^2*c^2*d^3*e^3+24*A*x*b*c^3*d
^4*e^2+76*B*x*b^2*c^2*d^4*e^2-35*B*x^2*b^3*c*d^2*e^4-15*B*x*b^3*c*d^3*e^3-128*B*
EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b*c^3*d^4*e^2*((c*x+b)/b)
^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+211*B*EllipticE(((c*x+b)/b)^(
1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b^3*c*d^2*e^4*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e
-c*d))^(1/2)*(-c*x/b)^(1/2)+422*B*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1
/2))*x*b^3*c*d^3*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/
2)-592*B*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^2*c^2*d^4*e^2*((
c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+256*B*EllipticE(((c*
x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b*c^3*d^5*e*((c*x+b)/b)^(1/2)*(-(e*x+d)*c
/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-296*B*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d
))^(1/2))*x^2*b^2*c^2*d^3*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c
*x/b)^(1/2)-43*B*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^4*d^3*e^3*
((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+128*B*EllipticE(((
c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*c^3*d^6*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(
b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+3*A*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(
1/2))*x^2*b^4*e^6*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+
3*A*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^4*d^2*e^4*((c*x+b)/b)^(
1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-128*B*EllipticF(((c*x+b)/b)^(1/
2),(b*e/(b*e-c*d))^(1/2))*b*c^3*d^6*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/
2)*(-c*x/b)^(1/2))/(c*x+b)/x/(b*e-c*d)/(e*x+d)^(5/2)/c/d/e^5

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x^2 + b*x)^(3/2)*(B*x + A)/(e*x + d)^(7/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((B*c*x^3 + A*b*x + (B*b + A*c)*x^2)*sqrt(c*x^2 + b*x)/((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(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^(3/2)*(B*x + A)/(e*x + d)^(7/2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError