3.1262 \(\int \frac{(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx\)

Optimal. Leaf size=413 \[ \frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} \left (5 A e \left (3 b^2 e^2-16 b c d e+16 c^2 d^2\right )-B d \left (39 b^2 e^2-152 b c d e+128 c^2 d^2\right )\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{15 \sqrt{c} e^5 \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (40 A c e (2 c d-b e)-B \left (3 b^2 e^2-88 b c d e+128 c^2 d^2\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{15 \sqrt{c} e^5 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}-\frac{2 \sqrt{b x+c x^2} (e x (-10 A c e-3 b B e+16 B c d)-5 A e (8 c d-3 b e)+4 B d (16 c d-9 b e))}{15 e^4 \sqrt{d+e x}}+\frac{2 \left (b x+c x^2\right )^{3/2} (-5 A e+8 B d+3 B e x)}{15 e^2 (d+e x)^{3/2}} \]

[Out]

(-2*(4*B*d*(16*c*d - 9*b*e) - 5*A*e*(8*c*d - 3*b*e) + e*(16*B*c*d - 3*b*B*e - 10
*A*c*e)*x)*Sqrt[b*x + c*x^2])/(15*e^4*Sqrt[d + e*x]) + (2*(8*B*d - 5*A*e + 3*B*e
*x)*(b*x + c*x^2)^(3/2))/(15*e^2*(d + e*x)^(3/2)) - (2*Sqrt[-b]*(40*A*c*e*(2*c*d
- b*e) - B*(128*c^2*d^2 - 88*b*c*d*e + 3*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*Sqrt
[c]*e^5*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (2*Sqrt[-b]*(5*A*e*(16*c^2*d^2 -
16*b*c*d*e + 3*b^2*e^2) - B*d*(128*c^2*d^2 - 152*b*c*d*e + 39*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 = 1.30608, antiderivative size = 413, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{\frac{e x}{d}+1} \left (5 A e \left (3 b^2 e^2-16 b c d e+16 c^2 d^2\right )-B d \left (39 b^2 e^2-152 b c d e+128 c^2 d^2\right )\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{15 \sqrt{c} e^5 \sqrt{b x+c x^2} \sqrt{d+e x}}-\frac{2 \sqrt{-b} \sqrt{x} \sqrt{\frac{c x}{b}+1} \sqrt{d+e x} \left (40 A c e (2 c d-b e)-B \left (3 b^2 e^2-88 b c d e+128 c^2 d^2\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{-b}}\right )|\frac{b e}{c d}\right )}{15 \sqrt{c} e^5 \sqrt{b x+c x^2} \sqrt{\frac{e x}{d}+1}}-\frac{2 \sqrt{b x+c x^2} (e x (-10 A c e-3 b B e+16 B c d)-5 A e (8 c d-3 b e)+4 B d (16 c d-9 b e))}{15 e^4 \sqrt{d+e x}}+\frac{2 \left (b x+c x^2\right )^{3/2} (-5 A e+8 B d+3 B e x)}{15 e^2 (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(b*x + c*x^2)^(3/2))/(d + e*x)^(5/2),x]

[Out]

(-2*(4*B*d*(16*c*d - 9*b*e) - 5*A*e*(8*c*d - 3*b*e) + e*(16*B*c*d - 3*b*B*e - 10
*A*c*e)*x)*Sqrt[b*x + c*x^2])/(15*e^4*Sqrt[d + e*x]) + (2*(8*B*d - 5*A*e + 3*B*e
*x)*(b*x + c*x^2)^(3/2))/(15*e^2*(d + e*x)^(3/2)) - (2*Sqrt[-b]*(40*A*c*e*(2*c*d
- b*e) - B*(128*c^2*d^2 - 88*b*c*d*e + 3*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*Sqrt
[c]*e^5*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (2*Sqrt[-b]*(5*A*e*(16*c^2*d^2 -
16*b*c*d*e + 3*b^2*e^2) - B*d*(128*c^2*d^2 - 152*b*c*d*e + 39*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 [A]  time = 137.445, size = 428, normalized size = 1.04 \[ - \frac{4 \left (b x + c x^{2}\right )^{\frac{3}{2}} \left (\frac{5 A e}{2} - 4 B d - \frac{3 B e x}{2}\right )}{15 e^{2} \left (d + e x\right )^{\frac{3}{2}}} - \frac{8 \sqrt{b x + c x^{2}} \left (\frac{15 A b e^{2}}{4} - 10 A c d e - 9 B b d e + 16 B c d^{2} - \frac{e x \left (10 A c e + 3 B b e - 16 B c d\right )}{4}\right )}{15 e^{4} \sqrt{d + e x}} + \frac{2 \sqrt{x} \sqrt{- b} \sqrt{1 + \frac{c x}{b}} \sqrt{1 + \frac{e x}{d}} \left (15 A b^{2} e^{3} - 80 A b c d e^{2} + 80 A c^{2} d^{2} e - 39 B b^{2} d e^{2} + 152 B b c d^{2} e - 128 B c^{2} d^{3}\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{- b}} \right )}\middle | \frac{b e}{c d}\right )}{15 \sqrt{c} e^{5} \sqrt{d + e x} \sqrt{b x + c x^{2}}} + \frac{2 \sqrt{x} \sqrt{- b} \sqrt{1 + \frac{c x}{b}} \sqrt{d + e x} \left (6 b c e \left (5 A e - 8 B d\right ) + \left (b e - 8 c d\right ) \left (10 A c e + 3 B b e - 16 B c d\right )\right ) E\left (\operatorname{asin}{\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{- b}} \right )}\middle | \frac{b e}{c d}\right )}{15 \sqrt{c} e^{5} \sqrt{1 + \frac{e x}{d}} \sqrt{b x + c x^{2}}} \]

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)**(5/2),x)

[Out]

-4*(b*x + c*x**2)**(3/2)*(5*A*e/2 - 4*B*d - 3*B*e*x/2)/(15*e**2*(d + e*x)**(3/2)
) - 8*sqrt(b*x + c*x**2)*(15*A*b*e**2/4 - 10*A*c*d*e - 9*B*b*d*e + 16*B*c*d**2 -
e*x*(10*A*c*e + 3*B*b*e - 16*B*c*d)/4)/(15*e**4*sqrt(d + e*x)) + 2*sqrt(x)*sqrt
(-b)*sqrt(1 + c*x/b)*sqrt(1 + e*x/d)*(15*A*b**2*e**3 - 80*A*b*c*d*e**2 + 80*A*c*
*2*d**2*e - 39*B*b**2*d*e**2 + 152*B*b*c*d**2*e - 128*B*c**2*d**3)*elliptic_f(as
in(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)) + 2*sqrt(x)*sqrt(-b)*sqrt(1 + c*x/b)*sqrt(d + e*x)*(6*b*c*e*(5*A*e -
8*B*d) + (b*e - 8*c*d)*(10*A*c*e + 3*B*b*e - 16*B*c*d))*elliptic_e(asin(sqrt(c)
*sqrt(x)/sqrt(-b)), b*e/(c*d))/(15*sqrt(c)*e**5*sqrt(1 + e*x/d)*sqrt(b*x + c*x**
2))

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Mathematica [C]  time = 3.52706, size = 436, normalized size = 1.06 \[ \frac{2 (x (b+c x))^{3/2} \left (i e x^{3/2} \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (5 A c e (8 c d-5 b e)+B \left (-3 b^2 e^2+52 b c d e-64 c^2 d^2\right )\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )-i e x^{3/2} \sqrt{\frac{b}{c}} \sqrt{\frac{b}{c x}+1} \sqrt{\frac{d}{e x}+1} \left (40 A c e (2 c d-b e)+B \left (-3 b^2 e^2+88 b c d e-128 c^2 d^2\right )\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b}{c}}}{\sqrt{x}}\right )|\frac{c d}{b e}\right )+\frac{(b+c x) (d+e x) \left (40 A c e (b e-2 c d)+B \left (3 b^2 e^2-88 b c d e+128 c^2 d^2\right )\right )}{c}+\frac{e x (b+c x) \left (5 A e \left (c \left (8 d^2+10 d e x+e^2 x^2\right )-b e (3 d+4 e x)\right )+B \left (b e \left (36 d^2+47 d e x+6 e^2 x^2\right )-c \left (64 d^3+80 d^2 e x+8 d e^2 x^2-3 e^3 x^3\right )\right )\right )}{d+e x}\right )}{15 e^5 x^2 (b+c x)^2 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(b*x + c*x^2)^(3/2))/(d + e*x)^(5/2),x]

[Out]

(2*(x*(b + c*x))^(3/2)*(((40*A*c*e*(-2*c*d + b*e) + B*(128*c^2*d^2 - 88*b*c*d*e
+ 3*b^2*e^2))*(b + c*x)*(d + e*x))/c + (e*x*(b + c*x)*(5*A*e*(-(b*e*(3*d + 4*e*x
)) + c*(8*d^2 + 10*d*e*x + e^2*x^2)) + B*(b*e*(36*d^2 + 47*d*e*x + 6*e^2*x^2) -
c*(64*d^3 + 80*d^2*e*x + 8*d*e^2*x^2 - 3*e^3*x^3))))/(d + e*x) - I*Sqrt[b/c]*e*(
40*A*c*e*(2*c*d - b*e) + B*(-128*c^2*d^2 + 88*b*c*d*e - 3*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*Sqrt[b/c]*e*(5*A*c*e*(8*c*d - 5*b*e) + B*(-64*c^2*d^2 + 52*b*c*d*e - 3*
b^2*e^2))*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*e^5*x^2*(b + c*x)^2*Sqrt[d + e*x])

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Maple [B]  time = 0.053, size = 2367, normalized size = 5.7 \[ \text{result too large to display} \]

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)^(5/2),x)

[Out]

2/15*(x*(c*x+b))^(1/2)*(15*A*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*
x*b^3*c*e^4*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-40*A*E
llipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^3*c*e^4*((c*x+b)/b)^(1/2)*
(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+15*A*EllipticF(((c*x+b)/b)^(1/2),(b*
e/(b*e-c*d))^(1/2))*b^3*c*d*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(
-c*x/b)^(1/2)-80*A*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^2*d^
2*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+80*A*Ellipti
cF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b*c^3*d^3*e*((c*x+b)/b)^(1/2)*(-(e*x
+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+50*A*x^3*c^4*d*e^3+6*B*x^3*b^2*c^2*e^4-80*
B*x^3*c^4*d^2*e^2-20*A*x^2*b^2*c^2*e^4+40*A*x^2*c^4*d^2*e^2-64*B*x^2*c^4*d^3*e+9
*B*x^4*b*c^3*e^4-8*B*x^4*c^4*d*e^3+39*B*x^3*b*c^3*d*e^3+3*B*x^5*c^4*e^4+5*A*x^4*
c^4*e^4-15*A*x^3*b*c^3*e^4-216*B*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/
2))*x*b^2*c^2*d^2*e^2*((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))*x*b*c^3*d^3*e*((c*x
+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-80*A*EllipticF(((c*x+b)
/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^2*c^2*d*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(
b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+80*A*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^
(1/2))*x*b*c^3*d^2*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(
1/2)+120*A*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^2*c^2*d*e^3*((
c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-80*A*EllipticE(((c*x
+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b*c^3*d^2*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*
c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-39*B*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d
))^(1/2))*x*b^3*c*d*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^
(1/2)+152*B*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^2*c^2*d^2*e^2
*((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))*x*b*c^3*d^3*e*((c*x+b)/b)^(1/2)*(-(e*x+d
)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+91*B*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c
*d))^(1/2))*x*b^3*c*d*e^3*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b
)^(1/2)-40*A*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c*d*e^3*((c*
x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+120*A*EllipticE(((c*x+
b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^2*d^2*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c
/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-80*A*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d)
)^(1/2))*b*c^3*d^3*e*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/
2)-39*B*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^3*c*d^2*e^2*((c*x+b
)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+152*B*EllipticF(((c*x+b)/
b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^2*d^3*e*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e
-c*d))^(1/2)*(-c*x/b)^(1/2)+91*B*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/
2))*b^3*c*d^2*e^2*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)-
216*B*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^2*c^2*d^3*e*((c*x+b)/
b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)+40*A*x*b*c^3*d^2*e^2+36*B*x
*b^2*c^2*d^2*e^2-64*B*x*b*c^3*d^3*e+35*A*x^2*b*c^3*d*e^3+47*B*x^2*b^2*c^2*d*e^3-
44*B*x^2*b*c^3*d^2*e^2-15*A*x*b^2*c^2*d*e^3-128*B*EllipticF(((c*x+b)/b)^(1/2),(b
*e/(b*e-c*d))^(1/2))*b*c^3*d^4*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-
c*x/b)^(1/2)-3*B*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*b^4*d*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^4*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*
e-c*d))^(1/2)*(-c*x/b)^(1/2)-3*B*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/
2))*x*b^4*e^4*((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/(e*x+d)^(3/2)/e^5/c^2

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

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)^(5/2),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x)^(3/2)*(B*x + A)/(e*x + d)^(5/2), x)

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

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)^(5/2),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^2*x^2 + 2*d*e
*x + d^2)*sqrt(e*x + d)), x)

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

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)**(5/2),x)

[Out]

Integral((x*(b + c*x))**(3/2)*(A + B*x)/(d + e*x)**(5/2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{2} + b x\right )}^{\frac{3}{2}}{\left (B x + A\right )}}{{\left (e x + d\right )}^{\frac{5}{2}}}\,{d x} \]

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)^(5/2),x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x)^(3/2)*(B*x + A)/(e*x + d)^(5/2), x)