3.1178 \(\int \frac{(A+B x) \left (b x+c x^2\right )^{3/2}}{(d+e x)^4} \, dx\)

Optimal. Leaf size=426 \[ -\frac{\sqrt{b x+c x^2} \left (A e \left (-b^2 e^2-6 b c d e+8 c^2 d^2\right )-2 c e x (B d (8 c d-7 b e)-A e (2 c d-b e))-B d \left (5 b^2 e^2-36 b c d e+32 c^2 d^2\right )\right )}{8 d e^4 (d+e x) (c d-b e)}+\frac{\left (B d \left (-5 b^3 e^3+60 b^2 c d e^2-120 b c^2 d^2 e+64 c^3 d^3\right )-A e \left (b^3 e^3+6 b^2 c d e^2-24 b c^2 d^2 e+16 c^3 d^3\right )\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{16 d^{3/2} e^5 (c d-b e)^{3/2}}-\frac{\sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) (-2 A c e-3 b B e+8 B c d)}{e^5}-\frac{\left (b x+c x^2\right )^{3/2} (3 e x (B d (4 c d-3 b e)-A e (2 c d-b e))+d (B d (8 c d-5 b e)-A e (b e+2 c d)))}{12 d e^2 (d+e x)^3 (c d-b e)} \]

[Out]

-((A*e*(8*c^2*d^2 - 6*b*c*d*e - b^2*e^2) - B*d*(32*c^2*d^2 - 36*b*c*d*e + 5*b^2*
e^2) - 2*c*e*(B*d*(8*c*d - 7*b*e) - A*e*(2*c*d - b*e))*x)*Sqrt[b*x + c*x^2])/(8*
d*e^4*(c*d - b*e)*(d + e*x)) - ((d*(B*d*(8*c*d - 5*b*e) - A*e*(2*c*d + b*e)) + 3
*e*(B*d*(4*c*d - 3*b*e) - A*e*(2*c*d - b*e))*x)*(b*x + c*x^2)^(3/2))/(12*d*e^2*(
c*d - b*e)*(d + e*x)^3) - (Sqrt[c]*(8*B*c*d - 3*b*B*e - 2*A*c*e)*ArcTanh[(Sqrt[c
]*x)/Sqrt[b*x + c*x^2]])/e^5 + ((B*d*(64*c^3*d^3 - 120*b*c^2*d^2*e + 60*b^2*c*d*
e^2 - 5*b^3*e^3) - A*e*(16*c^3*d^3 - 24*b*c^2*d^2*e + 6*b^2*c*d*e^2 + b^3*e^3))*
ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/
(16*d^(3/2)*e^5*(c*d - b*e)^(3/2))

_______________________________________________________________________________________

Rubi [A]  time = 1.29197, antiderivative size = 426, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{\sqrt{b x+c x^2} \left (A e \left (-b^2 e^2-6 b c d e+8 c^2 d^2\right )-2 c e x (B d (8 c d-7 b e)-A e (2 c d-b e))-B d \left (5 b^2 e^2-36 b c d e+32 c^2 d^2\right )\right )}{8 d e^4 (d+e x) (c d-b e)}+\frac{\left (B d \left (-5 b^3 e^3+60 b^2 c d e^2-120 b c^2 d^2 e+64 c^3 d^3\right )-A e \left (b^3 e^3+6 b^2 c d e^2-24 b c^2 d^2 e+16 c^3 d^3\right )\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{16 d^{3/2} e^5 (c d-b e)^{3/2}}-\frac{\sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right ) (-2 A c e-3 b B e+8 B c d)}{e^5}-\frac{\left (b x+c x^2\right )^{3/2} (3 e x (B d (4 c d-3 b e)-A e (2 c d-b e))+d (B d (8 c d-5 b e)-A e (b e+2 c d)))}{12 d e^2 (d+e x)^3 (c d-b e)} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(b*x + c*x^2)^(3/2))/(d + e*x)^4,x]

[Out]

-((A*e*(8*c^2*d^2 - 6*b*c*d*e - b^2*e^2) - B*d*(32*c^2*d^2 - 36*b*c*d*e + 5*b^2*
e^2) - 2*c*e*(B*d*(8*c*d - 7*b*e) - A*e*(2*c*d - b*e))*x)*Sqrt[b*x + c*x^2])/(8*
d*e^4*(c*d - b*e)*(d + e*x)) - ((d*(B*d*(8*c*d - 5*b*e) - A*e*(2*c*d + b*e)) + 3
*e*(B*d*(4*c*d - 3*b*e) - A*e*(2*c*d - b*e))*x)*(b*x + c*x^2)^(3/2))/(12*d*e^2*(
c*d - b*e)*(d + e*x)^3) - (Sqrt[c]*(8*B*c*d - 3*b*B*e - 2*A*c*e)*ArcTanh[(Sqrt[c
]*x)/Sqrt[b*x + c*x^2]])/e^5 + ((B*d*(64*c^3*d^3 - 120*b*c^2*d^2*e + 60*b^2*c*d*
e^2 - 5*b^3*e^3) - A*e*(16*c^3*d^3 - 24*b*c^2*d^2*e + 6*b^2*c*d*e^2 + b^3*e^3))*
ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/
(16*d^(3/2)*e^5*(c*d - b*e)^(3/2))

_______________________________________________________________________________________

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)**4,x)

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [A]  time = 3.10688, size = 376, normalized size = 0.88 \[ \frac{(x (b+c x))^{3/2} \left (\frac{e \sqrt{x} \left (\frac{A e \left (-3 b^2 e^2+44 b c d e-44 c^2 d^2\right )+B d \left (33 b^2 e^2-134 b c d e+104 c^2 d^2\right )}{d (d+e x) (c d-b e)}+\frac{B \left (26 b d e-40 c d^2\right )-14 A e (b e-2 c d)}{(d+e x)^2}+\frac{8 d (B d-A e) (c d-b e)}{(d+e x)^3}+24 B c\right )}{b+c x}-\frac{3 \left (B d \left (-5 b^3 e^3+60 b^2 c d e^2-120 b c^2 d^2 e+64 c^3 d^3\right )-A e \left (b^3 e^3+6 b^2 c d e^2-24 b c^2 d^2 e+16 c^3 d^3\right )\right ) \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )}{d^{3/2} (b+c x)^{3/2} (b e-c d)^{3/2}}+\frac{24 \sqrt{c} \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right ) (2 A c e+3 b B e-8 B c d)}{(b+c x)^{3/2}}\right )}{24 e^5 x^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(b*x + c*x^2)^(3/2))/(d + e*x)^4,x]

[Out]

((x*(b + c*x))^(3/2)*((e*Sqrt[x]*(24*B*c + (8*d*(B*d - A*e)*(c*d - b*e))/(d + e*
x)^3 + (-14*A*e*(-2*c*d + b*e) + B*(-40*c*d^2 + 26*b*d*e))/(d + e*x)^2 + (A*e*(-
44*c^2*d^2 + 44*b*c*d*e - 3*b^2*e^2) + B*d*(104*c^2*d^2 - 134*b*c*d*e + 33*b^2*e
^2))/(d*(c*d - b*e)*(d + e*x))))/(b + c*x) - (3*(B*d*(64*c^3*d^3 - 120*b*c^2*d^2
*e + 60*b^2*c*d*e^2 - 5*b^3*e^3) - A*e*(16*c^3*d^3 - 24*b*c^2*d^2*e + 6*b^2*c*d*
e^2 + b^3*e^3))*ArcTan[(Sqrt[-(c*d) + b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/(d
^(3/2)*(-(c*d) + b*e)^(3/2)*(b + c*x)^(3/2)) + (24*Sqrt[c]*(-8*B*c*d + 3*b*B*e +
2*A*c*e)*Log[c*Sqrt[x] + Sqrt[c]*Sqrt[b + c*x]])/(b + c*x)^(3/2)))/(24*e^5*x^(3
/2))

_______________________________________________________________________________________

Maple [B]  time = 0.035, size = 11396, normalized size = 26.8 \[ \text{output too large to display} \]

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)^4,x)

[Out]

result too large to display

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

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)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 6.54923, size = 1, normalized size = 0. \[ \text{result too large to display} \]

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)^4,x, algorithm="fricas")

[Out]

[-1/48*(24*(8*B*c^2*d^6 - (11*B*b*c + 2*A*c^2)*d^5*e + (3*B*b^2 + 2*A*b*c)*d^4*e
^2 + (8*B*c^2*d^3*e^3 - (11*B*b*c + 2*A*c^2)*d^2*e^4 + (3*B*b^2 + 2*A*b*c)*d*e^5
)*x^3 + 3*(8*B*c^2*d^4*e^2 - (11*B*b*c + 2*A*c^2)*d^3*e^3 + (3*B*b^2 + 2*A*b*c)*
d^2*e^4)*x^2 + 3*(8*B*c^2*d^5*e - (11*B*b*c + 2*A*c^2)*d^4*e^2 + (3*B*b^2 + 2*A*
b*c)*d^3*e^3)*x)*sqrt(c*d^2 - b*d*e)*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)
*sqrt(c)) - 2*(96*B*c^2*d^5*e + 3*A*b^2*d^2*e^4 - 12*(9*B*b*c + 2*A*c^2)*d^4*e^2
+ 3*(5*B*b^2 + 6*A*b*c)*d^3*e^3 + 24*(B*c^2*d^2*e^4 - B*b*c*d*e^5)*x^3 + (176*B
*c^2*d^3*e^3 - 3*A*b^2*e^6 - 2*(103*B*b*c + 22*A*c^2)*d^2*e^4 + 11*(3*B*b^2 + 4*
A*b*c)*d*e^5)*x^2 + 2*(120*B*c^2*d^4*e^2 + 4*A*b^2*d*e^5 - (137*B*b*c + 30*A*c^2
)*d^3*e^3 + (20*B*b^2 + 23*A*b*c)*d^2*e^4)*x)*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 + b
*x) - 3*(64*B*c^3*d^7 - A*b^3*d^3*e^4 - 8*(15*B*b*c^2 + 2*A*c^3)*d^6*e + 12*(5*B
*b^2*c + 2*A*b*c^2)*d^5*e^2 - (5*B*b^3 + 6*A*b^2*c)*d^4*e^3 + (64*B*c^3*d^4*e^3
- A*b^3*e^7 - 8*(15*B*b*c^2 + 2*A*c^3)*d^3*e^4 + 12*(5*B*b^2*c + 2*A*b*c^2)*d^2*
e^5 - (5*B*b^3 + 6*A*b^2*c)*d*e^6)*x^3 + 3*(64*B*c^3*d^5*e^2 - A*b^3*d*e^6 - 8*(
15*B*b*c^2 + 2*A*c^3)*d^4*e^3 + 12*(5*B*b^2*c + 2*A*b*c^2)*d^3*e^4 - (5*B*b^3 +
6*A*b^2*c)*d^2*e^5)*x^2 + 3*(64*B*c^3*d^6*e - A*b^3*d^2*e^5 - 8*(15*B*b*c^2 + 2*
A*c^3)*d^5*e^2 + 12*(5*B*b^2*c + 2*A*b*c^2)*d^4*e^3 - (5*B*b^3 + 6*A*b^2*c)*d^3*
e^4)*x)*log((2*(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x) + sqrt(c*d^2 - b*d*e)*(b*d + (2
*c*d - b*e)*x))/(e*x + d)))/((c*d^5*e^5 - b*d^4*e^6 + (c*d^2*e^8 - b*d*e^9)*x^3
+ 3*(c*d^3*e^7 - b*d^2*e^8)*x^2 + 3*(c*d^4*e^6 - b*d^3*e^7)*x)*sqrt(c*d^2 - b*d*
e)), -1/24*(12*(8*B*c^2*d^6 - (11*B*b*c + 2*A*c^2)*d^5*e + (3*B*b^2 + 2*A*b*c)*d
^4*e^2 + (8*B*c^2*d^3*e^3 - (11*B*b*c + 2*A*c^2)*d^2*e^4 + (3*B*b^2 + 2*A*b*c)*d
*e^5)*x^3 + 3*(8*B*c^2*d^4*e^2 - (11*B*b*c + 2*A*c^2)*d^3*e^3 + (3*B*b^2 + 2*A*b
*c)*d^2*e^4)*x^2 + 3*(8*B*c^2*d^5*e - (11*B*b*c + 2*A*c^2)*d^4*e^2 + (3*B*b^2 +
2*A*b*c)*d^3*e^3)*x)*sqrt(-c*d^2 + b*d*e)*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2 +
b*x)*sqrt(c)) - (96*B*c^2*d^5*e + 3*A*b^2*d^2*e^4 - 12*(9*B*b*c + 2*A*c^2)*d^4*
e^2 + 3*(5*B*b^2 + 6*A*b*c)*d^3*e^3 + 24*(B*c^2*d^2*e^4 - B*b*c*d*e^5)*x^3 + (17
6*B*c^2*d^3*e^3 - 3*A*b^2*e^6 - 2*(103*B*b*c + 22*A*c^2)*d^2*e^4 + 11*(3*B*b^2 +
4*A*b*c)*d*e^5)*x^2 + 2*(120*B*c^2*d^4*e^2 + 4*A*b^2*d*e^5 - (137*B*b*c + 30*A*
c^2)*d^3*e^3 + (20*B*b^2 + 23*A*b*c)*d^2*e^4)*x)*sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2
+ b*x) + 3*(64*B*c^3*d^7 - A*b^3*d^3*e^4 - 8*(15*B*b*c^2 + 2*A*c^3)*d^6*e + 12*
(5*B*b^2*c + 2*A*b*c^2)*d^5*e^2 - (5*B*b^3 + 6*A*b^2*c)*d^4*e^3 + (64*B*c^3*d^4*
e^3 - A*b^3*e^7 - 8*(15*B*b*c^2 + 2*A*c^3)*d^3*e^4 + 12*(5*B*b^2*c + 2*A*b*c^2)*
d^2*e^5 - (5*B*b^3 + 6*A*b^2*c)*d*e^6)*x^3 + 3*(64*B*c^3*d^5*e^2 - A*b^3*d*e^6 -
8*(15*B*b*c^2 + 2*A*c^3)*d^4*e^3 + 12*(5*B*b^2*c + 2*A*b*c^2)*d^3*e^4 - (5*B*b^
3 + 6*A*b^2*c)*d^2*e^5)*x^2 + 3*(64*B*c^3*d^6*e - A*b^3*d^2*e^5 - 8*(15*B*b*c^2
+ 2*A*c^3)*d^5*e^2 + 12*(5*B*b^2*c + 2*A*b*c^2)*d^4*e^3 - (5*B*b^3 + 6*A*b^2*c)*
d^3*e^4)*x)*arctan(-sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/((c*d - b*e)*x)))/((c
*d^5*e^5 - b*d^4*e^6 + (c*d^2*e^8 - b*d*e^9)*x^3 + 3*(c*d^3*e^7 - b*d^2*e^8)*x^2
+ 3*(c*d^4*e^6 - b*d^3*e^7)*x)*sqrt(-c*d^2 + b*d*e)), -1/48*(48*(8*B*c^2*d^6 -
(11*B*b*c + 2*A*c^2)*d^5*e + (3*B*b^2 + 2*A*b*c)*d^4*e^2 + (8*B*c^2*d^3*e^3 - (1
1*B*b*c + 2*A*c^2)*d^2*e^4 + (3*B*b^2 + 2*A*b*c)*d*e^5)*x^3 + 3*(8*B*c^2*d^4*e^2
- (11*B*b*c + 2*A*c^2)*d^3*e^3 + (3*B*b^2 + 2*A*b*c)*d^2*e^4)*x^2 + 3*(8*B*c^2*
d^5*e - (11*B*b*c + 2*A*c^2)*d^4*e^2 + (3*B*b^2 + 2*A*b*c)*d^3*e^3)*x)*sqrt(c*d^
2 - b*d*e)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)/(sqrt(-c)*x)) - 2*(96*B*c^2*d^5*e +
3*A*b^2*d^2*e^4 - 12*(9*B*b*c + 2*A*c^2)*d^4*e^2 + 3*(5*B*b^2 + 6*A*b*c)*d^3*e^
3 + 24*(B*c^2*d^2*e^4 - B*b*c*d*e^5)*x^3 + (176*B*c^2*d^3*e^3 - 3*A*b^2*e^6 - 2*
(103*B*b*c + 22*A*c^2)*d^2*e^4 + 11*(3*B*b^2 + 4*A*b*c)*d*e^5)*x^2 + 2*(120*B*c^
2*d^4*e^2 + 4*A*b^2*d*e^5 - (137*B*b*c + 30*A*c^2)*d^3*e^3 + (20*B*b^2 + 23*A*b*
c)*d^2*e^4)*x)*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x) - 3*(64*B*c^3*d^7 - A*b^3*d
^3*e^4 - 8*(15*B*b*c^2 + 2*A*c^3)*d^6*e + 12*(5*B*b^2*c + 2*A*b*c^2)*d^5*e^2 - (
5*B*b^3 + 6*A*b^2*c)*d^4*e^3 + (64*B*c^3*d^4*e^3 - A*b^3*e^7 - 8*(15*B*b*c^2 + 2
*A*c^3)*d^3*e^4 + 12*(5*B*b^2*c + 2*A*b*c^2)*d^2*e^5 - (5*B*b^3 + 6*A*b^2*c)*d*e
^6)*x^3 + 3*(64*B*c^3*d^5*e^2 - A*b^3*d*e^6 - 8*(15*B*b*c^2 + 2*A*c^3)*d^4*e^3 +
12*(5*B*b^2*c + 2*A*b*c^2)*d^3*e^4 - (5*B*b^3 + 6*A*b^2*c)*d^2*e^5)*x^2 + 3*(64
*B*c^3*d^6*e - A*b^3*d^2*e^5 - 8*(15*B*b*c^2 + 2*A*c^3)*d^5*e^2 + 12*(5*B*b^2*c
+ 2*A*b*c^2)*d^4*e^3 - (5*B*b^3 + 6*A*b^2*c)*d^3*e^4)*x)*log((2*(c*d^2 - b*d*e)*
sqrt(c*x^2 + b*x) + sqrt(c*d^2 - b*d*e)*(b*d + (2*c*d - b*e)*x))/(e*x + d)))/((c
*d^5*e^5 - b*d^4*e^6 + (c*d^2*e^8 - b*d*e^9)*x^3 + 3*(c*d^3*e^7 - b*d^2*e^8)*x^2
+ 3*(c*d^4*e^6 - b*d^3*e^7)*x)*sqrt(c*d^2 - b*d*e)), -1/24*(24*(8*B*c^2*d^6 - (
11*B*b*c + 2*A*c^2)*d^5*e + (3*B*b^2 + 2*A*b*c)*d^4*e^2 + (8*B*c^2*d^3*e^3 - (11
*B*b*c + 2*A*c^2)*d^2*e^4 + (3*B*b^2 + 2*A*b*c)*d*e^5)*x^3 + 3*(8*B*c^2*d^4*e^2
- (11*B*b*c + 2*A*c^2)*d^3*e^3 + (3*B*b^2 + 2*A*b*c)*d^2*e^4)*x^2 + 3*(8*B*c^2*d
^5*e - (11*B*b*c + 2*A*c^2)*d^4*e^2 + (3*B*b^2 + 2*A*b*c)*d^3*e^3)*x)*sqrt(-c*d^
2 + b*d*e)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)/(sqrt(-c)*x)) - (96*B*c^2*d^5*e + 3
*A*b^2*d^2*e^4 - 12*(9*B*b*c + 2*A*c^2)*d^4*e^2 + 3*(5*B*b^2 + 6*A*b*c)*d^3*e^3
+ 24*(B*c^2*d^2*e^4 - B*b*c*d*e^5)*x^3 + (176*B*c^2*d^3*e^3 - 3*A*b^2*e^6 - 2*(1
03*B*b*c + 22*A*c^2)*d^2*e^4 + 11*(3*B*b^2 + 4*A*b*c)*d*e^5)*x^2 + 2*(120*B*c^2*
d^4*e^2 + 4*A*b^2*d*e^5 - (137*B*b*c + 30*A*c^2)*d^3*e^3 + (20*B*b^2 + 23*A*b*c)
*d^2*e^4)*x)*sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x) + 3*(64*B*c^3*d^7 - A*b^3*d^
3*e^4 - 8*(15*B*b*c^2 + 2*A*c^3)*d^6*e + 12*(5*B*b^2*c + 2*A*b*c^2)*d^5*e^2 - (5
*B*b^3 + 6*A*b^2*c)*d^4*e^3 + (64*B*c^3*d^4*e^3 - A*b^3*e^7 - 8*(15*B*b*c^2 + 2*
A*c^3)*d^3*e^4 + 12*(5*B*b^2*c + 2*A*b*c^2)*d^2*e^5 - (5*B*b^3 + 6*A*b^2*c)*d*e^
6)*x^3 + 3*(64*B*c^3*d^5*e^2 - A*b^3*d*e^6 - 8*(15*B*b*c^2 + 2*A*c^3)*d^4*e^3 +
12*(5*B*b^2*c + 2*A*b*c^2)*d^3*e^4 - (5*B*b^3 + 6*A*b^2*c)*d^2*e^5)*x^2 + 3*(64*
B*c^3*d^6*e - A*b^3*d^2*e^5 - 8*(15*B*b*c^2 + 2*A*c^3)*d^5*e^2 + 12*(5*B*b^2*c +
2*A*b*c^2)*d^4*e^3 - (5*B*b^3 + 6*A*b^2*c)*d^3*e^4)*x)*arctan(-sqrt(-c*d^2 + b*
d*e)*sqrt(c*x^2 + b*x)/((c*d - b*e)*x)))/((c*d^5*e^5 - b*d^4*e^6 + (c*d^2*e^8 -
b*d*e^9)*x^3 + 3*(c*d^3*e^7 - b*d^2*e^8)*x^2 + 3*(c*d^4*e^6 - b*d^3*e^7)*x)*sqrt
(-c*d^2 + b*d*e))]

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x \left (b + c x\right )\right )^{\frac{3}{2}} \left (A + B x\right )}{\left (d + e x\right )^{4}}\, dx \]

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)**4,x)

[Out]

Integral((x*(b + c*x))**(3/2)*(A + B*x)/(d + e*x)**4, x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.694085, size = 4, normalized size = 0.01 \[ \mathit{sage}_{0} x \]

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)^4,x, algorithm="giac")

[Out]

sage0*x