3.1253 \(\int \frac{A+B x}{(d+e x)^{3/2} \left (b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=530 \[ -\frac{-5 A b e-8 A c d+4 b B d}{4 b^2 d^2 x (b+c x)^2 \sqrt{d+e x}}-\frac{3 \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (b^2 (-e) (4 B d-5 A e)-4 b c d (2 B d-3 A e)+16 A c^2 d^2\right )}{4 b^5 d^{7/2}}+\frac{c \left (b^2 e (4 B d-5 A e)-b c d (5 A e+6 B d)+12 A c^2 d^2\right )}{4 b^3 d^2 (b+c x)^2 \sqrt{d+e x} (c d-b e)}+\frac{3 c^{5/2} \left (3 b^2 c e (11 A e+8 B d)-4 b c^2 d (11 A e+2 B d)+16 A c^3 d^2-21 b^3 B e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{4 b^5 (c d-b e)^{7/2}}+\frac{c \left (b^3 \left (-e^2\right ) (4 B d-5 A e)+b^2 c d e (2 A e+21 B d)-12 b c^2 d^2 (3 A e+B d)+24 A c^3 d^3\right )}{4 b^4 d^2 (b+c x) \sqrt{d+e x} (c d-b e)^2}+\frac{3 e \left (b^4 e^3 (4 B d-5 A e)-b^3 c d e^2 (4 B d-3 A e)+b^2 c^2 d^2 e (5 A e+9 B d)-4 b c^3 d^3 (4 A e+B d)+8 A c^4 d^4\right )}{4 b^4 d^3 \sqrt{d+e x} (c d-b e)^3}-\frac{A}{2 b d x^2 (b+c x)^2 \sqrt{d+e x}} \]

[Out]

(3*e*(8*A*c^4*d^4 + b^4*e^3*(4*B*d - 5*A*e) - b^3*c*d*e^2*(4*B*d - 3*A*e) - 4*b*
c^3*d^3*(B*d + 4*A*e) + b^2*c^2*d^2*e*(9*B*d + 5*A*e)))/(4*b^4*d^3*(c*d - b*e)^3
*Sqrt[d + e*x]) + (c*(12*A*c^2*d^2 + b^2*e*(4*B*d - 5*A*e) - b*c*d*(6*B*d + 5*A*
e)))/(4*b^3*d^2*(c*d - b*e)*(b + c*x)^2*Sqrt[d + e*x]) - A/(2*b*d*x^2*(b + c*x)^
2*Sqrt[d + e*x]) - (4*b*B*d - 8*A*c*d - 5*A*b*e)/(4*b^2*d^2*x*(b + c*x)^2*Sqrt[d
+ e*x]) + (c*(24*A*c^3*d^3 - b^3*e^2*(4*B*d - 5*A*e) + b^2*c*d*e*(21*B*d + 2*A*
e) - 12*b*c^2*d^2*(B*d + 3*A*e)))/(4*b^4*d^2*(c*d - b*e)^2*(b + c*x)*Sqrt[d + e*
x]) - (3*(16*A*c^2*d^2 - b^2*e*(4*B*d - 5*A*e) - 4*b*c*d*(2*B*d - 3*A*e))*ArcTan
h[Sqrt[d + e*x]/Sqrt[d]])/(4*b^5*d^(7/2)) + (3*c^(5/2)*(16*A*c^3*d^2 - 21*b^3*B*
e^2 - 4*b*c^2*d*(2*B*d + 11*A*e) + 3*b^2*c*e*(8*B*d + 11*A*e))*ArcTanh[(Sqrt[c]*
Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(4*b^5*(c*d - b*e)^(7/2))

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Rubi [A]  time = 3.37777, antiderivative size = 530, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ -\frac{-5 A b e-8 A c d+4 b B d}{4 b^2 d^2 x (b+c x)^2 \sqrt{d+e x}}-\frac{3 \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (b^2 (-e) (4 B d-5 A e)-4 b c d (2 B d-3 A e)+16 A c^2 d^2\right )}{4 b^5 d^{7/2}}+\frac{c \left (b^2 e (4 B d-5 A e)-b c d (5 A e+6 B d)+12 A c^2 d^2\right )}{4 b^3 d^2 (b+c x)^2 \sqrt{d+e x} (c d-b e)}+\frac{3 c^{5/2} \left (3 b^2 c e (11 A e+8 B d)-4 b c^2 d (11 A e+2 B d)+16 A c^3 d^2-21 b^3 B e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{4 b^5 (c d-b e)^{7/2}}+\frac{c \left (b^3 \left (-e^2\right ) (4 B d-5 A e)+b^2 c d e (2 A e+21 B d)-12 b c^2 d^2 (3 A e+B d)+24 A c^3 d^3\right )}{4 b^4 d^2 (b+c x) \sqrt{d+e x} (c d-b e)^2}+\frac{3 e \left (b^4 e^3 (4 B d-5 A e)-b^3 c d e^2 (4 B d-3 A e)+b^2 c^2 d^2 e (5 A e+9 B d)-4 b c^3 d^3 (4 A e+B d)+8 A c^4 d^4\right )}{4 b^4 d^3 \sqrt{d+e x} (c d-b e)^3}-\frac{A}{2 b d x^2 (b+c x)^2 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(3*e*(8*A*c^4*d^4 + b^4*e^3*(4*B*d - 5*A*e) - b^3*c*d*e^2*(4*B*d - 3*A*e) - 4*b*
c^3*d^3*(B*d + 4*A*e) + b^2*c^2*d^2*e*(9*B*d + 5*A*e)))/(4*b^4*d^3*(c*d - b*e)^3
*Sqrt[d + e*x]) + (c*(12*A*c^2*d^2 + b^2*e*(4*B*d - 5*A*e) - b*c*d*(6*B*d + 5*A*
e)))/(4*b^3*d^2*(c*d - b*e)*(b + c*x)^2*Sqrt[d + e*x]) - A/(2*b*d*x^2*(b + c*x)^
2*Sqrt[d + e*x]) - (4*b*B*d - 8*A*c*d - 5*A*b*e)/(4*b^2*d^2*x*(b + c*x)^2*Sqrt[d
+ e*x]) + (c*(24*A*c^3*d^3 - b^3*e^2*(4*B*d - 5*A*e) + b^2*c*d*e*(21*B*d + 2*A*
e) - 12*b*c^2*d^2*(B*d + 3*A*e)))/(4*b^4*d^2*(c*d - b*e)^2*(b + c*x)*Sqrt[d + e*
x]) - (3*(16*A*c^2*d^2 - b^2*e*(4*B*d - 5*A*e) - 4*b*c*d*(2*B*d - 3*A*e))*ArcTan
h[Sqrt[d + e*x]/Sqrt[d]])/(4*b^5*d^(7/2)) + (3*c^(5/2)*(16*A*c^3*d^2 - 21*b^3*B*
e^2 - 4*b*c^2*d*(2*B*d + 11*A*e) + 3*b^2*c*e*(8*B*d + 11*A*e))*ArcTanh[(Sqrt[c]*
Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(4*b^5*(c*d - b*e)^(7/2))

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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)/(e*x+d)**(3/2)/(c*x**2+b*x)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 3.41354, size = 336, normalized size = 0.63 \[ \frac{1}{4} \left (\frac{3 \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (b^2 e (4 B d-5 A e)+4 b c d (2 B d-3 A e)-16 A c^2 d^2\right )}{b^5 d^{7/2}}+\frac{3 c^{5/2} \left (3 b^2 c e (11 A e+8 B d)-4 b c^2 d (11 A e+2 B d)+16 A c^3 d^2-21 b^3 B e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^5 (c d-b e)^{7/2}}+\sqrt{d+e x} \left (\frac{7 A b e+12 A c d-4 b B d}{b^4 d^3 x}+\frac{2 c^3 (A c-b B)}{b^3 (b+c x)^2 (c d-b e)^2}-\frac{2 A}{b^3 d^2 x^2}+\frac{c^3 \left (b c (19 A e+8 B d)-12 A c^2 d-15 b^2 B e\right )}{b^4 (b+c x) (b e-c d)^3}+\frac{8 e^4 (B d-A e)}{d^3 (d+e x) (c d-b e)^3}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[d + e*x]*((-2*A)/(b^3*d^2*x^2) + (-4*b*B*d + 12*A*c*d + 7*A*b*e)/(b^4*d^3*
x) + (2*c^3*(-(b*B) + A*c))/(b^3*(c*d - b*e)^2*(b + c*x)^2) + (c^3*(-12*A*c^2*d
- 15*b^2*B*e + b*c*(8*B*d + 19*A*e)))/(b^4*(-(c*d) + b*e)^3*(b + c*x)) + (8*e^4*
(B*d - A*e))/(d^3*(c*d - b*e)^3*(d + e*x))) + (3*(-16*A*c^2*d^2 + b^2*e*(4*B*d -
5*A*e) + 4*b*c*d*(2*B*d - 3*A*e))*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(b^5*d^(7/2))
+ (3*c^(5/2)*(16*A*c^3*d^2 - 21*b^3*B*e^2 - 4*b*c^2*d*(2*B*d + 11*A*e) + 3*b^2*
c*e*(8*B*d + 11*A*e))*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b^5*(c*
d - b*e)^(7/2)))/4

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Maple [B]  time = 0.054, size = 1022, normalized size = 1.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*e*c^5/(b*e-c*d)^3/b^3/(c*e*x+b*e)^2*B*(e*x+d)^(1/2)*d^2-33*e*c^5/(b*e-c*d)^3/
b^4/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*A*d+18*e*c^4
/(b*e-c*d)^3/b^3/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))
*B*d-3*e*c^6/(b*e-c*d)^3/b^4/(c*e*x+b*e)^2*(e*x+d)^(3/2)*A*d+2*e*c^5/(b*e-c*d)^3
/b^3/(c*e*x+b*e)^2*(e*x+d)^(3/2)*B*d-33/4*e^2*c^5/(b*e-c*d)^3/b^3/(c*e*x+b*e)^2*
A*(e*x+d)^(1/2)*d+3*e*c^6/(b*e-c*d)^3/b^4/(c*e*x+b*e)^2*A*(e*x+d)^(1/2)*d^2+2*e^
5/d^3/(b*e-c*d)^3/(e*x+d)^(1/2)*A+3*e/b^3/d^(5/2)*arctanh((e*x+d)^(1/2)/d^(1/2))
*B-2*e^4/d^2/(b*e-c*d)^3/(e*x+d)^(1/2)*B-9/4/b^3/d^2/x^2*(e*x+d)^(1/2)*A-12/b^5/
d^(3/2)*arctanh((e*x+d)^(1/2)/d^(1/2))*A*c^2+6/b^4/d^(3/2)*arctanh((e*x+d)^(1/2)
/d^(1/2))*B*c+7/4/b^3/d^3/x^2*A*(e*x+d)^(3/2)-15/4*e^2/b^3/d^(7/2)*arctanh((e*x+
d)^(1/2)/d^(1/2))*A-17/4*e^3*c^3/(b*e-c*d)^3/b/(c*e*x+b*e)^2*B*(e*x+d)^(1/2)+99/
4*e^2*c^4/(b*e-c*d)^3/b^3/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*
c)^(1/2))*A-63/4*e^2*c^3/(b*e-c*d)^3/b^2/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1
/2)/((b*e-c*d)*c)^(1/2))*B+3/e/b^4/d^2/x^2*A*(e*x+d)^(3/2)*c-3/e/b^4/d/x^2*(e*x+
d)^(1/2)*A*c+19/4*e^2*c^5/(b*e-c*d)^3/b^3/(c*e*x+b*e)^2*(e*x+d)^(3/2)*A-15/4*e^2
*c^4/(b*e-c*d)^3/b^2/(c*e*x+b*e)^2*(e*x+d)^(3/2)*B+21/4*e^3*c^4/(b*e-c*d)^3/b^2/
(c*e*x+b*e)^2*A*(e*x+d)^(1/2)+12*c^6/(b*e-c*d)^3/b^5/((b*e-c*d)*c)^(1/2)*arctan(
c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*A*d^2-6*c^5/(b*e-c*d)^3/b^4/((b*e-c*d)*c)^(
1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*B*d^2-1/e/b^3/d^2/x^2*B*(e*x+d)
^(3/2)+1/e/b^3/d/x^2*(e*x+d)^(1/2)*B-9*e/b^4/d^(5/2)*arctanh((e*x+d)^(1/2)/d^(1/
2))*A*c+25/4*e^2*c^4/(b*e-c*d)^3/b^2/(c*e*x+b*e)^2*B*(e*x+d)^(1/2)*d

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 113.627, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/8*(3*((8*(B*b*c^6 - 2*A*c^7)*d^5 - 4*(6*B*b^2*c^5 - 11*A*b*c^6)*d^4*e + 3*(7
*B*b^3*c^4 - 11*A*b^2*c^5)*d^3*e^2)*x^4 + 2*(8*(B*b^2*c^5 - 2*A*b*c^6)*d^5 - 4*(
6*B*b^3*c^4 - 11*A*b^2*c^5)*d^4*e + 3*(7*B*b^4*c^3 - 11*A*b^3*c^4)*d^3*e^2)*x^3
+ (8*(B*b^3*c^4 - 2*A*b^2*c^5)*d^5 - 4*(6*B*b^4*c^3 - 11*A*b^3*c^4)*d^4*e + 3*(7
*B*b^5*c^2 - 11*A*b^4*c^3)*d^3*e^2)*x^2)*sqrt(e*x + d)*sqrt(d)*sqrt(c/(c*d - b*e
))*log((c*e*x + 2*c*d - b*e + 2*(c*d - b*e)*sqrt(e*x + d)*sqrt(c/(c*d - b*e)))/(
c*x + b)) - 3*((5*A*b^5*c^2*e^5 + 8*(B*b*c^6 - 2*A*c^7)*d^5 - 4*(5*B*b^2*c^5 - 9
*A*b*c^6)*d^4*e + (12*B*b^3*c^4 - 17*A*b^2*c^5)*d^3*e^2 + (4*B*b^4*c^3 - 5*A*b^3
*c^4)*d^2*e^3 - (4*B*b^5*c^2 + 3*A*b^4*c^3)*d*e^4)*x^4 + 2*(5*A*b^6*c*e^5 + 8*(B
*b^2*c^5 - 2*A*b*c^6)*d^5 - 4*(5*B*b^3*c^4 - 9*A*b^2*c^5)*d^4*e + (12*B*b^4*c^3
- 17*A*b^3*c^4)*d^3*e^2 + (4*B*b^5*c^2 - 5*A*b^4*c^3)*d^2*e^3 - (4*B*b^6*c + 3*A
*b^5*c^2)*d*e^4)*x^3 + (5*A*b^7*e^5 + 8*(B*b^3*c^4 - 2*A*b^2*c^5)*d^5 - 4*(5*B*b
^4*c^3 - 9*A*b^3*c^4)*d^4*e + (12*B*b^5*c^2 - 17*A*b^4*c^3)*d^3*e^2 + (4*B*b^6*c
- 5*A*b^5*c^2)*d^2*e^3 - (4*B*b^7 + 3*A*b^6*c)*d*e^4)*x^2)*sqrt(e*x + d)*log(((
e*x + 2*d)*sqrt(d) + 2*sqrt(e*x + d)*d)/x) + 2*(2*A*b^4*c^3*d^5 - 6*A*b^5*c^2*d^
4*e + 6*A*b^6*c*d^3*e^2 - 2*A*b^7*d^2*e^3 + 3*(5*A*b^5*c^2*e^5 + 4*(B*b^2*c^5 -
2*A*b*c^6)*d^4*e - (9*B*b^3*c^4 - 16*A*b^2*c^5)*d^3*e^2 + (4*B*b^4*c^3 - 5*A*b^3
*c^4)*d^2*e^3 - (4*B*b^5*c^2 + 3*A*b^4*c^3)*d*e^4)*x^4 + (30*A*b^6*c*e^5 + 12*(B
*b^2*c^5 - 2*A*b*c^6)*d^5 - 3*(3*B*b^3*c^4 - 4*A*b^2*c^5)*d^4*e - 29*(B*b^4*c^3
- 2*A*b^3*c^4)*d^3*e^2 + (20*B*b^5*c^2 - 33*A*b^4*c^3)*d^2*e^3 - (24*B*b^6*c + 1
3*A*b^5*c^2)*d*e^4)*x^3 + (15*A*b^7*e^5 + 18*(B*b^3*c^4 - 2*A*b^2*c^5)*d^5 - (37
*B*b^4*c^3 - 65*A*b^3*c^4)*d^4*e + (12*B*b^5*c^2 - 7*A*b^4*c^3)*d^3*e^2 + (4*B*b
^6*c - 23*A*b^5*c^2)*d^2*e^3 - (12*B*b^7 - A*b^6*c)*d*e^4)*x^2 + (5*A*b^7*d*e^4
+ 4*(B*b^4*c^3 - 2*A*b^3*c^4)*d^5 - (12*B*b^5*c^2 - 19*A*b^4*c^3)*d^4*e + 3*(4*B
*b^6*c - 3*A*b^5*c^2)*d^3*e^2 - (4*B*b^7 + 7*A*b^6*c)*d^2*e^3)*x)*sqrt(d))/(((b^
5*c^5*d^6 - 3*b^6*c^4*d^5*e + 3*b^7*c^3*d^4*e^2 - b^8*c^2*d^3*e^3)*x^4 + 2*(b^6*
c^4*d^6 - 3*b^7*c^3*d^5*e + 3*b^8*c^2*d^4*e^2 - b^9*c*d^3*e^3)*x^3 + (b^7*c^3*d^
6 - 3*b^8*c^2*d^5*e + 3*b^9*c*d^4*e^2 - b^10*d^3*e^3)*x^2)*sqrt(e*x + d)*sqrt(d)
), -1/8*(6*((8*(B*b*c^6 - 2*A*c^7)*d^5 - 4*(6*B*b^2*c^5 - 11*A*b*c^6)*d^4*e + 3*
(7*B*b^3*c^4 - 11*A*b^2*c^5)*d^3*e^2)*x^4 + 2*(8*(B*b^2*c^5 - 2*A*b*c^6)*d^5 - 4
*(6*B*b^3*c^4 - 11*A*b^2*c^5)*d^4*e + 3*(7*B*b^4*c^3 - 11*A*b^3*c^4)*d^3*e^2)*x^
3 + (8*(B*b^3*c^4 - 2*A*b^2*c^5)*d^5 - 4*(6*B*b^4*c^3 - 11*A*b^3*c^4)*d^4*e + 3*
(7*B*b^5*c^2 - 11*A*b^4*c^3)*d^3*e^2)*x^2)*sqrt(e*x + d)*sqrt(d)*sqrt(-c/(c*d -
b*e))*arctan(-(c*d - b*e)*sqrt(-c/(c*d - b*e))/(sqrt(e*x + d)*c)) - 3*((5*A*b^5*
c^2*e^5 + 8*(B*b*c^6 - 2*A*c^7)*d^5 - 4*(5*B*b^2*c^5 - 9*A*b*c^6)*d^4*e + (12*B*
b^3*c^4 - 17*A*b^2*c^5)*d^3*e^2 + (4*B*b^4*c^3 - 5*A*b^3*c^4)*d^2*e^3 - (4*B*b^5
*c^2 + 3*A*b^4*c^3)*d*e^4)*x^4 + 2*(5*A*b^6*c*e^5 + 8*(B*b^2*c^5 - 2*A*b*c^6)*d^
5 - 4*(5*B*b^3*c^4 - 9*A*b^2*c^5)*d^4*e + (12*B*b^4*c^3 - 17*A*b^3*c^4)*d^3*e^2
+ (4*B*b^5*c^2 - 5*A*b^4*c^3)*d^2*e^3 - (4*B*b^6*c + 3*A*b^5*c^2)*d*e^4)*x^3 + (
5*A*b^7*e^5 + 8*(B*b^3*c^4 - 2*A*b^2*c^5)*d^5 - 4*(5*B*b^4*c^3 - 9*A*b^3*c^4)*d^
4*e + (12*B*b^5*c^2 - 17*A*b^4*c^3)*d^3*e^2 + (4*B*b^6*c - 5*A*b^5*c^2)*d^2*e^3
- (4*B*b^7 + 3*A*b^6*c)*d*e^4)*x^2)*sqrt(e*x + d)*log(((e*x + 2*d)*sqrt(d) + 2*s
qrt(e*x + d)*d)/x) + 2*(2*A*b^4*c^3*d^5 - 6*A*b^5*c^2*d^4*e + 6*A*b^6*c*d^3*e^2
- 2*A*b^7*d^2*e^3 + 3*(5*A*b^5*c^2*e^5 + 4*(B*b^2*c^5 - 2*A*b*c^6)*d^4*e - (9*B*
b^3*c^4 - 16*A*b^2*c^5)*d^3*e^2 + (4*B*b^4*c^3 - 5*A*b^3*c^4)*d^2*e^3 - (4*B*b^5
*c^2 + 3*A*b^4*c^3)*d*e^4)*x^4 + (30*A*b^6*c*e^5 + 12*(B*b^2*c^5 - 2*A*b*c^6)*d^
5 - 3*(3*B*b^3*c^4 - 4*A*b^2*c^5)*d^4*e - 29*(B*b^4*c^3 - 2*A*b^3*c^4)*d^3*e^2 +
(20*B*b^5*c^2 - 33*A*b^4*c^3)*d^2*e^3 - (24*B*b^6*c + 13*A*b^5*c^2)*d*e^4)*x^3
+ (15*A*b^7*e^5 + 18*(B*b^3*c^4 - 2*A*b^2*c^5)*d^5 - (37*B*b^4*c^3 - 65*A*b^3*c^
4)*d^4*e + (12*B*b^5*c^2 - 7*A*b^4*c^3)*d^3*e^2 + (4*B*b^6*c - 23*A*b^5*c^2)*d^2
*e^3 - (12*B*b^7 - A*b^6*c)*d*e^4)*x^2 + (5*A*b^7*d*e^4 + 4*(B*b^4*c^3 - 2*A*b^3
*c^4)*d^5 - (12*B*b^5*c^2 - 19*A*b^4*c^3)*d^4*e + 3*(4*B*b^6*c - 3*A*b^5*c^2)*d^
3*e^2 - (4*B*b^7 + 7*A*b^6*c)*d^2*e^3)*x)*sqrt(d))/(((b^5*c^5*d^6 - 3*b^6*c^4*d^
5*e + 3*b^7*c^3*d^4*e^2 - b^8*c^2*d^3*e^3)*x^4 + 2*(b^6*c^4*d^6 - 3*b^7*c^3*d^5*
e + 3*b^8*c^2*d^4*e^2 - b^9*c*d^3*e^3)*x^3 + (b^7*c^3*d^6 - 3*b^8*c^2*d^5*e + 3*
b^9*c*d^4*e^2 - b^10*d^3*e^3)*x^2)*sqrt(e*x + d)*sqrt(d)), -1/8*(3*((8*(B*b*c^6
- 2*A*c^7)*d^5 - 4*(6*B*b^2*c^5 - 11*A*b*c^6)*d^4*e + 3*(7*B*b^3*c^4 - 11*A*b^2*
c^5)*d^3*e^2)*x^4 + 2*(8*(B*b^2*c^5 - 2*A*b*c^6)*d^5 - 4*(6*B*b^3*c^4 - 11*A*b^2
*c^5)*d^4*e + 3*(7*B*b^4*c^3 - 11*A*b^3*c^4)*d^3*e^2)*x^3 + (8*(B*b^3*c^4 - 2*A*
b^2*c^5)*d^5 - 4*(6*B*b^4*c^3 - 11*A*b^3*c^4)*d^4*e + 3*(7*B*b^5*c^2 - 11*A*b^4*
c^3)*d^3*e^2)*x^2)*sqrt(e*x + d)*sqrt(-d)*sqrt(c/(c*d - b*e))*log((c*e*x + 2*c*d
- b*e + 2*(c*d - b*e)*sqrt(e*x + d)*sqrt(c/(c*d - b*e)))/(c*x + b)) + 6*((5*A*b
^5*c^2*e^5 + 8*(B*b*c^6 - 2*A*c^7)*d^5 - 4*(5*B*b^2*c^5 - 9*A*b*c^6)*d^4*e + (12
*B*b^3*c^4 - 17*A*b^2*c^5)*d^3*e^2 + (4*B*b^4*c^3 - 5*A*b^3*c^4)*d^2*e^3 - (4*B*
b^5*c^2 + 3*A*b^4*c^3)*d*e^4)*x^4 + 2*(5*A*b^6*c*e^5 + 8*(B*b^2*c^5 - 2*A*b*c^6)
*d^5 - 4*(5*B*b^3*c^4 - 9*A*b^2*c^5)*d^4*e + (12*B*b^4*c^3 - 17*A*b^3*c^4)*d^3*e
^2 + (4*B*b^5*c^2 - 5*A*b^4*c^3)*d^2*e^3 - (4*B*b^6*c + 3*A*b^5*c^2)*d*e^4)*x^3
+ (5*A*b^7*e^5 + 8*(B*b^3*c^4 - 2*A*b^2*c^5)*d^5 - 4*(5*B*b^4*c^3 - 9*A*b^3*c^4)
*d^4*e + (12*B*b^5*c^2 - 17*A*b^4*c^3)*d^3*e^2 + (4*B*b^6*c - 5*A*b^5*c^2)*d^2*e
^3 - (4*B*b^7 + 3*A*b^6*c)*d*e^4)*x^2)*sqrt(e*x + d)*arctan(d/(sqrt(e*x + d)*sqr
t(-d))) + 2*(2*A*b^4*c^3*d^5 - 6*A*b^5*c^2*d^4*e + 6*A*b^6*c*d^3*e^2 - 2*A*b^7*d
^2*e^3 + 3*(5*A*b^5*c^2*e^5 + 4*(B*b^2*c^5 - 2*A*b*c^6)*d^4*e - (9*B*b^3*c^4 - 1
6*A*b^2*c^5)*d^3*e^2 + (4*B*b^4*c^3 - 5*A*b^3*c^4)*d^2*e^3 - (4*B*b^5*c^2 + 3*A*
b^4*c^3)*d*e^4)*x^4 + (30*A*b^6*c*e^5 + 12*(B*b^2*c^5 - 2*A*b*c^6)*d^5 - 3*(3*B*
b^3*c^4 - 4*A*b^2*c^5)*d^4*e - 29*(B*b^4*c^3 - 2*A*b^3*c^4)*d^3*e^2 + (20*B*b^5*
c^2 - 33*A*b^4*c^3)*d^2*e^3 - (24*B*b^6*c + 13*A*b^5*c^2)*d*e^4)*x^3 + (15*A*b^7
*e^5 + 18*(B*b^3*c^4 - 2*A*b^2*c^5)*d^5 - (37*B*b^4*c^3 - 65*A*b^3*c^4)*d^4*e +
(12*B*b^5*c^2 - 7*A*b^4*c^3)*d^3*e^2 + (4*B*b^6*c - 23*A*b^5*c^2)*d^2*e^3 - (12*
B*b^7 - A*b^6*c)*d*e^4)*x^2 + (5*A*b^7*d*e^4 + 4*(B*b^4*c^3 - 2*A*b^3*c^4)*d^5 -
(12*B*b^5*c^2 - 19*A*b^4*c^3)*d^4*e + 3*(4*B*b^6*c - 3*A*b^5*c^2)*d^3*e^2 - (4*
B*b^7 + 7*A*b^6*c)*d^2*e^3)*x)*sqrt(-d))/(((b^5*c^5*d^6 - 3*b^6*c^4*d^5*e + 3*b^
7*c^3*d^4*e^2 - b^8*c^2*d^3*e^3)*x^4 + 2*(b^6*c^4*d^6 - 3*b^7*c^3*d^5*e + 3*b^8*
c^2*d^4*e^2 - b^9*c*d^3*e^3)*x^3 + (b^7*c^3*d^6 - 3*b^8*c^2*d^5*e + 3*b^9*c*d^4*
e^2 - b^10*d^3*e^3)*x^2)*sqrt(e*x + d)*sqrt(-d)), -1/4*(3*((8*(B*b*c^6 - 2*A*c^7
)*d^5 - 4*(6*B*b^2*c^5 - 11*A*b*c^6)*d^4*e + 3*(7*B*b^3*c^4 - 11*A*b^2*c^5)*d^3*
e^2)*x^4 + 2*(8*(B*b^2*c^5 - 2*A*b*c^6)*d^5 - 4*(6*B*b^3*c^4 - 11*A*b^2*c^5)*d^4
*e + 3*(7*B*b^4*c^3 - 11*A*b^3*c^4)*d^3*e^2)*x^3 + (8*(B*b^3*c^4 - 2*A*b^2*c^5)*
d^5 - 4*(6*B*b^4*c^3 - 11*A*b^3*c^4)*d^4*e + 3*(7*B*b^5*c^2 - 11*A*b^4*c^3)*d^3*
e^2)*x^2)*sqrt(e*x + d)*sqrt(-d)*sqrt(-c/(c*d - b*e))*arctan(-(c*d - b*e)*sqrt(-
c/(c*d - b*e))/(sqrt(e*x + d)*c)) + 3*((5*A*b^5*c^2*e^5 + 8*(B*b*c^6 - 2*A*c^7)*
d^5 - 4*(5*B*b^2*c^5 - 9*A*b*c^6)*d^4*e + (12*B*b^3*c^4 - 17*A*b^2*c^5)*d^3*e^2
+ (4*B*b^4*c^3 - 5*A*b^3*c^4)*d^2*e^3 - (4*B*b^5*c^2 + 3*A*b^4*c^3)*d*e^4)*x^4 +
2*(5*A*b^6*c*e^5 + 8*(B*b^2*c^5 - 2*A*b*c^6)*d^5 - 4*(5*B*b^3*c^4 - 9*A*b^2*c^5
)*d^4*e + (12*B*b^4*c^3 - 17*A*b^3*c^4)*d^3*e^2 + (4*B*b^5*c^2 - 5*A*b^4*c^3)*d^
2*e^3 - (4*B*b^6*c + 3*A*b^5*c^2)*d*e^4)*x^3 + (5*A*b^7*e^5 + 8*(B*b^3*c^4 - 2*A
*b^2*c^5)*d^5 - 4*(5*B*b^4*c^3 - 9*A*b^3*c^4)*d^4*e + (12*B*b^5*c^2 - 17*A*b^4*c
^3)*d^3*e^2 + (4*B*b^6*c - 5*A*b^5*c^2)*d^2*e^3 - (4*B*b^7 + 3*A*b^6*c)*d*e^4)*x
^2)*sqrt(e*x + d)*arctan(d/(sqrt(e*x + d)*sqrt(-d))) + (2*A*b^4*c^3*d^5 - 6*A*b^
5*c^2*d^4*e + 6*A*b^6*c*d^3*e^2 - 2*A*b^7*d^2*e^3 + 3*(5*A*b^5*c^2*e^5 + 4*(B*b^
2*c^5 - 2*A*b*c^6)*d^4*e - (9*B*b^3*c^4 - 16*A*b^2*c^5)*d^3*e^2 + (4*B*b^4*c^3 -
5*A*b^3*c^4)*d^2*e^3 - (4*B*b^5*c^2 + 3*A*b^4*c^3)*d*e^4)*x^4 + (30*A*b^6*c*e^5
+ 12*(B*b^2*c^5 - 2*A*b*c^6)*d^5 - 3*(3*B*b^3*c^4 - 4*A*b^2*c^5)*d^4*e - 29*(B*
b^4*c^3 - 2*A*b^3*c^4)*d^3*e^2 + (20*B*b^5*c^2 - 33*A*b^4*c^3)*d^2*e^3 - (24*B*b
^6*c + 13*A*b^5*c^2)*d*e^4)*x^3 + (15*A*b^7*e^5 + 18*(B*b^3*c^4 - 2*A*b^2*c^5)*d
^5 - (37*B*b^4*c^3 - 65*A*b^3*c^4)*d^4*e + (12*B*b^5*c^2 - 7*A*b^4*c^3)*d^3*e^2
+ (4*B*b^6*c - 23*A*b^5*c^2)*d^2*e^3 - (12*B*b^7 - A*b^6*c)*d*e^4)*x^2 + (5*A*b^
7*d*e^4 + 4*(B*b^4*c^3 - 2*A*b^3*c^4)*d^5 - (12*B*b^5*c^2 - 19*A*b^4*c^3)*d^4*e
+ 3*(4*B*b^6*c - 3*A*b^5*c^2)*d^3*e^2 - (4*B*b^7 + 7*A*b^6*c)*d^2*e^3)*x)*sqrt(-
d))/(((b^5*c^5*d^6 - 3*b^6*c^4*d^5*e + 3*b^7*c^3*d^4*e^2 - b^8*c^2*d^3*e^3)*x^4
+ 2*(b^6*c^4*d^6 - 3*b^7*c^3*d^5*e + 3*b^8*c^2*d^4*e^2 - b^9*c*d^3*e^3)*x^3 + (b
^7*c^3*d^6 - 3*b^8*c^2*d^5*e + 3*b^9*c*d^4*e^2 - b^10*d^3*e^3)*x^2)*sqrt(e*x + d
)*sqrt(-d))]

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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)/(e*x+d)**(3/2)/(c*x**2+b*x)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.348863, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done