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3.1248 \(\int \frac{(A+B x) (d+e x)^{7/2}}{\left (b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=381 \[ -\frac{(d+e x)^{5/2} (7 A b e-8 A c d+4 b B d)}{4 b^2 x (b+c x)^2}-\frac{d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (7 b^2 e (5 A e+4 B d)-12 b c d (7 A e+2 B d)+48 A c^2 d^2\right )}{4 b^5}-\frac{(d+e x)^{3/2} (c d-b e) \left (11 A b c e-12 A c^2 d-2 b^2 B e+6 b B c d\right )}{4 b^3 c (b+c x)^2}+\frac{(c d-b e)^{3/2} \left (-b^2 c e (A e+8 B d)-12 b c^2 d (A e+2 B d)+48 A c^3 d^2-3 b^3 B e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{4 b^5 c^{5/2}}+\frac{\sqrt{d+e x} (c d-b e) \left (b^2 c e (A e+5 B d)-12 b c^2 d (2 A e+B d)+24 A c^3 d^2+3 b^3 B e^2\right )}{4 b^4 c^2 (b+c x)}-\frac{A (d+e x)^{7/2}}{2 b x^2 (b+c x)^2} \]

[Out]

((c*d - b*e)*(24*A*c^3*d^2 + 3*b^3*B*e^2 + b^2*c*e*(5*B*d + A*e) - 12*b*c^2*d*(B
*d + 2*A*e))*Sqrt[d + e*x])/(4*b^4*c^2*(b + c*x)) - ((c*d - b*e)*(6*b*B*c*d - 12
*A*c^2*d - 2*b^2*B*e + 11*A*b*c*e)*(d + e*x)^(3/2))/(4*b^3*c*(b + c*x)^2) - ((4*
b*B*d - 8*A*c*d + 7*A*b*e)*(d + e*x)^(5/2))/(4*b^2*x*(b + c*x)^2) - (A*(d + e*x)
^(7/2))/(2*b*x^2*(b + c*x)^2) - (d^(3/2)*(48*A*c^2*d^2 + 7*b^2*e*(4*B*d + 5*A*e)
 - 12*b*c*d*(2*B*d + 7*A*e))*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(4*b^5) + ((c*d - b
*e)^(3/2)*(48*A*c^3*d^2 - 3*b^3*B*e^2 - 12*b*c^2*d*(2*B*d + A*e) - b^2*c*e*(8*B*
d + A*e))*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(4*b^5*c^(5/2))

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Rubi [A]  time = 1.95392, antiderivative size = 381, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{(d+e x)^{5/2} (7 A b e-8 A c d+4 b B d)}{4 b^2 x (b+c x)^2}-\frac{d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (7 b^2 e (5 A e+4 B d)-12 b c d (7 A e+2 B d)+48 A c^2 d^2\right )}{4 b^5}-\frac{(d+e x)^{3/2} (c d-b e) \left (11 A b c e-12 A c^2 d-2 b^2 B e+6 b B c d\right )}{4 b^3 c (b+c x)^2}+\frac{(c d-b e)^{3/2} \left (-b^2 c e (A e+8 B d)-12 b c^2 d (A e+2 B d)+48 A c^3 d^2-3 b^3 B e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{4 b^5 c^{5/2}}+\frac{\sqrt{d+e x} (c d-b e) \left (b^2 c e (A e+5 B d)-12 b c^2 d (2 A e+B d)+24 A c^3 d^2+3 b^3 B e^2\right )}{4 b^4 c^2 (b+c x)}-\frac{A (d+e x)^{7/2}}{2 b x^2 (b+c x)^2} \]

Antiderivative was successfully verified.

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

[Out]

((c*d - b*e)*(24*A*c^3*d^2 + 3*b^3*B*e^2 + b^2*c*e*(5*B*d + A*e) - 12*b*c^2*d*(B
*d + 2*A*e))*Sqrt[d + e*x])/(4*b^4*c^2*(b + c*x)) - ((c*d - b*e)*(6*b*B*c*d - 12
*A*c^2*d - 2*b^2*B*e + 11*A*b*c*e)*(d + e*x)^(3/2))/(4*b^3*c*(b + c*x)^2) - ((4*
b*B*d - 8*A*c*d + 7*A*b*e)*(d + e*x)^(5/2))/(4*b^2*x*(b + c*x)^2) - (A*(d + e*x)
^(7/2))/(2*b*x^2*(b + c*x)^2) - (d^(3/2)*(48*A*c^2*d^2 + 7*b^2*e*(4*B*d + 5*A*e)
 - 12*b*c*d*(2*B*d + 7*A*e))*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(4*b^5) + ((c*d - b
*e)^(3/2)*(48*A*c^3*d^2 - 3*b^3*B*e^2 - 12*b*c^2*d*(2*B*d + A*e) - b^2*c*e*(8*B*
d + A*e))*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(4*b^5*c^(5/2))

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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)**(7/2)/(c*x**2+b*x)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 1.13244, size = 288, normalized size = 0.76 \[ \frac{d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (-7 b^2 e (5 A e+4 B d)+12 b c d (7 A e+2 B d)-48 A c^2 d^2\right )+b \sqrt{d+e x} \left (\frac{(c d-b e)^2 \left (A b c e+12 A c^2 d-5 b^2 B e-8 b B c d\right )}{c^2 (b+c x)}+\frac{2 b (b B-A c) (b e-c d)^3}{c^2 (b+c x)^2}-\frac{d^2 (13 A b e-12 A c d+4 b B d)}{x}-\frac{2 A b d^3}{x^2}\right )-\frac{(c d-b e)^{3/2} \left (b^2 c e (A e+8 B d)+12 b c^2 d (A e+2 B d)-48 A c^3 d^2+3 b^3 B e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{c^{5/2}}}{4 b^5} \]

Antiderivative was successfully verified.

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

[Out]

(b*Sqrt[d + e*x]*((-2*A*b*d^3)/x^2 - (d^2*(4*b*B*d - 12*A*c*d + 13*A*b*e))/x + (
2*b*(b*B - A*c)*(-(c*d) + b*e)^3)/(c^2*(b + c*x)^2) + ((c*d - b*e)^2*(-8*b*B*c*d
 + 12*A*c^2*d - 5*b^2*B*e + A*b*c*e))/(c^2*(b + c*x))) + d^(3/2)*(-48*A*c^2*d^2
- 7*b^2*e*(4*B*d + 5*A*e) + 12*b*c*d*(2*B*d + 7*A*e))*ArcTanh[Sqrt[d + e*x]/Sqrt
[d]] - ((c*d - b*e)^(3/2)*(-48*A*c^3*d^2 + 3*b^3*B*e^2 + 12*b*c^2*d*(2*B*d + A*e
) + b^2*c*e*(8*B*d + A*e))*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/c^(
5/2))/(4*b^5)

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Maple [B]  time = 0.036, size = 1218, normalized size = 3.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-35/4*e^2*d^(3/2)/b^3*arctanh((e*x+d)^(1/2)/d^(1/2))*A-7*e*d^(5/2)/b^3*arctanh((
e*x+d)^(1/2)/d^(1/2))*B+1/4*e^4/b/(c*e*x+b*e)^2*(e*x+d)^(3/2)*A+1/2*e^3/b/(c*e*x
+b*e)^2*(e*x+d)^(3/2)*B*d+15/4*e^4/b/(c*e*x+b*e)^2*(e*x+d)^(1/2)*A*d+1/4*e^4/(c*
e*x+b*e)^2/c*(e*x+d)^(1/2)*B*d-12/b^5*c^3/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(
1/2)/((b*e-c*d)*c)^(1/2))*A*d^4+6/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^4*c^2+21*e*d^(5/2)/b^4*arctanh((e*x+d)^(1/2)/d^(1/2))*
A*c-1/e*d^3/b^3/x^2*(e*x+d)^(3/2)*B+1/e*d^4/b^3/x^2*(e*x+d)^(1/2)*B-3/4*e^5*b/(c
*e*x+b*e)^2/c^2*(e*x+d)^(1/2)*B+1/4*e^4/b/c/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)
^(1/2)/((b*e-c*d)*c)^(1/2))*A+15/4*e^3/b/(c*e*x+b*e)^2*(e*x+d)^(1/2)*B*d^2+5/2*e
^3/b^2/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*A*d+11/4*
e^2/b^2/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*B*d^2-13
/4*d^2/b^3/x^2*(e*x+d)^(3/2)*A+11/4*d^3/b^3/x^2*(e*x+d)^(1/2)*A+6*d^(7/2)/b^4*ar
ctanh((e*x+d)^(1/2)/d^(1/2))*B*c-5/4*e^4/(c*e*x+b*e)^2/c*(e*x+d)^(3/2)*B-1/4*e^5
/(c*e*x+b*e)^2/c*(e*x+d)^(1/2)*A+3/4*e^4/c^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*d^4/b^4/x^2*c*(e*x+d)^(1/2)*A+11/4*e^2/b^2/(c
*e*x+b*e)^2*c*(e*x+d)^(3/2)*B*d^2-2*e/b^3/(c*e*x+b*e)^2*(e*x+d)^(3/2)*B*d^3*c^2-
39/4*e^3/b^2/(c*e*x+b*e)^2*c*(e*x+d)^(1/2)*A*d^2-10*e/b^3*c/((b*e-c*d)*c)^(1/2)*
arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*B*d^3+3*e/b^4/(c*e*x+b*e)^2*c^3*(e*x
+d)^(3/2)*A*d^3+37/4*e^2/b^3/(c*e*x+b*e)^2*(e*x+d)^(1/2)*A*d^3*c^2-3*e/b^4/(c*e*
x+b*e)^2*c^3*(e*x+d)^(1/2)*A*d^4-21/4*e^2/b^2/(c*e*x+b*e)^2*c*(e*x+d)^(1/2)*B*d^
3-71/4*e^2/b^3*c/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))
*A*d^2+27*e/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^3*c^2+2*e/b^3/(c*e*x+b*e)^2*(e*x+d)^(1/2)*B*d^4*c^2+1/2*e^3/b/c/((b*e-c*d)*c
)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*B*d+5/2*e^3/b^2/(c*e*x+b*e)^
2*(e*x+d)^(3/2)*A*c*d-23/4*e^2/b^3/(c*e*x+b*e)^2*(e*x+d)^(3/2)*A*c^2*d^2+3/e*d^3
/b^4/x^2*c*(e*x+d)^(3/2)*A-12*d^(7/2)/b^5*arctanh((e*x+d)^(1/2)/d^(1/2))*A*c^2

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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)*(e*x + d)^(7/2)/(c*x^2 + b*x)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 36.3847, 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)*(e*x + d)^(7/2)/(c*x^2 + b*x)^3,x, algorithm="fricas")

[Out]

[-1/8*(((24*(B*b*c^5 - 2*A*c^6)*d^3 - 4*(4*B*b^2*c^4 - 15*A*b*c^5)*d^2*e - (5*B*
b^3*c^3 + 11*A*b^2*c^4)*d*e^2 - (3*B*b^4*c^2 + A*b^3*c^3)*e^3)*x^4 + 2*(24*(B*b^
2*c^4 - 2*A*b*c^5)*d^3 - 4*(4*B*b^3*c^3 - 15*A*b^2*c^4)*d^2*e - (5*B*b^4*c^2 + 1
1*A*b^3*c^3)*d*e^2 - (3*B*b^5*c + A*b^4*c^2)*e^3)*x^3 + (24*(B*b^3*c^3 - 2*A*b^2
*c^4)*d^3 - 4*(4*B*b^4*c^2 - 15*A*b^3*c^3)*d^2*e - (5*B*b^5*c + 11*A*b^4*c^2)*d*
e^2 - (3*B*b^6 + A*b^5*c)*e^3)*x^2)*sqrt((c*d - b*e)/c)*log((c*e*x + 2*c*d - b*e
 + 2*sqrt(e*x + d)*c*sqrt((c*d - b*e)/c))/(c*x + b)) - ((35*A*b^2*c^4*d*e^2 - 24
*(B*b*c^5 - 2*A*c^6)*d^3 + 28*(B*b^2*c^4 - 3*A*b*c^5)*d^2*e)*x^4 + 2*(35*A*b^3*c
^3*d*e^2 - 24*(B*b^2*c^4 - 2*A*b*c^5)*d^3 + 28*(B*b^3*c^3 - 3*A*b^2*c^4)*d^2*e)*
x^3 + (35*A*b^4*c^2*d*e^2 - 24*(B*b^3*c^3 - 2*A*b^2*c^4)*d^3 + 28*(B*b^4*c^2 - 3
*A*b^3*c^3)*d^2*e)*x^2)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) + 2
*(2*A*b^4*c^2*d^3 + (12*(B*b^2*c^4 - 2*A*b*c^5)*d^3 - (11*B*b^3*c^3 - 36*A*b^2*c
^4)*d^2*e - 2*(B*b^4*c^2 + 5*A*b^3*c^3)*d*e^2 + (5*B*b^5*c - A*b^4*c^2)*e^3)*x^3
 + (18*(B*b^3*c^3 - 2*A*b^2*c^4)*d^3 - (17*B*b^4*c^2 - 55*A*b^3*c^3)*d^2*e + 4*(
B*b^5*c - 4*A*b^4*c^2)*d*e^2 + (3*B*b^6 + A*b^5*c)*e^3)*x^2 + (13*A*b^4*c^2*d^2*
e + 4*(B*b^4*c^2 - 2*A*b^3*c^3)*d^3)*x)*sqrt(e*x + d))/(b^5*c^4*x^4 + 2*b^6*c^3*
x^3 + b^7*c^2*x^2), -1/8*(2*((24*(B*b*c^5 - 2*A*c^6)*d^3 - 4*(4*B*b^2*c^4 - 15*A
*b*c^5)*d^2*e - (5*B*b^3*c^3 + 11*A*b^2*c^4)*d*e^2 - (3*B*b^4*c^2 + A*b^3*c^3)*e
^3)*x^4 + 2*(24*(B*b^2*c^4 - 2*A*b*c^5)*d^3 - 4*(4*B*b^3*c^3 - 15*A*b^2*c^4)*d^2
*e - (5*B*b^4*c^2 + 11*A*b^3*c^3)*d*e^2 - (3*B*b^5*c + A*b^4*c^2)*e^3)*x^3 + (24
*(B*b^3*c^3 - 2*A*b^2*c^4)*d^3 - 4*(4*B*b^4*c^2 - 15*A*b^3*c^3)*d^2*e - (5*B*b^5
*c + 11*A*b^4*c^2)*d*e^2 - (3*B*b^6 + A*b^5*c)*e^3)*x^2)*sqrt(-(c*d - b*e)/c)*ar
ctan(sqrt(e*x + d)/sqrt(-(c*d - b*e)/c)) - ((35*A*b^2*c^4*d*e^2 - 24*(B*b*c^5 -
2*A*c^6)*d^3 + 28*(B*b^2*c^4 - 3*A*b*c^5)*d^2*e)*x^4 + 2*(35*A*b^3*c^3*d*e^2 - 2
4*(B*b^2*c^4 - 2*A*b*c^5)*d^3 + 28*(B*b^3*c^3 - 3*A*b^2*c^4)*d^2*e)*x^3 + (35*A*
b^4*c^2*d*e^2 - 24*(B*b^3*c^3 - 2*A*b^2*c^4)*d^3 + 28*(B*b^4*c^2 - 3*A*b^3*c^3)*
d^2*e)*x^2)*sqrt(d)*log((e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) + 2*(2*A*b^4*c^
2*d^3 + (12*(B*b^2*c^4 - 2*A*b*c^5)*d^3 - (11*B*b^3*c^3 - 36*A*b^2*c^4)*d^2*e -
2*(B*b^4*c^2 + 5*A*b^3*c^3)*d*e^2 + (5*B*b^5*c - A*b^4*c^2)*e^3)*x^3 + (18*(B*b^
3*c^3 - 2*A*b^2*c^4)*d^3 - (17*B*b^4*c^2 - 55*A*b^3*c^3)*d^2*e + 4*(B*b^5*c - 4*
A*b^4*c^2)*d*e^2 + (3*B*b^6 + A*b^5*c)*e^3)*x^2 + (13*A*b^4*c^2*d^2*e + 4*(B*b^4
*c^2 - 2*A*b^3*c^3)*d^3)*x)*sqrt(e*x + d))/(b^5*c^4*x^4 + 2*b^6*c^3*x^3 + b^7*c^
2*x^2), -1/8*(2*((35*A*b^2*c^4*d*e^2 - 24*(B*b*c^5 - 2*A*c^6)*d^3 + 28*(B*b^2*c^
4 - 3*A*b*c^5)*d^2*e)*x^4 + 2*(35*A*b^3*c^3*d*e^2 - 24*(B*b^2*c^4 - 2*A*b*c^5)*d
^3 + 28*(B*b^3*c^3 - 3*A*b^2*c^4)*d^2*e)*x^3 + (35*A*b^4*c^2*d*e^2 - 24*(B*b^3*c
^3 - 2*A*b^2*c^4)*d^3 + 28*(B*b^4*c^2 - 3*A*b^3*c^3)*d^2*e)*x^2)*sqrt(-d)*arctan
(sqrt(e*x + d)/sqrt(-d)) + ((24*(B*b*c^5 - 2*A*c^6)*d^3 - 4*(4*B*b^2*c^4 - 15*A*
b*c^5)*d^2*e - (5*B*b^3*c^3 + 11*A*b^2*c^4)*d*e^2 - (3*B*b^4*c^2 + A*b^3*c^3)*e^
3)*x^4 + 2*(24*(B*b^2*c^4 - 2*A*b*c^5)*d^3 - 4*(4*B*b^3*c^3 - 15*A*b^2*c^4)*d^2*
e - (5*B*b^4*c^2 + 11*A*b^3*c^3)*d*e^2 - (3*B*b^5*c + A*b^4*c^2)*e^3)*x^3 + (24*
(B*b^3*c^3 - 2*A*b^2*c^4)*d^3 - 4*(4*B*b^4*c^2 - 15*A*b^3*c^3)*d^2*e - (5*B*b^5*
c + 11*A*b^4*c^2)*d*e^2 - (3*B*b^6 + A*b^5*c)*e^3)*x^2)*sqrt((c*d - b*e)/c)*log(
(c*e*x + 2*c*d - b*e + 2*sqrt(e*x + d)*c*sqrt((c*d - b*e)/c))/(c*x + b)) + 2*(2*
A*b^4*c^2*d^3 + (12*(B*b^2*c^4 - 2*A*b*c^5)*d^3 - (11*B*b^3*c^3 - 36*A*b^2*c^4)*
d^2*e - 2*(B*b^4*c^2 + 5*A*b^3*c^3)*d*e^2 + (5*B*b^5*c - A*b^4*c^2)*e^3)*x^3 + (
18*(B*b^3*c^3 - 2*A*b^2*c^4)*d^3 - (17*B*b^4*c^2 - 55*A*b^3*c^3)*d^2*e + 4*(B*b^
5*c - 4*A*b^4*c^2)*d*e^2 + (3*B*b^6 + A*b^5*c)*e^3)*x^2 + (13*A*b^4*c^2*d^2*e +
4*(B*b^4*c^2 - 2*A*b^3*c^3)*d^3)*x)*sqrt(e*x + d))/(b^5*c^4*x^4 + 2*b^6*c^3*x^3
+ b^7*c^2*x^2), -1/4*(((35*A*b^2*c^4*d*e^2 - 24*(B*b*c^5 - 2*A*c^6)*d^3 + 28*(B*
b^2*c^4 - 3*A*b*c^5)*d^2*e)*x^4 + 2*(35*A*b^3*c^3*d*e^2 - 24*(B*b^2*c^4 - 2*A*b*
c^5)*d^3 + 28*(B*b^3*c^3 - 3*A*b^2*c^4)*d^2*e)*x^3 + (35*A*b^4*c^2*d*e^2 - 24*(B
*b^3*c^3 - 2*A*b^2*c^4)*d^3 + 28*(B*b^4*c^2 - 3*A*b^3*c^3)*d^2*e)*x^2)*sqrt(-d)*
arctan(sqrt(e*x + d)/sqrt(-d)) + ((24*(B*b*c^5 - 2*A*c^6)*d^3 - 4*(4*B*b^2*c^4 -
 15*A*b*c^5)*d^2*e - (5*B*b^3*c^3 + 11*A*b^2*c^4)*d*e^2 - (3*B*b^4*c^2 + A*b^3*c
^3)*e^3)*x^4 + 2*(24*(B*b^2*c^4 - 2*A*b*c^5)*d^3 - 4*(4*B*b^3*c^3 - 15*A*b^2*c^4
)*d^2*e - (5*B*b^4*c^2 + 11*A*b^3*c^3)*d*e^2 - (3*B*b^5*c + A*b^4*c^2)*e^3)*x^3
+ (24*(B*b^3*c^3 - 2*A*b^2*c^4)*d^3 - 4*(4*B*b^4*c^2 - 15*A*b^3*c^3)*d^2*e - (5*
B*b^5*c + 11*A*b^4*c^2)*d*e^2 - (3*B*b^6 + A*b^5*c)*e^3)*x^2)*sqrt(-(c*d - b*e)/
c)*arctan(sqrt(e*x + d)/sqrt(-(c*d - b*e)/c)) + (2*A*b^4*c^2*d^3 + (12*(B*b^2*c^
4 - 2*A*b*c^5)*d^3 - (11*B*b^3*c^3 - 36*A*b^2*c^4)*d^2*e - 2*(B*b^4*c^2 + 5*A*b^
3*c^3)*d*e^2 + (5*B*b^5*c - A*b^4*c^2)*e^3)*x^3 + (18*(B*b^3*c^3 - 2*A*b^2*c^4)*
d^3 - (17*B*b^4*c^2 - 55*A*b^3*c^3)*d^2*e + 4*(B*b^5*c - 4*A*b^4*c^2)*d*e^2 + (3
*B*b^6 + A*b^5*c)*e^3)*x^2 + (13*A*b^4*c^2*d^2*e + 4*(B*b^4*c^2 - 2*A*b^3*c^3)*d
^3)*x)*sqrt(e*x + d))/(b^5*c^4*x^4 + 2*b^6*c^3*x^3 + b^7*c^2*x^2)]

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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)**(7/2)/(c*x**2+b*x)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.322223, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done

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