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3.1246 \(\int \frac{A+B x}{(d+e x)^{7/2} \left (b x+c x^2\right )^2} \, dx\)

Optimal. Leaf size=464 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) (-7 A b e-4 A c d+2 b B d)}{b^3 d^{9/2}}-\frac{e \left (b^2 (-e) (2 B d-7 A e)-5 b c d (2 A e+B d)+10 A c^2 d^2\right )}{5 b^2 d^2 (d+e x)^{5/2} (c d-b e)^2}+\frac{c (A b e-2 A c d+b B d)}{b^2 d (b+c x) (d+e x)^{5/2} (c d-b e)}+\frac{c^{7/2} \left (11 A b c e-4 A c^2 d-9 b^2 B e+2 b B c d\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 (c d-b e)^{9/2}}-\frac{e \left (b^3 e^2 (2 B d-7 A e)-b^2 c d e (6 B d-17 A e)-3 b c^2 d^2 (3 A e+B d)+6 A c^3 d^3\right )}{3 b^2 d^3 (d+e x)^{3/2} (c d-b e)^3}-\frac{e \left (b^4 \left (-e^3\right ) (2 B d-7 A e)+8 b^3 c d e^2 (B d-3 A e)-2 b^2 c^2 d^2 e (6 B d-13 A e)-b c^3 d^3 (4 A e+B d)+2 A c^4 d^4\right )}{b^2 d^4 \sqrt{d+e x} (c d-b e)^4}-\frac{A}{b d x (b+c x) (d+e x)^{5/2}} \]

[Out]

-(e*(10*A*c^2*d^2 - b^2*e*(2*B*d - 7*A*e) - 5*b*c*d*(B*d + 2*A*e)))/(5*b^2*d^2*(
c*d - b*e)^2*(d + e*x)^(5/2)) + (c*(b*B*d - 2*A*c*d + A*b*e))/(b^2*d*(c*d - b*e)
*(b + c*x)*(d + e*x)^(5/2)) - A/(b*d*x*(b + c*x)*(d + e*x)^(5/2)) - (e*(6*A*c^3*
d^3 - b^2*c*d*e*(6*B*d - 17*A*e) + b^3*e^2*(2*B*d - 7*A*e) - 3*b*c^2*d^2*(B*d +
3*A*e)))/(3*b^2*d^3*(c*d - b*e)^3*(d + e*x)^(3/2)) - (e*(2*A*c^4*d^4 - 2*b^2*c^2
*d^2*e*(6*B*d - 13*A*e) - b^4*e^3*(2*B*d - 7*A*e) + 8*b^3*c*d*e^2*(B*d - 3*A*e)
- b*c^3*d^3*(B*d + 4*A*e)))/(b^2*d^4*(c*d - b*e)^4*Sqrt[d + e*x]) - ((2*b*B*d -
4*A*c*d - 7*A*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(b^3*d^(9/2)) + (c^(7/2)*(2*b
*B*c*d - 4*A*c^2*d - 9*b^2*B*e + 11*A*b*c*e)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqr
t[c*d - b*e]])/(b^3*(c*d - b*e)^(9/2))

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Rubi [A]  time = 2.99792, antiderivative size = 464, 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 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) (-7 A b e-4 A c d+2 b B d)}{b^3 d^{9/2}}-\frac{e \left (b^2 (-e) (2 B d-7 A e)-5 b c d (2 A e+B d)+10 A c^2 d^2\right )}{5 b^2 d^2 (d+e x)^{5/2} (c d-b e)^2}+\frac{c (A b e-2 A c d+b B d)}{b^2 d (b+c x) (d+e x)^{5/2} (c d-b e)}-\frac{c^{7/2} \left (-b c (11 A e+2 B d)+4 A c^2 d+9 b^2 B e\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 (c d-b e)^{9/2}}-\frac{e \left (b^3 e^2 (2 B d-7 A e)-b^2 c d e (6 B d-17 A e)-3 b c^2 d^2 (3 A e+B d)+6 A c^3 d^3\right )}{3 b^2 d^3 (d+e x)^{3/2} (c d-b e)^3}-\frac{e \left (b^4 \left (-e^3\right ) (2 B d-7 A e)+8 b^3 c d e^2 (B d-3 A e)-2 b^2 c^2 d^2 e (6 B d-13 A e)-b c^3 d^3 (4 A e+B d)+2 A c^4 d^4\right )}{b^2 d^4 \sqrt{d+e x} (c d-b e)^4}-\frac{A}{b d x (b+c x) (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

-(e*(10*A*c^2*d^2 - b^2*e*(2*B*d - 7*A*e) - 5*b*c*d*(B*d + 2*A*e)))/(5*b^2*d^2*(
c*d - b*e)^2*(d + e*x)^(5/2)) + (c*(b*B*d - 2*A*c*d + A*b*e))/(b^2*d*(c*d - b*e)
*(b + c*x)*(d + e*x)^(5/2)) - A/(b*d*x*(b + c*x)*(d + e*x)^(5/2)) - (e*(6*A*c^3*
d^3 - b^2*c*d*e*(6*B*d - 17*A*e) + b^3*e^2*(2*B*d - 7*A*e) - 3*b*c^2*d^2*(B*d +
3*A*e)))/(3*b^2*d^3*(c*d - b*e)^3*(d + e*x)^(3/2)) - (e*(2*A*c^4*d^4 - 2*b^2*c^2
*d^2*e*(6*B*d - 13*A*e) - b^4*e^3*(2*B*d - 7*A*e) + 8*b^3*c*d*e^2*(B*d - 3*A*e)
- b*c^3*d^3*(B*d + 4*A*e)))/(b^2*d^4*(c*d - b*e)^4*Sqrt[d + e*x]) - ((2*b*B*d -
4*A*c*d - 7*A*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(b^3*d^(9/2)) - (c^(7/2)*(4*A
*c^2*d + 9*b^2*B*e - b*c*(2*B*d + 11*A*e))*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[
c*d - b*e]])/(b^3*(c*d - b*e)^(9/2))

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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)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 2.84797, size = 340, normalized size = 0.73 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) (-7 A b e-4 A c d+2 b B d)}{b^3 d^{9/2}}+\sqrt{d+e x} \left (\frac{c^4 (b B-A c)}{b^2 (b+c x) (c d-b e)^4}+\frac{2 e^2 \left (A e \left (-3 b^2 e^2+10 b c d e-10 c^2 d^2\right )+B d \left (b^2 e^2-4 b c d e+6 c^2 d^2\right )\right )}{d^4 (d+e x) (c d-b e)^4}-\frac{A}{b^2 d^4 x}+\frac{2 e^2 (2 A e (b e-2 c d)+B d (3 c d-b e))}{3 d^3 (d+e x)^2 (c d-b e)^3}+\frac{2 e^2 (B d-A e)}{5 d^2 (d+e x)^3 (c d-b e)^2}\right )-\frac{c^{7/2} \left (-b c (11 A e+2 B d)+4 A c^2 d+9 b^2 B e\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 (c d-b e)^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[d + e*x]*(-(A/(b^2*d^4*x)) + (c^4*(b*B - A*c))/(b^2*(c*d - b*e)^4*(b + c*x)
) + (2*e^2*(B*d - A*e))/(5*d^2*(c*d - b*e)^2*(d + e*x)^3) + (2*e^2*(B*d*(3*c*d -
 b*e) + 2*A*e*(-2*c*d + b*e)))/(3*d^3*(c*d - b*e)^3*(d + e*x)^2) + (2*e^2*(A*e*(
-10*c^2*d^2 + 10*b*c*d*e - 3*b^2*e^2) + B*d*(6*c^2*d^2 - 4*b*c*d*e + b^2*e^2)))/
(d^4*(c*d - b*e)^4*(d + e*x))) - ((2*b*B*d - 4*A*c*d - 7*A*b*e)*ArcTanh[Sqrt[d +
 e*x]/Sqrt[d]])/(b^3*d^(9/2)) - (c^(7/2)*(4*A*c^2*d + 9*b^2*B*e - b*c*(2*B*d + 1
1*A*e))*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b^3*(c*d - b*e)^(9/2)
)

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Maple [A]  time = 0.054, size = 707, normalized size = 1.5 \[ \text{result too large to display} \]

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)^2,x)

[Out]

-2/5*e^3/d^2/(b*e-c*d)^2/(e*x+d)^(5/2)*A+2/5*e^2/d/(b*e-c*d)^2/(e*x+d)^(5/2)*B-4
/3*e^4/d^3/(b*e-c*d)^3/(e*x+d)^(3/2)*A*b+8/3*e^3/d^2/(b*e-c*d)^3/(e*x+d)^(3/2)*A
*c+2/3*e^3/d^2/(b*e-c*d)^3/(e*x+d)^(3/2)*B*b-2*e^2/d/(b*e-c*d)^3/(e*x+d)^(3/2)*B
*c-6*e^5/d^4/(b*e-c*d)^4/(e*x+d)^(1/2)*A*b^2+20*e^4/d^3/(b*e-c*d)^4/(e*x+d)^(1/2
)*A*b*c-20*e^3/d^2/(b*e-c*d)^4/(e*x+d)^(1/2)*A*c^2+2*e^4/d^3/(b*e-c*d)^4/(e*x+d)
^(1/2)*B*b^2-8*e^3/d^2/(b*e-c*d)^4/(e*x+d)^(1/2)*B*b*c+12*e^2/d/(b*e-c*d)^4/(e*x
+d)^(1/2)*B*c^2-e*c^5/(b*e-c*d)^4/b^2*(e*x+d)^(1/2)/(c*e*x+b*e)*A+e*c^4/(b*e-c*d
)^4/b*(e*x+d)^(1/2)/(c*e*x+b*e)*B-11*e*c^5/(b*e-c*d)^4/b^2/((b*e-c*d)*c)^(1/2)*a
rctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*A+4*c^6/(b*e-c*d)^4/b^3/((b*e-c*d)*c)
^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*A*d+9*e*c^4/(b*e-c*d)^4/b/((b
*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*B-2*c^5/(b*e-c*d)^4
/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-1/b^2/d
^4*A*(e*x+d)^(1/2)/x+7*e/b^2/d^(9/2)*arctanh((e*x+d)^(1/2)/d^(1/2))*A+4/b^3/d^(7
/2)*arctanh((e*x+d)^(1/2)/d^(1/2))*A*c-2/b^2/d^(7/2)*arctanh((e*x+d)^(1/2)/d^(1/
2))*B

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 145.64, 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)^2*(e*x + d)^(7/2)),x, algorithm="fricas")

[Out]

[1/30*(15*((2*(B*b*c^5 - 2*A*c^6)*d^5*e^2 - (9*B*b^2*c^4 - 11*A*b*c^5)*d^4*e^3)*
x^4 + (4*(B*b*c^5 - 2*A*c^6)*d^6*e - 2*(8*B*b^2*c^4 - 9*A*b*c^5)*d^5*e^2 - (9*B*
b^3*c^3 - 11*A*b^2*c^4)*d^4*e^3)*x^3 + (2*(B*b*c^5 - 2*A*c^6)*d^7 - (5*B*b^2*c^4
 - 3*A*b*c^5)*d^6*e - 2*(9*B*b^3*c^3 - 11*A*b^2*c^4)*d^5*e^2)*x^2 + (2*(B*b^2*c^
4 - 2*A*b*c^5)*d^7 - (9*B*b^3*c^3 - 11*A*b^2*c^4)*d^6*e)*x)*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)*sqr
t(c/(c*d - b*e)))/(c*x + b)) + 15*((7*A*b^5*c*e^7 - 2*(B*b*c^5 - 2*A*c^6)*d^5*e^
2 + (8*B*b^2*c^4 - 9*A*b*c^5)*d^4*e^3 - 4*(3*B*b^3*c^3 + A*b^2*c^4)*d^3*e^4 + 2*
(4*B*b^4*c^2 + 13*A*b^3*c^3)*d^2*e^5 - 2*(B*b^5*c + 12*A*b^4*c^2)*d*e^6)*x^4 + (
7*A*b^6*e^7 - 4*(B*b*c^5 - 2*A*c^6)*d^6*e + 14*(B*b^2*c^4 - A*b*c^5)*d^5*e^2 - (
16*B*b^3*c^3 + 17*A*b^2*c^4)*d^4*e^3 + 4*(B*b^4*c^2 + 12*A*b^3*c^3)*d^3*e^4 + 2*
(2*B*b^5*c - 11*A*b^4*c^2)*d^2*e^5 - 2*(B*b^6 + 5*A*b^5*c)*d*e^6)*x^3 + (14*A*b^
6*d*e^6 - 2*(B*b*c^5 - 2*A*c^6)*d^7 + (4*B*b^2*c^4 - A*b*c^5)*d^6*e + 2*(2*B*b^3
*c^3 - 11*A*b^2*c^4)*d^5*e^2 - 2*(8*B*b^4*c^2 - 9*A*b^3*c^3)*d^4*e^3 + 14*(B*b^5
*c + 2*A*b^4*c^2)*d^3*e^4 - (4*B*b^6 + 41*A*b^5*c)*d^2*e^5)*x^2 + (7*A*b^6*d^2*e
^5 - 2*(B*b^2*c^4 - 2*A*b*c^5)*d^7 + (8*B*b^3*c^3 - 9*A*b^2*c^4)*d^6*e - 4*(3*B*
b^4*c^2 + A*b^3*c^3)*d^5*e^2 + 2*(4*B*b^5*c + 13*A*b^4*c^2)*d^4*e^3 - 2*(B*b^6 +
 12*A*b^5*c)*d^3*e^4)*x)*sqrt(e*x + d)*log(((e*x + 2*d)*sqrt(d) + 2*sqrt(e*x + d
)*d)/x) - 2*(15*A*b^2*c^4*d^7 - 60*A*b^3*c^3*d^6*e + 90*A*b^4*c^2*d^5*e^2 - 60*A
*b^5*c*d^4*e^3 + 15*A*b^6*d^3*e^4 + 15*(7*A*b^5*c*e^7 - (B*b^2*c^4 - 2*A*b*c^5)*
d^4*e^3 - 4*(3*B*b^3*c^3 + A*b^2*c^4)*d^3*e^4 + 2*(4*B*b^4*c^2 + 13*A*b^3*c^3)*d
^2*e^5 - 2*(B*b^5*c + 12*A*b^4*c^2)*d*e^6)*x^4 + 5*(21*A*b^6*e^7 - 9*(B*b^2*c^4
- 2*A*b*c^5)*d^5*e^2 - 3*(26*B*b^3*c^3 + 11*A*b^2*c^4)*d^4*e^3 + 10*(2*B*b^4*c^2
 + 17*A*b^3*c^3)*d^3*e^4 + 10*(B*b^5*c - 9*A*b^4*c^2)*d^2*e^5 - (6*B*b^6 + 23*A*
b^5*c)*d*e^6)*x^3 + (245*A*b^6*d*e^6 - 45*(B*b^2*c^4 - 2*A*b*c^5)*d^6*e - 27*(8*
B*b^3*c^3 + 5*A*b^2*c^4)*d^5*e^2 - 218*(B*b^4*c^2 - 2*A*b^3*c^3)*d^4*e^3 + 2*(11
7*B*b^5*c + 179*A*b^4*c^2)*d^3*e^4 - 7*(10*B*b^6 + 97*A*b^5*c)*d^2*e^5)*x^2 - (1
5*A*b^2*c^4*d^6*e - 161*A*b^6*d^2*e^5 + 15*(B*b^2*c^4 - 2*A*b*c^5)*d^7 + 18*(12*
B*b^4*c^2 + 5*A*b^3*c^3)*d^5*e^2 - 4*(43*B*b^5*c + 139*A*b^4*c^2)*d^4*e^3 + (46*
B*b^6 + 537*A*b^5*c)*d^3*e^4)*x)*sqrt(d))/(((b^3*c^5*d^8*e^2 - 4*b^4*c^4*d^7*e^3
 + 6*b^5*c^3*d^6*e^4 - 4*b^6*c^2*d^5*e^5 + b^7*c*d^4*e^6)*x^4 + (2*b^3*c^5*d^9*e
 - 7*b^4*c^4*d^8*e^2 + 8*b^5*c^3*d^7*e^3 - 2*b^6*c^2*d^6*e^4 - 2*b^7*c*d^5*e^5 +
 b^8*d^4*e^6)*x^3 + (b^3*c^5*d^10 - 2*b^4*c^4*d^9*e - 2*b^5*c^3*d^8*e^2 + 8*b^6*
c^2*d^7*e^3 - 7*b^7*c*d^6*e^4 + 2*b^8*d^5*e^5)*x^2 + (b^4*c^4*d^10 - 4*b^5*c^3*d
^9*e + 6*b^6*c^2*d^8*e^2 - 4*b^7*c*d^7*e^3 + b^8*d^6*e^4)*x)*sqrt(e*x + d)*sqrt(
d)), 1/30*(30*((2*(B*b*c^5 - 2*A*c^6)*d^5*e^2 - (9*B*b^2*c^4 - 11*A*b*c^5)*d^4*e
^3)*x^4 + (4*(B*b*c^5 - 2*A*c^6)*d^6*e - 2*(8*B*b^2*c^4 - 9*A*b*c^5)*d^5*e^2 - (
9*B*b^3*c^3 - 11*A*b^2*c^4)*d^4*e^3)*x^3 + (2*(B*b*c^5 - 2*A*c^6)*d^7 - (5*B*b^2
*c^4 - 3*A*b*c^5)*d^6*e - 2*(9*B*b^3*c^3 - 11*A*b^2*c^4)*d^5*e^2)*x^2 + (2*(B*b^
2*c^4 - 2*A*b*c^5)*d^7 - (9*B*b^3*c^3 - 11*A*b^2*c^4)*d^6*e)*x)*sqrt(e*x + d)*sq
rt(d)*sqrt(-c/(c*d - b*e))*arctan(-(c*d - b*e)*sqrt(-c/(c*d - b*e))/(sqrt(e*x +
d)*c)) + 15*((7*A*b^5*c*e^7 - 2*(B*b*c^5 - 2*A*c^6)*d^5*e^2 + (8*B*b^2*c^4 - 9*A
*b*c^5)*d^4*e^3 - 4*(3*B*b^3*c^3 + A*b^2*c^4)*d^3*e^4 + 2*(4*B*b^4*c^2 + 13*A*b^
3*c^3)*d^2*e^5 - 2*(B*b^5*c + 12*A*b^4*c^2)*d*e^6)*x^4 + (7*A*b^6*e^7 - 4*(B*b*c
^5 - 2*A*c^6)*d^6*e + 14*(B*b^2*c^4 - A*b*c^5)*d^5*e^2 - (16*B*b^3*c^3 + 17*A*b^
2*c^4)*d^4*e^3 + 4*(B*b^4*c^2 + 12*A*b^3*c^3)*d^3*e^4 + 2*(2*B*b^5*c - 11*A*b^4*
c^2)*d^2*e^5 - 2*(B*b^6 + 5*A*b^5*c)*d*e^6)*x^3 + (14*A*b^6*d*e^6 - 2*(B*b*c^5 -
 2*A*c^6)*d^7 + (4*B*b^2*c^4 - A*b*c^5)*d^6*e + 2*(2*B*b^3*c^3 - 11*A*b^2*c^4)*d
^5*e^2 - 2*(8*B*b^4*c^2 - 9*A*b^3*c^3)*d^4*e^3 + 14*(B*b^5*c + 2*A*b^4*c^2)*d^3*
e^4 - (4*B*b^6 + 41*A*b^5*c)*d^2*e^5)*x^2 + (7*A*b^6*d^2*e^5 - 2*(B*b^2*c^4 - 2*
A*b*c^5)*d^7 + (8*B*b^3*c^3 - 9*A*b^2*c^4)*d^6*e - 4*(3*B*b^4*c^2 + A*b^3*c^3)*d
^5*e^2 + 2*(4*B*b^5*c + 13*A*b^4*c^2)*d^4*e^3 - 2*(B*b^6 + 12*A*b^5*c)*d^3*e^4)*
x)*sqrt(e*x + d)*log(((e*x + 2*d)*sqrt(d) + 2*sqrt(e*x + d)*d)/x) - 2*(15*A*b^2*
c^4*d^7 - 60*A*b^3*c^3*d^6*e + 90*A*b^4*c^2*d^5*e^2 - 60*A*b^5*c*d^4*e^3 + 15*A*
b^6*d^3*e^4 + 15*(7*A*b^5*c*e^7 - (B*b^2*c^4 - 2*A*b*c^5)*d^4*e^3 - 4*(3*B*b^3*c
^3 + A*b^2*c^4)*d^3*e^4 + 2*(4*B*b^4*c^2 + 13*A*b^3*c^3)*d^2*e^5 - 2*(B*b^5*c +
12*A*b^4*c^2)*d*e^6)*x^4 + 5*(21*A*b^6*e^7 - 9*(B*b^2*c^4 - 2*A*b*c^5)*d^5*e^2 -
 3*(26*B*b^3*c^3 + 11*A*b^2*c^4)*d^4*e^3 + 10*(2*B*b^4*c^2 + 17*A*b^3*c^3)*d^3*e
^4 + 10*(B*b^5*c - 9*A*b^4*c^2)*d^2*e^5 - (6*B*b^6 + 23*A*b^5*c)*d*e^6)*x^3 + (2
45*A*b^6*d*e^6 - 45*(B*b^2*c^4 - 2*A*b*c^5)*d^6*e - 27*(8*B*b^3*c^3 + 5*A*b^2*c^
4)*d^5*e^2 - 218*(B*b^4*c^2 - 2*A*b^3*c^3)*d^4*e^3 + 2*(117*B*b^5*c + 179*A*b^4*
c^2)*d^3*e^4 - 7*(10*B*b^6 + 97*A*b^5*c)*d^2*e^5)*x^2 - (15*A*b^2*c^4*d^6*e - 16
1*A*b^6*d^2*e^5 + 15*(B*b^2*c^4 - 2*A*b*c^5)*d^7 + 18*(12*B*b^4*c^2 + 5*A*b^3*c^
3)*d^5*e^2 - 4*(43*B*b^5*c + 139*A*b^4*c^2)*d^4*e^3 + (46*B*b^6 + 537*A*b^5*c)*d
^3*e^4)*x)*sqrt(d))/(((b^3*c^5*d^8*e^2 - 4*b^4*c^4*d^7*e^3 + 6*b^5*c^3*d^6*e^4 -
 4*b^6*c^2*d^5*e^5 + b^7*c*d^4*e^6)*x^4 + (2*b^3*c^5*d^9*e - 7*b^4*c^4*d^8*e^2 +
 8*b^5*c^3*d^7*e^3 - 2*b^6*c^2*d^6*e^4 - 2*b^7*c*d^5*e^5 + b^8*d^4*e^6)*x^3 + (b
^3*c^5*d^10 - 2*b^4*c^4*d^9*e - 2*b^5*c^3*d^8*e^2 + 8*b^6*c^2*d^7*e^3 - 7*b^7*c*
d^6*e^4 + 2*b^8*d^5*e^5)*x^2 + (b^4*c^4*d^10 - 4*b^5*c^3*d^9*e + 6*b^6*c^2*d^8*e
^2 - 4*b^7*c*d^7*e^3 + b^8*d^6*e^4)*x)*sqrt(e*x + d)*sqrt(d)), 1/30*(15*((2*(B*b
*c^5 - 2*A*c^6)*d^5*e^2 - (9*B*b^2*c^4 - 11*A*b*c^5)*d^4*e^3)*x^4 + (4*(B*b*c^5
- 2*A*c^6)*d^6*e - 2*(8*B*b^2*c^4 - 9*A*b*c^5)*d^5*e^2 - (9*B*b^3*c^3 - 11*A*b^2
*c^4)*d^4*e^3)*x^3 + (2*(B*b*c^5 - 2*A*c^6)*d^7 - (5*B*b^2*c^4 - 3*A*b*c^5)*d^6*
e - 2*(9*B*b^3*c^3 - 11*A*b^2*c^4)*d^5*e^2)*x^2 + (2*(B*b^2*c^4 - 2*A*b*c^5)*d^7
 - (9*B*b^3*c^3 - 11*A*b^2*c^4)*d^6*e)*x)*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)) - 30*((7*A*b^5*c*e^7 - 2*(B*b*c^5 - 2*A*c^6)*d^5*e^2 + (8*B*b^2*c^4
- 9*A*b*c^5)*d^4*e^3 - 4*(3*B*b^3*c^3 + A*b^2*c^4)*d^3*e^4 + 2*(4*B*b^4*c^2 + 13
*A*b^3*c^3)*d^2*e^5 - 2*(B*b^5*c + 12*A*b^4*c^2)*d*e^6)*x^4 + (7*A*b^6*e^7 - 4*(
B*b*c^5 - 2*A*c^6)*d^6*e + 14*(B*b^2*c^4 - A*b*c^5)*d^5*e^2 - (16*B*b^3*c^3 + 17
*A*b^2*c^4)*d^4*e^3 + 4*(B*b^4*c^2 + 12*A*b^3*c^3)*d^3*e^4 + 2*(2*B*b^5*c - 11*A
*b^4*c^2)*d^2*e^5 - 2*(B*b^6 + 5*A*b^5*c)*d*e^6)*x^3 + (14*A*b^6*d*e^6 - 2*(B*b*
c^5 - 2*A*c^6)*d^7 + (4*B*b^2*c^4 - A*b*c^5)*d^6*e + 2*(2*B*b^3*c^3 - 11*A*b^2*c
^4)*d^5*e^2 - 2*(8*B*b^4*c^2 - 9*A*b^3*c^3)*d^4*e^3 + 14*(B*b^5*c + 2*A*b^4*c^2)
*d^3*e^4 - (4*B*b^6 + 41*A*b^5*c)*d^2*e^5)*x^2 + (7*A*b^6*d^2*e^5 - 2*(B*b^2*c^4
 - 2*A*b*c^5)*d^7 + (8*B*b^3*c^3 - 9*A*b^2*c^4)*d^6*e - 4*(3*B*b^4*c^2 + A*b^3*c
^3)*d^5*e^2 + 2*(4*B*b^5*c + 13*A*b^4*c^2)*d^4*e^3 - 2*(B*b^6 + 12*A*b^5*c)*d^3*
e^4)*x)*sqrt(e*x + d)*arctan(d/(sqrt(e*x + d)*sqrt(-d))) - 2*(15*A*b^2*c^4*d^7 -
 60*A*b^3*c^3*d^6*e + 90*A*b^4*c^2*d^5*e^2 - 60*A*b^5*c*d^4*e^3 + 15*A*b^6*d^3*e
^4 + 15*(7*A*b^5*c*e^7 - (B*b^2*c^4 - 2*A*b*c^5)*d^4*e^3 - 4*(3*B*b^3*c^3 + A*b^
2*c^4)*d^3*e^4 + 2*(4*B*b^4*c^2 + 13*A*b^3*c^3)*d^2*e^5 - 2*(B*b^5*c + 12*A*b^4*
c^2)*d*e^6)*x^4 + 5*(21*A*b^6*e^7 - 9*(B*b^2*c^4 - 2*A*b*c^5)*d^5*e^2 - 3*(26*B*
b^3*c^3 + 11*A*b^2*c^4)*d^4*e^3 + 10*(2*B*b^4*c^2 + 17*A*b^3*c^3)*d^3*e^4 + 10*(
B*b^5*c - 9*A*b^4*c^2)*d^2*e^5 - (6*B*b^6 + 23*A*b^5*c)*d*e^6)*x^3 + (245*A*b^6*
d*e^6 - 45*(B*b^2*c^4 - 2*A*b*c^5)*d^6*e - 27*(8*B*b^3*c^3 + 5*A*b^2*c^4)*d^5*e^
2 - 218*(B*b^4*c^2 - 2*A*b^3*c^3)*d^4*e^3 + 2*(117*B*b^5*c + 179*A*b^4*c^2)*d^3*
e^4 - 7*(10*B*b^6 + 97*A*b^5*c)*d^2*e^5)*x^2 - (15*A*b^2*c^4*d^6*e - 161*A*b^6*d
^2*e^5 + 15*(B*b^2*c^4 - 2*A*b*c^5)*d^7 + 18*(12*B*b^4*c^2 + 5*A*b^3*c^3)*d^5*e^
2 - 4*(43*B*b^5*c + 139*A*b^4*c^2)*d^4*e^3 + (46*B*b^6 + 537*A*b^5*c)*d^3*e^4)*x
)*sqrt(-d))/(((b^3*c^5*d^8*e^2 - 4*b^4*c^4*d^7*e^3 + 6*b^5*c^3*d^6*e^4 - 4*b^6*c
^2*d^5*e^5 + b^7*c*d^4*e^6)*x^4 + (2*b^3*c^5*d^9*e - 7*b^4*c^4*d^8*e^2 + 8*b^5*c
^3*d^7*e^3 - 2*b^6*c^2*d^6*e^4 - 2*b^7*c*d^5*e^5 + b^8*d^4*e^6)*x^3 + (b^3*c^5*d
^10 - 2*b^4*c^4*d^9*e - 2*b^5*c^3*d^8*e^2 + 8*b^6*c^2*d^7*e^3 - 7*b^7*c*d^6*e^4
+ 2*b^8*d^5*e^5)*x^2 + (b^4*c^4*d^10 - 4*b^5*c^3*d^9*e + 6*b^6*c^2*d^8*e^2 - 4*b
^7*c*d^7*e^3 + b^8*d^6*e^4)*x)*sqrt(e*x + d)*sqrt(-d)), 1/15*(15*((2*(B*b*c^5 -
2*A*c^6)*d^5*e^2 - (9*B*b^2*c^4 - 11*A*b*c^5)*d^4*e^3)*x^4 + (4*(B*b*c^5 - 2*A*c
^6)*d^6*e - 2*(8*B*b^2*c^4 - 9*A*b*c^5)*d^5*e^2 - (9*B*b^3*c^3 - 11*A*b^2*c^4)*d
^4*e^3)*x^3 + (2*(B*b*c^5 - 2*A*c^6)*d^7 - (5*B*b^2*c^4 - 3*A*b*c^5)*d^6*e - 2*(
9*B*b^3*c^3 - 11*A*b^2*c^4)*d^5*e^2)*x^2 + (2*(B*b^2*c^4 - 2*A*b*c^5)*d^7 - (9*B
*b^3*c^3 - 11*A*b^2*c^4)*d^6*e)*x)*sqrt(e*x + d)*sqrt(-d)*sqrt(-c/(c*d - b*e))*a
rctan(-(c*d - b*e)*sqrt(-c/(c*d - b*e))/(sqrt(e*x + d)*c)) - 15*((7*A*b^5*c*e^7
- 2*(B*b*c^5 - 2*A*c^6)*d^5*e^2 + (8*B*b^2*c^4 - 9*A*b*c^5)*d^4*e^3 - 4*(3*B*b^3
*c^3 + A*b^2*c^4)*d^3*e^4 + 2*(4*B*b^4*c^2 + 13*A*b^3*c^3)*d^2*e^5 - 2*(B*b^5*c
+ 12*A*b^4*c^2)*d*e^6)*x^4 + (7*A*b^6*e^7 - 4*(B*b*c^5 - 2*A*c^6)*d^6*e + 14*(B*
b^2*c^4 - A*b*c^5)*d^5*e^2 - (16*B*b^3*c^3 + 17*A*b^2*c^4)*d^4*e^3 + 4*(B*b^4*c^
2 + 12*A*b^3*c^3)*d^3*e^4 + 2*(2*B*b^5*c - 11*A*b^4*c^2)*d^2*e^5 - 2*(B*b^6 + 5*
A*b^5*c)*d*e^6)*x^3 + (14*A*b^6*d*e^6 - 2*(B*b*c^5 - 2*A*c^6)*d^7 + (4*B*b^2*c^4
 - A*b*c^5)*d^6*e + 2*(2*B*b^3*c^3 - 11*A*b^2*c^4)*d^5*e^2 - 2*(8*B*b^4*c^2 - 9*
A*b^3*c^3)*d^4*e^3 + 14*(B*b^5*c + 2*A*b^4*c^2)*d^3*e^4 - (4*B*b^6 + 41*A*b^5*c)
*d^2*e^5)*x^2 + (7*A*b^6*d^2*e^5 - 2*(B*b^2*c^4 - 2*A*b*c^5)*d^7 + (8*B*b^3*c^3
- 9*A*b^2*c^4)*d^6*e - 4*(3*B*b^4*c^2 + A*b^3*c^3)*d^5*e^2 + 2*(4*B*b^5*c + 13*A
*b^4*c^2)*d^4*e^3 - 2*(B*b^6 + 12*A*b^5*c)*d^3*e^4)*x)*sqrt(e*x + d)*arctan(d/(s
qrt(e*x + d)*sqrt(-d))) - (15*A*b^2*c^4*d^7 - 60*A*b^3*c^3*d^6*e + 90*A*b^4*c^2*
d^5*e^2 - 60*A*b^5*c*d^4*e^3 + 15*A*b^6*d^3*e^4 + 15*(7*A*b^5*c*e^7 - (B*b^2*c^4
 - 2*A*b*c^5)*d^4*e^3 - 4*(3*B*b^3*c^3 + A*b^2*c^4)*d^3*e^4 + 2*(4*B*b^4*c^2 + 1
3*A*b^3*c^3)*d^2*e^5 - 2*(B*b^5*c + 12*A*b^4*c^2)*d*e^6)*x^4 + 5*(21*A*b^6*e^7 -
 9*(B*b^2*c^4 - 2*A*b*c^5)*d^5*e^2 - 3*(26*B*b^3*c^3 + 11*A*b^2*c^4)*d^4*e^3 + 1
0*(2*B*b^4*c^2 + 17*A*b^3*c^3)*d^3*e^4 + 10*(B*b^5*c - 9*A*b^4*c^2)*d^2*e^5 - (6
*B*b^6 + 23*A*b^5*c)*d*e^6)*x^3 + (245*A*b^6*d*e^6 - 45*(B*b^2*c^4 - 2*A*b*c^5)*
d^6*e - 27*(8*B*b^3*c^3 + 5*A*b^2*c^4)*d^5*e^2 - 218*(B*b^4*c^2 - 2*A*b^3*c^3)*d
^4*e^3 + 2*(117*B*b^5*c + 179*A*b^4*c^2)*d^3*e^4 - 7*(10*B*b^6 + 97*A*b^5*c)*d^2
*e^5)*x^2 - (15*A*b^2*c^4*d^6*e - 161*A*b^6*d^2*e^5 + 15*(B*b^2*c^4 - 2*A*b*c^5)
*d^7 + 18*(12*B*b^4*c^2 + 5*A*b^3*c^3)*d^5*e^2 - 4*(43*B*b^5*c + 139*A*b^4*c^2)*
d^4*e^3 + (46*B*b^6 + 537*A*b^5*c)*d^3*e^4)*x)*sqrt(-d))/(((b^3*c^5*d^8*e^2 - 4*
b^4*c^4*d^7*e^3 + 6*b^5*c^3*d^6*e^4 - 4*b^6*c^2*d^5*e^5 + b^7*c*d^4*e^6)*x^4 + (
2*b^3*c^5*d^9*e - 7*b^4*c^4*d^8*e^2 + 8*b^5*c^3*d^7*e^3 - 2*b^6*c^2*d^6*e^4 - 2*
b^7*c*d^5*e^5 + b^8*d^4*e^6)*x^3 + (b^3*c^5*d^10 - 2*b^4*c^4*d^9*e - 2*b^5*c^3*d
^8*e^2 + 8*b^6*c^2*d^7*e^3 - 7*b^7*c*d^6*e^4 + 2*b^8*d^5*e^5)*x^2 + (b^4*c^4*d^1
0 - 4*b^5*c^3*d^9*e + 6*b^6*c^2*d^8*e^2 - 4*b^7*c*d^7*e^3 + b^8*d^6*e^4)*x)*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)**(7/2)/(c*x**2+b*x)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.351501, 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)^2*(e*x + d)^(7/2)),x, algorithm="giac")

[Out]

Done

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