3.1824 \(\int \frac{A+B x}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^2} \, dx\)

Optimal. Leaf size=339 \[ -\frac{21 b^{3/2} e^2 (5 a B e-11 A b e+6 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 (b d-a e)^{13/2}}+\frac{21 b e^2 (5 a B e-11 A b e+6 b B d)}{8 \sqrt{d+e x} (b d-a e)^6}+\frac{7 e^2 (5 a B e-11 A b e+6 b B d)}{8 (d+e x)^{3/2} (b d-a e)^5}+\frac{21 e^2 (5 a B e-11 A b e+6 b B d)}{40 b (d+e x)^{5/2} (b d-a e)^4}+\frac{3 e (5 a B e-11 A b e+6 b B d)}{8 b (a+b x) (d+e x)^{5/2} (b d-a e)^3}-\frac{5 a B e-11 A b e+6 b B d}{12 b (a+b x)^2 (d+e x)^{5/2} (b d-a e)^2}-\frac{A b-a B}{3 b (a+b x)^3 (d+e x)^{5/2} (b d-a e)} \]

[Out]

(21*e^2*(6*b*B*d - 11*A*b*e + 5*a*B*e))/(40*b*(b*d - a*e)^4*(d + e*x)^(5/2)) - (
A*b - a*B)/(3*b*(b*d - a*e)*(a + b*x)^3*(d + e*x)^(5/2)) - (6*b*B*d - 11*A*b*e +
5*a*B*e)/(12*b*(b*d - a*e)^2*(a + b*x)^2*(d + e*x)^(5/2)) + (3*e*(6*b*B*d - 11*
A*b*e + 5*a*B*e))/(8*b*(b*d - a*e)^3*(a + b*x)*(d + e*x)^(5/2)) + (7*e^2*(6*b*B*
d - 11*A*b*e + 5*a*B*e))/(8*(b*d - a*e)^5*(d + e*x)^(3/2)) + (21*b*e^2*(6*b*B*d
- 11*A*b*e + 5*a*B*e))/(8*(b*d - a*e)^6*Sqrt[d + e*x]) - (21*b^(3/2)*e^2*(6*b*B*
d - 11*A*b*e + 5*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(8*(b*
d - a*e)^(13/2))

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Rubi [A]  time = 0.919048, antiderivative size = 339, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{21 b^{3/2} e^2 (5 a B e-11 A b e+6 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 (b d-a e)^{13/2}}+\frac{21 b e^2 (5 a B e-11 A b e+6 b B d)}{8 \sqrt{d+e x} (b d-a e)^6}+\frac{7 e^2 (5 a B e-11 A b e+6 b B d)}{8 (d+e x)^{3/2} (b d-a e)^5}+\frac{21 e^2 (5 a B e-11 A b e+6 b B d)}{40 b (d+e x)^{5/2} (b d-a e)^4}+\frac{3 e (5 a B e-11 A b e+6 b B d)}{8 b (a+b x) (d+e x)^{5/2} (b d-a e)^3}-\frac{5 a B e-11 A b e+6 b B d}{12 b (a+b x)^2 (d+e x)^{5/2} (b d-a e)^2}-\frac{A b-a B}{3 b (a+b x)^3 (d+e x)^{5/2} (b d-a e)} \]

Antiderivative was successfully verified.

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

[Out]

(21*e^2*(6*b*B*d - 11*A*b*e + 5*a*B*e))/(40*b*(b*d - a*e)^4*(d + e*x)^(5/2)) - (
A*b - a*B)/(3*b*(b*d - a*e)*(a + b*x)^3*(d + e*x)^(5/2)) - (6*b*B*d - 11*A*b*e +
5*a*B*e)/(12*b*(b*d - a*e)^2*(a + b*x)^2*(d + e*x)^(5/2)) + (3*e*(6*b*B*d - 11*
A*b*e + 5*a*B*e))/(8*b*(b*d - a*e)^3*(a + b*x)*(d + e*x)^(5/2)) + (7*e^2*(6*b*B*
d - 11*A*b*e + 5*a*B*e))/(8*(b*d - a*e)^5*(d + e*x)^(3/2)) + (21*b*e^2*(6*b*B*d
- 11*A*b*e + 5*a*B*e))/(8*(b*d - a*e)^6*Sqrt[d + e*x]) - (21*b^(3/2)*e^2*(6*b*B*
d - 11*A*b*e + 5*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(8*(b*
d - a*e)^(13/2))

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Rubi in Sympy [A]  time = 164.911, size = 332, normalized size = 0.98 \[ - \frac{21 b^{\frac{3}{2}} e^{2} \left (11 A b e - 5 B a e - 6 B b d\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{8 \left (a e - b d\right )^{\frac{13}{2}}} - \frac{21 b e^{2} \left (11 A b e - 5 B a e - 6 B b d\right )}{8 \sqrt{d + e x} \left (a e - b d\right )^{6}} + \frac{7 e^{2} \left (11 A b e - 5 B a e - 6 B b d\right )}{8 \left (d + e x\right )^{\frac{3}{2}} \left (a e - b d\right )^{5}} - \frac{21 e^{2} \left (11 A b e - 5 B a e - 6 B b d\right )}{40 b \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )^{4}} + \frac{3 e \left (11 A b e - 5 B a e - 6 B b d\right )}{8 b \left (a + b x\right ) \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )^{3}} + \frac{11 A b e - 5 B a e - 6 B b d}{12 b \left (a + b x\right )^{2} \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )^{2}} + \frac{A b - B a}{3 b \left (a + b x\right )^{3} \left (d + e x\right )^{\frac{5}{2}} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/(e*x+d)**(7/2)/(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

-21*b**(3/2)*e**2*(11*A*b*e - 5*B*a*e - 6*B*b*d)*atan(sqrt(b)*sqrt(d + e*x)/sqrt
(a*e - b*d))/(8*(a*e - b*d)**(13/2)) - 21*b*e**2*(11*A*b*e - 5*B*a*e - 6*B*b*d)/
(8*sqrt(d + e*x)*(a*e - b*d)**6) + 7*e**2*(11*A*b*e - 5*B*a*e - 6*B*b*d)/(8*(d +
e*x)**(3/2)*(a*e - b*d)**5) - 21*e**2*(11*A*b*e - 5*B*a*e - 6*B*b*d)/(40*b*(d +
e*x)**(5/2)*(a*e - b*d)**4) + 3*e*(11*A*b*e - 5*B*a*e - 6*B*b*d)/(8*b*(a + b*x)
*(d + e*x)**(5/2)*(a*e - b*d)**3) + (11*A*b*e - 5*B*a*e - 6*B*b*d)/(12*b*(a + b*
x)**2*(d + e*x)**(5/2)*(a*e - b*d)**2) + (A*b - B*a)/(3*b*(a + b*x)**3*(d + e*x)
**(5/2)*(a*e - b*d))

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Mathematica [A]  time = 1.88316, size = 283, normalized size = 0.83 \[ -\frac{21 b^{3/2} e^2 (5 a B e-11 A b e+6 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{8 (b d-a e)^{13/2}}-\frac{\sqrt{d+e x} \left (\frac{15 b^2 e (-41 a B e+71 A b e-30 b B d)}{a+b x}+\frac{10 b^2 (b d-a e) (17 a B e-23 A b e+6 b B d)}{(a+b x)^2}+\frac{40 b^2 (A b-a B) (b d-a e)^2}{(a+b x)^3}+\frac{480 b e^2 (-2 a B e+5 A b e-3 b B d)}{d+e x}-\frac{80 e^2 (a e-b d) (-a B e+4 A b e-3 b B d)}{(d+e x)^2}+\frac{48 e^2 (b d-a e)^2 (A e-B d)}{(d+e x)^3}\right )}{120 (b d-a e)^6} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[d + e*x]*((40*b^2*(A*b - a*B)*(b*d - a*e)^2)/(a + b*x)^3 + (10*b^2*(b*d -
a*e)*(6*b*B*d - 23*A*b*e + 17*a*B*e))/(a + b*x)^2 + (15*b^2*e*(-30*b*B*d + 71*A
*b*e - 41*a*B*e))/(a + b*x) + (48*e^2*(b*d - a*e)^2*(-(B*d) + A*e))/(d + e*x)^3
- (80*e^2*(-(b*d) + a*e)*(-3*b*B*d + 4*A*b*e - a*B*e))/(d + e*x)^2 + (480*b*e^2*
(-3*b*B*d + 5*A*b*e - 2*a*B*e))/(d + e*x)))/(120*(b*d - a*e)^6) - (21*b^(3/2)*e^
2*(6*b*B*d - 11*A*b*e + 5*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]
])/(8*(b*d - a*e)^(13/2))

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Maple [B]  time = 0.049, size = 935, normalized size = 2.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-19/2*e^4/(a*e-b*d)^6*b^3/(b*e*x+a*e)^3*(e*x+d)^(1/2)*B*a^2*d+2/5*e^2/(a*e-b*d)^
4/(e*x+d)^(5/2)*B*d+8/3*e^3/(a*e-b*d)^5/(e*x+d)^(3/2)*A*b-2/3*e^3/(a*e-b*d)^5/(e
*x+d)^(3/2)*a*B-20*e^3*b^2/(a*e-b*d)^6/(e*x+d)^(1/2)*A+8*e^3*b/(a*e-b*d)^6/(e*x+
d)^(1/2)*a*B-231/8*e^3/(a*e-b*d)^6*b^3/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*
b/(b*(a*e-b*d))^(1/2))*A-71/8*e^3/(a*e-b*d)^6*b^5/(b*e*x+a*e)^3*(e*x+d)^(5/2)*A+
12*e^2*b^2/(a*e-b*d)^6/(e*x+d)^(1/2)*B*d-2*e^2/(a*e-b*d)^5/(e*x+d)^(3/2)*B*b*d+8
9/4*e^4/(a*e-b*d)^6*b^4/(b*e*x+a*e)^3*(e*x+d)^(1/2)*A*a*d-13/8*e^3/(a*e-b*d)^6*b
^4/(b*e*x+a*e)^3*(e*x+d)^(1/2)*B*a*d^2+41/8*e^3/(a*e-b*d)^6*b^4/(b*e*x+a*e)^3*(e
*x+d)^(5/2)*B*a-59/3*e^4/(a*e-b*d)^6*b^4/(b*e*x+a*e)^3*A*(e*x+d)^(3/2)*a+59/3*e^
3/(a*e-b*d)^6*b^5/(b*e*x+a*e)^3*A*(e*x+d)^(3/2)*d+35/3*e^4/(a*e-b*d)^6*b^3/(b*e*
x+a*e)^3*B*(e*x+d)^(3/2)*a^2-11/3*e^3/(a*e-b*d)^6*b^4/(b*e*x+a*e)^3*B*(e*x+d)^(3
/2)*a*d-89/8*e^5/(a*e-b*d)^6*b^3/(b*e*x+a*e)^3*(e*x+d)^(1/2)*A*a^2-89/8*e^3/(a*e
-b*d)^6*b^5/(b*e*x+a*e)^3*(e*x+d)^(1/2)*A*d^2+55/8*e^5/(a*e-b*d)^6*b^2/(b*e*x+a*
e)^3*(e*x+d)^(1/2)*B*a^3+105/8*e^3/(a*e-b*d)^6*b^2/(b*(a*e-b*d))^(1/2)*arctan((e
*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a*B+15/4*e^2/(a*e-b*d)^6*b^5/(b*e*x+a*e)^3*(e
*x+d)^(5/2)*B*d-8*e^2/(a*e-b*d)^6*b^5/(b*e*x+a*e)^3*B*(e*x+d)^(3/2)*d^2+17/4*e^2
/(a*e-b*d)^6*b^5/(b*e*x+a*e)^3*(e*x+d)^(1/2)*B*d^3+63/4*e^2/(a*e-b*d)^6*b^3/(b*(
a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*B*d-2/5*e^3/(a*e-b*d
)^4/(e*x+d)^(5/2)*A

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.338637, 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)/((b^2*x^2 + 2*a*b*x + a^2)^2*(e*x + d)^(7/2)),x, algorithm="fricas")

[Out]

[-1/240*(96*A*a^5*e^5 + 40*(B*a*b^4 + 2*A*b^5)*d^5 - 20*(26*B*a^2*b^3 + 31*A*a*b
^4)*d^4*e - 6*(851*B*a^3*b^2 - 445*A*a^2*b^3)*d^3*e^2 - 32*(44*B*a^4*b - 173*A*a
^3*b^2)*d^2*e^3 + 64*(B*a^5 - 13*A*a^4*b)*d*e^4 - 630*(6*B*b^5*d*e^4 + (5*B*a*b^
4 - 11*A*b^5)*e^5)*x^5 - 210*(42*B*b^5*d^2*e^3 + (83*B*a*b^4 - 77*A*b^5)*d*e^4 +
8*(5*B*a^2*b^3 - 11*A*a*b^4)*e^5)*x^4 - 42*(138*B*b^5*d^3*e^2 + (679*B*a*b^4 -
253*A*b^5)*d^2*e^3 + 2*(334*B*a^2*b^3 - 517*A*a*b^4)*d*e^4 + 33*(5*B*a^3*b^2 - 1
1*A*a^2*b^3)*e^5)*x^3 - 18*(30*B*b^5*d^4*e + (901*B*a*b^4 - 55*A*b^5)*d^3*e^2 +
2*(914*B*a^2*b^3 - 803*A*a*b^4)*d^2*e^3 + 3*(337*B*a^3*b^2 - 671*A*a^2*b^3)*d*e^
4 + 16*(5*B*a^4*b - 11*A*a^3*b^2)*e^5)*x^2 + 315*(6*B*a^3*b^2*d^3*e^2 + (5*B*a^4
*b - 11*A*a^3*b^2)*d^2*e^3 + (6*B*b^5*d*e^4 + (5*B*a*b^4 - 11*A*b^5)*e^5)*x^5 +
(12*B*b^5*d^2*e^3 + 2*(14*B*a*b^4 - 11*A*b^5)*d*e^4 + 3*(5*B*a^2*b^3 - 11*A*a*b^
4)*e^5)*x^4 + (6*B*b^5*d^3*e^2 + (41*B*a*b^4 - 11*A*b^5)*d^2*e^3 + 6*(8*B*a^2*b^
3 - 11*A*a*b^4)*d*e^4 + 3*(5*B*a^3*b^2 - 11*A*a^2*b^3)*e^5)*x^3 + (18*B*a*b^4*d^
3*e^2 + 3*(17*B*a^2*b^3 - 11*A*a*b^4)*d^2*e^3 + 6*(6*B*a^3*b^2 - 11*A*a^2*b^3)*d
*e^4 + (5*B*a^4*b - 11*A*a^3*b^2)*e^5)*x^2 + (18*B*a^2*b^3*d^3*e^2 + 3*(9*B*a^3*
b^2 - 11*A*a^2*b^3)*d^2*e^3 + 2*(5*B*a^4*b - 11*A*a^3*b^2)*d*e^4)*x)*sqrt(e*x +
d)*sqrt(b/(b*d - a*e))*log((b*e*x + 2*b*d - a*e + 2*(b*d - a*e)*sqrt(e*x + d)*sq
rt(b/(b*d - a*e)))/(b*x + a)) + 2*(60*B*b^5*d^5 - 10*(73*B*a*b^4 + 11*A*b^5)*d^4
*e - 2*(3682*B*a^2*b^3 - 715*A*a*b^4)*d^3*e^2 - 3*(2569*B*a^3*b^2 - 4103*A*a^2*b
^3)*d^2*e^3 - 32*(52*B*a^4*b - 121*A*a^3*b^2)*d*e^4 + 16*(5*B*a^5 - 11*A*a^4*b)*
e^5)*x)/((a^3*b^6*d^8 - 6*a^4*b^5*d^7*e + 15*a^5*b^4*d^6*e^2 - 20*a^6*b^3*d^5*e^
3 + 15*a^7*b^2*d^4*e^4 - 6*a^8*b*d^3*e^5 + a^9*d^2*e^6 + (b^9*d^6*e^2 - 6*a*b^8*
d^5*e^3 + 15*a^2*b^7*d^4*e^4 - 20*a^3*b^6*d^3*e^5 + 15*a^4*b^5*d^2*e^6 - 6*a^5*b
^4*d*e^7 + a^6*b^3*e^8)*x^5 + (2*b^9*d^7*e - 9*a*b^8*d^6*e^2 + 12*a^2*b^7*d^5*e^
3 + 5*a^3*b^6*d^4*e^4 - 30*a^4*b^5*d^3*e^5 + 33*a^5*b^4*d^2*e^6 - 16*a^6*b^3*d*e
^7 + 3*a^7*b^2*e^8)*x^4 + (b^9*d^8 - 18*a^2*b^7*d^6*e^2 + 52*a^3*b^6*d^5*e^3 - 6
0*a^4*b^5*d^4*e^4 + 24*a^5*b^4*d^3*e^5 + 10*a^6*b^3*d^2*e^6 - 12*a^7*b^2*d*e^7 +
3*a^8*b*e^8)*x^3 + (3*a*b^8*d^8 - 12*a^2*b^7*d^7*e + 10*a^3*b^6*d^6*e^2 + 24*a^
4*b^5*d^5*e^3 - 60*a^5*b^4*d^4*e^4 + 52*a^6*b^3*d^3*e^5 - 18*a^7*b^2*d^2*e^6 + a
^9*e^8)*x^2 + (3*a^2*b^7*d^8 - 16*a^3*b^6*d^7*e + 33*a^4*b^5*d^6*e^2 - 30*a^5*b^
4*d^5*e^3 + 5*a^6*b^3*d^4*e^4 + 12*a^7*b^2*d^3*e^5 - 9*a^8*b*d^2*e^6 + 2*a^9*d*e
^7)*x)*sqrt(e*x + d)), -1/120*(48*A*a^5*e^5 + 20*(B*a*b^4 + 2*A*b^5)*d^5 - 10*(2
6*B*a^2*b^3 + 31*A*a*b^4)*d^4*e - 3*(851*B*a^3*b^2 - 445*A*a^2*b^3)*d^3*e^2 - 16
*(44*B*a^4*b - 173*A*a^3*b^2)*d^2*e^3 + 32*(B*a^5 - 13*A*a^4*b)*d*e^4 - 315*(6*B
*b^5*d*e^4 + (5*B*a*b^4 - 11*A*b^5)*e^5)*x^5 - 105*(42*B*b^5*d^2*e^3 + (83*B*a*b
^4 - 77*A*b^5)*d*e^4 + 8*(5*B*a^2*b^3 - 11*A*a*b^4)*e^5)*x^4 - 21*(138*B*b^5*d^3
*e^2 + (679*B*a*b^4 - 253*A*b^5)*d^2*e^3 + 2*(334*B*a^2*b^3 - 517*A*a*b^4)*d*e^4
+ 33*(5*B*a^3*b^2 - 11*A*a^2*b^3)*e^5)*x^3 - 9*(30*B*b^5*d^4*e + (901*B*a*b^4 -
55*A*b^5)*d^3*e^2 + 2*(914*B*a^2*b^3 - 803*A*a*b^4)*d^2*e^3 + 3*(337*B*a^3*b^2
- 671*A*a^2*b^3)*d*e^4 + 16*(5*B*a^4*b - 11*A*a^3*b^2)*e^5)*x^2 + 315*(6*B*a^3*b
^2*d^3*e^2 + (5*B*a^4*b - 11*A*a^3*b^2)*d^2*e^3 + (6*B*b^5*d*e^4 + (5*B*a*b^4 -
11*A*b^5)*e^5)*x^5 + (12*B*b^5*d^2*e^3 + 2*(14*B*a*b^4 - 11*A*b^5)*d*e^4 + 3*(5*
B*a^2*b^3 - 11*A*a*b^4)*e^5)*x^4 + (6*B*b^5*d^3*e^2 + (41*B*a*b^4 - 11*A*b^5)*d^
2*e^3 + 6*(8*B*a^2*b^3 - 11*A*a*b^4)*d*e^4 + 3*(5*B*a^3*b^2 - 11*A*a^2*b^3)*e^5)
*x^3 + (18*B*a*b^4*d^3*e^2 + 3*(17*B*a^2*b^3 - 11*A*a*b^4)*d^2*e^3 + 6*(6*B*a^3*
b^2 - 11*A*a^2*b^3)*d*e^4 + (5*B*a^4*b - 11*A*a^3*b^2)*e^5)*x^2 + (18*B*a^2*b^3*
d^3*e^2 + 3*(9*B*a^3*b^2 - 11*A*a^2*b^3)*d^2*e^3 + 2*(5*B*a^4*b - 11*A*a^3*b^2)*
d*e^4)*x)*sqrt(e*x + d)*sqrt(-b/(b*d - a*e))*arctan(-(b*d - a*e)*sqrt(-b/(b*d -
a*e))/(sqrt(e*x + d)*b)) + (60*B*b^5*d^5 - 10*(73*B*a*b^4 + 11*A*b^5)*d^4*e - 2*
(3682*B*a^2*b^3 - 715*A*a*b^4)*d^3*e^2 - 3*(2569*B*a^3*b^2 - 4103*A*a^2*b^3)*d^2
*e^3 - 32*(52*B*a^4*b - 121*A*a^3*b^2)*d*e^4 + 16*(5*B*a^5 - 11*A*a^4*b)*e^5)*x)
/((a^3*b^6*d^8 - 6*a^4*b^5*d^7*e + 15*a^5*b^4*d^6*e^2 - 20*a^6*b^3*d^5*e^3 + 15*
a^7*b^2*d^4*e^4 - 6*a^8*b*d^3*e^5 + a^9*d^2*e^6 + (b^9*d^6*e^2 - 6*a*b^8*d^5*e^3
+ 15*a^2*b^7*d^4*e^4 - 20*a^3*b^6*d^3*e^5 + 15*a^4*b^5*d^2*e^6 - 6*a^5*b^4*d*e^
7 + a^6*b^3*e^8)*x^5 + (2*b^9*d^7*e - 9*a*b^8*d^6*e^2 + 12*a^2*b^7*d^5*e^3 + 5*a
^3*b^6*d^4*e^4 - 30*a^4*b^5*d^3*e^5 + 33*a^5*b^4*d^2*e^6 - 16*a^6*b^3*d*e^7 + 3*
a^7*b^2*e^8)*x^4 + (b^9*d^8 - 18*a^2*b^7*d^6*e^2 + 52*a^3*b^6*d^5*e^3 - 60*a^4*b
^5*d^4*e^4 + 24*a^5*b^4*d^3*e^5 + 10*a^6*b^3*d^2*e^6 - 12*a^7*b^2*d*e^7 + 3*a^8*
b*e^8)*x^3 + (3*a*b^8*d^8 - 12*a^2*b^7*d^7*e + 10*a^3*b^6*d^6*e^2 + 24*a^4*b^5*d
^5*e^3 - 60*a^5*b^4*d^4*e^4 + 52*a^6*b^3*d^3*e^5 - 18*a^7*b^2*d^2*e^6 + a^9*e^8)
*x^2 + (3*a^2*b^7*d^8 - 16*a^3*b^6*d^7*e + 33*a^4*b^5*d^6*e^2 - 30*a^5*b^4*d^5*e
^3 + 5*a^6*b^3*d^4*e^4 + 12*a^7*b^2*d^3*e^5 - 9*a^8*b*d^2*e^6 + 2*a^9*d*e^7)*x)*
sqrt(e*x + 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)/(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.310331, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done