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3.1833 \(\int \frac{A+B x}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx\)

Optimal. Leaf size=400 \[ \frac{231 e^4 (3 a B e-13 A b e+10 b B d)}{128 \sqrt{d+e x} (b d-a e)^7}+\frac{77 e^4 (3 a B e-13 A b e+10 b B d)}{128 b (d+e x)^{3/2} (b d-a e)^6}-\frac{231 \sqrt{b} e^4 (3 a B e-13 A b e+10 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 (b d-a e)^{15/2}}+\frac{231 e^3 (3 a B e-13 A b e+10 b B d)}{640 b (a+b x) (d+e x)^{3/2} (b d-a e)^5}-\frac{33 e^2 (3 a B e-13 A b e+10 b B d)}{320 b (a+b x)^2 (d+e x)^{3/2} (b d-a e)^4}+\frac{11 e (3 a B e-13 A b e+10 b B d)}{240 b (a+b x)^3 (d+e x)^{3/2} (b d-a e)^3}-\frac{3 a B e-13 A b e+10 b B d}{40 b (a+b x)^4 (d+e x)^{3/2} (b d-a e)^2}-\frac{A b-a B}{5 b (a+b x)^5 (d+e x)^{3/2} (b d-a e)} \]

[Out]

(77*e^4*(10*b*B*d - 13*A*b*e + 3*a*B*e))/(128*b*(b*d - a*e)^6*(d + e*x)^(3/2)) -
 (A*b - a*B)/(5*b*(b*d - a*e)*(a + b*x)^5*(d + e*x)^(3/2)) - (10*b*B*d - 13*A*b*
e + 3*a*B*e)/(40*b*(b*d - a*e)^2*(a + b*x)^4*(d + e*x)^(3/2)) + (11*e*(10*b*B*d
- 13*A*b*e + 3*a*B*e))/(240*b*(b*d - a*e)^3*(a + b*x)^3*(d + e*x)^(3/2)) - (33*e
^2*(10*b*B*d - 13*A*b*e + 3*a*B*e))/(320*b*(b*d - a*e)^4*(a + b*x)^2*(d + e*x)^(
3/2)) + (231*e^3*(10*b*B*d - 13*A*b*e + 3*a*B*e))/(640*b*(b*d - a*e)^5*(a + b*x)
*(d + e*x)^(3/2)) + (231*e^4*(10*b*B*d - 13*A*b*e + 3*a*B*e))/(128*(b*d - a*e)^7
*Sqrt[d + e*x]) - (231*Sqrt[b]*e^4*(10*b*B*d - 13*A*b*e + 3*a*B*e)*ArcTanh[(Sqrt
[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(128*(b*d - a*e)^(15/2))

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Rubi [A]  time = 1.14584, antiderivative size = 400, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{231 e^4 (3 a B e-13 A b e+10 b B d)}{128 \sqrt{d+e x} (b d-a e)^7}+\frac{77 e^4 (3 a B e-13 A b e+10 b B d)}{128 b (d+e x)^{3/2} (b d-a e)^6}-\frac{231 \sqrt{b} e^4 (3 a B e-13 A b e+10 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 (b d-a e)^{15/2}}+\frac{231 e^3 (3 a B e-13 A b e+10 b B d)}{640 b (a+b x) (d+e x)^{3/2} (b d-a e)^5}-\frac{33 e^2 (3 a B e-13 A b e+10 b B d)}{320 b (a+b x)^2 (d+e x)^{3/2} (b d-a e)^4}+\frac{11 e (3 a B e-13 A b e+10 b B d)}{240 b (a+b x)^3 (d+e x)^{3/2} (b d-a e)^3}-\frac{3 a B e-13 A b e+10 b B d}{40 b (a+b x)^4 (d+e x)^{3/2} (b d-a e)^2}-\frac{A b-a B}{5 b (a+b x)^5 (d+e x)^{3/2} (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/((d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^3),x]

[Out]

(77*e^4*(10*b*B*d - 13*A*b*e + 3*a*B*e))/(128*b*(b*d - a*e)^6*(d + e*x)^(3/2)) -
 (A*b - a*B)/(5*b*(b*d - a*e)*(a + b*x)^5*(d + e*x)^(3/2)) - (10*b*B*d - 13*A*b*
e + 3*a*B*e)/(40*b*(b*d - a*e)^2*(a + b*x)^4*(d + e*x)^(3/2)) + (11*e*(10*b*B*d
- 13*A*b*e + 3*a*B*e))/(240*b*(b*d - a*e)^3*(a + b*x)^3*(d + e*x)^(3/2)) - (33*e
^2*(10*b*B*d - 13*A*b*e + 3*a*B*e))/(320*b*(b*d - a*e)^4*(a + b*x)^2*(d + e*x)^(
3/2)) + (231*e^3*(10*b*B*d - 13*A*b*e + 3*a*B*e))/(640*b*(b*d - a*e)^5*(a + b*x)
*(d + e*x)^(3/2)) + (231*e^4*(10*b*B*d - 13*A*b*e + 3*a*B*e))/(128*(b*d - a*e)^7
*Sqrt[d + e*x]) - (231*Sqrt[b]*e^4*(10*b*B*d - 13*A*b*e + 3*a*B*e)*ArcTanh[(Sqrt
[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(128*(b*d - a*e)^(15/2))

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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)**(5/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Timed out

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Mathematica [A]  time = 2.92831, size = 316, normalized size = 0.79 \[ \frac{-\frac{3465 \sqrt{b} e^4 (3 a B e-13 A b e+10 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{(b d-a e)^{15/2}}-\frac{\sqrt{d+e x} \left (\frac{3840 e^4 (-a B e+6 A b e-5 b B d)}{d+e x}-\frac{1280 e^4 (a e-b d) (A e-B d)}{(d+e x)^2}+\frac{15 b e^3 (-437 a B e+1467 A b e-1030 b B d)}{a+b x}-\frac{10 b e^2 (b d-a e) (-309 a B e+827 A b e-518 b B d)}{(a+b x)^2}+\frac{8 b e (b d-a e)^2 (-213 a B e+443 A b e-230 b B d)}{(a+b x)^3}+\frac{48 b (b d-a e)^3 (19 a B e-29 A b e+10 b B d)}{(a+b x)^4}+\frac{384 b (A b-a B) (b d-a e)^4}{(a+b x)^5}\right )}{(b d-a e)^7}}{1920} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/((d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^3),x]

[Out]

(-((Sqrt[d + e*x]*((384*b*(A*b - a*B)*(b*d - a*e)^4)/(a + b*x)^5 + (48*b*(b*d -
a*e)^3*(10*b*B*d - 29*A*b*e + 19*a*B*e))/(a + b*x)^4 + (8*b*e*(b*d - a*e)^2*(-23
0*b*B*d + 443*A*b*e - 213*a*B*e))/(a + b*x)^3 - (10*b*e^2*(b*d - a*e)*(-518*b*B*
d + 827*A*b*e - 309*a*B*e))/(a + b*x)^2 + (15*b*e^3*(-1030*b*B*d + 1467*A*b*e -
437*a*B*e))/(a + b*x) - (1280*e^4*(-(b*d) + a*e)*(-(B*d) + A*e))/(d + e*x)^2 + (
3840*e^4*(-5*b*B*d + 6*A*b*e - a*B*e))/(d + e*x)))/(b*d - a*e)^7) - (3465*Sqrt[b
]*e^4*(10*b*B*d - 13*A*b*e + 3*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d -
 a*e]])/(b*d - a*e)^(15/2))/1920

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Maple [B]  time = 0.065, size = 1653, normalized size = 4.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/(e*x+d)^(5/2)/(b^2*x^2+2*a*b*x+a^2)^3,x)

[Out]

-2/3*e^5/(a*e-b*d)^6/(e*x+d)^(3/2)*A+531/64*e^7/(a*e-b*d)^7*b^3/(b*e*x+a*e)^5*(e
*x+d)^(1/2)*B*a^3*d^2-363/8*e^6/(a*e-b*d)^7*b^4/(b*e*x+a*e)^5*(e*x+d)^(1/2)*B*a^
2*d^3+3349/96*e^4/(a*e-b*d)^7*b^6/(b*e*x+a*e)^5*B*(e*x+d)^(7/2)*d^2+12*e^5/(a*e-
b*d)^7/(e*x+d)^(1/2)*A*b-2*e^5/(a*e-b*d)^7/(e*x+d)^(1/2)*a*B+2/3*e^4/(a*e-b*d)^6
/(e*x+d)^(3/2)*B*d-10*e^4/(a*e-b*d)^7/(e*x+d)^(1/2)*B*b*d+3003/128*e^5/(a*e-b*d)
^7*b^2/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*A+1467/12
8*e^5/(a*e-b*d)^7*b^6/(b*e*x+a*e)^5*(e*x+d)^(9/2)*A-6823/64*e^5/(a*e-b*d)^7*b^5/
(b*e*x+a*e)^5*(e*x+d)^(3/2)*B*a*d^3+1327/15*e^5/(a*e-b*d)^7*b^5/(b*e*x+a*e)^5*(e
*x+d)^(5/2)*B*a*d^2-2506/15*e^6/(a*e-b*d)^7*b^5/(b*e*x+a*e)^5*(e*x+d)^(5/2)*A*a*
d-12131/64*e^7/(a*e-b*d)^7*b^4/(b*e*x+a*e)^5*(e*x+d)^(3/2)*A*a^2*d+12131/64*e^6/
(a*e-b*d)^7*b^5/(b*e*x+a*e)^5*(e*x+d)^(3/2)*A*a*d^2+3793/192*e^7/(a*e-b*d)^7*b^3
/(b*e*x+a*e)^5*(e*x+d)^(3/2)*B*a^3*d+4169/64*e^6/(a*e-b*d)^7*b^4/(b*e*x+a*e)^5*(
e*x+d)^(3/2)*B*a^2*d^2+5277/128*e^5/(a*e-b*d)^7*b^5/(b*e*x+a*e)^5*(e*x+d)^(1/2)*
B*a*d^4-3767/192*e^5/(a*e-b*d)^7*b^5/(b*e*x+a*e)^5*B*(e*x+d)^(7/2)*a*d-74/15*e^6
/(a*e-b*d)^7*b^4/(b*e*x+a*e)^5*(e*x+d)^(5/2)*B*a^2*d-2373/32*e^8/(a*e-b*d)^7*b^3
/(b*e*x+a*e)^5*(e*x+d)^(1/2)*A*a^3*d+7119/64*e^7/(a*e-b*d)^7*b^4/(b*e*x+a*e)^5*(
e*x+d)^(1/2)*A*a^2*d^2-2373/32*e^6/(a*e-b*d)^7*b^5/(b*e*x+a*e)^5*(e*x+d)^(1/2)*A
*a*d^3+921/64*e^8/(a*e-b*d)^7*b^2/(b*e*x+a*e)^5*(e*x+d)^(1/2)*B*a^4*d-131/5*e^7/
(a*e-b*d)^7*b^3/(b*e*x+a*e)^5*(e*x+d)^(5/2)*B*a^3+12131/192*e^8/(a*e-b*d)^7*b^3/
(b*e*x+a*e)^5*(e*x+d)^(3/2)*A*a^3-12131/192*e^5/(a*e-b*d)^7*b^6/(b*e*x+a*e)^5*(e
*x+d)^(3/2)*A*d^3-1327/64*e^8/(a*e-b*d)^7*b^2/(b*e*x+a*e)^5*(e*x+d)^(3/2)*B*a^4+
2373/128*e^9/(a*e-b*d)^7*b^2/(b*e*x+a*e)^5*(e*x+d)^(1/2)*A*a^4+2373/128*e^5/(a*e
-b*d)^7*b^6/(b*e*x+a*e)^5*(e*x+d)^(1/2)*A*d^4+9629/192*e^6/(a*e-b*d)^7*b^5/(b*e*
x+a*e)^5*A*(e*x+d)^(7/2)*a-9629/192*e^5/(a*e-b*d)^7*b^6/(b*e*x+a*e)^5*A*(e*x+d)^
(7/2)*d-977/64*e^6/(a*e-b*d)^7*b^4/(b*e*x+a*e)^5*B*(e*x+d)^(7/2)*a^2-1155/64*e^4
/(a*e-b*d)^7*b^2/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))
*B*d+1253/15*e^5/(a*e-b*d)^7*b^6/(b*e*x+a*e)^5*(e*x+d)^(5/2)*A*d^2-693/128*e^5/(
a*e-b*d)^7*b/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a*B
-843/128*e^9/(a*e-b*d)^7*b/(b*e*x+a*e)^5*(e*x+d)^(1/2)*B*a^5-437/128*e^5/(a*e-b*
d)^7*b^5/(b*e*x+a*e)^5*(e*x+d)^(9/2)*B*a+1253/15*e^7/(a*e-b*d)^7*b^4/(b*e*x+a*e)
^5*(e*x+d)^(5/2)*A*a^2-515/64*e^4/(a*e-b*d)^7*b^6/(b*e*x+a*e)^5*(e*x+d)^(9/2)*B*
d-172/3*e^4/(a*e-b*d)^7*b^6/(b*e*x+a*e)^5*(e*x+d)^(5/2)*B*d^3+4075/96*e^4/(a*e-b
*d)^7*b^6/(b*e*x+a*e)^5*(e*x+d)^(3/2)*B*d^4-765/64*e^4/(a*e-b*d)^7*b^6/(b*e*x+a*
e)^5*(e*x+d)^(1/2)*B*d^5

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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)^3*(e*x + d)^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.402562, 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)^3*(e*x + d)^(5/2)),x, algorithm="fricas")

[Out]

[1/3840*(2560*A*a^6*e^6 - 192*(B*a*b^5 + 4*A*b^6)*d^6 + 32*(52*B*a^2*b^4 + 183*A
*a*b^5)*d^5*e - 56*(127*B*a^3*b^3 + 358*A*a^2*b^4)*d^4*e^2 + 140*(174*B*a^4*b^2
+ 301*A*a^3*b^3)*d^3*e^3 + 10*(6625*B*a^5*b - 7119*A*a^4*b^2)*d^2*e^4 + 2560*(2*
B*a^6 - 19*A*a^5*b)*d*e^5 + 6930*(10*B*b^6*d*e^5 + (3*B*a*b^5 - 13*A*b^6)*e^6)*x
^6 + 4620*(20*B*b^6*d^2*e^4 + 2*(38*B*a*b^5 - 13*A*b^6)*d*e^5 + 7*(3*B*a^2*b^4 -
 13*A*a*b^5)*e^6)*x^5 + 462*(30*B*b^6*d^3*e^3 + 13*(73*B*a*b^5 - 3*A*b^6)*d^2*e^
4 + 2*(781*B*a^2*b^4 - 611*A*a*b^5)*d*e^5 + 128*(3*B*a^3*b^3 - 13*A*a^2*b^4)*e^6
)*x^4 - 132*(30*B*b^6*d^4*e^2 - 3*(167*B*a*b^5 + 13*A*b^6)*d^3*e^3 - (6223*B*a^2
*b^4 - 663*A*a*b^5)*d^2*e^4 - (5771*B*a^3*b^3 - 7891*A*a^2*b^4)*d*e^5 - 395*(3*B
*a^4*b^2 - 13*A*a^3*b^3)*e^6)*x^3 + 22*(80*B*b^6*d^5*e - 4*(209*B*a*b^5 + 26*A*b
^6)*d^4*e^2 + 2*(2811*B*a^2*b^4 + 559*A*a*b^5)*d^3*e^3 + 4*(8566*B*a^3*b^3 - 191
1*A*a^2*b^4)*d^2*e^4 + 50*(388*B*a^4*b^2 - 845*A*a^3*b^3)*d*e^5 + 965*(3*B*a^5*b
 - 13*A*a^4*b^2)*e^6)*x^2 + 3465*(10*B*a^5*b*d^2*e^4 + (3*B*a^6 - 13*A*a^5*b)*d*
e^5 + (10*B*b^6*d*e^5 + (3*B*a*b^5 - 13*A*b^6)*e^6)*x^6 + (10*B*b^6*d^2*e^4 + (5
3*B*a*b^5 - 13*A*b^6)*d*e^5 + 5*(3*B*a^2*b^4 - 13*A*a*b^5)*e^6)*x^5 + 5*(10*B*a*
b^5*d^2*e^4 + (23*B*a^2*b^4 - 13*A*a*b^5)*d*e^5 + 2*(3*B*a^3*b^3 - 13*A*a^2*b^4)
*e^6)*x^4 + 10*(10*B*a^2*b^4*d^2*e^4 + 13*(B*a^3*b^3 - A*a^2*b^4)*d*e^5 + (3*B*a
^4*b^2 - 13*A*a^3*b^3)*e^6)*x^3 + 5*(20*B*a^3*b^3*d^2*e^4 + 2*(8*B*a^4*b^2 - 13*
A*a^3*b^3)*d*e^5 + (3*B*a^5*b - 13*A*a^4*b^2)*e^6)*x^2 + (50*B*a^4*b^2*d^2*e^4 +
 5*(5*B*a^5*b - 13*A*a^4*b^2)*d*e^5 + (3*B*a^6 - 13*A*a^5*b)*e^6)*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)) - 4*(240*B*b^6*d^6 - 8*(251*B*a*b^5 + 39*A*b^6)*d^
5*e + 2*(4133*B*a^2*b^4 + 1352*A*a*b^5)*d^4*e^2 - 7*(3969*B*a^3*b^3 + 1651*A*a^2
*b^4)*d^3*e^3 - 5*(16657*B*a^4*b^2 - 7917*A*a^3*b^3)*d^2*e^4 - 5*(5729*B*a^5*b -
 19279*A*a^4*b^2)*d*e^5 - 640*(3*B*a^6 - 13*A*a^5*b)*e^6)*x)/((a^5*b^7*d^8 - 7*a
^6*b^6*d^7*e + 21*a^7*b^5*d^6*e^2 - 35*a^8*b^4*d^5*e^3 + 35*a^9*b^3*d^4*e^4 - 21
*a^10*b^2*d^3*e^5 + 7*a^11*b*d^2*e^6 - a^12*d*e^7 + (b^12*d^7*e - 7*a*b^11*d^6*e
^2 + 21*a^2*b^10*d^5*e^3 - 35*a^3*b^9*d^4*e^4 + 35*a^4*b^8*d^3*e^5 - 21*a^5*b^7*
d^2*e^6 + 7*a^6*b^6*d*e^7 - a^7*b^5*e^8)*x^6 + (b^12*d^8 - 2*a*b^11*d^7*e - 14*a
^2*b^10*d^6*e^2 + 70*a^3*b^9*d^5*e^3 - 140*a^4*b^8*d^4*e^4 + 154*a^5*b^7*d^3*e^5
 - 98*a^6*b^6*d^2*e^6 + 34*a^7*b^5*d*e^7 - 5*a^8*b^4*e^8)*x^5 + 5*(a*b^11*d^8 -
5*a^2*b^10*d^7*e + 7*a^3*b^9*d^6*e^2 + 7*a^4*b^8*d^5*e^3 - 35*a^5*b^7*d^4*e^4 +
49*a^6*b^6*d^3*e^5 - 35*a^7*b^5*d^2*e^6 + 13*a^8*b^4*d*e^7 - 2*a^9*b^3*e^8)*x^4
+ 10*(a^2*b^10*d^8 - 6*a^3*b^9*d^7*e + 14*a^4*b^8*d^6*e^2 - 14*a^5*b^7*d^5*e^3 +
 14*a^7*b^5*d^3*e^5 - 14*a^8*b^4*d^2*e^6 + 6*a^9*b^3*d*e^7 - a^10*b^2*e^8)*x^3 +
 5*(2*a^3*b^9*d^8 - 13*a^4*b^8*d^7*e + 35*a^5*b^7*d^6*e^2 - 49*a^6*b^6*d^5*e^3 +
 35*a^7*b^5*d^4*e^4 - 7*a^8*b^4*d^3*e^5 - 7*a^9*b^3*d^2*e^6 + 5*a^10*b^2*d*e^7 -
 a^11*b*e^8)*x^2 + (5*a^4*b^8*d^8 - 34*a^5*b^7*d^7*e + 98*a^6*b^6*d^6*e^2 - 154*
a^7*b^5*d^5*e^3 + 140*a^8*b^4*d^4*e^4 - 70*a^9*b^3*d^3*e^5 + 14*a^10*b^2*d^2*e^6
 + 2*a^11*b*d*e^7 - a^12*e^8)*x)*sqrt(e*x + d)), 1/1920*(1280*A*a^6*e^6 - 96*(B*
a*b^5 + 4*A*b^6)*d^6 + 16*(52*B*a^2*b^4 + 183*A*a*b^5)*d^5*e - 28*(127*B*a^3*b^3
 + 358*A*a^2*b^4)*d^4*e^2 + 70*(174*B*a^4*b^2 + 301*A*a^3*b^3)*d^3*e^3 + 5*(6625
*B*a^5*b - 7119*A*a^4*b^2)*d^2*e^4 + 1280*(2*B*a^6 - 19*A*a^5*b)*d*e^5 + 3465*(1
0*B*b^6*d*e^5 + (3*B*a*b^5 - 13*A*b^6)*e^6)*x^6 + 2310*(20*B*b^6*d^2*e^4 + 2*(38
*B*a*b^5 - 13*A*b^6)*d*e^5 + 7*(3*B*a^2*b^4 - 13*A*a*b^5)*e^6)*x^5 + 231*(30*B*b
^6*d^3*e^3 + 13*(73*B*a*b^5 - 3*A*b^6)*d^2*e^4 + 2*(781*B*a^2*b^4 - 611*A*a*b^5)
*d*e^5 + 128*(3*B*a^3*b^3 - 13*A*a^2*b^4)*e^6)*x^4 - 66*(30*B*b^6*d^4*e^2 - 3*(1
67*B*a*b^5 + 13*A*b^6)*d^3*e^3 - (6223*B*a^2*b^4 - 663*A*a*b^5)*d^2*e^4 - (5771*
B*a^3*b^3 - 7891*A*a^2*b^4)*d*e^5 - 395*(3*B*a^4*b^2 - 13*A*a^3*b^3)*e^6)*x^3 +
11*(80*B*b^6*d^5*e - 4*(209*B*a*b^5 + 26*A*b^6)*d^4*e^2 + 2*(2811*B*a^2*b^4 + 55
9*A*a*b^5)*d^3*e^3 + 4*(8566*B*a^3*b^3 - 1911*A*a^2*b^4)*d^2*e^4 + 50*(388*B*a^4
*b^2 - 845*A*a^3*b^3)*d*e^5 + 965*(3*B*a^5*b - 13*A*a^4*b^2)*e^6)*x^2 - 3465*(10
*B*a^5*b*d^2*e^4 + (3*B*a^6 - 13*A*a^5*b)*d*e^5 + (10*B*b^6*d*e^5 + (3*B*a*b^5 -
 13*A*b^6)*e^6)*x^6 + (10*B*b^6*d^2*e^4 + (53*B*a*b^5 - 13*A*b^6)*d*e^5 + 5*(3*B
*a^2*b^4 - 13*A*a*b^5)*e^6)*x^5 + 5*(10*B*a*b^5*d^2*e^4 + (23*B*a^2*b^4 - 13*A*a
*b^5)*d*e^5 + 2*(3*B*a^3*b^3 - 13*A*a^2*b^4)*e^6)*x^4 + 10*(10*B*a^2*b^4*d^2*e^4
 + 13*(B*a^3*b^3 - A*a^2*b^4)*d*e^5 + (3*B*a^4*b^2 - 13*A*a^3*b^3)*e^6)*x^3 + 5*
(20*B*a^3*b^3*d^2*e^4 + 2*(8*B*a^4*b^2 - 13*A*a^3*b^3)*d*e^5 + (3*B*a^5*b - 13*A
*a^4*b^2)*e^6)*x^2 + (50*B*a^4*b^2*d^2*e^4 + 5*(5*B*a^5*b - 13*A*a^4*b^2)*d*e^5
+ (3*B*a^6 - 13*A*a^5*b)*e^6)*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)) - 2*(240*B*b^6*d^6 - 8*(251*B*a*
b^5 + 39*A*b^6)*d^5*e + 2*(4133*B*a^2*b^4 + 1352*A*a*b^5)*d^4*e^2 - 7*(3969*B*a^
3*b^3 + 1651*A*a^2*b^4)*d^3*e^3 - 5*(16657*B*a^4*b^2 - 7917*A*a^3*b^3)*d^2*e^4 -
 5*(5729*B*a^5*b - 19279*A*a^4*b^2)*d*e^5 - 640*(3*B*a^6 - 13*A*a^5*b)*e^6)*x)/(
(a^5*b^7*d^8 - 7*a^6*b^6*d^7*e + 21*a^7*b^5*d^6*e^2 - 35*a^8*b^4*d^5*e^3 + 35*a^
9*b^3*d^4*e^4 - 21*a^10*b^2*d^3*e^5 + 7*a^11*b*d^2*e^6 - a^12*d*e^7 + (b^12*d^7*
e - 7*a*b^11*d^6*e^2 + 21*a^2*b^10*d^5*e^3 - 35*a^3*b^9*d^4*e^4 + 35*a^4*b^8*d^3
*e^5 - 21*a^5*b^7*d^2*e^6 + 7*a^6*b^6*d*e^7 - a^7*b^5*e^8)*x^6 + (b^12*d^8 - 2*a
*b^11*d^7*e - 14*a^2*b^10*d^6*e^2 + 70*a^3*b^9*d^5*e^3 - 140*a^4*b^8*d^4*e^4 + 1
54*a^5*b^7*d^3*e^5 - 98*a^6*b^6*d^2*e^6 + 34*a^7*b^5*d*e^7 - 5*a^8*b^4*e^8)*x^5
+ 5*(a*b^11*d^8 - 5*a^2*b^10*d^7*e + 7*a^3*b^9*d^6*e^2 + 7*a^4*b^8*d^5*e^3 - 35*
a^5*b^7*d^4*e^4 + 49*a^6*b^6*d^3*e^5 - 35*a^7*b^5*d^2*e^6 + 13*a^8*b^4*d*e^7 - 2
*a^9*b^3*e^8)*x^4 + 10*(a^2*b^10*d^8 - 6*a^3*b^9*d^7*e + 14*a^4*b^8*d^6*e^2 - 14
*a^5*b^7*d^5*e^3 + 14*a^7*b^5*d^3*e^5 - 14*a^8*b^4*d^2*e^6 + 6*a^9*b^3*d*e^7 - a
^10*b^2*e^8)*x^3 + 5*(2*a^3*b^9*d^8 - 13*a^4*b^8*d^7*e + 35*a^5*b^7*d^6*e^2 - 49
*a^6*b^6*d^5*e^3 + 35*a^7*b^5*d^4*e^4 - 7*a^8*b^4*d^3*e^5 - 7*a^9*b^3*d^2*e^6 +
5*a^10*b^2*d*e^7 - a^11*b*e^8)*x^2 + (5*a^4*b^8*d^8 - 34*a^5*b^7*d^7*e + 98*a^6*
b^6*d^6*e^2 - 154*a^7*b^5*d^5*e^3 + 140*a^8*b^4*d^4*e^4 - 70*a^9*b^3*d^3*e^5 + 1
4*a^10*b^2*d^2*e^6 + 2*a^11*b*d*e^7 - a^12*e^8)*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)**(5/2)/(b**2*x**2+2*a*b*x+a**2)**3,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.332505, 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)^3*(e*x + d)^(5/2)),x, algorithm="giac")

[Out]

Done

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