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3.1834 \(\int \frac{A+B x}{(d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx\)

Optimal. Leaf size=435 \[ -\frac{3003 b^{3/2} e^4 (a B e-3 A b e+2 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 (b d-a e)^{17/2}}+\frac{3003 b e^4 (a B e-3 A b e+2 b B d)}{128 \sqrt{d+e x} (b d-a e)^8}+\frac{1001 e^4 (a B e-3 A b e+2 b B d)}{128 (d+e x)^{3/2} (b d-a e)^7}+\frac{3003 e^4 (a B e-3 A b e+2 b B d)}{640 b (d+e x)^{5/2} (b d-a e)^6}+\frac{429 e^3 (a B e-3 A b e+2 b B d)}{128 b (a+b x) (d+e x)^{5/2} (b d-a e)^5}-\frac{143 e^2 (a B e-3 A b e+2 b B d)}{192 b (a+b x)^2 (d+e x)^{5/2} (b d-a e)^4}+\frac{13 e (a B e-3 A b e+2 b B d)}{48 b (a+b x)^3 (d+e x)^{5/2} (b d-a e)^3}-\frac{a B e-3 A b e+2 b B d}{8 b (a+b x)^4 (d+e x)^{5/2} (b d-a e)^2}-\frac{A b-a B}{5 b (a+b x)^5 (d+e x)^{5/2} (b d-a e)} \]

[Out]

(3003*e^4*(2*b*B*d - 3*A*b*e + a*B*e))/(640*b*(b*d - a*e)^6*(d + e*x)^(5/2)) - (
A*b - a*B)/(5*b*(b*d - a*e)*(a + b*x)^5*(d + e*x)^(5/2)) - (2*b*B*d - 3*A*b*e +
a*B*e)/(8*b*(b*d - a*e)^2*(a + b*x)^4*(d + e*x)^(5/2)) + (13*e*(2*b*B*d - 3*A*b*
e + a*B*e))/(48*b*(b*d - a*e)^3*(a + b*x)^3*(d + e*x)^(5/2)) - (143*e^2*(2*b*B*d
 - 3*A*b*e + a*B*e))/(192*b*(b*d - a*e)^4*(a + b*x)^2*(d + e*x)^(5/2)) + (429*e^
3*(2*b*B*d - 3*A*b*e + a*B*e))/(128*b*(b*d - a*e)^5*(a + b*x)*(d + e*x)^(5/2)) +
 (1001*e^4*(2*b*B*d - 3*A*b*e + a*B*e))/(128*(b*d - a*e)^7*(d + e*x)^(3/2)) + (3
003*b*e^4*(2*b*B*d - 3*A*b*e + a*B*e))/(128*(b*d - a*e)^8*Sqrt[d + e*x]) - (3003
*b^(3/2)*e^4*(2*b*B*d - 3*A*b*e + a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*
d - a*e]])/(128*(b*d - a*e)^(17/2))

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Rubi [A]  time = 1.22599, antiderivative size = 435, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 6, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{3003 b^{3/2} e^4 (a B e-3 A b e+2 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{128 (b d-a e)^{17/2}}+\frac{3003 b e^4 (a B e-3 A b e+2 b B d)}{128 \sqrt{d+e x} (b d-a e)^8}+\frac{1001 e^4 (a B e-3 A b e+2 b B d)}{128 (d+e x)^{3/2} (b d-a e)^7}+\frac{3003 e^4 (a B e-3 A b e+2 b B d)}{640 b (d+e x)^{5/2} (b d-a e)^6}+\frac{429 e^3 (a B e-3 A b e+2 b B d)}{128 b (a+b x) (d+e x)^{5/2} (b d-a e)^5}-\frac{143 e^2 (a B e-3 A b e+2 b B d)}{192 b (a+b x)^2 (d+e x)^{5/2} (b d-a e)^4}+\frac{13 e (a B e-3 A b e+2 b B d)}{48 b (a+b x)^3 (d+e x)^{5/2} (b d-a e)^3}-\frac{a B e-3 A b e+2 b B d}{8 b (a+b x)^4 (d+e x)^{5/2} (b d-a e)^2}-\frac{A b-a B}{5 b (a+b x)^5 (d+e x)^{5/2} (b d-a e)} \]

Antiderivative was successfully verified.

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

[Out]

(3003*e^4*(2*b*B*d - 3*A*b*e + a*B*e))/(640*b*(b*d - a*e)^6*(d + e*x)^(5/2)) - (
A*b - a*B)/(5*b*(b*d - a*e)*(a + b*x)^5*(d + e*x)^(5/2)) - (2*b*B*d - 3*A*b*e +
a*B*e)/(8*b*(b*d - a*e)^2*(a + b*x)^4*(d + e*x)^(5/2)) + (13*e*(2*b*B*d - 3*A*b*
e + a*B*e))/(48*b*(b*d - a*e)^3*(a + b*x)^3*(d + e*x)^(5/2)) - (143*e^2*(2*b*B*d
 - 3*A*b*e + a*B*e))/(192*b*(b*d - a*e)^4*(a + b*x)^2*(d + e*x)^(5/2)) + (429*e^
3*(2*b*B*d - 3*A*b*e + a*B*e))/(128*b*(b*d - a*e)^5*(a + b*x)*(d + e*x)^(5/2)) +
 (1001*e^4*(2*b*B*d - 3*A*b*e + a*B*e))/(128*(b*d - a*e)^7*(d + e*x)^(3/2)) + (3
003*b*e^4*(2*b*B*d - 3*A*b*e + a*B*e))/(128*(b*d - a*e)^8*Sqrt[d + e*x]) - (3003
*b^(3/2)*e^4*(2*b*B*d - 3*A*b*e + a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*
d - a*e]])/(128*(b*d - a*e)^(17/2))

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

[Out]

Timed out

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Mathematica [A]  time = 4.14628, size = 364, normalized size = 0.84 \[ \frac{-\frac{45045 b^{3/2} e^4 (a B e-3 A b e+2 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{(b d-a e)^{17/2}}-\frac{\sqrt{d+e x} \left (\frac{45 b^2 e^3 (-489 a B e+1211 A b e-722 b B d)}{a+b x}-\frac{10 b^2 e^2 (b d-a e) (-827 a B e+1713 A b e-886 b B d)}{(a+b x)^2}+\frac{8 b^2 e (b d-a e)^2 (-443 a B e+753 A b e-310 b B d)}{(a+b x)^3}+\frac{48 b^2 (b d-a e)^3 (29 a B e-39 A b e+10 b B d)}{(a+b x)^4}+\frac{384 b^2 (A b-a B) (b d-a e)^4}{(a+b x)^5}+\frac{11520 b e^4 (-2 a B e+7 A b e-5 b B d)}{d+e x}-\frac{1280 e^4 (a e-b d) (-a B e+6 A b e-5 b B d)}{(d+e x)^2}+\frac{768 e^4 (b d-a e)^2 (A e-B d)}{(d+e x)^3}\right )}{(b d-a e)^8}}{1920} \]

Antiderivative was successfully verified.

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

[Out]

(-((Sqrt[d + e*x]*((384*b^2*(A*b - a*B)*(b*d - a*e)^4)/(a + b*x)^5 + (48*b^2*(b*
d - a*e)^3*(10*b*B*d - 39*A*b*e + 29*a*B*e))/(a + b*x)^4 + (8*b^2*e*(b*d - a*e)^
2*(-310*b*B*d + 753*A*b*e - 443*a*B*e))/(a + b*x)^3 - (10*b^2*e^2*(b*d - a*e)*(-
886*b*B*d + 1713*A*b*e - 827*a*B*e))/(a + b*x)^2 + (45*b^2*e^3*(-722*b*B*d + 121
1*A*b*e - 489*a*B*e))/(a + b*x) + (768*e^4*(b*d - a*e)^2*(-(B*d) + A*e))/(d + e*
x)^3 - (1280*e^4*(-(b*d) + a*e)*(-5*b*B*d + 6*A*b*e - a*B*e))/(d + e*x)^2 + (115
20*b*e^4*(-5*b*B*d + 7*A*b*e - 2*a*B*e))/(d + e*x)))/(b*d - a*e)^8) - (45045*b^(
3/2)*e^4*(2*b*B*d - 3*A*b*e + a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d -
a*e]])/(b*d - a*e)^(17/2))/1920

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Maple [B]  time = 0.067, size = 1735, normalized size = 4. \[ \text{result too large to display} \]

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)^3,x)

[Out]

-252/5*e^6/(a*e-b*d)^8*b^5/(b*e*x+a*e)^5*(e*x+d)^(5/2)*B*a^2*d+28329/64*e^7/(a*e
-b*d)^8*b^5/(b*e*x+a*e)^5*(e*x+d)^(3/2)*A*a^2*d-28329/64*e^6/(a*e-b*d)^8*b^6/(b*
e*x+a*e)^5*(e*x+d)^(3/2)*A*a*d^2+1029/16*e^6/(a*e-b*d)^8*b^5/(b*e*x+a*e)^5*(e*x+
d)^(1/2)*B*a^2*d^3+2002/5*e^6/(a*e-b*d)^8*b^6/(b*e*x+a*e)^5*(e*x+d)^(5/2)*A*a*d-
9443/128*e^5/(a*e-b*d)^8*b^6/(b*e*x+a*e)^5*(e*x+d)^(1/2)*B*a*d^4+4253/192*e^5/(a
*e-b*d)^8*b^6/(b*e*x+a*e)^5*B*(e*x+d)^(7/2)*a*d-749/5*e^5/(a*e-b*d)^8*b^6/(b*e*x
+a*e)^5*(e*x+d)^(5/2)*B*a*d^2-42*e^5*b^2/(a*e-b*d)^8/(e*x+d)^(1/2)*A+4*e^5/(a*e-
b*d)^7/(e*x+d)^(3/2)*A*b-2/3*e^5/(a*e-b*d)^7/(e*x+d)^(3/2)*a*B-3633/128*e^5/(a*e
-b*d)^8*b^7/(b*e*x+a*e)^5*(e*x+d)^(9/2)*A-10/3*e^4/(a*e-b*d)^7/(e*x+d)^(3/2)*B*b
*d+30*e^4*b^2/(a*e-b*d)^8/(e*x+d)^(1/2)*B*d+12*e^5*b/(a*e-b*d)^8/(e*x+d)^(1/2)*a
*B-9009/128*e^5/(a*e-b*d)^8*b^3/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a
*e-b*d))^(1/2))*A-4067/64*e^6/(a*e-b*d)^8*b^5/(b*e*x+a*e)^5*(e*x+d)^(3/2)*B*a^2*
d^2+36463/192*e^5/(a*e-b*d)^8*b^6/(b*e*x+a*e)^5*(e*x+d)^(3/2)*B*a*d^3+5327/32*e^
8/(a*e-b*d)^8*b^4/(b*e*x+a*e)^5*(e*x+d)^(1/2)*A*a^3*d-15981/64*e^7/(a*e-b*d)^8*b
^5/(b*e*x+a*e)^5*(e*x+d)^(1/2)*A*a^2*d^2+5327/32*e^6/(a*e-b*d)^8*b^6/(b*e*x+a*e)
^5*(e*x+d)^(1/2)*A*a*d^3-3269/64*e^8/(a*e-b*d)^8*b^3/(b*e*x+a*e)^5*(e*x+d)^(1/2)
*B*a^4*d+1211/64*e^7/(a*e-b*d)^8*b^4/(b*e*x+a*e)^5*(e*x+d)^(1/2)*B*a^3*d^2+2/5*e
^4/(a*e-b*d)^6/(e*x+d)^(5/2)*B*d-2/5*e^5/(a*e-b*d)^6/(e*x+d)^(5/2)*A-5327/128*e^
9/(a*e-b*d)^8*b^3/(b*e*x+a*e)^5*(e*x+d)^(1/2)*A*a^4-5327/128*e^5/(a*e-b*d)^8*b^7
/(b*e*x+a*e)^5*(e*x+d)^(1/2)*A*d^4-7837/64*e^6/(a*e-b*d)^8*b^6/(b*e*x+a*e)^5*A*(
e*x+d)^(7/2)*a-6941/96*e^4/(a*e-b*d)^8*b^7/(b*e*x+a*e)^5*B*(e*x+d)^(7/2)*d^2+783
7/64*e^5/(a*e-b*d)^8*b^7/(b*e*x+a*e)^5*A*(e*x+d)^(7/2)*d+9629/192*e^6/(a*e-b*d)^
8*b^5/(b*e*x+a*e)^5*B*(e*x+d)^(7/2)*a^2+3003/128*e^5/(a*e-b*d)^8*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+2373/128*e^9/(a*e-b*d)^
8*b^2/(b*e*x+a*e)^5*(e*x+d)^(1/2)*B*a^5+1467/128*e^5/(a*e-b*d)^8*b^6/(b*e*x+a*e)
^5*(e*x+d)^(9/2)*B*a-1001/5*e^7/(a*e-b*d)^8*b^5/(b*e*x+a*e)^5*(e*x+d)^(5/2)*A*a^
2-1001/5*e^5/(a*e-b*d)^8*b^7/(b*e*x+a*e)^5*(e*x+d)^(5/2)*A*d^2+1253/15*e^7/(a*e-
b*d)^8*b^4/(b*e*x+a*e)^5*(e*x+d)^(5/2)*B*a^3-9443/64*e^8/(a*e-b*d)^8*b^4/(b*e*x+
a*e)^5*(e*x+d)^(3/2)*A*a^3+9443/64*e^5/(a*e-b*d)^8*b^7/(b*e*x+a*e)^5*(e*x+d)^(3/
2)*A*d^3-20195/192*e^7/(a*e-b*d)^8*b^4/(b*e*x+a*e)^5*(e*x+d)^(3/2)*B*a^3*d+3003/
64*e^4/(a*e-b*d)^8*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+1083/64*e^4/(a*e-b*d)^8*b^7/(b*e*x+a*e)^5*(e*x+d)^(9/2)*B*d+350/3*e^4
/(a*e-b*d)^8*b^7/(b*e*x+a*e)^5*(e*x+d)^(5/2)*B*d^3-8099/96*e^4/(a*e-b*d)^8*b^7/(
b*e*x+a*e)^5*(e*x+d)^(3/2)*B*d^4+1477/64*e^4/(a*e-b*d)^8*b^7/(b*e*x+a*e)^5*(e*x+
d)^(1/2)*B*d^5+12131/192*e^8/(a*e-b*d)^8*b^3/(b*e*x+a*e)^5*(e*x+d)^(3/2)*B*a^4

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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)^(7/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.408368, 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)^(7/2)),x, algorithm="fricas")

[Out]

[-1/3840*(1536*A*a^7*e^7 + 192*(B*a*b^6 + 4*A*b^7)*d^7 - 32*(62*B*a^2*b^5 + 213*
A*a*b^6)*d^6*e + 56*(187*B*a^3*b^4 + 498*A*a^2*b^5)*d^5*e^2 - 140*(332*B*a^4*b^3
 + 519*A*a^3*b^4)*d^4*e^3 - 2*(100363*B*a^5*b^2 - 79905*A*a^4*b^3)*d^3*e^4 - 204
8*(16*B*a^6*b - 87*A*a^5*b^2)*d^2*e^5 + 1024*(B*a^7 - 18*A*a^6*b)*d*e^6 - 90090*
(2*B*b^7*d*e^6 + (B*a*b^6 - 3*A*b^7)*e^7)*x^7 - 210210*(2*B*b^7*d^2*e^5 + (5*B*a
*b^6 - 3*A*b^7)*d*e^6 + 2*(B*a^2*b^5 - 3*A*a*b^6)*e^7)*x^6 - 6006*(46*B*b^7*d^3*
e^4 + 3*(117*B*a*b^6 - 23*A*b^7)*d^2*e^5 + 12*(35*B*a^2*b^5 - 41*A*a*b^6)*d*e^6
+ 128*(B*a^3*b^4 - 3*A*a^2*b^5)*e^7)*x^5 - 4290*(6*B*b^7*d^4*e^3 + (307*B*a*b^6
- 9*A*b^7)*d^3*e^4 + 12*(83*B*a^2*b^5 - 38*A*a*b^6)*d^2*e^5 + 6*(123*B*a^3*b^4 -
 211*A*a^2*b^5)*d*e^6 + 158*(B*a^4*b^3 - 3*A*a^3*b^4)*e^7)*x^4 + 1430*(4*B*b^7*d
^5*e^2 - 2*(43*B*a*b^6 + 3*A*b^7)*d^4*e^3 - 4*(434*B*a^2*b^5 - 33*A*a*b^6)*d^3*e
^4 - 2*(1547*B*a^3*b^4 - 1269*A*a^2*b^5)*d^2*e^5 - 2*(755*B*a^4*b^3 - 1686*A*a^3
*b^4)*d*e^6 - 193*(B*a^5*b^2 - 3*A*a^4*b^3)*e^7)*x^3 - 130*(16*B*b^7*d^6*e - 12*
(17*B*a*b^6 + 2*A*b^7)*d^5*e^2 + 6*(295*B*a^2*b^5 + 53*A*a*b^6)*d^4*e^3 + 2*(883
7*B*a^3*b^4 - 1407*A*a^2*b^5)*d^3*e^4 + 6*(3091*B*a^4*b^3 - 4184*A*a^3*b^4)*d^2*
e^5 + 3*(1867*B*a^5*b^2 - 5089*A*a^4*b^3)*d*e^6 + 256*(B*a^6*b - 3*A*a^5*b^2)*e^
7)*x^2 + 45045*(2*B*a^5*b^2*d^3*e^4 + (B*a^6*b - 3*A*a^5*b^2)*d^2*e^5 + (2*B*b^7
*d*e^6 + (B*a*b^6 - 3*A*b^7)*e^7)*x^7 + (4*B*b^7*d^2*e^5 + 6*(2*B*a*b^6 - A*b^7)
*d*e^6 + 5*(B*a^2*b^5 - 3*A*a*b^6)*e^7)*x^6 + (2*B*b^7*d^3*e^4 + 3*(7*B*a*b^6 -
A*b^7)*d^2*e^5 + 30*(B*a^2*b^5 - A*a*b^6)*d*e^6 + 10*(B*a^3*b^4 - 3*A*a^2*b^5)*e
^7)*x^5 + 5*(2*B*a*b^6*d^3*e^4 + 3*(3*B*a^2*b^5 - A*a*b^6)*d^2*e^5 + 4*(2*B*a^3*
b^4 - 3*A*a^2*b^5)*d*e^6 + 2*(B*a^4*b^3 - 3*A*a^3*b^4)*e^7)*x^4 + 5*(4*B*a^2*b^5
*d^3*e^4 + 2*(5*B*a^3*b^4 - 3*A*a^2*b^5)*d^2*e^5 + 6*(B*a^4*b^3 - 2*A*a^3*b^4)*d
*e^6 + (B*a^5*b^2 - 3*A*a^4*b^3)*e^7)*x^3 + (20*B*a^3*b^4*d^3*e^4 + 30*(B*a^4*b^
3 - A*a^3*b^4)*d^2*e^5 + 6*(2*B*a^5*b^2 - 5*A*a^4*b^3)*d*e^6 + (B*a^6*b - 3*A*a^
5*b^2)*e^7)*x^2 + (10*B*a^4*b^3*d^3*e^4 + 3*(3*B*a^5*b^2 - 5*A*a^4*b^3)*d^2*e^5
+ 2*(B*a^6*b - 3*A*a^5*b^2)*d*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)*sqrt(b/(b*d - a*e)))/(b*x + a)) +
 10*(96*B*b^7*d^7 - 16*(59*B*a*b^6 + 9*A*b^7)*d^6*e + 12*(395*B*a^2*b^5 + 124*A*
a*b^6)*d^5*e^2 - 42*(491*B*a^3*b^4 + 187*A*a^2*b^5)*d^4*e^3 - 2*(51487*B*a^4*b^3
 - 17430*A*a^3*b^4)*d^3*e^4 - 3*(20687*B*a^5*b^2 - 45677*A*a^4*b^3)*d^2*e^5 - 15
36*(5*B*a^6*b - 16*A*a^5*b^2)*d*e^6 + 256*(B*a^7 - 3*A*a^6*b)*e^7)*x)/((a^5*b^8*
d^10 - 8*a^6*b^7*d^9*e + 28*a^7*b^6*d^8*e^2 - 56*a^8*b^5*d^7*e^3 + 70*a^9*b^4*d^
6*e^4 - 56*a^10*b^3*d^5*e^5 + 28*a^11*b^2*d^4*e^6 - 8*a^12*b*d^3*e^7 + a^13*d^2*
e^8 + (b^13*d^8*e^2 - 8*a*b^12*d^7*e^3 + 28*a^2*b^11*d^6*e^4 - 56*a^3*b^10*d^5*e
^5 + 70*a^4*b^9*d^4*e^6 - 56*a^5*b^8*d^3*e^7 + 28*a^6*b^7*d^2*e^8 - 8*a^7*b^6*d*
e^9 + a^8*b^5*e^10)*x^7 + (2*b^13*d^9*e - 11*a*b^12*d^8*e^2 + 16*a^2*b^11*d^7*e^
3 + 28*a^3*b^10*d^6*e^4 - 140*a^4*b^9*d^5*e^5 + 238*a^5*b^8*d^4*e^6 - 224*a^6*b^
7*d^3*e^7 + 124*a^7*b^6*d^2*e^8 - 38*a^8*b^5*d*e^9 + 5*a^9*b^4*e^10)*x^6 + (b^13
*d^10 + 2*a*b^12*d^9*e - 42*a^2*b^11*d^8*e^2 + 144*a^3*b^10*d^7*e^3 - 210*a^4*b^
9*d^6*e^4 + 84*a^5*b^8*d^5*e^5 + 168*a^6*b^7*d^4*e^6 - 288*a^7*b^6*d^3*e^7 + 201
*a^8*b^5*d^2*e^8 - 70*a^9*b^4*d*e^9 + 10*a^10*b^3*e^10)*x^5 + 5*(a*b^12*d^10 - 4
*a^2*b^11*d^9*e - 2*a^3*b^10*d^8*e^2 + 40*a^4*b^9*d^7*e^3 - 98*a^5*b^8*d^6*e^4 +
 112*a^6*b^7*d^5*e^5 - 56*a^7*b^6*d^4*e^6 - 8*a^8*b^5*d^3*e^7 + 25*a^9*b^4*d^2*e
^8 - 12*a^10*b^3*d*e^9 + 2*a^11*b^2*e^10)*x^4 + 5*(2*a^2*b^11*d^10 - 12*a^3*b^10
*d^9*e + 25*a^4*b^9*d^8*e^2 - 8*a^5*b^8*d^7*e^3 - 56*a^6*b^7*d^6*e^4 + 112*a^7*b
^6*d^5*e^5 - 98*a^8*b^5*d^4*e^6 + 40*a^9*b^4*d^3*e^7 - 2*a^10*b^3*d^2*e^8 - 4*a^
11*b^2*d*e^9 + a^12*b*e^10)*x^3 + (10*a^3*b^10*d^10 - 70*a^4*b^9*d^9*e + 201*a^5
*b^8*d^8*e^2 - 288*a^6*b^7*d^7*e^3 + 168*a^7*b^6*d^6*e^4 + 84*a^8*b^5*d^5*e^5 -
210*a^9*b^4*d^4*e^6 + 144*a^10*b^3*d^3*e^7 - 42*a^11*b^2*d^2*e^8 + 2*a^12*b*d*e^
9 + a^13*e^10)*x^2 + (5*a^4*b^9*d^10 - 38*a^5*b^8*d^9*e + 124*a^6*b^7*d^8*e^2 -
224*a^7*b^6*d^7*e^3 + 238*a^8*b^5*d^6*e^4 - 140*a^9*b^4*d^5*e^5 + 28*a^10*b^3*d^
4*e^6 + 16*a^11*b^2*d^3*e^7 - 11*a^12*b*d^2*e^8 + 2*a^13*d*e^9)*x)*sqrt(e*x + d)
), -1/1920*(768*A*a^7*e^7 + 96*(B*a*b^6 + 4*A*b^7)*d^7 - 16*(62*B*a^2*b^5 + 213*
A*a*b^6)*d^6*e + 28*(187*B*a^3*b^4 + 498*A*a^2*b^5)*d^5*e^2 - 70*(332*B*a^4*b^3
+ 519*A*a^3*b^4)*d^4*e^3 - (100363*B*a^5*b^2 - 79905*A*a^4*b^3)*d^3*e^4 - 1024*(
16*B*a^6*b - 87*A*a^5*b^2)*d^2*e^5 + 512*(B*a^7 - 18*A*a^6*b)*d*e^6 - 45045*(2*B
*b^7*d*e^6 + (B*a*b^6 - 3*A*b^7)*e^7)*x^7 - 105105*(2*B*b^7*d^2*e^5 + (5*B*a*b^6
 - 3*A*b^7)*d*e^6 + 2*(B*a^2*b^5 - 3*A*a*b^6)*e^7)*x^6 - 3003*(46*B*b^7*d^3*e^4
+ 3*(117*B*a*b^6 - 23*A*b^7)*d^2*e^5 + 12*(35*B*a^2*b^5 - 41*A*a*b^6)*d*e^6 + 12
8*(B*a^3*b^4 - 3*A*a^2*b^5)*e^7)*x^5 - 2145*(6*B*b^7*d^4*e^3 + (307*B*a*b^6 - 9*
A*b^7)*d^3*e^4 + 12*(83*B*a^2*b^5 - 38*A*a*b^6)*d^2*e^5 + 6*(123*B*a^3*b^4 - 211
*A*a^2*b^5)*d*e^6 + 158*(B*a^4*b^3 - 3*A*a^3*b^4)*e^7)*x^4 + 715*(4*B*b^7*d^5*e^
2 - 2*(43*B*a*b^6 + 3*A*b^7)*d^4*e^3 - 4*(434*B*a^2*b^5 - 33*A*a*b^6)*d^3*e^4 -
2*(1547*B*a^3*b^4 - 1269*A*a^2*b^5)*d^2*e^5 - 2*(755*B*a^4*b^3 - 1686*A*a^3*b^4)
*d*e^6 - 193*(B*a^5*b^2 - 3*A*a^4*b^3)*e^7)*x^3 - 65*(16*B*b^7*d^6*e - 12*(17*B*
a*b^6 + 2*A*b^7)*d^5*e^2 + 6*(295*B*a^2*b^5 + 53*A*a*b^6)*d^4*e^3 + 2*(8837*B*a^
3*b^4 - 1407*A*a^2*b^5)*d^3*e^4 + 6*(3091*B*a^4*b^3 - 4184*A*a^3*b^4)*d^2*e^5 +
3*(1867*B*a^5*b^2 - 5089*A*a^4*b^3)*d*e^6 + 256*(B*a^6*b - 3*A*a^5*b^2)*e^7)*x^2
 + 45045*(2*B*a^5*b^2*d^3*e^4 + (B*a^6*b - 3*A*a^5*b^2)*d^2*e^5 + (2*B*b^7*d*e^6
 + (B*a*b^6 - 3*A*b^7)*e^7)*x^7 + (4*B*b^7*d^2*e^5 + 6*(2*B*a*b^6 - A*b^7)*d*e^6
 + 5*(B*a^2*b^5 - 3*A*a*b^6)*e^7)*x^6 + (2*B*b^7*d^3*e^4 + 3*(7*B*a*b^6 - A*b^7)
*d^2*e^5 + 30*(B*a^2*b^5 - A*a*b^6)*d*e^6 + 10*(B*a^3*b^4 - 3*A*a^2*b^5)*e^7)*x^
5 + 5*(2*B*a*b^6*d^3*e^4 + 3*(3*B*a^2*b^5 - A*a*b^6)*d^2*e^5 + 4*(2*B*a^3*b^4 -
3*A*a^2*b^5)*d*e^6 + 2*(B*a^4*b^3 - 3*A*a^3*b^4)*e^7)*x^4 + 5*(4*B*a^2*b^5*d^3*e
^4 + 2*(5*B*a^3*b^4 - 3*A*a^2*b^5)*d^2*e^5 + 6*(B*a^4*b^3 - 2*A*a^3*b^4)*d*e^6 +
 (B*a^5*b^2 - 3*A*a^4*b^3)*e^7)*x^3 + (20*B*a^3*b^4*d^3*e^4 + 30*(B*a^4*b^3 - A*
a^3*b^4)*d^2*e^5 + 6*(2*B*a^5*b^2 - 5*A*a^4*b^3)*d*e^6 + (B*a^6*b - 3*A*a^5*b^2)
*e^7)*x^2 + (10*B*a^4*b^3*d^3*e^4 + 3*(3*B*a^5*b^2 - 5*A*a^4*b^3)*d^2*e^5 + 2*(B
*a^6*b - 3*A*a^5*b^2)*d*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)) + 5*(96*B*b^7*d^7 - 16*(59*B*a*b^
6 + 9*A*b^7)*d^6*e + 12*(395*B*a^2*b^5 + 124*A*a*b^6)*d^5*e^2 - 42*(491*B*a^3*b^
4 + 187*A*a^2*b^5)*d^4*e^3 - 2*(51487*B*a^4*b^3 - 17430*A*a^3*b^4)*d^3*e^4 - 3*(
20687*B*a^5*b^2 - 45677*A*a^4*b^3)*d^2*e^5 - 1536*(5*B*a^6*b - 16*A*a^5*b^2)*d*e
^6 + 256*(B*a^7 - 3*A*a^6*b)*e^7)*x)/((a^5*b^8*d^10 - 8*a^6*b^7*d^9*e + 28*a^7*b
^6*d^8*e^2 - 56*a^8*b^5*d^7*e^3 + 70*a^9*b^4*d^6*e^4 - 56*a^10*b^3*d^5*e^5 + 28*
a^11*b^2*d^4*e^6 - 8*a^12*b*d^3*e^7 + a^13*d^2*e^8 + (b^13*d^8*e^2 - 8*a*b^12*d^
7*e^3 + 28*a^2*b^11*d^6*e^4 - 56*a^3*b^10*d^5*e^5 + 70*a^4*b^9*d^4*e^6 - 56*a^5*
b^8*d^3*e^7 + 28*a^6*b^7*d^2*e^8 - 8*a^7*b^6*d*e^9 + a^8*b^5*e^10)*x^7 + (2*b^13
*d^9*e - 11*a*b^12*d^8*e^2 + 16*a^2*b^11*d^7*e^3 + 28*a^3*b^10*d^6*e^4 - 140*a^4
*b^9*d^5*e^5 + 238*a^5*b^8*d^4*e^6 - 224*a^6*b^7*d^3*e^7 + 124*a^7*b^6*d^2*e^8 -
 38*a^8*b^5*d*e^9 + 5*a^9*b^4*e^10)*x^6 + (b^13*d^10 + 2*a*b^12*d^9*e - 42*a^2*b
^11*d^8*e^2 + 144*a^3*b^10*d^7*e^3 - 210*a^4*b^9*d^6*e^4 + 84*a^5*b^8*d^5*e^5 +
168*a^6*b^7*d^4*e^6 - 288*a^7*b^6*d^3*e^7 + 201*a^8*b^5*d^2*e^8 - 70*a^9*b^4*d*e
^9 + 10*a^10*b^3*e^10)*x^5 + 5*(a*b^12*d^10 - 4*a^2*b^11*d^9*e - 2*a^3*b^10*d^8*
e^2 + 40*a^4*b^9*d^7*e^3 - 98*a^5*b^8*d^6*e^4 + 112*a^6*b^7*d^5*e^5 - 56*a^7*b^6
*d^4*e^6 - 8*a^8*b^5*d^3*e^7 + 25*a^9*b^4*d^2*e^8 - 12*a^10*b^3*d*e^9 + 2*a^11*b
^2*e^10)*x^4 + 5*(2*a^2*b^11*d^10 - 12*a^3*b^10*d^9*e + 25*a^4*b^9*d^8*e^2 - 8*a
^5*b^8*d^7*e^3 - 56*a^6*b^7*d^6*e^4 + 112*a^7*b^6*d^5*e^5 - 98*a^8*b^5*d^4*e^6 +
 40*a^9*b^4*d^3*e^7 - 2*a^10*b^3*d^2*e^8 - 4*a^11*b^2*d*e^9 + a^12*b*e^10)*x^3 +
 (10*a^3*b^10*d^10 - 70*a^4*b^9*d^9*e + 201*a^5*b^8*d^8*e^2 - 288*a^6*b^7*d^7*e^
3 + 168*a^7*b^6*d^6*e^4 + 84*a^8*b^5*d^5*e^5 - 210*a^9*b^4*d^4*e^6 + 144*a^10*b^
3*d^3*e^7 - 42*a^11*b^2*d^2*e^8 + 2*a^12*b*d*e^9 + a^13*e^10)*x^2 + (5*a^4*b^9*d
^10 - 38*a^5*b^8*d^9*e + 124*a^6*b^7*d^8*e^2 - 224*a^7*b^6*d^7*e^3 + 238*a^8*b^5
*d^6*e^4 - 140*a^9*b^4*d^5*e^5 + 28*a^10*b^3*d^4*e^6 + 16*a^11*b^2*d^3*e^7 - 11*
a^12*b*d^2*e^8 + 2*a^13*d*e^9)*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)**3,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.328036, 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)^(7/2)),x, algorithm="giac")

[Out]

Done

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