3.1882 \(\int \frac{A+B x}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=496 \[ -\frac{105 e^3 (a+b x) (3 a B e-11 A b e+8 b B d)}{64 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^6}-\frac{35 e^3 (a+b x) (3 a B e-11 A b e+8 b B d)}{64 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^5}+\frac{105 \sqrt{b} e^3 (a+b x) (3 a B e-11 A b e+8 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{13/2}}-\frac{21 e^2 (3 a B e-11 A b e+8 b B d)}{64 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^4}+\frac{3 e (3 a B e-11 A b e+8 b B d)}{32 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^3}-\frac{3 a B e-11 A b e+8 b B d}{24 b (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^2}-\frac{A b-a B}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)} \]

[Out]

(-21*e^2*(8*b*B*d - 11*A*b*e + 3*a*B*e))/(64*b*(b*d - a*e)^4*(d + e*x)^(3/2)*Sqr
t[a^2 + 2*a*b*x + b^2*x^2]) - (A*b - a*B)/(4*b*(b*d - a*e)*(a + b*x)^3*(d + e*x)
^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (8*b*B*d - 11*A*b*e + 3*a*B*e)/(24*b*(b*
d - a*e)^2*(a + b*x)^2*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (3*e*(8*
b*B*d - 11*A*b*e + 3*a*B*e))/(32*b*(b*d - a*e)^3*(a + b*x)*(d + e*x)^(3/2)*Sqrt[
a^2 + 2*a*b*x + b^2*x^2]) - (35*e^3*(8*b*B*d - 11*A*b*e + 3*a*B*e)*(a + b*x))/(6
4*b*(b*d - a*e)^5*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (105*e^3*(8*b
*B*d - 11*A*b*e + 3*a*B*e)*(a + b*x))/(64*(b*d - a*e)^6*Sqrt[d + e*x]*Sqrt[a^2 +
2*a*b*x + b^2*x^2]) + (105*Sqrt[b]*e^3*(8*b*B*d - 11*A*b*e + 3*a*B*e)*(a + b*x)
*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(64*(b*d - a*e)^(13/2)*Sqrt[a
^2 + 2*a*b*x + b^2*x^2])

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Rubi [A]  time = 1.31278, antiderivative size = 496, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171 \[ -\frac{105 e^3 (a+b x) (3 a B e-11 A b e+8 b B d)}{64 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^6}-\frac{35 e^3 (a+b x) (3 a B e-11 A b e+8 b B d)}{64 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^5}+\frac{105 \sqrt{b} e^3 (a+b x) (3 a B e-11 A b e+8 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{13/2}}-\frac{21 e^2 (3 a B e-11 A b e+8 b B d)}{64 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^4}+\frac{3 e (3 a B e-11 A b e+8 b B d)}{32 b (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^3}-\frac{3 a B e-11 A b e+8 b B d}{24 b (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^2}-\frac{A b-a B}{4 b (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)} \]

Antiderivative was successfully verified.

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

[Out]

(-21*e^2*(8*b*B*d - 11*A*b*e + 3*a*B*e))/(64*b*(b*d - a*e)^4*(d + e*x)^(3/2)*Sqr
t[a^2 + 2*a*b*x + b^2*x^2]) - (A*b - a*B)/(4*b*(b*d - a*e)*(a + b*x)^3*(d + e*x)
^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (8*b*B*d - 11*A*b*e + 3*a*B*e)/(24*b*(b*
d - a*e)^2*(a + b*x)^2*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (3*e*(8*
b*B*d - 11*A*b*e + 3*a*B*e))/(32*b*(b*d - a*e)^3*(a + b*x)*(d + e*x)^(3/2)*Sqrt[
a^2 + 2*a*b*x + b^2*x^2]) - (35*e^3*(8*b*B*d - 11*A*b*e + 3*a*B*e)*(a + b*x))/(6
4*b*(b*d - a*e)^5*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (105*e^3*(8*b
*B*d - 11*A*b*e + 3*a*B*e)*(a + b*x))/(64*(b*d - a*e)^6*Sqrt[d + e*x]*Sqrt[a^2 +
2*a*b*x + b^2*x^2]) + (105*Sqrt[b]*e^3*(8*b*B*d - 11*A*b*e + 3*a*B*e)*(a + b*x)
*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(64*(b*d - a*e)^(13/2)*Sqrt[a
^2 + 2*a*b*x + b^2*x^2])

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Rubi in Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RecursionError} \]

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)**(5/2),x)

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 3.05034, size = 295, normalized size = 0.59 \[ \frac{(a+b x) \left (\frac{105 \sqrt{b} e^3 (3 a B e-11 A b e+8 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{(b d-a e)^{13/2}}+\frac{\sqrt{d+e x} \left (\frac{384 e^3 (-a B e+5 A b e-4 b B d)}{d+e x}+\frac{128 e^3 (b d-a e) (A e-B d)}{(d+e x)^2}+\frac{3 b e^2 (-187 a B e+515 A b e-328 b B d)}{a+b x}-\frac{2 b e (b d-a e) (-123 a B e+259 A b e-136 b B d)}{(a+b x)^2}-\frac{8 b (b d-a e)^2 (15 a B e-23 A b e+8 b B d)}{(a+b x)^3}-\frac{48 b (A b-a B) (b d-a e)^3}{(a+b x)^4}\right )}{3 (b d-a e)^6}\right )}{64 \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x)*((Sqrt[d + e*x]*((-48*b*(A*b - a*B)*(b*d - a*e)^3)/(a + b*x)^4 - (8*b
*(b*d - a*e)^2*(8*b*B*d - 23*A*b*e + 15*a*B*e))/(a + b*x)^3 - (2*b*e*(b*d - a*e)
*(-136*b*B*d + 259*A*b*e - 123*a*B*e))/(a + b*x)^2 + (3*b*e^2*(-328*b*B*d + 515*
A*b*e - 187*a*B*e))/(a + b*x) + (128*e^3*(b*d - a*e)*(-(B*d) + A*e))/(d + e*x)^2
+ (384*e^3*(-4*b*B*d + 5*A*b*e - a*B*e))/(d + e*x)))/(3*(b*d - a*e)^6) + (105*S
qrt[b]*e^3*(8*b*B*d - 11*A*b*e + 3*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b
*d - a*e]])/(b*d - a*e)^(13/2)))/(64*Sqrt[(a + b*x)^2])

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Maple [B]  time = 0.055, size = 1860, normalized size = 3.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/192*(-945*B*(b*(a*e-b*d))^(1/2)*x^5*a*b^4*e^5-2520*B*(b*(a*e-b*d))^(1/2)*x^5*b
^5*d*e^4+12705*A*(b*(a*e-b*d))^(1/2)*x^4*a*b^4*e^5+4620*A*(b*(a*e-b*d))^(1/2)*x^
4*b^5*d*e^4-3465*B*(b*(a*e-b*d))^(1/2)*x^4*a^2*b^3*e^5-3360*B*(b*(a*e-b*d))^(1/2
)*x^4*b^5*d^2*e^3+16863*A*(b*(a*e-b*d))^(1/2)*x^3*a^2*b^3*e^5+693*A*(b*(a*e-b*d)
)^(1/2)*x^3*b^5*d^2*e^3+3465*A*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+
d)^(3/2)*a^4*b^2*e^4-4599*B*(b*(a*e-b*d))^(1/2)*x^3*a^3*b^2*e^5-504*B*(b*(a*e-b*
d))^(1/2)*x^3*b^5*d^3*e^2-945*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x
+d)^(3/2)*a^5*b*e^4+9207*A*(b*(a*e-b*d))^(1/2)*x^2*a^3*b^2*e^5-198*A*(b*(a*e-b*d
))^(1/2)*x^2*b^5*d^3*e^2-2511*B*(b*(a*e-b*d))^(1/2)*x^2*a^4*b*e^5+144*B*(b*(a*e-
b*d))^(1/2)*x^2*b^5*d^4*e+1408*A*(b*(a*e-b*d))^(1/2)*x*a^4*b*e^5+88*A*(b*(a*e-b*
d))^(1/2)*x*b^5*d^4*e-128*A*(b*(a*e-b*d))^(1/2)*a^5*e^5-48*A*(b*(a*e-b*d))^(1/2)
*b^5*d^5+136*B*(b*(a*e-b*d))^(1/2)*a^2*b^3*d^4*e-12960*B*(b*(a*e-b*d))^(1/2)*x^2
*a^3*b^2*d*e^4-17433*B*(b*(a*e-b*d))^(1/2)*x^2*a^2*b^3*d^2*e^3-1890*B*(b*(a*e-b*
d))^(1/2)*x^2*a*b^4*d^3*e^2+12782*A*(b*(a*e-b*d))^(1/2)*x*a^3*b^2*d*e^4+3795*A*(
b*(a*e-b*d))^(1/2)*x*a^2*b^3*d^2*e^3-748*A*(b*(a*e-b*d))^(1/2)*x*a*b^4*d^3*e^2-9
45*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+d)^(3/2)*x^4*a*b^5*e^4-252
0*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+d)^(3/2)*x^4*b^6*d*e^3+1386
0*A*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+d)^(3/2)*x^3*a*b^5*e^4-3780
*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+d)^(3/2)*x^3*a^2*b^4*e^4+207
90*A*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+d)^(3/2)*x^2*a^2*b^4*e^4-5
670*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+d)^(3/2)*x^2*a^3*b^3*e^4+
13860*A*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+d)^(3/2)*x*a^3*b^3*e^4-
10500*B*(b*(a*e-b*d))^(1/2)*x^4*a*b^4*d*e^4-3780*B*arctan((e*x+d)^(1/2)*b/(b*(a*
e-b*d))^(1/2))*(e*x+d)^(3/2)*x*a^4*b^2*e^4+17094*A*(b*(a*e-b*d))^(1/2)*x^3*a*b^4
*d*e^4-16926*B*(b*(a*e-b*d))^(1/2)*x^3*a^2*b^3*d*e^4-12621*B*(b*(a*e-b*d))^(1/2)
*x^3*a*b^4*d^2*e^3-2520*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+d)^(3
/2)*a^4*b^2*d*e^3+22968*A*(b*(a*e-b*d))^(1/2)*x^2*a^2*b^3*d*e^4+2673*A*(b*(a*e-b
*d))^(1/2)*x^2*a*b^4*d^2*e^3-4510*B*(b*(a*e-b*d))^(1/2)*x*a^4*b*d*e^4-10331*B*(b
*(a*e-b*d))^(1/2)*x*a^3*b^2*d^2*e^3-2556*B*(b*(a*e-b*d))^(1/2)*x*a^2*b^3*d^3*e^2
+520*B*(b*(a*e-b*d))^(1/2)*x*a*b^4*d^4*e-256*B*(b*(a*e-b*d))^(1/2)*a^5*d*e^4-16*
B*(b*(a*e-b*d))^(1/2)*a*b^4*d^5-15120*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/
2))*(e*x+d)^(3/2)*x^2*a^2*b^4*d*e^3-10080*B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))
^(1/2))*(e*x+d)^(3/2)*x*a^3*b^3*d*e^3+3465*A*(b*(a*e-b*d))^(1/2)*x^5*b^5*e^5-384
*B*(b*(a*e-b*d))^(1/2)*x*a^5*e^5-64*B*(b*(a*e-b*d))^(1/2)*x*b^5*d^5+2048*A*(b*(a
*e-b*d))^(1/2)*a^4*b*d*e^4+2295*A*(b*(a*e-b*d))^(1/2)*a^3*b^2*d^2*e^3-1030*A*(b*
(a*e-b*d))^(1/2)*a^2*b^3*d^3*e^2+328*A*(b*(a*e-b*d))^(1/2)*a*b^4*d^4*e-2639*B*(b
*(a*e-b*d))^(1/2)*a^4*b*d^2*e^3-690*B*(b*(a*e-b*d))^(1/2)*a^3*b^2*d^3*e^2-10080*
B*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+d)^(3/2)*x^3*a*b^5*d*e^3+3465
*A*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+d)^(3/2)*x^4*b^6*e^4)*(b*x+a
)/(b*(a*e-b*d))^(1/2)/(e*x+d)^(3/2)/(a*e-b*d)^6/((b*x+a)^2)^(5/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.349212, 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)^(5/2)*(e*x + d)^(5/2)),x, algorithm="fricas")

[Out]

[-1/384*(256*A*a^5*e^5 + 32*(B*a*b^4 + 3*A*b^5)*d^5 - 16*(17*B*a^2*b^3 + 41*A*a*
b^4)*d^4*e + 20*(69*B*a^3*b^2 + 103*A*a^2*b^3)*d^3*e^2 + 2*(2639*B*a^4*b - 2295*
A*a^3*b^2)*d^2*e^3 + 512*(B*a^5 - 8*A*a^4*b)*d*e^4 + 630*(8*B*b^5*d*e^4 + (3*B*a
*b^4 - 11*A*b^5)*e^5)*x^5 + 210*(32*B*b^5*d^2*e^3 + 4*(25*B*a*b^4 - 11*A*b^5)*d*
e^4 + 11*(3*B*a^2*b^3 - 11*A*a*b^4)*e^5)*x^4 + 42*(24*B*b^5*d^3*e^2 + (601*B*a*b
^4 - 33*A*b^5)*d^2*e^3 + 2*(403*B*a^2*b^3 - 407*A*a*b^4)*d*e^4 + 73*(3*B*a^3*b^2
- 11*A*a^2*b^3)*e^5)*x^3 - 18*(16*B*b^5*d^4*e - 2*(105*B*a*b^4 + 11*A*b^5)*d^3*
e^2 - (1937*B*a^2*b^3 - 297*A*a*b^4)*d^2*e^3 - 8*(180*B*a^3*b^2 - 319*A*a^2*b^3)
*d*e^4 - 93*(3*B*a^4*b - 11*A*a^3*b^2)*e^5)*x^2 + 315*(8*B*a^4*b*d^2*e^3 + (3*B*
a^5 - 11*A*a^4*b)*d*e^4 + (8*B*b^5*d*e^4 + (3*B*a*b^4 - 11*A*b^5)*e^5)*x^5 + (8*
B*b^5*d^2*e^3 + (35*B*a*b^4 - 11*A*b^5)*d*e^4 + 4*(3*B*a^2*b^3 - 11*A*a*b^4)*e^5
)*x^4 + 2*(16*B*a*b^4*d^2*e^3 + 2*(15*B*a^2*b^3 - 11*A*a*b^4)*d*e^4 + 3*(3*B*a^3
*b^2 - 11*A*a^2*b^3)*e^5)*x^3 + 2*(24*B*a^2*b^3*d^2*e^3 + (25*B*a^3*b^2 - 33*A*a
^2*b^3)*d*e^4 + 2*(3*B*a^4*b - 11*A*a^3*b^2)*e^5)*x^2 + (32*B*a^3*b^2*d^2*e^3 +
4*(5*B*a^4*b - 11*A*a^3*b^2)*d*e^4 + (3*B*a^5 - 11*A*a^4*b)*e^5)*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)*sqr
t(b/(b*d - a*e)))/(b*x + a)) + 2*(64*B*b^5*d^5 - 8*(65*B*a*b^4 + 11*A*b^5)*d^4*e
+ 4*(639*B*a^2*b^3 + 187*A*a*b^4)*d^3*e^2 + (10331*B*a^3*b^2 - 3795*A*a^2*b^3)*
d^2*e^3 + 22*(205*B*a^4*b - 581*A*a^3*b^2)*d*e^4 + 128*(3*B*a^5 - 11*A*a^4*b)*e^
5)*x)/((a^4*b^6*d^7 - 6*a^5*b^5*d^6*e + 15*a^6*b^4*d^5*e^2 - 20*a^7*b^3*d^4*e^3
+ 15*a^8*b^2*d^3*e^4 - 6*a^9*b*d^2*e^5 + a^10*d*e^6 + (b^10*d^6*e - 6*a*b^9*d^5*
e^2 + 15*a^2*b^8*d^4*e^3 - 20*a^3*b^7*d^3*e^4 + 15*a^4*b^6*d^2*e^5 - 6*a^5*b^5*d
*e^6 + a^6*b^4*e^7)*x^5 + (b^10*d^7 - 2*a*b^9*d^6*e - 9*a^2*b^8*d^5*e^2 + 40*a^3
*b^7*d^4*e^3 - 65*a^4*b^6*d^3*e^4 + 54*a^5*b^5*d^2*e^5 - 23*a^6*b^4*d*e^6 + 4*a^
7*b^3*e^7)*x^4 + 2*(2*a*b^9*d^7 - 9*a^2*b^8*d^6*e + 12*a^3*b^7*d^5*e^2 + 5*a^4*b
^6*d^4*e^3 - 30*a^5*b^5*d^3*e^4 + 33*a^6*b^4*d^2*e^5 - 16*a^7*b^3*d*e^6 + 3*a^8*
b^2*e^7)*x^3 + 2*(3*a^2*b^8*d^7 - 16*a^3*b^7*d^6*e + 33*a^4*b^6*d^5*e^2 - 30*a^5
*b^5*d^4*e^3 + 5*a^6*b^4*d^3*e^4 + 12*a^7*b^3*d^2*e^5 - 9*a^8*b^2*d*e^6 + 2*a^9*
b*e^7)*x^2 + (4*a^3*b^7*d^7 - 23*a^4*b^6*d^6*e + 54*a^5*b^5*d^5*e^2 - 65*a^6*b^4
*d^4*e^3 + 40*a^7*b^3*d^3*e^4 - 9*a^8*b^2*d^2*e^5 - 2*a^9*b*d*e^6 + a^10*e^7)*x)
*sqrt(e*x + d)), -1/192*(128*A*a^5*e^5 + 16*(B*a*b^4 + 3*A*b^5)*d^5 - 8*(17*B*a^
2*b^3 + 41*A*a*b^4)*d^4*e + 10*(69*B*a^3*b^2 + 103*A*a^2*b^3)*d^3*e^2 + (2639*B*
a^4*b - 2295*A*a^3*b^2)*d^2*e^3 + 256*(B*a^5 - 8*A*a^4*b)*d*e^4 + 315*(8*B*b^5*d
*e^4 + (3*B*a*b^4 - 11*A*b^5)*e^5)*x^5 + 105*(32*B*b^5*d^2*e^3 + 4*(25*B*a*b^4 -
11*A*b^5)*d*e^4 + 11*(3*B*a^2*b^3 - 11*A*a*b^4)*e^5)*x^4 + 21*(24*B*b^5*d^3*e^2
+ (601*B*a*b^4 - 33*A*b^5)*d^2*e^3 + 2*(403*B*a^2*b^3 - 407*A*a*b^4)*d*e^4 + 73
*(3*B*a^3*b^2 - 11*A*a^2*b^3)*e^5)*x^3 - 9*(16*B*b^5*d^4*e - 2*(105*B*a*b^4 + 11
*A*b^5)*d^3*e^2 - (1937*B*a^2*b^3 - 297*A*a*b^4)*d^2*e^3 - 8*(180*B*a^3*b^2 - 31
9*A*a^2*b^3)*d*e^4 - 93*(3*B*a^4*b - 11*A*a^3*b^2)*e^5)*x^2 - 315*(8*B*a^4*b*d^2
*e^3 + (3*B*a^5 - 11*A*a^4*b)*d*e^4 + (8*B*b^5*d*e^4 + (3*B*a*b^4 - 11*A*b^5)*e^
5)*x^5 + (8*B*b^5*d^2*e^3 + (35*B*a*b^4 - 11*A*b^5)*d*e^4 + 4*(3*B*a^2*b^3 - 11*
A*a*b^4)*e^5)*x^4 + 2*(16*B*a*b^4*d^2*e^3 + 2*(15*B*a^2*b^3 - 11*A*a*b^4)*d*e^4
+ 3*(3*B*a^3*b^2 - 11*A*a^2*b^3)*e^5)*x^3 + 2*(24*B*a^2*b^3*d^2*e^3 + (25*B*a^3*
b^2 - 33*A*a^2*b^3)*d*e^4 + 2*(3*B*a^4*b - 11*A*a^3*b^2)*e^5)*x^2 + (32*B*a^3*b^
2*d^2*e^3 + 4*(5*B*a^4*b - 11*A*a^3*b^2)*d*e^4 + (3*B*a^5 - 11*A*a^4*b)*e^5)*x)*
sqrt(e*x + d)*sqrt(-b/(b*d - a*e))*arctan(-(b*d - a*e)*sqrt(-b/(b*d - a*e))/(sqr
t(e*x + d)*b)) + (64*B*b^5*d^5 - 8*(65*B*a*b^4 + 11*A*b^5)*d^4*e + 4*(639*B*a^2*
b^3 + 187*A*a*b^4)*d^3*e^2 + (10331*B*a^3*b^2 - 3795*A*a^2*b^3)*d^2*e^3 + 22*(20
5*B*a^4*b - 581*A*a^3*b^2)*d*e^4 + 128*(3*B*a^5 - 11*A*a^4*b)*e^5)*x)/((a^4*b^6*
d^7 - 6*a^5*b^5*d^6*e + 15*a^6*b^4*d^5*e^2 - 20*a^7*b^3*d^4*e^3 + 15*a^8*b^2*d^3
*e^4 - 6*a^9*b*d^2*e^5 + a^10*d*e^6 + (b^10*d^6*e - 6*a*b^9*d^5*e^2 + 15*a^2*b^8
*d^4*e^3 - 20*a^3*b^7*d^3*e^4 + 15*a^4*b^6*d^2*e^5 - 6*a^5*b^5*d*e^6 + a^6*b^4*e
^7)*x^5 + (b^10*d^7 - 2*a*b^9*d^6*e - 9*a^2*b^8*d^5*e^2 + 40*a^3*b^7*d^4*e^3 - 6
5*a^4*b^6*d^3*e^4 + 54*a^5*b^5*d^2*e^5 - 23*a^6*b^4*d*e^6 + 4*a^7*b^3*e^7)*x^4 +
2*(2*a*b^9*d^7 - 9*a^2*b^8*d^6*e + 12*a^3*b^7*d^5*e^2 + 5*a^4*b^6*d^4*e^3 - 30*
a^5*b^5*d^3*e^4 + 33*a^6*b^4*d^2*e^5 - 16*a^7*b^3*d*e^6 + 3*a^8*b^2*e^7)*x^3 + 2
*(3*a^2*b^8*d^7 - 16*a^3*b^7*d^6*e + 33*a^4*b^6*d^5*e^2 - 30*a^5*b^5*d^4*e^3 + 5
*a^6*b^4*d^3*e^4 + 12*a^7*b^3*d^2*e^5 - 9*a^8*b^2*d*e^6 + 2*a^9*b*e^7)*x^2 + (4*
a^3*b^7*d^7 - 23*a^4*b^6*d^6*e + 54*a^5*b^5*d^5*e^2 - 65*a^6*b^4*d^4*e^3 + 40*a^
7*b^3*d^3*e^4 - 9*a^8*b^2*d^2*e^5 - 2*a^9*b*d*e^6 + a^10*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)**(5/2)/(b**2*x**2+2*a*b*x+a**2)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.396007, 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)^(5/2)*(e*x + d)^(5/2)),x, algorithm="giac")

[Out]

Done