3.1718 \(\int \frac{1}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=381 \[ \frac{1155 b e^4 (a+b x)}{64 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^6}+\frac{385 e^4 (a+b x)}{64 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^5}+\frac{231 e^3}{64 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^4}-\frac{33 e^2}{32 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^3}+\frac{11 e}{24 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^2}-\frac{1}{4 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)}-\frac{1155 b^{3/2} e^4 (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{13/2}} \]

[Out]

(231*e^3)/(64*(b*d - a*e)^4*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - 1/(
4*(b*d - a*e)*(a + b*x)^3*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (11*e
)/(24*(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]) -
(33*e^2)/(32*(b*d - a*e)^3*(a + b*x)*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x
^2]) + (385*e^4*(a + b*x))/(64*(b*d - a*e)^5*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2]) + (1155*b*e^4*(a + b*x))/(64*(b*d - a*e)^6*Sqrt[d + e*x]*Sqrt[a^2 +
2*a*b*x + b^2*x^2]) - (1155*b^(3/2)*e^4*(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 = 0.79807, antiderivative size = 381, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{1155 b e^4 (a+b x)}{64 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^6}+\frac{385 e^4 (a+b x)}{64 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^5}+\frac{231 e^3}{64 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^4}-\frac{33 e^2}{32 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^3}+\frac{11 e}{24 (a+b x)^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^2}-\frac{1}{4 (a+b x)^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)}-\frac{1155 b^{3/2} e^4 (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{64 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{13/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]

[Out]

(231*e^3)/(64*(b*d - a*e)^4*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - 1/(
4*(b*d - a*e)*(a + b*x)^3*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (11*e
)/(24*(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]) -
(33*e^2)/(32*(b*d - a*e)^3*(a + b*x)*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x
^2]) + (385*e^4*(a + b*x))/(64*(b*d - a*e)^5*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2]) + (1155*b*e^4*(a + b*x))/(64*(b*d - a*e)^6*Sqrt[d + e*x]*Sqrt[a^2 +
2*a*b*x + b^2*x^2]) - (1155*b^(3/2)*e^4*(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(1/(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 = 1.45935, size = 213, normalized size = 0.56 \[ \frac{(a+b x)^5 \left (\frac{\sqrt{d+e x} \left (-\frac{518 b^2 e^2 (b d-a e)}{(a+b x)^2}+\frac{184 b^2 e (b d-a e)^2}{(a+b x)^3}-\frac{48 b^2 (b d-a e)^3}{(a+b x)^4}+\frac{1545 b^2 e^3}{a+b x}+\frac{128 e^4 (b d-a e)}{(d+e x)^2}+\frac{1920 b e^4}{d+e x}\right )}{3 (b d-a e)^6}-\frac{1155 b^{3/2} e^4 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{(b d-a e)^{13/2}}\right )}{64 \left ((a+b x)^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2)),x]

[Out]

((a + b*x)^5*((Sqrt[d + e*x]*((-48*b^2*(b*d - a*e)^3)/(a + b*x)^4 + (184*b^2*e*(
b*d - a*e)^2)/(a + b*x)^3 - (518*b^2*e^2*(b*d - a*e))/(a + b*x)^2 + (1545*b^2*e^
3)/(a + b*x) + (128*e^4*(b*d - a*e))/(d + e*x)^2 + (1920*b*e^4)/(d + e*x)))/(3*(
b*d - a*e)^6) - (1155*b^(3/2)*e^4*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e
]])/(b*d - a*e)^(13/2)))/(64*((a + b*x)^2)^(5/2))

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Maple [B]  time = 0.039, size = 763, normalized size = 2. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^(5/2)/(b^2*x^2+2*a*b*x+a^2)^(5/2),x)

[Out]

1/192*(3795*(b*(a*e-b*d))^(1/2)*x*a^2*b^3*d^2*e^3+2295*(b*(a*e-b*d))^(1/2)*a^3*b
^2*d^2*e^3-1030*(b*(a*e-b*d))^(1/2)*a^2*b^3*d^3*e^2+328*(b*(a*e-b*d))^(1/2)*a*b^
4*d^4*e-128*(b*(a*e-b*d))^(1/2)*a^5*e^5-48*(b*(a*e-b*d))^(1/2)*b^5*d^5+13860*arc
tan((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+20790*arcta
n((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+13860*arcta
n((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+d)^(3/2)*x*a^3*b^3*e^4-748*(b*(a*e-b
*d))^(1/2)*x*a*b^4*d^3*e^2+17094*(b*(a*e-b*d))^(1/2)*x^3*a*b^4*d*e^4+22968*(b*(a
*e-b*d))^(1/2)*x^2*a^2*b^3*d*e^4+12782*(b*(a*e-b*d))^(1/2)*x*a^3*b^2*d*e^4+3465*
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+12705*(b*(
a*e-b*d))^(1/2)*x^4*a*b^4*e^5+693*(b*(a*e-b*d))^(1/2)*x^3*b^5*d^2*e^3-198*(b*(a*
e-b*d))^(1/2)*x^2*b^5*d^3*e^2+88*(b*(a*e-b*d))^(1/2)*x*b^5*d^4*e+4620*(b*(a*e-b*
d))^(1/2)*x^4*b^5*d*e^4+16863*(b*(a*e-b*d))^(1/2)*x^3*a^2*b^3*e^5+3465*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+9207*(b*(a*e-b*d))^
(1/2)*x^2*a^3*b^2*e^5+1408*(b*(a*e-b*d))^(1/2)*x*a^4*b*e^5+3465*(b*(a*e-b*d))^(1
/2)*x^5*b^5*e^5+2673*(b*(a*e-b*d))^(1/2)*x^2*a*b^4*d^2*e^3+2048*(b*(a*e-b*d))^(1
/2)*a^4*b*d*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(1/((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 = 0.260535, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(e*x + d)^(5/2)),x, algorithm="fricas")

[Out]

[1/384*(6930*b^5*e^5*x^5 - 96*b^5*d^5 + 656*a*b^4*d^4*e - 2060*a^2*b^3*d^3*e^2 +
4590*a^3*b^2*d^2*e^3 + 4096*a^4*b*d*e^4 - 256*a^5*e^5 + 2310*(4*b^5*d*e^4 + 11*
a*b^4*e^5)*x^4 + 462*(3*b^5*d^2*e^3 + 74*a*b^4*d*e^4 + 73*a^2*b^3*e^5)*x^3 - 198
*(2*b^5*d^3*e^2 - 27*a*b^4*d^2*e^3 - 232*a^2*b^3*d*e^4 - 93*a^3*b^2*e^5)*x^2 + 3
465*(b^5*e^5*x^5 + a^4*b*d*e^4 + (b^5*d*e^4 + 4*a*b^4*e^5)*x^4 + 2*(2*a*b^4*d*e^
4 + 3*a^2*b^3*e^5)*x^3 + 2*(3*a^2*b^3*d*e^4 + 2*a^3*b^2*e^5)*x^2 + (4*a^3*b^2*d*
e^4 + 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)*sqrt(b/(b*d - a*e)))/(b*x + a)) + 22*(8*b^5*d^4*e -
68*a*b^4*d^3*e^2 + 345*a^2*b^3*d^2*e^3 + 1162*a^3*b^2*d*e^4 + 128*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*(3465*b^5*e^5*x^5 - 48*b^5*d^5 + 328*a*b^4*d^4*e - 1030*a^2*b^
3*d^3*e^2 + 2295*a^3*b^2*d^2*e^3 + 2048*a^4*b*d*e^4 - 128*a^5*e^5 + 1155*(4*b^5*
d*e^4 + 11*a*b^4*e^5)*x^4 + 231*(3*b^5*d^2*e^3 + 74*a*b^4*d*e^4 + 73*a^2*b^3*e^5
)*x^3 - 99*(2*b^5*d^3*e^2 - 27*a*b^4*d^2*e^3 - 232*a^2*b^3*d*e^4 - 93*a^3*b^2*e^
5)*x^2 - 3465*(b^5*e^5*x^5 + a^4*b*d*e^4 + (b^5*d*e^4 + 4*a*b^4*e^5)*x^4 + 2*(2*
a*b^4*d*e^4 + 3*a^2*b^3*e^5)*x^3 + 2*(3*a^2*b^3*d*e^4 + 2*a^3*b^2*e^5)*x^2 + (4*
a^3*b^2*d*e^4 + 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))/(sqrt(e*x + d)*b)) + 11*(8*b^5*d^4*e - 68*a*b^4*d^3*e^
2 + 345*a^2*b^3*d^2*e^3 + 1162*a^3*b^2*d*e^4 + 128*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))]

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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(1/(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 = 0.272942, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^2 + 2*a*b*x + a^2)^(5/2)*(e*x + d)^(5/2)),x, algorithm="giac")

[Out]

Done