3.1707 \(\int \frac{1}{(d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=275 \[ \frac{35 b e^2 (a+b x)}{4 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^4}+\frac{35 e^2 (a+b x)}{12 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^3}+\frac{7 e}{4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^2}-\frac{1}{2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)}-\frac{35 b^{3/2} e^2 (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{9/2}} \]

[Out]

(7*e)/(4*(b*d - a*e)^2*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - 1/(2*(b*
d - a*e)*(a + b*x)*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (35*e^2*(a +
b*x))/(12*(b*d - a*e)^3*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (35*b*
e^2*(a + b*x))/(4*(b*d - a*e)^4*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (
35*b^(3/2)*e^2*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(4*(b
*d - a*e)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi [A]  time = 0.426437, antiderivative size = 275, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{35 b e^2 (a+b x)}{4 \sqrt{a^2+2 a b x+b^2 x^2} \sqrt{d+e x} (b d-a e)^4}+\frac{35 e^2 (a+b x)}{12 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^3}+\frac{7 e}{4 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^2}-\frac{1}{2 (a+b x) \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)}-\frac{35 b^{3/2} e^2 (a+b x) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{4 \sqrt{a^2+2 a b x+b^2 x^2} (b d-a e)^{9/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2)),x]

[Out]

(7*e)/(4*(b*d - a*e)^2*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - 1/(2*(b*
d - a*e)*(a + b*x)*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (35*e^2*(a +
b*x))/(12*(b*d - a*e)^3*(d + e*x)^(3/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) + (35*b*
e^2*(a + b*x))/(4*(b*d - a*e)^4*Sqrt[d + e*x]*Sqrt[a^2 + 2*a*b*x + b^2*x^2]) - (
35*b^(3/2)*e^2*(a + b*x)*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(4*(b
*d - a*e)^(9/2)*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)**(3/2),x)

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.894357, size = 163, normalized size = 0.59 \[ \frac{(a+b x)^3 \left (\frac{\sqrt{d+e x} \left (-\frac{6 b^2 (b d-a e)}{(a+b x)^2}+\frac{33 b^2 e}{a+b x}+\frac{8 e^2 (b d-a e)}{(d+e x)^2}+\frac{72 b e^2}{d+e x}\right )}{3 (b d-a e)^4}-\frac{35 b^{3/2} e^2 \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{(b d-a e)^{9/2}}\right )}{4 \left ((a+b x)^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2)),x]

[Out]

((a + b*x)^3*((Sqrt[d + e*x]*((-6*b^2*(b*d - a*e))/(a + b*x)^2 + (33*b^2*e)/(a +
b*x) + (8*e^2*(b*d - a*e))/(d + e*x)^2 + (72*b*e^2)/(d + e*x)))/(3*(b*d - a*e)^
4) - (35*b^(3/2)*e^2*ArcTanh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/(b*d - a*
e)^(9/2)))/(4*((a + b*x)^2)^(3/2))

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Maple [B]  time = 0.031, size = 388, normalized size = 1.4 \[{\frac{bx+a}{12\, \left ( ae-bd \right ) ^{4}} \left ( 105\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) \left ( ex+d \right ) ^{3/2}{x}^{2}{b}^{4}{e}^{2}+210\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) \left ( ex+d \right ) ^{3/2}xa{b}^{3}{e}^{2}+105\,\sqrt{b \left ( ae-bd \right ) }{x}^{3}{b}^{3}{e}^{3}+105\,\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) \left ( ex+d \right ) ^{3/2}{a}^{2}{b}^{2}{e}^{2}+175\,\sqrt{b \left ( ae-bd \right ) }{x}^{2}a{b}^{2}{e}^{3}+140\,\sqrt{b \left ( ae-bd \right ) }{x}^{2}{b}^{3}d{e}^{2}+56\,\sqrt{b \left ( ae-bd \right ) }x{a}^{2}b{e}^{3}+238\,\sqrt{b \left ( ae-bd \right ) }xa{b}^{2}d{e}^{2}+21\,\sqrt{b \left ( ae-bd \right ) }x{b}^{3}{d}^{2}e-8\,\sqrt{b \left ( ae-bd \right ) }{a}^{3}{e}^{3}+80\,\sqrt{b \left ( ae-bd \right ) }{a}^{2}bd{e}^{2}+39\,\sqrt{b \left ( ae-bd \right ) }a{b}^{2}{d}^{2}e-6\,\sqrt{b \left ( ae-bd \right ) }{b}^{3}{d}^{3} \right ){\frac{1}{\sqrt{b \left ( ae-bd \right ) }}} \left ( ex+d \right ) ^{-{\frac{3}{2}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]

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)^(3/2),x)

[Out]

1/12*(105*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+d)^(3/2)*x^2*b^4*e^2+
210*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e*x+d)^(3/2)*x*a*b^3*e^2+105*(b
*(a*e-b*d))^(1/2)*x^3*b^3*e^3+105*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*(e
*x+d)^(3/2)*a^2*b^2*e^2+175*(b*(a*e-b*d))^(1/2)*x^2*a*b^2*e^3+140*(b*(a*e-b*d))^
(1/2)*x^2*b^3*d*e^2+56*(b*(a*e-b*d))^(1/2)*x*a^2*b*e^3+238*(b*(a*e-b*d))^(1/2)*x
*a*b^2*d*e^2+21*(b*(a*e-b*d))^(1/2)*x*b^3*d^2*e-8*(b*(a*e-b*d))^(1/2)*a^3*e^3+80
*(b*(a*e-b*d))^(1/2)*a^2*b*d*e^2+39*(b*(a*e-b*d))^(1/2)*a*b^2*d^2*e-6*(b*(a*e-b*
d))^(1/2)*b^3*d^3)*(b*x+a)/(b*(a*e-b*d))^(1/2)/(e*x+d)^(3/2)/(a*e-b*d)^4/((b*x+a
)^2)^(3/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.244438, 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)^(3/2)*(e*x + d)^(5/2)),x, algorithm="fricas")

[Out]

[1/24*(210*b^3*e^3*x^3 - 12*b^3*d^3 + 78*a*b^2*d^2*e + 160*a^2*b*d*e^2 - 16*a^3*
e^3 + 70*(4*b^3*d*e^2 + 5*a*b^2*e^3)*x^2 + 105*(b^3*e^3*x^3 + a^2*b*d*e^2 + (b^3
*d*e^2 + 2*a*b^2*e^3)*x^2 + (2*a*b^2*d*e^2 + a^2*b*e^3)*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)) + 14*(3*b^3*d^2*e + 34*a*b^2*d*e^2 + 8*a^2*b*e^3)*x)/((a^2*b
^4*d^5 - 4*a^3*b^3*d^4*e + 6*a^4*b^2*d^3*e^2 - 4*a^5*b*d^2*e^3 + a^6*d*e^4 + (b^
6*d^4*e - 4*a*b^5*d^3*e^2 + 6*a^2*b^4*d^2*e^3 - 4*a^3*b^3*d*e^4 + a^4*b^2*e^5)*x
^3 + (b^6*d^5 - 2*a*b^5*d^4*e - 2*a^2*b^4*d^3*e^2 + 8*a^3*b^3*d^2*e^3 - 7*a^4*b^
2*d*e^4 + 2*a^5*b*e^5)*x^2 + (2*a*b^5*d^5 - 7*a^2*b^4*d^4*e + 8*a^3*b^3*d^3*e^2
- 2*a^4*b^2*d^2*e^3 - 2*a^5*b*d*e^4 + a^6*e^5)*x)*sqrt(e*x + d)), 1/12*(105*b^3*
e^3*x^3 - 6*b^3*d^3 + 39*a*b^2*d^2*e + 80*a^2*b*d*e^2 - 8*a^3*e^3 + 35*(4*b^3*d*
e^2 + 5*a*b^2*e^3)*x^2 - 105*(b^3*e^3*x^3 + a^2*b*d*e^2 + (b^3*d*e^2 + 2*a*b^2*e
^3)*x^2 + (2*a*b^2*d*e^2 + a^2*b*e^3)*x)*sqrt(e*x + d)*sqrt(-b/(b*d - a*e))*arct
an(-(b*d - a*e)*sqrt(-b/(b*d - a*e))/(sqrt(e*x + d)*b)) + 7*(3*b^3*d^2*e + 34*a*
b^2*d*e^2 + 8*a^2*b*e^3)*x)/((a^2*b^4*d^5 - 4*a^3*b^3*d^4*e + 6*a^4*b^2*d^3*e^2
- 4*a^5*b*d^2*e^3 + a^6*d*e^4 + (b^6*d^4*e - 4*a*b^5*d^3*e^2 + 6*a^2*b^4*d^2*e^3
- 4*a^3*b^3*d*e^4 + a^4*b^2*e^5)*x^3 + (b^6*d^5 - 2*a*b^5*d^4*e - 2*a^2*b^4*d^3
*e^2 + 8*a^3*b^3*d^2*e^3 - 7*a^4*b^2*d*e^4 + 2*a^5*b*e^5)*x^2 + (2*a*b^5*d^5 - 7
*a^2*b^4*d^4*e + 8*a^3*b^3*d^3*e^2 - 2*a^4*b^2*d^2*e^3 - 2*a^5*b*d*e^4 + a^6*e^5
)*x)*sqrt(e*x + d))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (d + e x\right )^{\frac{5}{2}} \left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}}\, dx \]

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)**(3/2),x)

[Out]

Integral(1/((d + e*x)**(5/2)*((a + b*x)**2)**(3/2)), x)

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GIAC/XCAS [A]  time = 0.253998, size = 864, normalized size = 3.14 \[ -\frac{35 \, b^{2} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right ) e^{2}}{4 \,{\left (b^{4} d^{4}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - 4 \, a b^{3} d^{3} e{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) + 6 \, a^{2} b^{2} d^{2} e^{2}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - 4 \, a^{3} b d e^{3}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) + a^{4} e^{4}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )\right )} \sqrt{-b^{2} d + a b e}} - \frac{2 \,{\left (9 \,{\left (x e + d\right )} b e^{2} + b d e^{2} - a e^{3}\right )}}{3 \,{\left (b^{4} d^{4}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - 4 \, a b^{3} d^{3} e{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) + 6 \, a^{2} b^{2} d^{2} e^{2}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - 4 \, a^{3} b d e^{3}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) + a^{4} e^{4}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )\right )}{\left (x e + d\right )}^{\frac{3}{2}}} - \frac{11 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{3} e^{2} - 13 \, \sqrt{x e + d} b^{3} d e^{2} + 13 \, \sqrt{x e + d} a b^{2} e^{3}}{4 \,{\left (b^{4} d^{4}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - 4 \, a b^{3} d^{3} e{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) + 6 \, a^{2} b^{2} d^{2} e^{2}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) - 4 \, a^{3} b d e^{3}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right ) + a^{4} e^{4}{\rm sign}\left (-{\left (x e + d\right )} b e + b d e - a e^{2}\right )\right )}{\left ({\left (x e + d\right )} b - b d + a e\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^2 + 2*a*b*x + a^2)^(3/2)*(e*x + d)^(5/2)),x, algorithm="giac")

[Out]

-35/4*b^2*arctan(sqrt(x*e + d)*b/sqrt(-b^2*d + a*b*e))*e^2/((b^4*d^4*sign(-(x*e
+ d)*b*e + b*d*e - a*e^2) - 4*a*b^3*d^3*e*sign(-(x*e + d)*b*e + b*d*e - a*e^2) +
6*a^2*b^2*d^2*e^2*sign(-(x*e + d)*b*e + b*d*e - a*e^2) - 4*a^3*b*d*e^3*sign(-(x
*e + d)*b*e + b*d*e - a*e^2) + a^4*e^4*sign(-(x*e + d)*b*e + b*d*e - a*e^2))*sqr
t(-b^2*d + a*b*e)) - 2/3*(9*(x*e + d)*b*e^2 + b*d*e^2 - a*e^3)/((b^4*d^4*sign(-(
x*e + d)*b*e + b*d*e - a*e^2) - 4*a*b^3*d^3*e*sign(-(x*e + d)*b*e + b*d*e - a*e^
2) + 6*a^2*b^2*d^2*e^2*sign(-(x*e + d)*b*e + b*d*e - a*e^2) - 4*a^3*b*d*e^3*sign
(-(x*e + d)*b*e + b*d*e - a*e^2) + a^4*e^4*sign(-(x*e + d)*b*e + b*d*e - a*e^2))
*(x*e + d)^(3/2)) - 1/4*(11*(x*e + d)^(3/2)*b^3*e^2 - 13*sqrt(x*e + d)*b^3*d*e^2
+ 13*sqrt(x*e + d)*a*b^2*e^3)/((b^4*d^4*sign(-(x*e + d)*b*e + b*d*e - a*e^2) -
4*a*b^3*d^3*e*sign(-(x*e + d)*b*e + b*d*e - a*e^2) + 6*a^2*b^2*d^2*e^2*sign(-(x*
e + d)*b*e + b*d*e - a*e^2) - 4*a^3*b*d*e^3*sign(-(x*e + d)*b*e + b*d*e - a*e^2)
+ a^4*e^4*sign(-(x*e + d)*b*e + b*d*e - a*e^2))*((x*e + d)*b - b*d + a*e)^2)