3.1943 \(\int \frac{x}{\left (a+\frac{b}{x^2}\right )^{5/2}} \, dx\)

Optimal. Leaf size=92 \[ -\frac{5 b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{2 a^{7/2}}+\frac{5 x^2 \sqrt{a+\frac{b}{x^2}}}{2 a^3}-\frac{5 x^2}{3 a^2 \sqrt{a+\frac{b}{x^2}}}-\frac{x^2}{3 a \left (a+\frac{b}{x^2}\right )^{3/2}} \]

[Out]

-x^2/(3*a*(a + b/x^2)^(3/2)) - (5*x^2)/(3*a^2*Sqrt[a + b/x^2]) + (5*Sqrt[a + b/x
^2]*x^2)/(2*a^3) - (5*b*ArcTanh[Sqrt[a + b/x^2]/Sqrt[a]])/(2*a^(7/2))

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Rubi [A]  time = 0.136101, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ -\frac{5 b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )}{2 a^{7/2}}+\frac{5 x^2 \sqrt{a+\frac{b}{x^2}}}{2 a^3}-\frac{5 x^2}{3 a^2 \sqrt{a+\frac{b}{x^2}}}-\frac{x^2}{3 a \left (a+\frac{b}{x^2}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[x/(a + b/x^2)^(5/2),x]

[Out]

-x^2/(3*a*(a + b/x^2)^(3/2)) - (5*x^2)/(3*a^2*Sqrt[a + b/x^2]) + (5*Sqrt[a + b/x
^2]*x^2)/(2*a^3) - (5*b*ArcTanh[Sqrt[a + b/x^2]/Sqrt[a]])/(2*a^(7/2))

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Rubi in Sympy [A]  time = 12.8146, size = 83, normalized size = 0.9 \[ - \frac{x^{2}}{3 a \left (a + \frac{b}{x^{2}}\right )^{\frac{3}{2}}} - \frac{5 x^{2}}{3 a^{2} \sqrt{a + \frac{b}{x^{2}}}} + \frac{5 x^{2} \sqrt{a + \frac{b}{x^{2}}}}{2 a^{3}} - \frac{5 b \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x^{2}}}}{\sqrt{a}} \right )}}{2 a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(a+b/x**2)**(5/2),x)

[Out]

-x**2/(3*a*(a + b/x**2)**(3/2)) - 5*x**2/(3*a**2*sqrt(a + b/x**2)) + 5*x**2*sqrt
(a + b/x**2)/(2*a**3) - 5*b*atanh(sqrt(a + b/x**2)/sqrt(a))/(2*a**(7/2))

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Mathematica [A]  time = 0.0807643, size = 97, normalized size = 1.05 \[ \frac{\sqrt{a} x \left (3 a^2 x^4+20 a b x^2+15 b^2\right )-15 b \left (a x^2+b\right )^{3/2} \log \left (\sqrt{a} \sqrt{a x^2+b}+a x\right )}{6 a^{7/2} x \sqrt{a+\frac{b}{x^2}} \left (a x^2+b\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[x/(a + b/x^2)^(5/2),x]

[Out]

(Sqrt[a]*x*(15*b^2 + 20*a*b*x^2 + 3*a^2*x^4) - 15*b*(b + a*x^2)^(3/2)*Log[a*x +
Sqrt[a]*Sqrt[b + a*x^2]])/(6*a^(7/2)*Sqrt[a + b/x^2]*x*(b + a*x^2))

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Maple [A]  time = 0.015, size = 85, normalized size = 0.9 \[{\frac{a{x}^{2}+b}{6\,{x}^{5}} \left ( 3\,{x}^{5}{a}^{7/2}+20\,{a}^{5/2}{x}^{3}b+15\,{a}^{3/2}x{b}^{2}-15\,\ln \left ( \sqrt{a}x+\sqrt{a{x}^{2}+b} \right ) \left ( a{x}^{2}+b \right ) ^{3/2}ab \right ) \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{-{\frac{5}{2}}}{a}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(a+b/x^2)^(5/2),x)

[Out]

1/6*(a*x^2+b)*(3*x^5*a^(7/2)+20*a^(5/2)*x^3*b+15*a^(3/2)*x*b^2-15*ln(a^(1/2)*x+(
a*x^2+b)^(1/2))*(a*x^2+b)^(3/2)*a*b)/((a*x^2+b)/x^2)^(5/2)/x^5/a^(9/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.271512, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (a^{2} b x^{4} + 2 \, a b^{2} x^{2} + b^{3}\right )} \sqrt{a} \log \left (2 \, a x^{2} \sqrt{\frac{a x^{2} + b}{x^{2}}} -{\left (2 \, a x^{2} + b\right )} \sqrt{a}\right ) + 2 \,{\left (3 \, a^{3} x^{6} + 20 \, a^{2} b x^{4} + 15 \, a b^{2} x^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{12 \,{\left (a^{6} x^{4} + 2 \, a^{5} b x^{2} + a^{4} b^{2}\right )}}, \frac{15 \,{\left (a^{2} b x^{4} + 2 \, a b^{2} x^{2} + b^{3}\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{-a}}{\sqrt{\frac{a x^{2} + b}{x^{2}}}}\right ) +{\left (3 \, a^{3} x^{6} + 20 \, a^{2} b x^{4} + 15 \, a b^{2} x^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{6 \,{\left (a^{6} x^{4} + 2 \, a^{5} b x^{2} + a^{4} b^{2}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/12*(15*(a^2*b*x^4 + 2*a*b^2*x^2 + b^3)*sqrt(a)*log(2*a*x^2*sqrt((a*x^2 + b)/x
^2) - (2*a*x^2 + b)*sqrt(a)) + 2*(3*a^3*x^6 + 20*a^2*b*x^4 + 15*a*b^2*x^2)*sqrt(
(a*x^2 + b)/x^2))/(a^6*x^4 + 2*a^5*b*x^2 + a^4*b^2), 1/6*(15*(a^2*b*x^4 + 2*a*b^
2*x^2 + b^3)*sqrt(-a)*arctan(sqrt(-a)/sqrt((a*x^2 + b)/x^2)) + (3*a^3*x^6 + 20*a
^2*b*x^4 + 15*a*b^2*x^2)*sqrt((a*x^2 + b)/x^2))/(a^6*x^4 + 2*a^5*b*x^2 + a^4*b^2
)]

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Sympy [A]  time = 18.7551, size = 819, normalized size = 8.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(a+b/x**2)**(5/2),x)

[Out]

6*a**17*x**8*sqrt(1 + b/(a*x**2))/(12*a**(39/2)*x**6 + 36*a**(37/2)*b*x**4 + 36*
a**(35/2)*b**2*x**2 + 12*a**(33/2)*b**3) + 46*a**16*b*x**6*sqrt(1 + b/(a*x**2))/
(12*a**(39/2)*x**6 + 36*a**(37/2)*b*x**4 + 36*a**(35/2)*b**2*x**2 + 12*a**(33/2)
*b**3) + 15*a**16*b*x**6*log(b/(a*x**2))/(12*a**(39/2)*x**6 + 36*a**(37/2)*b*x**
4 + 36*a**(35/2)*b**2*x**2 + 12*a**(33/2)*b**3) - 30*a**16*b*x**6*log(sqrt(1 + b
/(a*x**2)) + 1)/(12*a**(39/2)*x**6 + 36*a**(37/2)*b*x**4 + 36*a**(35/2)*b**2*x**
2 + 12*a**(33/2)*b**3) + 70*a**15*b**2*x**4*sqrt(1 + b/(a*x**2))/(12*a**(39/2)*x
**6 + 36*a**(37/2)*b*x**4 + 36*a**(35/2)*b**2*x**2 + 12*a**(33/2)*b**3) + 45*a**
15*b**2*x**4*log(b/(a*x**2))/(12*a**(39/2)*x**6 + 36*a**(37/2)*b*x**4 + 36*a**(3
5/2)*b**2*x**2 + 12*a**(33/2)*b**3) - 90*a**15*b**2*x**4*log(sqrt(1 + b/(a*x**2)
) + 1)/(12*a**(39/2)*x**6 + 36*a**(37/2)*b*x**4 + 36*a**(35/2)*b**2*x**2 + 12*a*
*(33/2)*b**3) + 30*a**14*b**3*x**2*sqrt(1 + b/(a*x**2))/(12*a**(39/2)*x**6 + 36*
a**(37/2)*b*x**4 + 36*a**(35/2)*b**2*x**2 + 12*a**(33/2)*b**3) + 45*a**14*b**3*x
**2*log(b/(a*x**2))/(12*a**(39/2)*x**6 + 36*a**(37/2)*b*x**4 + 36*a**(35/2)*b**2
*x**2 + 12*a**(33/2)*b**3) - 90*a**14*b**3*x**2*log(sqrt(1 + b/(a*x**2)) + 1)/(1
2*a**(39/2)*x**6 + 36*a**(37/2)*b*x**4 + 36*a**(35/2)*b**2*x**2 + 12*a**(33/2)*b
**3) + 15*a**13*b**4*log(b/(a*x**2))/(12*a**(39/2)*x**6 + 36*a**(37/2)*b*x**4 +
36*a**(35/2)*b**2*x**2 + 12*a**(33/2)*b**3) - 30*a**13*b**4*log(sqrt(1 + b/(a*x*
*2)) + 1)/(12*a**(39/2)*x**6 + 36*a**(37/2)*b*x**4 + 36*a**(35/2)*b**2*x**2 + 12
*a**(33/2)*b**3)

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GIAC/XCAS [A]  time = 0.272632, size = 151, normalized size = 1.64 \[ \frac{1}{6} \, b{\left (\frac{2 \,{\left (a + \frac{6 \,{\left (a x^{2} + b\right )}}{x^{2}}\right )} x^{2}}{{\left (a x^{2} + b\right )} a^{3} \sqrt{\frac{a x^{2} + b}{x^{2}}}} + \frac{15 \, \arctan \left (\frac{\sqrt{\frac{a x^{2} + b}{x^{2}}}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{3}} - \frac{3 \, \sqrt{\frac{a x^{2} + b}{x^{2}}}}{{\left (a - \frac{a x^{2} + b}{x^{2}}\right )} a^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*b*(2*(a + 6*(a*x^2 + b)/x^2)*x^2/((a*x^2 + b)*a^3*sqrt((a*x^2 + b)/x^2)) + 1
5*arctan(sqrt((a*x^2 + b)/x^2)/sqrt(-a))/(sqrt(-a)*a^3) - 3*sqrt((a*x^2 + b)/x^2
)/((a - (a*x^2 + b)/x^2)*a^3))