3.1933 \(\int \frac{1}{\left (a+\frac{b}{x^2}\right )^{3/2} x^7} \, dx\)

Optimal. Leaf size=54 \[ \frac{a^2}{b^3 \sqrt{a+\frac{b}{x^2}}}+\frac{2 a \sqrt{a+\frac{b}{x^2}}}{b^3}-\frac{\left (a+\frac{b}{x^2}\right )^{3/2}}{3 b^3} \]

[Out]

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

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Rubi [A]  time = 0.0905849, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{a^2}{b^3 \sqrt{a+\frac{b}{x^2}}}+\frac{2 a \sqrt{a+\frac{b}{x^2}}}{b^3}-\frac{\left (a+\frac{b}{x^2}\right )^{3/2}}{3 b^3} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b/x^2)^(3/2)*x^7),x]

[Out]

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

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Rubi in Sympy [A]  time = 10.4471, size = 48, normalized size = 0.89 \[ \frac{a^{2}}{b^{3} \sqrt{a + \frac{b}{x^{2}}}} + \frac{2 a \sqrt{a + \frac{b}{x^{2}}}}{b^{3}} - \frac{\left (a + \frac{b}{x^{2}}\right )^{\frac{3}{2}}}{3 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(a+b/x**2)**(3/2)/x**7,x)

[Out]

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

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Mathematica [A]  time = 0.033243, size = 42, normalized size = 0.78 \[ \frac{8 a^2 x^4+4 a b x^2-b^2}{3 b^3 x^4 \sqrt{a+\frac{b}{x^2}}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b/x^2)^(3/2)*x^7),x]

[Out]

(-b^2 + 4*a*b*x^2 + 8*a^2*x^4)/(3*b^3*Sqrt[a + b/x^2]*x^4)

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Maple [A]  time = 0.008, size = 50, normalized size = 0.9 \[{\frac{ \left ( a{x}^{2}+b \right ) \left ( 8\,{x}^{4}{a}^{2}+4\,ab{x}^{2}-{b}^{2} \right ) }{3\,{b}^{3}{x}^{6}} \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(a+b/x^2)^(3/2)/x^7,x)

[Out]

1/3*(a*x^2+b)*(8*a^2*x^4+4*a*b*x^2-b^2)/x^6/b^3/((a*x^2+b)/x^2)^(3/2)

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Maxima [A]  time = 1.43702, size = 62, normalized size = 1.15 \[ -\frac{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}}}{3 \, b^{3}} + \frac{2 \, \sqrt{a + \frac{b}{x^{2}}} a}{b^{3}} + \frac{a^{2}}{\sqrt{a + \frac{b}{x^{2}}} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)^(3/2)*x^7),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.237157, size = 73, normalized size = 1.35 \[ \frac{{\left (8 \, a^{2} x^{4} + 4 \, a b x^{2} - b^{2}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{3 \,{\left (a b^{3} x^{4} + b^{4} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)^(3/2)*x^7),x, algorithm="fricas")

[Out]

1/3*(8*a^2*x^4 + 4*a*b*x^2 - b^2)*sqrt((a*x^2 + b)/x^2)/(a*b^3*x^4 + b^4*x^2)

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Sympy [A]  time = 12.939, size = 423, normalized size = 7.83 \[ \frac{8 a^{\frac{9}{2}} b^{\frac{7}{2}} x^{6} \sqrt{\frac{a x^{2}}{b} + 1}}{3 a^{\frac{7}{2}} b^{6} x^{7} + 6 a^{\frac{5}{2}} b^{7} x^{5} + 3 a^{\frac{3}{2}} b^{8} x^{3}} + \frac{12 a^{\frac{7}{2}} b^{\frac{9}{2}} x^{4} \sqrt{\frac{a x^{2}}{b} + 1}}{3 a^{\frac{7}{2}} b^{6} x^{7} + 6 a^{\frac{5}{2}} b^{7} x^{5} + 3 a^{\frac{3}{2}} b^{8} x^{3}} + \frac{3 a^{\frac{5}{2}} b^{\frac{11}{2}} x^{2} \sqrt{\frac{a x^{2}}{b} + 1}}{3 a^{\frac{7}{2}} b^{6} x^{7} + 6 a^{\frac{5}{2}} b^{7} x^{5} + 3 a^{\frac{3}{2}} b^{8} x^{3}} - \frac{a^{\frac{3}{2}} b^{\frac{13}{2}} \sqrt{\frac{a x^{2}}{b} + 1}}{3 a^{\frac{7}{2}} b^{6} x^{7} + 6 a^{\frac{5}{2}} b^{7} x^{5} + 3 a^{\frac{3}{2}} b^{8} x^{3}} - \frac{8 a^{5} b^{3} x^{7}}{3 a^{\frac{7}{2}} b^{6} x^{7} + 6 a^{\frac{5}{2}} b^{7} x^{5} + 3 a^{\frac{3}{2}} b^{8} x^{3}} - \frac{16 a^{4} b^{4} x^{5}}{3 a^{\frac{7}{2}} b^{6} x^{7} + 6 a^{\frac{5}{2}} b^{7} x^{5} + 3 a^{\frac{3}{2}} b^{8} x^{3}} - \frac{8 a^{3} b^{5} x^{3}}{3 a^{\frac{7}{2}} b^{6} x^{7} + 6 a^{\frac{5}{2}} b^{7} x^{5} + 3 a^{\frac{3}{2}} b^{8} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(a+b/x**2)**(3/2)/x**7,x)

[Out]

8*a**(9/2)*b**(7/2)*x**6*sqrt(a*x**2/b + 1)/(3*a**(7/2)*b**6*x**7 + 6*a**(5/2)*b
**7*x**5 + 3*a**(3/2)*b**8*x**3) + 12*a**(7/2)*b**(9/2)*x**4*sqrt(a*x**2/b + 1)/
(3*a**(7/2)*b**6*x**7 + 6*a**(5/2)*b**7*x**5 + 3*a**(3/2)*b**8*x**3) + 3*a**(5/2
)*b**(11/2)*x**2*sqrt(a*x**2/b + 1)/(3*a**(7/2)*b**6*x**7 + 6*a**(5/2)*b**7*x**5
+ 3*a**(3/2)*b**8*x**3) - a**(3/2)*b**(13/2)*sqrt(a*x**2/b + 1)/(3*a**(7/2)*b**
6*x**7 + 6*a**(5/2)*b**7*x**5 + 3*a**(3/2)*b**8*x**3) - 8*a**5*b**3*x**7/(3*a**(
7/2)*b**6*x**7 + 6*a**(5/2)*b**7*x**5 + 3*a**(3/2)*b**8*x**3) - 16*a**4*b**4*x**
5/(3*a**(7/2)*b**6*x**7 + 6*a**(5/2)*b**7*x**5 + 3*a**(3/2)*b**8*x**3) - 8*a**3*
b**5*x**3/(3*a**(7/2)*b**6*x**7 + 6*a**(5/2)*b**7*x**5 + 3*a**(3/2)*b**8*x**3)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{3}{2}} x^{7}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((a + b/x^2)^(3/2)*x^7),x, algorithm="giac")

[Out]

integrate(1/((a + b/x^2)^(3/2)*x^7), x)