3.858 \(\int \frac{1}{x^7 \left (a+b x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=68 \[ \frac{4 b^2 x^2}{3 a^3 \sqrt{a+b x^4}}+\frac{2 b}{3 a^2 x^2 \sqrt{a+b x^4}}-\frac{1}{6 a x^6 \sqrt{a+b x^4}} \]

[Out]

-1/(6*a*x^6*Sqrt[a + b*x^4]) + (2*b)/(3*a^2*x^2*Sqrt[a + b*x^4]) + (4*b^2*x^2)/(
3*a^3*Sqrt[a + b*x^4])

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Rubi [A]  time = 0.0608464, antiderivative size = 68, 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 \[ \frac{4 b^2 x^2}{3 a^3 \sqrt{a+b x^4}}+\frac{2 b}{3 a^2 x^2 \sqrt{a+b x^4}}-\frac{1}{6 a x^6 \sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^7*(a + b*x^4)^(3/2)),x]

[Out]

-1/(6*a*x^6*Sqrt[a + b*x^4]) + (2*b)/(3*a^2*x^2*Sqrt[a + b*x^4]) + (4*b^2*x^2)/(
3*a^3*Sqrt[a + b*x^4])

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Rubi in Sympy [A]  time = 6.48862, size = 63, normalized size = 0.93 \[ - \frac{1}{6 a x^{6} \sqrt{a + b x^{4}}} + \frac{2 b}{3 a^{2} x^{2} \sqrt{a + b x^{4}}} + \frac{4 b^{2} x^{2}}{3 a^{3} \sqrt{a + b x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**7/(b*x**4+a)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.0382332, size = 42, normalized size = 0.62 \[ \frac{-a^2+4 a b x^4+8 b^2 x^8}{6 a^3 x^6 \sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^7*(a + b*x^4)^(3/2)),x]

[Out]

(-a^2 + 4*a*b*x^4 + 8*b^2*x^8)/(6*a^3*x^6*Sqrt[a + b*x^4])

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Maple [A]  time = 0.007, size = 37, normalized size = 0.5 \[ -{\frac{-8\,{b}^{2}{x}^{8}-4\,ab{x}^{4}+{a}^{2}}{6\,{a}^{3}{x}^{6}}{\frac{1}{\sqrt{b{x}^{4}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^7/(b*x^4+a)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.43974, size = 76, normalized size = 1.12 \[ \frac{b^{2} x^{2}}{2 \, \sqrt{b x^{4} + a} a^{3}} + \frac{\frac{6 \, \sqrt{b x^{4} + a} b}{x^{2}} - \frac{{\left (b x^{4} + a\right )}^{\frac{3}{2}}}{x^{6}}}{6 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(3/2)*x^7),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.280812, size = 68, normalized size = 1. \[ \frac{{\left (8 \, b^{2} x^{8} + 4 \, a b x^{4} - a^{2}\right )} \sqrt{b x^{4} + a}}{6 \,{\left (a^{3} b x^{10} + a^{4} x^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(3/2)*x^7),x, algorithm="fricas")

[Out]

1/6*(8*b^2*x^8 + 4*a*b*x^4 - a^2)*sqrt(b*x^4 + a)/(a^3*b*x^10 + a^4*x^6)

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Sympy [A]  time = 6.35944, size = 233, normalized size = 3.43 \[ - \frac{a^{3} b^{\frac{9}{2}} \sqrt{\frac{a}{b x^{4}} + 1}}{6 a^{5} b^{4} x^{4} + 12 a^{4} b^{5} x^{8} + 6 a^{3} b^{6} x^{12}} + \frac{3 a^{2} b^{\frac{11}{2}} x^{4} \sqrt{\frac{a}{b x^{4}} + 1}}{6 a^{5} b^{4} x^{4} + 12 a^{4} b^{5} x^{8} + 6 a^{3} b^{6} x^{12}} + \frac{12 a b^{\frac{13}{2}} x^{8} \sqrt{\frac{a}{b x^{4}} + 1}}{6 a^{5} b^{4} x^{4} + 12 a^{4} b^{5} x^{8} + 6 a^{3} b^{6} x^{12}} + \frac{8 b^{\frac{15}{2}} x^{12} \sqrt{\frac{a}{b x^{4}} + 1}}{6 a^{5} b^{4} x^{4} + 12 a^{4} b^{5} x^{8} + 6 a^{3} b^{6} x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**7/(b*x**4+a)**(3/2),x)

[Out]

-a**3*b**(9/2)*sqrt(a/(b*x**4) + 1)/(6*a**5*b**4*x**4 + 12*a**4*b**5*x**8 + 6*a*
*3*b**6*x**12) + 3*a**2*b**(11/2)*x**4*sqrt(a/(b*x**4) + 1)/(6*a**5*b**4*x**4 +
12*a**4*b**5*x**8 + 6*a**3*b**6*x**12) + 12*a*b**(13/2)*x**8*sqrt(a/(b*x**4) + 1
)/(6*a**5*b**4*x**4 + 12*a**4*b**5*x**8 + 6*a**3*b**6*x**12) + 8*b**(15/2)*x**12
*sqrt(a/(b*x**4) + 1)/(6*a**5*b**4*x**4 + 12*a**4*b**5*x**8 + 6*a**3*b**6*x**12)

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GIAC/XCAS [A]  time = 0.239089, size = 65, normalized size = 0.96 \[ -\frac{{\left (b + \frac{a}{x^{4}}\right )}^{\frac{3}{2}} - 6 \, \sqrt{b + \frac{a}{x^{4}}} b}{6 \, a^{3}} - \frac{x^{2}}{256 \, \sqrt{b x^{4} + a} a^{3} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(3/2)*x^7),x, algorithm="giac")

[Out]

-1/6*((b + a/x^4)^(3/2) - 6*sqrt(b + a/x^4)*b)/a^3 - 1/256*x^2/(sqrt(b*x^4 + a)*
a^3*b^2)