3.817 \(\int \frac{1}{x^7 \sqrt{a+b x^4}} \, dx\)

Optimal. Leaf size=44 \[ \frac{b \sqrt{a+b x^4}}{3 a^2 x^2}-\frac{\sqrt{a+b x^4}}{6 a x^6} \]

[Out]

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

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Rubi [A]  time = 0.0406231, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{b \sqrt{a+b x^4}}{3 a^2 x^2}-\frac{\sqrt{a+b x^4}}{6 a x^6} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^7*Sqrt[a + b*x^4]),x]

[Out]

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

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Rubi in Sympy [A]  time = 4.34652, size = 36, normalized size = 0.82 \[ - \frac{\sqrt{a + b x^{4}}}{6 a x^{6}} + \frac{b \sqrt{a + b x^{4}}}{3 a^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.023233, size = 29, normalized size = 0.66 \[ -\frac{\left (a-2 b x^4\right ) \sqrt{a+b x^4}}{6 a^2 x^6} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 26, normalized size = 0.6 \[ -{\frac{-2\,b{x}^{4}+a}{6\,{a}^{2}{x}^{6}}\sqrt{b{x}^{4}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^7/(b*x^4+a)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.43673, size = 47, normalized size = 1.07 \[ \frac{\frac{3 \, \sqrt{b x^{4} + a} b}{x^{2}} - \frac{{\left (b x^{4} + a\right )}^{\frac{3}{2}}}{x^{6}}}{6 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.233565, size = 36, normalized size = 0.82 \[ \frac{{\left (2 \, b x^{4} - a\right )} \sqrt{b x^{4} + a}}{6 \, a^{2} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.59953, size = 44, normalized size = 1. \[ - \frac{\sqrt{b} \sqrt{\frac{a}{b x^{4}} + 1}}{6 a x^{4}} + \frac{b^{\frac{3}{2}} \sqrt{\frac{a}{b x^{4}} + 1}}{3 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.220643, size = 36, normalized size = 0.82 \[ -\frac{{\left (b + \frac{a}{x^{4}}\right )}^{\frac{3}{2}} - 3 \, \sqrt{b + \frac{a}{x^{4}}} b}{6 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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