3.1091 \(\int \frac{1}{x^7 \sqrt [4]{a+b x^4}} \, dx\)

Optimal. Leaf size=128 \[ \frac{b^{3/2} \sqrt [4]{\frac{b x^4}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{4 a^{3/2} \sqrt [4]{a+b x^4}}-\frac{b^2 x^2}{4 a^2 \sqrt [4]{a+b x^4}}+\frac{b \left (a+b x^4\right )^{3/4}}{4 a^2 x^2}-\frac{\left (a+b x^4\right )^{3/4}}{6 a x^6} \]

[Out]

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

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Rubi [A]  time = 0.172179, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{b^{3/2} \sqrt [4]{\frac{b x^4}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{4 a^{3/2} \sqrt [4]{a+b x^4}}-\frac{b^2 x^2}{4 a^2 \sqrt [4]{a+b x^4}}+\frac{b \left (a+b x^4\right )^{3/4}}{4 a^2 x^2}-\frac{\left (a+b x^4\right )^{3/4}}{6 a x^6} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{\left (a + b x^{4}\right )^{\frac{3}{4}}}{6 a x^{6}} - \frac{b^{2} \int ^{x^{2}} \frac{1}{\sqrt [4]{a + b x^{2}}}\, dx}{8 a^{2}} + \frac{b \left (a + b x^{4}\right )^{\frac{3}{4}}}{4 a^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.059074, size = 83, normalized size = 0.65 \[ \frac{-4 a^2-3 b^2 x^8 \sqrt [4]{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^4}{a}\right )+2 a b x^4+6 b^2 x^8}{24 a^2 x^6 \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

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

[Out]

(-4*a^2 + 2*a*b*x^4 + 6*b^2*x^8 - 3*b^2*x^8*(1 + (b*x^4)/a)^(1/4)*Hypergeometric
2F1[1/4, 1/2, 3/2, -((b*x^4)/a)])/(24*a^2*x^6*(a + b*x^4)^(1/4))

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Maple [F]  time = 0.043, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{7}}{\frac{1}{\sqrt [4]{b{x}^{4}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^7/(b*x^4+a)^(1/4),x)

[Out]

int(1/x^7/(b*x^4+a)^(1/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} x^{7}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((b*x^4 + a)^(1/4)*x^7), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} x^{7}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/((b*x^4 + a)^(1/4)*x^7), x)

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Sympy [A]  time = 4.494, size = 32, normalized size = 0.25 \[ - \frac{{{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, \frac{1}{4} \\ - \frac{1}{2} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{6 \sqrt [4]{a} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-hyper((-3/2, 1/4), (-1/2,), b*x**4*exp_polar(I*pi)/a)/(6*a**(1/4)*x**6)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} x^{7}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((b*x^4 + a)^(1/4)*x^7), x)