3.1150 \(\int \frac{1}{x^7 \left (a+b x^4\right )^{5/4}} \, dx\)

Optimal. Leaf size=104 \[ \frac{7 b^{3/2} \sqrt [4]{\frac{b x^4}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{4 a^{5/2} \sqrt [4]{a+b x^4}}+\frac{7 b}{12 a^2 x^2 \sqrt [4]{a+b x^4}}-\frac{1}{6 a x^6 \sqrt [4]{a+b x^4}} \]

[Out]

-1/(6*a*x^6*(a + b*x^4)^(1/4)) + (7*b)/(12*a^2*x^2*(a + b*x^4)^(1/4)) + (7*b^(3/
2)*(1 + (b*x^4)/a)^(1/4)*EllipticE[ArcTan[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(4*a^(5/
2)*(a + b*x^4)^(1/4))

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Rubi [A]  time = 0.150067, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{7 b^{3/2} \sqrt [4]{\frac{b x^4}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{4 a^{5/2} \sqrt [4]{a+b x^4}}+\frac{7 b}{12 a^2 x^2 \sqrt [4]{a+b x^4}}-\frac{1}{6 a x^6 \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^7*(a + b*x^4)^(5/4)),x]

[Out]

-1/(6*a*x^6*(a + b*x^4)^(1/4)) + (7*b)/(12*a^2*x^2*(a + b*x^4)^(1/4)) + (7*b^(3/
2)*(1 + (b*x^4)/a)^(1/4)*EllipticE[ArcTan[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(4*a^(5/
2)*(a + b*x^4)^(1/4))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{1}{6 a x^{6} \sqrt [4]{a + b x^{4}}} + \frac{7 b^{2} \int ^{x^{2}} \frac{1}{\left (a + b x^{2}\right )^{\frac{5}{4}}}\, dx}{8 a^{2}} + \frac{7 b}{12 a^{2} x^{2} \sqrt [4]{a + b x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**7/(b*x**4+a)**(5/4),x)

[Out]

-1/(6*a*x**6*(a + b*x**4)**(1/4)) + 7*b**2*Integral((a + b*x**2)**(-5/4), (x, x*
*2))/(8*a**2) + 7*b/(12*a**2*x**2*(a + b*x**4)**(1/4))

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Mathematica [C]  time = 0.0731702, size = 83, normalized size = 0.8 \[ \frac{-4 a^2-21 b^2 x^8 \sqrt [4]{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^4}{a}\right )+14 a b x^4+42 b^2 x^8}{24 a^3 x^6 \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^7*(a + b*x^4)^(5/4)),x]

[Out]

(-4*a^2 + 14*a*b*x^4 + 42*b^2*x^8 - 21*b^2*x^8*(1 + (b*x^4)/a)^(1/4)*Hypergeomet
ric2F1[1/4, 1/2, 3/2, -((b*x^4)/a)])/(24*a^3*x^6*(a + b*x^4)^(1/4))

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Maple [F]  time = 0.076, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{7}} \left ( b{x}^{4}+a \right ) ^{-{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^7/(b*x^4+a)^(5/4),x)

[Out]

int(1/x^7/(b*x^4+a)^(5/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{5}{4}} x^{7}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(5/4)*x^7),x, algorithm="maxima")

[Out]

integrate(1/((b*x^4 + a)^(5/4)*x^7), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b x^{11} + a x^{7}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(5/4)*x^7),x, algorithm="fricas")

[Out]

integral(1/((b*x^11 + a*x^7)*(b*x^4 + a)^(1/4)), x)

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Sympy [A]  time = 8.29714, size = 32, normalized size = 0.31 \[ - \frac{{{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, \frac{5}{4} \\ - \frac{1}{2} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{6 a^{\frac{5}{4}} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**7/(b*x**4+a)**(5/4),x)

[Out]

-hyper((-3/2, 5/4), (-1/2,), b*x**4*exp_polar(I*pi)/a)/(6*a**(5/4)*x**6)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{5}{4}} x^{7}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(5/4)*x^7),x, algorithm="giac")

[Out]

integrate(1/((b*x^4 + a)^(5/4)*x^7), x)