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