Optimal. Leaf size=104 \[ \frac{b x^2}{2 a \sqrt [4]{a+b x^4}}-\frac{\left (a+b x^4\right )^{3/4}}{2 a x^2}-\frac{\sqrt{b} \sqrt [4]{\frac{b x^4}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{2 \sqrt{a} \sqrt [4]{a+b x^4}} \]
[Out]
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Rubi [A] time = 0.128741, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{b x^2}{2 a \sqrt [4]{a+b x^4}}-\frac{\left (a+b x^4\right )^{3/4}}{2 a x^2}-\frac{\sqrt{b} \sqrt [4]{\frac{b x^4}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{2 \sqrt{a} \sqrt [4]{a+b x^4}} \]
Antiderivative was successfully verified.
[In] Int[1/(x^3*(a + b*x^4)^(1/4)),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{b \int ^{x^{2}} \frac{1}{\left (a + b x^{2}\right )^{\frac{5}{4}}}\, dx}{4} + \frac{b x^{2}}{2 a \sqrt [4]{a + b x^{4}}} - \frac{\left (a + b x^{4}\right )^{\frac{3}{4}}}{2 a x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**3/(b*x**4+a)**(1/4),x)
[Out]
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Mathematica [C] time = 0.0491708, size = 69, normalized size = 0.66 \[ \frac{b x^4 \sqrt [4]{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^4}{a}\right )-2 \left (a+b x^4\right )}{4 a x^2 \sqrt [4]{a+b x^4}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^3*(a + b*x^4)^(1/4)),x]
[Out]
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Maple [F] time = 0.039, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{3}}{\frac{1}{\sqrt [4]{b{x}^{4}+a}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^3/(b*x^4+a)^(1/4),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^4 + a)^(1/4)*x^3),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^4 + a)^(1/4)*x^3),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.68967, size = 31, normalized size = 0.3 \[ - \frac{{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{1}{4} \\ \frac{1}{2} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{2 \sqrt [4]{a} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**3/(b*x**4+a)**(1/4),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^4 + a)^(1/4)*x^3),x, algorithm="giac")
[Out]