Optimal. Leaf size=128 \[ -\frac{8 a^{5/2} \sqrt [4]{\frac{b x^4}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{3 b^{7/2} \sqrt [4]{a+b x^4}}+\frac{4 a^2 x^2}{3 b^3 \sqrt [4]{a+b x^4}}-\frac{2 a x^6}{9 b^2 \sqrt [4]{a+b x^4}}+\frac{x^{10}}{9 b \sqrt [4]{a+b x^4}} \]
[Out]
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Rubi [A] time = 0.202789, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{8 a^{5/2} \sqrt [4]{\frac{b x^4}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{3 b^{7/2} \sqrt [4]{a+b x^4}}+\frac{4 a^2 x^2}{3 b^3 \sqrt [4]{a+b x^4}}-\frac{2 a x^6}{9 b^2 \sqrt [4]{a+b x^4}}+\frac{x^{10}}{9 b \sqrt [4]{a+b x^4}} \]
Antiderivative was successfully verified.
[In] Int[x^13/(a + b*x^4)^(5/4),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{4 a^{2} x^{2}}{3 b^{3} \sqrt [4]{a + b x^{4}}} + \frac{4 a^{2} \int ^{x^{2}} \frac{1}{\sqrt [4]{a + b x^{2}}}\, dx}{3 b^{3}} - \frac{2 a x^{6}}{9 b^{2} \sqrt [4]{a + b x^{4}}} + \frac{x^{10}}{9 b \sqrt [4]{a + b x^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**13/(b*x**4+a)**(5/4),x)
[Out]
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Mathematica [C] time = 0.0736067, size = 79, normalized size = 0.62 \[ \frac{x^2 \left (12 a^2 \sqrt [4]{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^4}{a}\right )-12 a^2-2 a b x^4+b^2 x^8\right )}{9 b^3 \sqrt [4]{a+b x^4}} \]
Antiderivative was successfully verified.
[In] Integrate[x^13/(a + b*x^4)^(5/4),x]
[Out]
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Maple [F] time = 0.081, size = 0, normalized size = 0. \[ \int{{x}^{13} \left ( b{x}^{4}+a \right ) ^{-{\frac{5}{4}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^13/(b*x^4+a)^(5/4),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{13}}{{\left (b x^{4} + a\right )}^{\frac{5}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^13/(b*x^4 + a)^(5/4),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{13}}{{\left (b x^{4} + a\right )}^{\frac{5}{4}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^13/(b*x^4 + a)^(5/4),x, algorithm="fricas")
[Out]
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Sympy [A] time = 8.51519, size = 27, normalized size = 0.21 \[ \frac{x^{14}{{}_{2}F_{1}\left (\begin{matrix} \frac{5}{4}, \frac{7}{2} \\ \frac{9}{2} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{14 a^{\frac{5}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**13/(b*x**4+a)**(5/4),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{13}}{{\left (b x^{4} + a\right )}^{\frac{5}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^13/(b*x^4 + a)^(5/4),x, algorithm="giac")
[Out]