Optimal. Leaf size=128 \[ -\frac{40 a^{7/2} \left (\frac{b x^4}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{77 b^{7/2} \left (a+b x^4\right )^{3/4}}+\frac{20 a^2 x^2 \sqrt [4]{a+b x^4}}{77 b^3}-\frac{10 a x^6 \sqrt [4]{a+b x^4}}{77 b^2}+\frac{x^{10} \sqrt [4]{a+b x^4}}{11 b} \]
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Rubi [A] time = 0.197787, 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 \[ -\frac{40 a^{7/2} \left (\frac{b x^4}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{77 b^{7/2} \left (a+b x^4\right )^{3/4}}+\frac{20 a^2 x^2 \sqrt [4]{a+b x^4}}{77 b^3}-\frac{10 a x^6 \sqrt [4]{a+b x^4}}{77 b^2}+\frac{x^{10} \sqrt [4]{a+b x^4}}{11 b} \]
Antiderivative was successfully verified.
[In] Int[x^13/(a + b*x^4)^(3/4),x]
[Out]
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Rubi in Sympy [A] time = 19.623, size = 116, normalized size = 0.91 \[ - \frac{40 a^{\frac{7}{2}} \left (1 + \frac{b x^{4}}{a}\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{77 b^{\frac{7}{2}} \left (a + b x^{4}\right )^{\frac{3}{4}}} + \frac{20 a^{2} x^{2} \sqrt [4]{a + b x^{4}}}{77 b^{3}} - \frac{10 a x^{6} \sqrt [4]{a + b x^{4}}}{77 b^{2}} + \frac{x^{10} \sqrt [4]{a + b x^{4}}}{11 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**13/(b*x**4+a)**(3/4),x)
[Out]
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Mathematica [C] time = 0.0729296, size = 91, normalized size = 0.71 \[ \frac{x^2 \left (-20 a^3 \left (\frac{b x^4}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{b x^4}{a}\right )+20 a^3+10 a^2 b x^4-3 a b^2 x^8+7 b^3 x^{12}\right )}{77 b^3 \left (a+b x^4\right )^{3/4}} \]
Antiderivative was successfully verified.
[In] Integrate[x^13/(a + b*x^4)^(3/4),x]
[Out]
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Maple [F] time = 0.038, size = 0, normalized size = 0. \[ \int{{x}^{13} \left ( b{x}^{4}+a \right ) ^{-{\frac{3}{4}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^13/(b*x^4+a)^(3/4),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{13}}{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^13/(b*x^4 + a)^(3/4),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{13}}{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^13/(b*x^4 + a)^(3/4),x, algorithm="fricas")
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Sympy [A] time = 8.4009, size = 27, normalized size = 0.21 \[ \frac{x^{14}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{7}{2} \\ \frac{9}{2} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{14 a^{\frac{3}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**13/(b*x**4+a)**(3/4),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{13}}{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^13/(b*x^4 + a)^(3/4),x, algorithm="giac")
[Out]