Optimal. Leaf size=80 \[ \frac{3 a \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 b^{7/4}}-\frac{3 a \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 b^{7/4}}+\frac{x^3 \sqrt [4]{a+b x^4}}{4 b} \]
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Rubi [A] time = 0.0796956, antiderivative size = 80, 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 \[ \frac{3 a \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 b^{7/4}}-\frac{3 a \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 b^{7/4}}+\frac{x^3 \sqrt [4]{a+b x^4}}{4 b} \]
Antiderivative was successfully verified.
[In] Int[x^6/(a + b*x^4)^(3/4),x]
[Out]
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Rubi in Sympy [A] time = 10.6395, size = 73, normalized size = 0.91 \[ \frac{3 a \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{8 b^{\frac{7}{4}}} - \frac{3 a \operatorname{atanh}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{8 b^{\frac{7}{4}}} + \frac{x^{3} \sqrt [4]{a + b x^{4}}}{4 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**6/(b*x**4+a)**(3/4),x)
[Out]
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Mathematica [C] time = 0.0482144, size = 64, normalized size = 0.8 \[ \frac{x^3 \left (-a \left (\frac{b x^4}{a}+1\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right )+a+b x^4\right )}{4 b \left (a+b x^4\right )^{3/4}} \]
Antiderivative was successfully verified.
[In] Integrate[x^6/(a + b*x^4)^(3/4),x]
[Out]
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Maple [F] time = 0.05, size = 0, normalized size = 0. \[ \int{{x}^{6} \left ( b{x}^{4}+a \right ) ^{-{\frac{3}{4}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^6/(b*x^4+a)^(3/4),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^6/(b*x^4 + a)^(3/4),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.26681, size = 254, normalized size = 3.18 \[ \frac{4 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} x^{3} + 12 \, b \left (\frac{a^{4}}{b^{7}}\right )^{\frac{1}{4}} \arctan \left (\frac{b^{2} x \left (\frac{a^{4}}{b^{7}}\right )^{\frac{1}{4}}}{x \sqrt{\frac{b^{4} x^{2} \sqrt{\frac{a^{4}}{b^{7}}} + \sqrt{b x^{4} + a} a^{2}}{x^{2}}} +{\left (b x^{4} + a\right )}^{\frac{1}{4}} a}\right ) - 3 \, b \left (\frac{a^{4}}{b^{7}}\right )^{\frac{1}{4}} \log \left (\frac{3 \,{\left (b^{2} x \left (\frac{a^{4}}{b^{7}}\right )^{\frac{1}{4}} +{\left (b x^{4} + a\right )}^{\frac{1}{4}} a\right )}}{x}\right ) + 3 \, b \left (\frac{a^{4}}{b^{7}}\right )^{\frac{1}{4}} \log \left (-\frac{3 \,{\left (b^{2} x \left (\frac{a^{4}}{b^{7}}\right )^{\frac{1}{4}} -{\left (b x^{4} + a\right )}^{\frac{1}{4}} a\right )}}{x}\right )}{16 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^6/(b*x^4 + a)^(3/4),x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.61782, size = 37, normalized size = 0.46 \[ \frac{x^{7} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac{3}{4}} \Gamma \left (\frac{11}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**6/(b*x**4+a)**(3/4),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{6}}{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^6/(b*x^4 + a)^(3/4),x, algorithm="giac")
[Out]