Optimal. Leaf size=126 \[ -\frac{3 a^{5/2} x \sqrt [4]{\frac{a}{b x^4}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{40 b^{3/2} \sqrt [4]{a+b x^4}}-\frac{3 a^2 x^3}{40 b \sqrt [4]{a+b x^4}}+\frac{1}{10} x^7 \left (a+b x^4\right )^{3/4}+\frac{a x^3 \left (a+b x^4\right )^{3/4}}{20 b} \]
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Rubi [A] time = 0.194994, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.467 \[ -\frac{3 a^{5/2} x \sqrt [4]{\frac{a}{b x^4}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{40 b^{3/2} \sqrt [4]{a+b x^4}}-\frac{3 a^2 x^3}{40 b \sqrt [4]{a+b x^4}}+\frac{1}{10} x^7 \left (a+b x^4\right )^{3/4}+\frac{a x^3 \left (a+b x^4\right )^{3/4}}{20 b} \]
Antiderivative was successfully verified.
[In] Int[x^6*(a + b*x^4)^(3/4),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{3 a^{3} x \sqrt [4]{\frac{a}{b x^{4}} + 1} \int ^{\frac{1}{x^{2}}} \frac{1}{\sqrt [4]{\frac{a x^{2}}{b} + 1}}\, dx}{80 b^{2} \sqrt [4]{a + b x^{4}}} - \frac{3 a^{3}}{40 b^{2} x \sqrt [4]{a + b x^{4}}} - \frac{3 a^{2} x^{3}}{40 b \sqrt [4]{a + b x^{4}}} + \frac{a x^{3} \left (a + b x^{4}\right )^{\frac{3}{4}}}{20 b} + \frac{x^{7} \left (a + b x^{4}\right )^{\frac{3}{4}}}{10} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**6*(b*x**4+a)**(3/4),x)
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Mathematica [C] time = 0.0529185, size = 78, normalized size = 0.62 \[ \frac{x^3 \left (-a^2 \sqrt [4]{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right )+a^2+3 a b x^4+2 b^2 x^8\right )}{20 b \sqrt [4]{a+b x^4}} \]
Antiderivative was successfully verified.
[In] Integrate[x^6*(a + b*x^4)^(3/4),x]
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Maple [F] time = 0.04, 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)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{4} + a\right )}^{\frac{3}{4}} x^{6}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^(3/4)*x^6,x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x^{4} + a\right )}^{\frac{3}{4}} x^{6}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^(3/4)*x^6,x, algorithm="fricas")
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Sympy [A] time = 6.92388, size = 39, normalized size = 0.31 \[ \frac{a^{\frac{3}{4}} x^{7} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{11}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**6*(b*x**4+a)**(3/4),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{4} + a\right )}^{\frac{3}{4}} x^{6}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^(3/4)*x^6,x, algorithm="giac")
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