Optimal. Leaf size=99 \[ \frac{a^{3/2} x \sqrt [4]{\frac{a}{b x^4}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{4 \sqrt{b} \sqrt [4]{a+b x^4}}+\frac{1}{6} x^3 \left (a+b x^4\right )^{3/4}+\frac{a x^3}{4 \sqrt [4]{a+b x^4}} \]
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Rubi [A] time = 0.14688, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ \frac{a^{3/2} x \sqrt [4]{\frac{a}{b x^4}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{4 \sqrt{b} \sqrt [4]{a+b x^4}}+\frac{1}{6} x^3 \left (a+b x^4\right )^{3/4}+\frac{a x^3}{4 \sqrt [4]{a+b x^4}} \]
Antiderivative was successfully verified.
[In] Int[x^2*(a + b*x^4)^(3/4),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{a^{2} x \sqrt [4]{\frac{a}{b x^{4}} + 1} \int ^{\frac{1}{x^{2}}} \frac{1}{\left (\frac{a x^{2}}{b} + 1\right )^{\frac{5}{4}}}\, dx}{8 b \sqrt [4]{a + b x^{4}}} + \frac{a x^{3}}{4 \sqrt [4]{a + b x^{4}}} + \frac{x^{3} \left (a + b x^{4}\right )^{\frac{3}{4}}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2*(b*x**4+a)**(3/4),x)
[Out]
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Mathematica [C] time = 0.0438901, size = 60, normalized size = 0.61 \[ \frac{x^3 \left (a \sqrt [4]{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right )+a+b x^4\right )}{6 \sqrt [4]{a+b x^4}} \]
Antiderivative was successfully verified.
[In] Integrate[x^2*(a + b*x^4)^(3/4),x]
[Out]
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Maple [F] time = 0.036, size = 0, normalized size = 0. \[ \int{x}^{2} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2*(b*x^4+a)^(3/4),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{4} + a\right )}^{\frac{3}{4}} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^(3/4)*x^2,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x^{4} + a\right )}^{\frac{3}{4}} x^{2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^(3/4)*x^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.67509, size = 39, normalized size = 0.39 \[ \frac{a^{\frac{3}{4}} x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{7}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2*(b*x**4+a)**(3/4),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{4} + a\right )}^{\frac{3}{4}} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^(3/4)*x^2,x, algorithm="giac")
[Out]