Optimal. Leaf size=98 \[ -\frac{3 a^{3/2} \sqrt [4]{\frac{b x^4}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{5 \sqrt{b} \sqrt [4]{a+b x^4}}+\frac{1}{5} x^2 \left (a+b x^4\right )^{3/4}+\frac{3 a x^2}{5 \sqrt [4]{a+b x^4}} \]
[Out]
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Rubi [A] time = 0.122223, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385 \[ -\frac{3 a^{3/2} \sqrt [4]{\frac{b x^4}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{5 \sqrt{b} \sqrt [4]{a+b x^4}}+\frac{1}{5} x^2 \left (a+b x^4\right )^{3/4}+\frac{3 a x^2}{5 \sqrt [4]{a+b x^4}} \]
Antiderivative was successfully verified.
[In] Int[x*(a + b*x^4)^(3/4),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{3 a^{2} \int ^{x^{2}} \frac{1}{\left (a + b x^{2}\right )^{\frac{5}{4}}}\, dx}{10} + \frac{3 a x^{2}}{5 \sqrt [4]{a + b x^{4}}} + \frac{x^{2} \left (a + b x^{4}\right )^{\frac{3}{4}}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(b*x**4+a)**(3/4),x)
[Out]
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Mathematica [C] time = 0.0571778, size = 64, normalized size = 0.65 \[ \frac{x^2 \left (3 a \sqrt [4]{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^4}{a}\right )+2 \left (a+b x^4\right )\right )}{10 \sqrt [4]{a+b x^4}} \]
Antiderivative was successfully verified.
[In] Integrate[x*(a + b*x^4)^(3/4),x]
[Out]
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Maple [F] time = 0.033, size = 0, normalized size = 0. \[ \int x \left ( b{x}^{4}+a \right ) ^{{\frac{3}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(b*x^4+a)^(3/4),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{4} + a\right )}^{\frac{3}{4}} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^(3/4)*x,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x^{4} + a\right )}^{\frac{3}{4}} x, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^(3/4)*x,x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.14084, size = 29, normalized size = 0.3 \[ \frac{a^{\frac{3}{4}} x^{2}{{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(b*x**4+a)**(3/4),x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^4 + a)^(3/4)*x,x, algorithm="giac")
[Out]