Optimal. Leaf size=122 \[ -\frac{8 a^{5/2} \sqrt [4]{\frac{b x^2}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{15 b^{5/2} \sqrt [4]{a+b x^2}}+\frac{8 a^2 x}{15 b^2 \sqrt [4]{a+b x^2}}-\frac{4 a x \left (a+b x^2\right )^{3/4}}{15 b^2}+\frac{2 x^3 \left (a+b x^2\right )^{3/4}}{9 b} \]
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Rubi [A] time = 0.121214, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{8 a^{5/2} \sqrt [4]{\frac{b x^2}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{15 b^{5/2} \sqrt [4]{a+b x^2}}+\frac{8 a^2 x}{15 b^2 \sqrt [4]{a+b x^2}}-\frac{4 a x \left (a+b x^2\right )^{3/4}}{15 b^2}+\frac{2 x^3 \left (a+b x^2\right )^{3/4}}{9 b} \]
Antiderivative was successfully verified.
[In] Int[x^4/(a + b*x^2)^(1/4),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{4 a^{3} \int \frac{1}{\left (a + b x^{2}\right )^{\frac{5}{4}}}\, dx}{15 b^{2}} + \frac{8 a^{2} x}{15 b^{2} \sqrt [4]{a + b x^{2}}} - \frac{4 a x \left (a + b x^{2}\right )^{\frac{3}{4}}}{15 b^{2}} + \frac{2 x^{3} \left (a + b x^{2}\right )^{\frac{3}{4}}}{9 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**4/(b*x**2+a)**(1/4),x)
[Out]
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Mathematica [C] time = 0.056046, size = 79, normalized size = 0.65 \[ \frac{2 \left (6 a^2 x \sqrt [4]{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right )-6 a^2 x-a b x^3+5 b^2 x^5\right )}{45 b^2 \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[x^4/(a + b*x^2)^(1/4),x]
[Out]
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Maple [F] time = 0.04, size = 0, normalized size = 0. \[ \int{{x}^{4}{\frac{1}{\sqrt [4]{b{x}^{2}+a}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4/(b*x^2+a)^(1/4),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{{\left (b x^{2} + a\right )}^{\frac{1}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(b*x^2 + a)^(1/4),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{4}}{{\left (b x^{2} + a\right )}^{\frac{1}{4}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(b*x^2 + a)^(1/4),x, algorithm="fricas")
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Sympy [A] time = 2.57385, size = 27, normalized size = 0.22 \[ \frac{x^{5}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{5 \sqrt [4]{a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**4/(b*x**2+a)**(1/4),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{{\left (b x^{2} + a\right )}^{\frac{1}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(b*x^2 + a)^(1/4),x, algorithm="giac")
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