Optimal. Leaf size=121 \[ \frac{8 a^{7/2} \left (\frac{b x^2}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{77 b^{5/2} \left (a+b x^2\right )^{3/4}}-\frac{4 a^2 x \sqrt [4]{a+b x^2}}{77 b^2}+\frac{2}{11} x^5 \sqrt [4]{a+b x^2}+\frac{2 a x^3 \sqrt [4]{a+b x^2}}{77 b} \]
[Out]
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Rubi [A] time = 0.148729, antiderivative size = 121, 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 \[ \frac{8 a^{7/2} \left (\frac{b x^2}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{77 b^{5/2} \left (a+b x^2\right )^{3/4}}-\frac{4 a^2 x \sqrt [4]{a+b x^2}}{77 b^2}+\frac{2}{11} x^5 \sqrt [4]{a+b x^2}+\frac{2 a x^3 \sqrt [4]{a+b x^2}}{77 b} \]
Antiderivative was successfully verified.
[In] Int[x^4*(a + b*x^2)^(1/4),x]
[Out]
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Rubi in Sympy [A] time = 16.1499, size = 110, normalized size = 0.91 \[ \frac{8 a^{\frac{7}{2}} \left (1 + \frac{b x^{2}}{a}\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{77 b^{\frac{5}{2}} \left (a + b x^{2}\right )^{\frac{3}{4}}} - \frac{4 a^{2} x \sqrt [4]{a + b x^{2}}}{77 b^{2}} + \frac{2 a x^{3} \sqrt [4]{a + b x^{2}}}{77 b} + \frac{2 x^{5} \sqrt [4]{a + b x^{2}}}{11} \]
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 = 0.0644206, size = 89, normalized size = 0.74 \[ \frac{2 x \left (2 a^3 \left (\frac{b x^2}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{b x^2}{a}\right )-2 a^3-a^2 b x^2+8 a b^2 x^4+7 b^3 x^6\right )}{77 b^2 \left (a+b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
[In] Integrate[x^4*(a + b*x^2)^(1/4),x]
[Out]
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Maple [F] time = 0.049, size = 0, normalized size = 0. \[ \int{x}^{4}\sqrt [4]{b{x}^{2}+a}\, dx \]
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. \[ \int{\left (b x^{2} + a\right )}^{\frac{1}{4}} x^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(1/4)*x^4,x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x^{2} + a\right )}^{\frac{1}{4}} x^{4}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(1/4)*x^4,x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.01684, size = 29, normalized size = 0.24 \[ \frac{\sqrt [4]{a} x^{5}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**4*(b*x**2+a)**(1/4),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{2} + a\right )}^{\frac{1}{4}} x^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(1/4)*x^4,x, algorithm="giac")
[Out]