Optimal. Leaf size=108 \[ \frac{\left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{a} b^{5/4} \sqrt{a+b x^4}}-\frac{x}{2 b \sqrt{a+b x^4}} \]
[Out]
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Rubi [A] time = 0.0736015, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{\left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 \sqrt [4]{a} b^{5/4} \sqrt{a+b x^4}}-\frac{x}{2 b \sqrt{a+b x^4}} \]
Antiderivative was successfully verified.
[In] Int[x^4/(a + b*x^4)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 7.60235, size = 94, normalized size = 0.87 \[ - \frac{x}{2 b \sqrt{a + b x^{4}}} + \frac{\sqrt{\frac{a + b x^{4}}{\left (\sqrt{a} + \sqrt{b} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{4 \sqrt [4]{a} b^{\frac{5}{4}} \sqrt{a + b x^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**4/(b*x**4+a)**(3/2),x)
[Out]
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Mathematica [C] time = 0.124246, size = 102, normalized size = 0.94 \[ -\frac{i \sqrt{\frac{b x^4}{a}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )+x \sqrt{\frac{i \sqrt{b}}{\sqrt{a}}}}{2 b \sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \sqrt{a+b x^4}} \]
Antiderivative was successfully verified.
[In] Integrate[x^4/(a + b*x^4)^(3/2),x]
[Out]
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Maple [C] time = 0.018, size = 94, normalized size = 0.9 \[ -{\frac{x}{2\,b}{\frac{1}{\sqrt{ \left ({x}^{4}+{\frac{a}{b}} \right ) b}}}}+{\frac{1}{2\,b}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4/(b*x^4+a)^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{{\left (b x^{4} + a\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(b*x^4 + a)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{4}}{{\left (b x^{4} + a\right )}^{\frac{3}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(b*x^4 + a)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.39853, size = 37, normalized size = 0.34 \[ \frac{x^{5} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{5}{4}, \frac{3}{2} \\ \frac{9}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac{3}{2}} \Gamma \left (\frac{9}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**4/(b*x**4+a)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{{\left (b x^{4} + a\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/(b*x^4 + a)^(3/2),x, algorithm="giac")
[Out]