Optimal. Leaf size=140 \[ -\frac{x^7}{2 \sqrt{x^4+1}}+\frac{7}{10} \sqrt{x^4+1} x^3-\frac{21 \sqrt{x^4+1} x}{10 \left (x^2+1\right )}-\frac{21 \left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{20 \sqrt{x^4+1}}+\frac{21 \left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{10 \sqrt{x^4+1}} \]
[Out]
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Rubi [A] time = 0.0904643, antiderivative size = 140, 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 \[ -\frac{x^7}{2 \sqrt{x^4+1}}+\frac{7}{10} \sqrt{x^4+1} x^3-\frac{21 \sqrt{x^4+1} x}{10 \left (x^2+1\right )}-\frac{21 \left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{20 \sqrt{x^4+1}}+\frac{21 \left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{10 \sqrt{x^4+1}} \]
Antiderivative was successfully verified.
[In] Int[x^10/(1 + x^4)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 8.76336, size = 128, normalized size = 0.91 \[ - \frac{x^{7}}{2 \sqrt{x^{4} + 1}} + \frac{7 x^{3} \sqrt{x^{4} + 1}}{10} - \frac{21 x \sqrt{x^{4} + 1}}{10 \left (x^{2} + 1\right )} + \frac{21 \sqrt{\frac{x^{4} + 1}{\left (x^{2} + 1\right )^{2}}} \left (x^{2} + 1\right ) E\left (2 \operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{10 \sqrt{x^{4} + 1}} - \frac{21 \sqrt{\frac{x^{4} + 1}{\left (x^{2} + 1\right )^{2}}} \left (x^{2} + 1\right ) F\left (2 \operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{20 \sqrt{x^{4} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**10/(x**4+1)**(3/2),x)
[Out]
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Mathematica [C] time = 0.0724765, size = 75, normalized size = 0.54 \[ \frac{1}{10} \left (\frac{2 x^7}{\sqrt{x^4+1}}+\frac{7 x^3}{\sqrt{x^4+1}}-21 (-1)^{3/4} F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )+21 (-1)^{3/4} E\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x^10/(1 + x^4)^(3/2),x]
[Out]
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Maple [C] time = 0.012, size = 107, normalized size = 0.8 \[{\frac{{x}^{3}}{2}{\frac{1}{\sqrt{{x}^{4}+1}}}}+{\frac{{x}^{3}}{5}\sqrt{{x}^{4}+1}}-{\frac{{\frac{21\,i}{10}} \left ({\it EllipticF} \left ( x \left ({\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) ,i \right ) -{\it EllipticE} \left ( x \left ({\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) ,i \right ) \right ) }{{\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2}}\sqrt{1-i{x}^{2}}\sqrt{1+i{x}^{2}}{\frac{1}{\sqrt{{x}^{4}+1}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^10/(x^4+1)^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{10}}{{\left (x^{4} + 1\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^10/(x^4 + 1)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{10}}{{\left (x^{4} + 1\right )}^{\frac{3}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^10/(x^4 + 1)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.10445, size = 29, normalized size = 0.21 \[ \frac{x^{11} \Gamma \left (\frac{11}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, \frac{11}{4} \\ \frac{15}{4} \end{matrix}\middle |{x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac{15}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**10/(x**4+1)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{10}}{{\left (x^{4} + 1\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^10/(x^4 + 1)^(3/2),x, algorithm="giac")
[Out]