Optimal. Leaf size=281 \[ \frac{\sqrt{x^3+1} x^2 \sqrt{\frac{a}{x^4}}}{x+\sqrt{3}+1}-\sqrt{x^3+1} x \sqrt{\frac{a}{x^4}}+\frac{\sqrt{2} (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} x^2 \sqrt{\frac{a}{x^4}} F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^3+1}}-\frac{\sqrt [4]{3} \sqrt{2-\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} x^2 \sqrt{\frac{a}{x^4}} E\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{2 \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^3+1}} \]
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Rubi [A] time = 0.0730181, antiderivative size = 281, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {15, 325, 303, 218, 1877} \[ \frac{\sqrt{x^3+1} x^2 \sqrt{\frac{a}{x^4}}}{x+\sqrt{3}+1}-\sqrt{x^3+1} x \sqrt{\frac{a}{x^4}}+\frac{\sqrt{2} (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} x^2 \sqrt{\frac{a}{x^4}} F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^3+1}}-\frac{\sqrt [4]{3} \sqrt{2-\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} x^2 \sqrt{\frac{a}{x^4}} E\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{2 \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^3+1}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 325
Rule 303
Rule 218
Rule 1877
Rubi steps
\begin{align*} \int \frac{\sqrt{\frac{a}{x^4}}}{\sqrt{1+x^3}} \, dx &=\left (\sqrt{\frac{a}{x^4}} x^2\right ) \int \frac{1}{x^2 \sqrt{1+x^3}} \, dx\\ &=-\sqrt{\frac{a}{x^4}} x \sqrt{1+x^3}+\frac{1}{2} \left (\sqrt{\frac{a}{x^4}} x^2\right ) \int \frac{x}{\sqrt{1+x^3}} \, dx\\ &=-\sqrt{\frac{a}{x^4}} x \sqrt{1+x^3}+\frac{1}{2} \left (\sqrt{\frac{a}{x^4}} x^2\right ) \int \frac{1-\sqrt{3}+x}{\sqrt{1+x^3}} \, dx+\left (\sqrt{\frac{1}{2} \left (2-\sqrt{3}\right )} \sqrt{\frac{a}{x^4}} x^2\right ) \int \frac{1}{\sqrt{1+x^3}} \, dx\\ &=-\sqrt{\frac{a}{x^4}} x \sqrt{1+x^3}+\frac{\sqrt{\frac{a}{x^4}} x^2 \sqrt{1+x^3}}{1+\sqrt{3}+x}-\frac{\sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt{\frac{a}{x^4}} x^2 (1+x) \sqrt{\frac{1-x+x^2}{\left (1+\sqrt{3}+x\right )^2}} E\left (\sin ^{-1}\left (\frac{1-\sqrt{3}+x}{1+\sqrt{3}+x}\right )|-7-4 \sqrt{3}\right )}{2 \sqrt{\frac{1+x}{\left (1+\sqrt{3}+x\right )^2}} \sqrt{1+x^3}}+\frac{\sqrt{2} \sqrt{\frac{a}{x^4}} x^2 (1+x) \sqrt{\frac{1-x+x^2}{\left (1+\sqrt{3}+x\right )^2}} F\left (\sin ^{-1}\left (\frac{1-\sqrt{3}+x}{1+\sqrt{3}+x}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{\frac{1+x}{\left (1+\sqrt{3}+x\right )^2}} \sqrt{1+x^3}}\\ \end{align*}
Mathematica [C] time = 0.0049953, size = 27, normalized size = 0.1 \[ x \left (-\sqrt{\frac{a}{x^4}}\right ) \, _2F_1\left (-\frac{1}{3},\frac{1}{2};\frac{2}{3};-x^3\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 353, normalized size = 1.3 \begin{align*}{\frac{x}{2}\sqrt{{\frac{a}{{x}^{4}}}} \left ( i\sqrt{3}\sqrt{{\frac{i\sqrt{3}-2\,x+1}{3+i\sqrt{3}}}}\sqrt{{\frac{i\sqrt{3}+2\,x-1}{i\sqrt{3}-3}}}{\it EllipticF} \left ( \sqrt{-2\,{\frac{1+x}{i\sqrt{3}-3}}},\sqrt{-{\frac{i\sqrt{3}-3}{3+i\sqrt{3}}}} \right ) \sqrt{-2\,{\frac{1+x}{i\sqrt{3}-3}}}x-6\,\sqrt{{\frac{i\sqrt{3}-2\,x+1}{3+i\sqrt{3}}}}\sqrt{{\frac{i\sqrt{3}+2\,x-1}{i\sqrt{3}-3}}}{\it EllipticE} \left ( \sqrt{-2\,{\frac{1+x}{i\sqrt{3}-3}}},\sqrt{-{\frac{i\sqrt{3}-3}{3+i\sqrt{3}}}} \right ) \sqrt{-2\,{\frac{1+x}{i\sqrt{3}-3}}}x+3\,\sqrt{{\frac{i\sqrt{3}-2\,x+1}{3+i\sqrt{3}}}}\sqrt{{\frac{i\sqrt{3}+2\,x-1}{i\sqrt{3}-3}}}{\it EllipticF} \left ( \sqrt{-2\,{\frac{1+x}{i\sqrt{3}-3}}},\sqrt{-{\frac{i\sqrt{3}-3}{3+i\sqrt{3}}}} \right ) \sqrt{-2\,{\frac{1+x}{i\sqrt{3}-3}}}x-2\,{x}^{3}-2 \right ){\frac{1}{\sqrt{{x}^{3}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\frac{a}{x^{4}}}}{\sqrt{x^{3} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{\frac{a}{x^{4}}}}{\sqrt{x^{3} + 1}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\frac{a}{x^{4}}}}{\sqrt{\left (x + 1\right ) \left (x^{2} - x + 1\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\frac{a}{x^{4}}}}{\sqrt{x^{3} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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