Optimal. Leaf size=281 \[ -\sqrt {x^3+1} x \sqrt {\frac {a}{x^4}}+\frac {\sqrt {x^3+1} x^2 \sqrt {\frac {a}{x^4}}}{x+\sqrt {3}+1}+\frac {\sqrt {2} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} x^2 \sqrt {\frac {a}{x^4}} F\left (\sin ^{-1}\left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} x^2 \sqrt {\frac {a}{x^4}} E\left (\sin ^{-1}\left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{2 \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}} \]
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Rubi [A] time = 0.07, antiderivative size = 281, normalized size of antiderivative = 1.00, 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 218
Rule 303
Rule 325
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*}
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Mathematica [C] time = 0.00, size = 27, normalized size = 0.10 \[ x \left (-\sqrt {\frac {a}{x^4}}\right ) \, _2F_1\left (-\frac {1}{3},\frac {1}{2};\frac {2}{3};-x^3\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {\frac {a}{x^{4}}}}{\sqrt {x^{3} + 1}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\frac {a}{x^{4}}}}{\sqrt {x^{3} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 353, normalized size = 1.26 \[ \frac {\sqrt {\frac {a}{x^{4}}}\, \left (-2 x^{3}-6 \sqrt {-\frac {2 \left (x +1\right )}{-3+i \sqrt {3}}}\, \sqrt {\frac {-2 x +i \sqrt {3}+1}{3+i \sqrt {3}}}\, \sqrt {\frac {2 x +i \sqrt {3}-1}{-3+i \sqrt {3}}}\, x \EllipticE \left (\sqrt {-\frac {2 \left (x +1\right )}{-3+i \sqrt {3}}}, \sqrt {-\frac {-3+i \sqrt {3}}{3+i \sqrt {3}}}\right )+i \sqrt {3}\, \sqrt {-\frac {2 \left (x +1\right )}{-3+i \sqrt {3}}}\, \sqrt {\frac {-2 x +i \sqrt {3}+1}{3+i \sqrt {3}}}\, \sqrt {\frac {2 x +i \sqrt {3}-1}{-3+i \sqrt {3}}}\, x \EllipticF \left (\sqrt {-\frac {2 \left (x +1\right )}{-3+i \sqrt {3}}}, \sqrt {-\frac {-3+i \sqrt {3}}{3+i \sqrt {3}}}\right )+3 \sqrt {-\frac {2 \left (x +1\right )}{-3+i \sqrt {3}}}\, \sqrt {\frac {-2 x +i \sqrt {3}+1}{3+i \sqrt {3}}}\, \sqrt {\frac {2 x +i \sqrt {3}-1}{-3+i \sqrt {3}}}\, x \EllipticF \left (\sqrt {-\frac {2 \left (x +1\right )}{-3+i \sqrt {3}}}, \sqrt {-\frac {-3+i \sqrt {3}}{3+i \sqrt {3}}}\right )-2\right ) x}{2 \sqrt {x^{3}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\frac {a}{x^{4}}}}{\sqrt {x^{3} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sqrt {\frac {a}{x^4}}}{\sqrt {x^3+1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\frac {a}{x^{4}}}}{\sqrt {\left (x + 1\right ) \left (x^{2} - x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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