Optimal. Leaf size=52 \[ -\frac {2 \tanh ^{-1}\left (\frac {\left (\sqrt [3]{a}-\sqrt [3]{b} x\right )^2}{3 \sqrt [6]{a} \sqrt {a-b x^3}}\right )}{3 \sqrt [6]{a} \sqrt [3]{b}} \]
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Rubi [A]
time = 0.09, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {2163, 212}
\begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\left (\sqrt [3]{a}-\sqrt [3]{b} x\right )^2}{3 \sqrt [6]{a} \sqrt {a-b x^3}}\right )}{3 \sqrt [6]{a} \sqrt [3]{b}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 2163
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{a}-\sqrt [3]{b} x}{\left (2 \sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {a-b x^3}} \, dx &=-\frac {\left (2 \sqrt [3]{a}\right ) \text {Subst}\left (\int \frac {1}{9-a x^2} \, dx,x,\frac {\left (1-\frac {\sqrt [3]{b} x}{\sqrt [3]{a}}\right )^2}{\sqrt {a-b x^3}}\right )}{\sqrt [3]{b}}\\ &=-\frac {2 \tanh ^{-1}\left (\frac {\left (\sqrt [3]{a}-\sqrt [3]{b} x\right )^2}{3 \sqrt [6]{a} \sqrt {a-b x^3}}\right )}{3 \sqrt [6]{a} \sqrt [3]{b}}\\ \end {align*}
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Mathematica [A]
time = 3.61, size = 50, normalized size = 0.96 \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {3 \sqrt [6]{a} \sqrt {a-b x^3}}{\left (\sqrt [3]{a}-\sqrt [3]{b} x\right )^2}\right )}{3 \sqrt [6]{a} \sqrt [3]{b}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {a^{\frac {1}{3}}-b^{\frac {1}{3}} x}{\left (2 a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) \sqrt {-b \,x^{3}+a}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 134 vs.
\(2 (37) = 74\).
time = 0.85, size = 630, normalized size = 12.12 \begin {gather*} \left [\frac {1}{6} \, a^{\frac {1}{3}} \sqrt {\frac {1}{a b^{\frac {2}{3}}}} \log \left (\frac {b^{6} x^{18} - 7800 \, a b^{5} x^{15} + 535272 \, a^{2} b^{4} x^{12} - 5147264 \, a^{3} b^{3} x^{9} + 10516992 \, a^{4} b^{2} x^{6} - 5922816 \, a^{5} b x^{3} + 557056 \, a^{6} + 144 \, {\left (7 \, b^{5} x^{16} - 1169 \, a b^{4} x^{13} + 20266 \, a^{2} b^{3} x^{10} - 66976 \, a^{3} b^{2} x^{7} + 58112 \, a^{4} b x^{4} - 10240 \, a^{5} x\right )} a^{\frac {2}{3}} b^{\frac {1}{3}} - 72 \, {\left (b^{5} x^{17} - 581 \, a b^{4} x^{14} + 19108 \, a^{2} b^{3} x^{11} - 106336 \, a^{3} b^{2} x^{8} + 137984 \, a^{4} b x^{5} - 50176 \, a^{5} x^{2}\right )} a^{\frac {1}{3}} b^{\frac {2}{3}} - 12 \, {\left ({\left (b^{5} x^{16} - 1568 \, a b^{4} x^{13} + 72520 \, a^{2} b^{3} x^{10} - 498304 \, a^{3} b^{2} x^{7} + 625664 \, a^{4} b x^{4} - 139264 \, a^{5} x\right )} \sqrt {-b x^{3} + a} a^{\frac {2}{3}} b^{\frac {2}{3}} + 6 \, {\left (41 \, a b^{5} x^{14} - 4268 \, a^{2} b^{4} x^{11} + 52896 \, a^{3} b^{3} x^{8} - 116480 \, a^{4} b^{2} x^{5} + 48128 \, a^{5} b x^{2}\right )} \sqrt {-b x^{3} + a} a^{\frac {1}{3}} - {\left (25 \, a b^{5} x^{15} - 7202 \, a^{2} b^{4} x^{12} + 167392 \, a^{3} b^{3} x^{9} - 647296 \, a^{4} b^{2} x^{6} + 468992 \, a^{5} b x^{3} - 40960 \, a^{6}\right )} \sqrt {-b x^{3} + a} b^{\frac {1}{3}}\right )} \sqrt {\frac {1}{a b^{\frac {2}{3}}}}}{b^{6} x^{18} + 48 \, a b^{5} x^{15} + 960 \, a^{2} b^{4} x^{12} + 10240 \, a^{3} b^{3} x^{9} + 61440 \, a^{4} b^{2} x^{6} + 196608 \, a^{5} b x^{3} + 262144 \, a^{6}}\right ), \frac {1}{3} \, a^{\frac {1}{3}} \sqrt {-\frac {1}{a b^{\frac {2}{3}}}} \arctan \left (\frac {{\left ({\left (11 \, b x^{4} + 16 \, a x\right )} \sqrt {-b x^{3} + a} a^{\frac {2}{3}} b^{\frac {2}{3}} - {\left (b^{2} x^{5} - 28 \, a b x^{2}\right )} \sqrt {-b x^{3} + a} a^{\frac {1}{3}} + {\left (17 \, a b x^{3} + 10 \, a^{2}\right )} \sqrt {-b x^{3} + a} b^{\frac {1}{3}}\right )} \sqrt {-\frac {1}{a b^{\frac {2}{3}}}}}{6 \, {\left (b^{2} x^{6} - 2 \, a b x^{3} + a^{2}\right )}}\right )\right ] \end {gather*}
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 \begin {gather*} - \int \left (- \frac {\sqrt [3]{a}}{2 \sqrt [3]{a} \sqrt {a - b x^{3}} + \sqrt [3]{b} x \sqrt {a - b x^{3}}}\right )\, dx - \int \frac {\sqrt [3]{b} x}{2 \sqrt [3]{a} \sqrt {a - b x^{3}} + \sqrt [3]{b} x \sqrt {a - b x^{3}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.59, size = 67, normalized size = 1.29 \begin {gather*} \frac {\ln \left (\frac {\left (\sqrt {a-b\,x^3}-\sqrt {a}\right )\,{\left (\sqrt {a-b\,x^3}+\sqrt {a}+2\,a^{1/6}\,b^{1/3}\,x\right )}^3}{x^3\,{\left (b^{1/3}\,x+2\,a^{1/3}\right )}^3}\right )}{3\,a^{1/6}\,b^{1/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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