Optimal. Leaf size=53 \[ \frac {2 \tan ^{-1}\left (\frac {\left (\sqrt [3]{a}+\sqrt [3]{b} x\right )^2}{3 \sqrt [6]{a} \sqrt {-a-b x^3}}\right )}{3 \sqrt [6]{a} \sqrt [3]{b}} \]
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Rubi [A]
time = 0.10, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {2163, 209}
\begin {gather*} \frac {2 \text {ArcTan}\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 209
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 \tan ^{-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 = 3.66, size = 51, normalized size = 0.96 \begin {gather*} -\frac {2 \tan ^{-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. 138 vs.
\(2 (37) = 74\).
time = 0.83, size = 641, normalized size = 12.09 \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 \frac {\sqrt [3]{a}}{- 2 \sqrt [3]{a} \sqrt {- a - b x^{3}} + \sqrt [3]{b} x \sqrt {- a - b x^{3}}}\, 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 = 5.37, size = 78, normalized size = 1.47 \begin {gather*} \frac {\ln \left (\frac {\left (\sqrt {-b\,x^3-a}-\sqrt {a}\,1{}\mathrm {i}\right )\,{\left (2\,a^{1/6}\,b^{1/3}\,x-\sqrt {a}+\sqrt {-b\,x^3-a}\,1{}\mathrm {i}\right )}^3}{x^3\,{\left (b^{1/3}\,x-2\,a^{1/3}\right )}^3}\right )\,1{}\mathrm {i}}{3\,a^{1/6}\,b^{1/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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