Optimal. Leaf size=98 \[ \frac {\tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{a+b}}+\frac {\log \left (1-x^3\right )}{6 \sqrt [3]{a+b}}-\frac {\log \left (\sqrt [3]{a+b} x-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a+b}} \]
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Rubi [A]
time = 0.02, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {384}
\begin {gather*} \frac {\text {ArcTan}\left (\frac {\frac {2 x \sqrt [3]{a+b}}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{a+b}}+\frac {\log \left (1-x^3\right )}{6 \sqrt [3]{a+b}}-\frac {\log \left (x \sqrt [3]{a+b}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a+b}} \end {gather*}
Antiderivative was successfully verified.
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Rule 384
Rubi steps
\begin {align*} \int \frac {1}{\left (1-x^3\right ) \sqrt [3]{a+b x^3}} \, dx &=\text {Subst}\left (\int \frac {1}{1-(a+b) x^3} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \frac {1}{1-\sqrt [3]{a+b} x} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )+\frac {1}{3} \text {Subst}\left (\int \frac {2+\sqrt [3]{a+b} x}{1+\sqrt [3]{a+b} x+(a+b)^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )\\ &=-\frac {\log \left (1-\frac {\sqrt [3]{a+b} x}{\sqrt [3]{a+b x^3}}\right )}{3 \sqrt [3]{a+b}}+\frac {1}{2} \text {Subst}\left (\int \frac {1}{1+\sqrt [3]{a+b} x+(a+b)^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )+\frac {\text {Subst}\left (\int \frac {\sqrt [3]{a+b}+2 (a+b)^{2/3} x}{1+\sqrt [3]{a+b} x+(a+b)^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )}{6 \sqrt [3]{a+b}}\\ &=-\frac {\log \left (1-\frac {\sqrt [3]{a+b} x}{\sqrt [3]{a+b x^3}}\right )}{3 \sqrt [3]{a+b}}+\frac {\log \left (1+\frac {(a+b)^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac {\sqrt [3]{a+b} x}{\sqrt [3]{a+b x^3}}\right )}{6 \sqrt [3]{a+b}}-\frac {\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{a+b} x}{\sqrt [3]{a+b x^3}}\right )}{\sqrt [3]{a+b}}\\ &=\frac {\tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{a+b}}-\frac {\log \left (1-\frac {\sqrt [3]{a+b} x}{\sqrt [3]{a+b x^3}}\right )}{3 \sqrt [3]{a+b}}+\frac {\log \left (1+\frac {(a+b)^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac {\sqrt [3]{a+b} x}{\sqrt [3]{a+b x^3}}\right )}{6 \sqrt [3]{a+b}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.90, size = 189, normalized size = 1.93 \begin {gather*} \frac {-2 \sqrt {-6+6 i \sqrt {3}} \tan ^{-1}\left (\frac {3 \sqrt [3]{a+b} x}{\sqrt {3} \sqrt [3]{a+b} x-\left (3 i+\sqrt {3}\right ) \sqrt [3]{a+b x^3}}\right )+\left (1+i \sqrt {3}\right ) \left (2 \log \left (2 \sqrt [3]{a+b} x+\left (1+i \sqrt {3}\right ) \sqrt [3]{a+b x^3}\right )-\log \left (\left (-\sqrt [3]{a+b} x+\sqrt [3]{a+b x^3}\right ) \left (2 i \sqrt [3]{a+b} x+\left (i+\sqrt {3}\right ) \sqrt [3]{a+b x^3}\right )\right )\right )}{12 \sqrt [3]{a+b}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (-x^{3}+1\right ) \left (b \,x^{3}+a \right )^{\frac {1}{3}}}\, dx\]
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. 559 vs.
\(2 (78) = 156\).
time = 134.08, size = 1252, normalized size = 12.78 \begin {gather*} \left [\frac {3 \, \sqrt {\frac {1}{3}} {\left (a + b\right )} \sqrt {\frac {{\left (-a - b\right )}^{\frac {1}{3}}}{a + b}} \log \left (-\frac {{\left (a^{3} - 27 \, a^{2} b - 108 \, a b^{2} - 81 \, b^{3}\right )} x^{9} - 3 \, {\left (10 \, a^{3} + 54 \, a^{2} b + 45 \, a b^{2}\right )} x^{6} - 3 \, {\left (17 \, a^{3} + 18 \, a^{2} b\right )} x^{3} - a^{3} + 9 \, {\left ({\left (2 \, a^{2} + 3 \, a b\right )} x^{7} - {\left (a^{2} + 3 \, a b\right )} x^{4} - a^{2} x\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}} {\left (-a - b\right )}^{\frac {1}{3}} + 9 \, {\left ({\left (a^{2} + 9 \, a b + 9 \, b^{2}\right )} x^{8} + {\left (7 \, a^{2} + 9 \, a b\right )} x^{5} + a^{2} x^{2}\right )} {\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (-a - b\right )}^{\frac {2}{3}} + 3 \, \sqrt {\frac {1}{3}} {\left (3 \, {\left ({\left (4 \, a^{2} + 21 \, a b + 18 \, b^{2}\right )} x^{7} + {\left (13 \, a^{2} + 15 \, a b\right )} x^{4} + a^{2} x\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}} {\left (-a - b\right )}^{\frac {2}{3}} + 3 \, {\left ({\left (a^{3} - 2 \, a^{2} b - 12 \, a b^{2} - 9 \, b^{3}\right )} x^{8} - 5 \, {\left (a^{3} + 4 \, a^{2} b + 3 \, a b^{2}\right )} x^{5} - 5 \, {\left (a^{3} + a^{2} b\right )} x^{2}\right )} {\left (b x^{3} + a\right )}^{\frac {1}{3}} + {\left ({\left (a^{3} + 27 \, a^{2} b + 54 \, a b^{2} + 27 \, b^{3}\right )} x^{9} + 3 \, {\left (8 \, a^{3} + 18 \, a^{2} b + 9 \, a b^{2}\right )} x^{6} + 3 \, a^{3} x^{3} - a^{3}\right )} {\left (-a - b\right )}^{\frac {1}{3}}\right )} \sqrt {\frac {{\left (-a - b\right )}^{\frac {1}{3}}}{a + b}}}{x^{9} - 3 \, x^{6} + 3 \, x^{3} - 1}\right ) - 2 \, {\left (-a - b\right )}^{\frac {2}{3}} \log \left (-\frac {3 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (a + b\right )} {\left (-a - b\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} {\left (a + b\right )} x + {\left (a x^{3} - a\right )} {\left (-a - b\right )}^{\frac {2}{3}}}{x^{3} - 1}\right ) + {\left (-a - b\right )}^{\frac {2}{3}} \log \left (\frac {3 \, {\left ({\left (2 \, a + 3 \, b\right )} x^{4} + a x\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}} {\left (-a - b\right )}^{\frac {2}{3}} + 3 \, {\left ({\left (a^{2} + 4 \, a b + 3 \, b^{2}\right )} x^{5} + 2 \, {\left (a^{2} + a b\right )} x^{2}\right )} {\left (b x^{3} + a\right )}^{\frac {1}{3}} - {\left ({\left (a^{2} + 9 \, a b + 9 \, b^{2}\right )} x^{6} + {\left (7 \, a^{2} + 9 \, a b\right )} x^{3} + a^{2}\right )} {\left (-a - b\right )}^{\frac {1}{3}}}{x^{6} - 2 \, x^{3} + 1}\right )}{18 \, {\left (a + b\right )}}, \frac {6 \, \sqrt {\frac {1}{3}} {\left (a + b\right )} \sqrt {-\frac {{\left (-a - b\right )}^{\frac {1}{3}}}{a + b}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (6 \, {\left ({\left (2 \, a^{2} + 3 \, a b\right )} x^{7} - {\left (a^{2} + 3 \, a b\right )} x^{4} - a^{2} x\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}} {\left (-a - b\right )}^{\frac {2}{3}} - 6 \, {\left ({\left (a^{3} + 10 \, a^{2} b + 18 \, a b^{2} + 9 \, b^{3}\right )} x^{8} + {\left (7 \, a^{3} + 16 \, a^{2} b + 9 \, a b^{2}\right )} x^{5} + {\left (a^{3} + a^{2} b\right )} x^{2}\right )} {\left (b x^{3} + a\right )}^{\frac {1}{3}} - {\left ({\left (a^{3} - 9 \, a^{2} b - 36 \, a b^{2} - 27 \, b^{3}\right )} x^{9} - 3 \, {\left (4 \, a^{3} + 18 \, a^{2} b + 15 \, a b^{2}\right )} x^{6} - 3 \, {\left (5 \, a^{3} + 6 \, a^{2} b\right )} x^{3} - a^{3}\right )} {\left (-a - b\right )}^{\frac {1}{3}}\right )} \sqrt {-\frac {{\left (-a - b\right )}^{\frac {1}{3}}}{a + b}}}{{\left (a^{3} + 27 \, a^{2} b + 54 \, a b^{2} + 27 \, b^{3}\right )} x^{9} + 3 \, {\left (8 \, a^{3} + 18 \, a^{2} b + 9 \, a b^{2}\right )} x^{6} + 3 \, a^{3} x^{3} - a^{3}}\right ) - 2 \, {\left (-a - b\right )}^{\frac {2}{3}} \log \left (-\frac {3 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (a + b\right )} {\left (-a - b\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} {\left (a + b\right )} x + {\left (a x^{3} - a\right )} {\left (-a - b\right )}^{\frac {2}{3}}}{x^{3} - 1}\right ) + {\left (-a - b\right )}^{\frac {2}{3}} \log \left (\frac {3 \, {\left ({\left (2 \, a + 3 \, b\right )} x^{4} + a x\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}} {\left (-a - b\right )}^{\frac {2}{3}} + 3 \, {\left ({\left (a^{2} + 4 \, a b + 3 \, b^{2}\right )} x^{5} + 2 \, {\left (a^{2} + a b\right )} x^{2}\right )} {\left (b x^{3} + a\right )}^{\frac {1}{3}} - {\left ({\left (a^{2} + 9 \, a b + 9 \, b^{2}\right )} x^{6} + {\left (7 \, a^{2} + 9 \, a b\right )} x^{3} + a^{2}\right )} {\left (-a - b\right )}^{\frac {1}{3}}}{x^{6} - 2 \, x^{3} + 1}\right )}{18 \, {\left (a + b\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 {1}{x^{3} \sqrt [3]{a + b x^{3}} - \sqrt [3]{a + b x^{3}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {1}{\left (x^3-1\right )\,{\left (b\,x^3+a\right )}^{1/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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