Optimal. Leaf size=264 \[ -\frac {\sqrt [3]{a} d \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b^{2/3} \left (b^{2/3} c^2+\sqrt [3]{a} \sqrt [3]{b} c d+a^{2/3} d^2\right )}+\frac {\sqrt [3]{a} d \left (\sqrt [3]{b} c+\sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{2/3} \left (b c^3-a d^3\right )}-\frac {c^2 \log (c+d x)}{b c^3-a d^3}-\frac {\sqrt [3]{a} d \left (\sqrt [3]{b} c+\sqrt [3]{a} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{2/3} \left (b c^3-a d^3\right )}+\frac {c^2 \log \left (a+b x^3\right )}{3 \left (b c^3-a d^3\right )} \]
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Rubi [A]
time = 0.34, antiderivative size = 264, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {6857, 1885,
1874, 31, 648, 631, 210, 642, 266} \begin {gather*} -\frac {\sqrt [3]{a} d \text {ArcTan}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b^{2/3} \left (a^{2/3} d^2+\sqrt [3]{a} \sqrt [3]{b} c d+b^{2/3} c^2\right )}-\frac {\sqrt [3]{a} d \left (\sqrt [3]{a} d+\sqrt [3]{b} c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{2/3} \left (b c^3-a d^3\right )}+\frac {\sqrt [3]{a} d \left (\sqrt [3]{a} d+\sqrt [3]{b} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{2/3} \left (b c^3-a d^3\right )}+\frac {c^2 \log \left (a+b x^3\right )}{3 \left (b c^3-a d^3\right )}-\frac {c^2 \log (c+d x)}{b c^3-a d^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 210
Rule 266
Rule 631
Rule 642
Rule 648
Rule 1874
Rule 1885
Rule 6857
Rubi steps
\begin {align*} \int \frac {x^2}{(c+d x) \left (a+b x^3\right )} \, dx &=\int \left (-\frac {c^2 d}{\left (b c^3-a d^3\right ) (c+d x)}+\frac {a c d-a d^2 x+b c^2 x^2}{\left (b c^3-a d^3\right ) \left (a+b x^3\right )}\right ) \, dx\\ &=-\frac {c^2 \log (c+d x)}{b c^3-a d^3}+\frac {\int \frac {a c d-a d^2 x+b c^2 x^2}{a+b x^3} \, dx}{b c^3-a d^3}\\ &=-\frac {c^2 \log (c+d x)}{b c^3-a d^3}+\frac {\int \frac {a c d-a d^2 x}{a+b x^3} \, dx}{b c^3-a d^3}+\frac {\left (b c^2\right ) \int \frac {x^2}{a+b x^3} \, dx}{b c^3-a d^3}\\ &=-\frac {c^2 \log (c+d x)}{b c^3-a d^3}+\frac {c^2 \log \left (a+b x^3\right )}{3 \left (b c^3-a d^3\right )}+\frac {\int \frac {\sqrt [3]{a} \left (2 a \sqrt [3]{b} c d-a^{4/3} d^2\right )+\sqrt [3]{b} \left (-a \sqrt [3]{b} c d-a^{4/3} d^2\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{3 a^{2/3} \sqrt [3]{b} \left (b c^3-a d^3\right )}+\frac {\left (\sqrt [3]{a} d \left (c+\frac {\sqrt [3]{a} d}{\sqrt [3]{b}}\right )\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 \left (b c^3-a d^3\right )}\\ &=\frac {\sqrt [3]{a} d \left (\sqrt [3]{b} c+\sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{2/3} \left (b c^3-a d^3\right )}-\frac {c^2 \log (c+d x)}{b c^3-a d^3}+\frac {c^2 \log \left (a+b x^3\right )}{3 \left (b c^3-a d^3\right )}+\frac {\left (a^{2/3} d\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 \sqrt [3]{b} \left (b^{2/3} c^2+\sqrt [3]{a} \sqrt [3]{b} c d+a^{2/3} d^2\right )}-\frac {\left (\sqrt [3]{a} d \left (\sqrt [3]{b} c+\sqrt [3]{a} d\right )\right ) \int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 b^{2/3} \left (b c^3-a d^3\right )}\\ &=\frac {\sqrt [3]{a} d \left (\sqrt [3]{b} c+\sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{2/3} \left (b c^3-a d^3\right )}-\frac {c^2 \log (c+d x)}{b c^3-a d^3}-\frac {\sqrt [3]{a} d \left (\sqrt [3]{b} c+\sqrt [3]{a} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{2/3} \left (b c^3-a d^3\right )}+\frac {c^2 \log \left (a+b x^3\right )}{3 \left (b c^3-a d^3\right )}+\frac {\left (\sqrt [3]{a} d\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{b^{2/3} \left (b^{2/3} c^2+\sqrt [3]{a} \sqrt [3]{b} c d+a^{2/3} d^2\right )}\\ &=-\frac {\sqrt [3]{a} d \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b^{2/3} \left (b^{2/3} c^2+\sqrt [3]{a} \sqrt [3]{b} c d+a^{2/3} d^2\right )}+\frac {\sqrt [3]{a} d \left (\sqrt [3]{b} c+\sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{2/3} \left (b c^3-a d^3\right )}-\frac {c^2 \log (c+d x)}{b c^3-a d^3}-\frac {\sqrt [3]{a} d \left (\sqrt [3]{b} c+\sqrt [3]{a} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{2/3} \left (b c^3-a d^3\right )}+\frac {c^2 \log \left (a+b x^3\right )}{3 \left (b c^3-a d^3\right )}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 228, normalized size = 0.86 \begin {gather*} \frac {2 \sqrt {3} \sqrt [3]{a} d \left (-\sqrt [3]{b} c+\sqrt [3]{a} d\right ) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )+2 \sqrt [3]{a} d \left (\sqrt [3]{b} c+\sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-6 b^{2/3} c^2 \log (c+d x)-\sqrt [3]{a} \sqrt [3]{b} c d \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-a^{2/3} d^2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+2 b^{2/3} c^2 \log \left (a+b x^3\right )}{6 b^{2/3} \left (b c^3-a d^3\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.23, size = 245, normalized size = 0.93
method | result | size |
risch | \(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (1+\left (a \,b^{2} d^{3}-b^{3} c^{3}\right ) \textit {\_Z}^{3}+3 b^{2} c^{2} \textit {\_Z}^{2}-3 b c \textit {\_Z} \right )}{\sum }\textit {\_R} \ln \left (\left (\left (-4 a \,b^{2} d^{4}-2 b^{3} c^{3} d \right ) \textit {\_R}^{3}-3 \textit {\_R}^{2} b^{2} c^{2} d +8 \textit {\_R} b c d -3 d \right ) x +\left (-5 a \,b^{2} c \,d^{3}-b^{3} c^{4}\right ) \textit {\_R}^{3}+\left (d^{3} a b -b^{2} c^{3}\right ) \textit {\_R}^{2}+5 b \,c^{2} \textit {\_R} -3 c \right )\right )}{3}+\frac {c^{2} \ln \left (-d x -c \right )}{d^{3} a -b \,c^{3}}\) | \(177\) |
default | \(\frac {-a c d \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )+a \,d^{2} \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )-\frac {c^{2} \ln \left (b \,x^{3}+a \right )}{3}}{d^{3} a -b \,c^{3}}+\frac {c^{2} \ln \left (d x +c \right )}{d^{3} a -b \,c^{3}}\) | \(245\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 279, normalized size = 1.06 \begin {gather*} -\frac {c^{2} \log \left (d x + c\right )}{b c^{3} - a d^{3}} - \frac {\sqrt {3} {\left (a d^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} - a c d \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, {\left (b^{2} c^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}} - a b d^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {{\left (2 \, b c^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} - a d^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}} - a c d\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, {\left (b^{2} c^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}} - a b d^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}} + \frac {{\left (b c^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} + a d^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}} + a c d\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, {\left (b^{2} c^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}} - a b d^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains complex when optimal does not.
time = 1.28, size = 5975, normalized size = 22.63 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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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Giac [A]
time = 3.47, size = 320, normalized size = 1.21 \begin {gather*} -\frac {c^{2} d \log \left ({\left | d x + c \right |}\right )}{b c^{3} d - a d^{4}} + \frac {c^{2} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, {\left (b c^{3} - a d^{3}\right )}} + \frac {\left (-a b^{2}\right )^{\frac {1}{3}} d \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{\sqrt {3} b^{2} c^{2} - \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} b c d + \sqrt {3} \left (-a b^{2}\right )^{\frac {2}{3}} d^{2}} + \frac {{\left (a b^{2} c^{3} d^{2} \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a^{2} b d^{5} \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a b^{2} c^{4} d + a^{2} b c d^{4}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, {\left (a b^{3} c^{6} - 2 \, a^{2} b^{2} c^{3} d^{3} + a^{3} b d^{6}\right )}} + \frac {{\left (\left (-a b^{2}\right )^{\frac {1}{3}} b c d - \left (-a b^{2}\right )^{\frac {2}{3}} d^{2}\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, {\left (b^{3} c^{3} - a b^{2} d^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.50, size = 570, normalized size = 2.16 \begin {gather*} \left (\sum _{k=1}^3\ln \left (-a\,b\,d\,\left (c+d\,x+{\mathrm {root}\left (27\,a\,b^2\,d^3\,z^3-27\,b^3\,c^3\,z^3+27\,b^2\,c^2\,z^2-9\,b\,c\,z+1,z,k\right )}^2\,b^2\,c^3\,3+{\mathrm {root}\left (27\,a\,b^2\,d^3\,z^3-27\,b^3\,c^3\,z^3+27\,b^2\,c^2\,z^2-9\,b\,c\,z+1,z,k\right )}^3\,b^3\,c^4\,9-\mathrm {root}\left (27\,a\,b^2\,d^3\,z^3-27\,b^3\,c^3\,z^3+27\,b^2\,c^2\,z^2-9\,b\,c\,z+1,z,k\right )\,b\,c^2\,5-{\mathrm {root}\left (27\,a\,b^2\,d^3\,z^3-27\,b^3\,c^3\,z^3+27\,b^2\,c^2\,z^2-9\,b\,c\,z+1,z,k\right )}^2\,a\,b\,d^3\,3-\mathrm {root}\left (27\,a\,b^2\,d^3\,z^3-27\,b^3\,c^3\,z^3+27\,b^2\,c^2\,z^2-9\,b\,c\,z+1,z,k\right )\,b\,c\,d\,x\,8+{\mathrm {root}\left (27\,a\,b^2\,d^3\,z^3-27\,b^3\,c^3\,z^3+27\,b^2\,c^2\,z^2-9\,b\,c\,z+1,z,k\right )}^3\,a\,b^2\,c\,d^3\,45+{\mathrm {root}\left (27\,a\,b^2\,d^3\,z^3-27\,b^3\,c^3\,z^3+27\,b^2\,c^2\,z^2-9\,b\,c\,z+1,z,k\right )}^3\,a\,b^2\,d^4\,x\,36+{\mathrm {root}\left (27\,a\,b^2\,d^3\,z^3-27\,b^3\,c^3\,z^3+27\,b^2\,c^2\,z^2-9\,b\,c\,z+1,z,k\right )}^2\,b^2\,c^2\,d\,x\,9+{\mathrm {root}\left (27\,a\,b^2\,d^3\,z^3-27\,b^3\,c^3\,z^3+27\,b^2\,c^2\,z^2-9\,b\,c\,z+1,z,k\right )}^3\,b^3\,c^3\,d\,x\,18\right )\right )\,\mathrm {root}\left (27\,a\,b^2\,d^3\,z^3-27\,b^3\,c^3\,z^3+27\,b^2\,c^2\,z^2-9\,b\,c\,z+1,z,k\right )\right )+\frac {c^2\,\ln \left (c+d\,x\right )}{a\,d^3-b\,c^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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