Optimal. Leaf size=292 \[ \frac {\log \left (-\frac {d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right ) \log (c+d x)}{3 b}+\frac {\log \left (-\frac {d \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right ) \log (c+d x)}{3 b}+\frac {\log \left (\frac {\sqrt [3]{-1} d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right ) \log (c+d x)}{3 b}+\frac {\text {Li}_2\left (\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 b}+\frac {\text {Li}_2\left (\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right )}{3 b}+\frac {\text {Li}_2\left (\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )}{3 b} \]
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Rubi [A]
time = 0.20, antiderivative size = 292, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {266, 2463,
2441, 2440, 2438} \begin {gather*} \frac {\text {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 b}+\frac {\text {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{-1} \sqrt [3]{a} d+\sqrt [3]{b} c}\right )}{3 b}+\frac {\text {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )}{3 b}+\frac {\log (c+d x) \log \left (-\frac {d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 b}+\frac {\log (c+d x) \log \left (-\frac {d \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )}{3 b}+\frac {\log (c+d x) \log \left (\frac {\sqrt [3]{-1} d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{-1} \sqrt [3]{a} d+\sqrt [3]{b} c}\right )}{3 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 266
Rule 2438
Rule 2440
Rule 2441
Rule 2463
Rubi steps
\begin {align*} \int \frac {x^2 \log (c+d x)}{a+b x^3} \, dx &=\int \left (\frac {\log (c+d x)}{3 b^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {\log (c+d x)}{3 b^{2/3} \left (-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {\log (c+d x)}{3 b^{2/3} \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}\right ) \, dx\\ &=\frac {\int \frac {\log (c+d x)}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 b^{2/3}}+\frac {\int \frac {\log (c+d x)}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 b^{2/3}}+\frac {\int \frac {\log (c+d x)}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 b^{2/3}}\\ &=\frac {\log \left (-\frac {d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right ) \log (c+d x)}{3 b}+\frac {\log \left (-\frac {d \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right ) \log (c+d x)}{3 b}+\frac {\log \left (\frac {\sqrt [3]{-1} d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right ) \log (c+d x)}{3 b}-\frac {d \int \frac {\log \left (\frac {d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{-\sqrt [3]{b} c+\sqrt [3]{a} d}\right )}{c+d x} \, dx}{3 b}-\frac {d \int \frac {\log \left (\frac {d \left (-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{-\sqrt [3]{b} c-\sqrt [3]{-1} \sqrt [3]{a} d}\right )}{c+d x} \, dx}{3 b}-\frac {d \int \frac {\log \left (\frac {d \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{-\sqrt [3]{b} c+(-1)^{2/3} \sqrt [3]{a} d}\right )}{c+d x} \, dx}{3 b}\\ &=\frac {\log \left (-\frac {d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right ) \log (c+d x)}{3 b}+\frac {\log \left (-\frac {d \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right ) \log (c+d x)}{3 b}+\frac {\log \left (\frac {\sqrt [3]{-1} d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right ) \log (c+d x)}{3 b}-\frac {\text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt [3]{b} x}{-\sqrt [3]{b} c+\sqrt [3]{a} d}\right )}{x} \, dx,x,c+d x\right )}{3 b}-\frac {\text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt [3]{b} x}{-\sqrt [3]{b} c-\sqrt [3]{-1} \sqrt [3]{a} d}\right )}{x} \, dx,x,c+d x\right )}{3 b}-\frac {\text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt [3]{b} x}{-\sqrt [3]{b} c+(-1)^{2/3} \sqrt [3]{a} d}\right )}{x} \, dx,x,c+d x\right )}{3 b}\\ &=\frac {\log \left (-\frac {d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right ) \log (c+d x)}{3 b}+\frac {\log \left (-\frac {d \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right ) \log (c+d x)}{3 b}+\frac {\log \left (\frac {\sqrt [3]{-1} d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right ) \log (c+d x)}{3 b}+\frac {\text {Li}_2\left (\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 b}+\frac {\text {Li}_2\left (\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right )}{3 b}+\frac {\text {Li}_2\left (\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )}{3 b}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 297, normalized size = 1.02 \begin {gather*} \frac {\log \left (-\frac {d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right ) \log (c+d x)}{3 b}+\frac {\log \left (-\frac {(-1)^{2/3} d \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right ) \log (c+d x)}{3 b}+\frac {\log \left (\frac {\sqrt [3]{-1} d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right ) \log (c+d x)}{3 b}+\frac {\text {Li}_2\left (\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 b}+\frac {\text {Li}_2\left (\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right )}{3 b}+\frac {\text {Li}_2\left (\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )}{3 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.51, size = 77, normalized size = 0.26
method | result | size |
derivativedivides | \(\frac {\munderset {\textit {\_R1} =\RootOf \left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\left (\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\dilog \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )\right )}{3 b}\) | \(77\) |
default | \(\frac {\munderset {\textit {\_R1} =\RootOf \left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\left (\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\dilog \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )\right )}{3 b}\) | \(77\) |
risch | \(\frac {\munderset {\textit {\_R1} =\RootOf \left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\left (\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\dilog \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )\right )}{3 b}\) | \(77\) |
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 [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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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 [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 = -0.00 \begin {gather*} \int \frac {x^2\,\ln \left (c+d\,x\right )}{b\,x^3+a} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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