Optimal. Leaf size=371 \[ -\frac {c^2 x}{3 b d^2}+\frac {c x^2}{6 b d}-\frac {x^3}{9 b}+\frac {c^3 \log (c+d x)}{3 b d^3}+\frac {x^3 \log (c+d x)}{3 b}-\frac {a \log \left (-\frac {d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right ) \log (c+d x)}{3 b^2}-\frac {a \log \left (-\frac {d \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right ) \log (c+d x)}{3 b^2}-\frac {a \log \left (\frac {\sqrt [3]{-1} d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right ) \log (c+d x)}{3 b^2}-\frac {a \text {Li}_2\left (\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 b^2}-\frac {a \text {Li}_2\left (\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right )}{3 b^2}-\frac {a \text {Li}_2\left (\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )}{3 b^2} \]
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Rubi [A]
time = 0.42, antiderivative size = 371, normalized size of antiderivative = 1.00, number of steps
used = 16, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {272, 45, 2463,
2442, 266, 2441, 2440, 2438} \begin {gather*} -\frac {a \text {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 b^2}-\frac {a \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^2}-\frac {a \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^2}-\frac {a \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^2}-\frac {a \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^2}-\frac {a \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^2}+\frac {c^3 \log (c+d x)}{3 b d^3}-\frac {c^2 x}{3 b d^2}+\frac {x^3 \log (c+d x)}{3 b}+\frac {c x^2}{6 b d}-\frac {x^3}{9 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 266
Rule 272
Rule 2438
Rule 2440
Rule 2441
Rule 2442
Rule 2463
Rubi steps
\begin {align*} \int \frac {x^5 \log (c+d x)}{a+b x^3} \, dx &=\int \left (\frac {x^2 \log (c+d x)}{b}-\frac {a x^2 \log (c+d x)}{b \left (a+b x^3\right )}\right ) \, dx\\ &=\frac {\int x^2 \log (c+d x) \, dx}{b}-\frac {a \int \frac {x^2 \log (c+d x)}{a+b x^3} \, dx}{b}\\ &=\frac {x^3 \log (c+d x)}{3 b}-\frac {a \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}{b}-\frac {d \int \frac {x^3}{c+d x} \, dx}{3 b}\\ &=\frac {x^3 \log (c+d x)}{3 b}-\frac {a \int \frac {\log (c+d x)}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 b^{5/3}}-\frac {a \int \frac {\log (c+d x)}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 b^{5/3}}-\frac {a \int \frac {\log (c+d x)}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 b^{5/3}}-\frac {d \int \left (\frac {c^2}{d^3}-\frac {c x}{d^2}+\frac {x^2}{d}-\frac {c^3}{d^3 (c+d x)}\right ) \, dx}{3 b}\\ &=-\frac {c^2 x}{3 b d^2}+\frac {c x^2}{6 b d}-\frac {x^3}{9 b}+\frac {c^3 \log (c+d x)}{3 b d^3}+\frac {x^3 \log (c+d x)}{3 b}-\frac {a \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^2}-\frac {a \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^2}-\frac {a \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^2}+\frac {(a 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^2}+\frac {(a 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^2}+\frac {(a 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^2}\\ &=-\frac {c^2 x}{3 b d^2}+\frac {c x^2}{6 b d}-\frac {x^3}{9 b}+\frac {c^3 \log (c+d x)}{3 b d^3}+\frac {x^3 \log (c+d x)}{3 b}-\frac {a \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^2}-\frac {a \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^2}-\frac {a \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^2}+\frac {a \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^2}+\frac {a \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^2}+\frac {a \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^2}\\ &=-\frac {c^2 x}{3 b d^2}+\frac {c x^2}{6 b d}-\frac {x^3}{9 b}+\frac {c^3 \log (c+d x)}{3 b d^3}+\frac {x^3 \log (c+d x)}{3 b}-\frac {a \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^2}-\frac {a \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^2}-\frac {a \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^2}-\frac {a \text {Li}_2\left (\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 b^2}-\frac {a \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^2}-\frac {a \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^2}\\ \end {align*}
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Mathematica [A]
time = 0.24, size = 345, normalized size = 0.93 \begin {gather*} -\frac {6 b c^2 d x-3 b c d^2 x^2+2 b d^3 x^3-6 b c^3 \log (c+d x)-6 b d^3 x^3 \log (c+d x)+6 a d^3 \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 ) \log (c+d x)+6 a d^3 \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)+6 a d^3 \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)+6 a d^3 \text {Li}_2\left (\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )+6 a d^3 \text {Li}_2\left (\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right )+6 a d^3 \text {Li}_2\left (\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )}{18 b^2 d^3} \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 = 0.48, size = 170, normalized size = 0.46
method | result | size |
risch | \(-\frac {c^{2} x}{3 b \,d^{2}}-\frac {11 c^{3}}{18 d^{3} b}+\frac {c \,x^{2}}{6 b d}+\frac {x^{3} \ln \left (d x +c \right )}{3 b}+\frac {c^{3} \ln \left (d x +c \right )}{3 b \,d^{3}}-\frac {x^{3}}{9 b}-\frac {\left (\munderset {\textit {\_R1} =\RootOf \left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\left (\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\dilog \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )\right )\right ) a}{3 b^{2}}\) | \(153\) |
derivativedivides | \(\frac {\frac {\left (c^{2} \left (\left (d x +c \right ) \ln \left (d x +c \right )-d x -c \right )-2 \left (\frac {\left (d x +c \right )^{2} \ln \left (d x +c \right )}{2}-\frac {\left (d x +c \right )^{2}}{4}\right ) c +\frac {\left (d x +c \right )^{3} \ln \left (d x +c \right )}{3}-\frac {\left (d x +c \right )^{3}}{9}\right ) d^{3}}{b}-\frac {\left (\munderset {\textit {\_R1} =\RootOf \left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\left (\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\dilog \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )\right )\right ) a \,d^{6}}{3 b^{2}}}{d^{6}}\) | \(170\) |
default | \(\frac {\frac {\left (c^{2} \left (\left (d x +c \right ) \ln \left (d x +c \right )-d x -c \right )-2 \left (\frac {\left (d x +c \right )^{2} \ln \left (d x +c \right )}{2}-\frac {\left (d x +c \right )^{2}}{4}\right ) c +\frac {\left (d x +c \right )^{3} \ln \left (d x +c \right )}{3}-\frac {\left (d x +c \right )^{3}}{9}\right ) d^{3}}{b}-\frac {\left (\munderset {\textit {\_R1} =\RootOf \left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\left (\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\dilog \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )\right )\right ) a \,d^{6}}{3 b^{2}}}{d^{6}}\) | \(170\) |
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^5\,\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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