Optimal. Leaf size=398 \[ \frac {\sqrt [3]{b} \text {Li}_2\left (\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^{4/3}}+\frac {(-1)^{2/3} \sqrt [3]{b} \text {Li}_2\left (\frac {(-1)^{2/3} \sqrt [3]{b} (c+d x)}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^{4/3}}-\frac {\sqrt [3]{-1} \sqrt [3]{b} \text {Li}_2\left (\frac {\sqrt [3]{-1} \sqrt [3]{b} (c+d x)}{\sqrt [3]{-1} \sqrt [3]{b} c+\sqrt [3]{a} d}\right )}{3 a^{4/3}}+\frac {\sqrt [3]{b} \log (c+d x) \log \left (-\frac {d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^{4/3}}-\frac {\sqrt [3]{-1} \sqrt [3]{b} \log (c+d x) \log \left (\frac {d \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} c}\right )}{3 a^{4/3}}+\frac {(-1)^{2/3} \sqrt [3]{b} \log (c+d x) \log \left (-\frac {d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^{4/3}}+\frac {d \log (x)}{a c}-\frac {d \log (c+d x)}{a c}-\frac {\log (c+d x)}{a x} \]
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Rubi [A] time = 0.49, antiderivative size = 398, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 14, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.737, Rules used = {325, 292, 31, 634, 617, 204, 628, 2416, 2395, 36, 29, 2394, 2393, 2391} \[ \frac {\sqrt [3]{b} \text {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^{4/3}}+\frac {(-1)^{2/3} \sqrt [3]{b} \text {PolyLog}\left (2,\frac {(-1)^{2/3} \sqrt [3]{b} (c+d x)}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^{4/3}}-\frac {\sqrt [3]{-1} \sqrt [3]{b} \text {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{b} (c+d x)}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} c}\right )}{3 a^{4/3}}+\frac {\sqrt [3]{b} \log (c+d x) \log \left (-\frac {d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^{4/3}}-\frac {\sqrt [3]{-1} \sqrt [3]{b} \log (c+d x) \log \left (\frac {d \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} c}\right )}{3 a^{4/3}}+\frac {(-1)^{2/3} \sqrt [3]{b} \log (c+d x) \log \left (-\frac {d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^{4/3}}+\frac {d \log (x)}{a c}-\frac {d \log (c+d x)}{a c}-\frac {\log (c+d x)}{a x} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 204
Rule 292
Rule 325
Rule 617
Rule 628
Rule 634
Rule 2391
Rule 2393
Rule 2394
Rule 2395
Rule 2416
Rubi steps
\begin {align*} \int \frac {\log (c+d x)}{x^2 \left (a+b x^3\right )} \, dx &=\int \left (\frac {\log (c+d x)}{a x^2}-\frac {b x \log (c+d x)}{a \left (a+b x^3\right )}\right ) \, dx\\ &=\frac {\int \frac {\log (c+d x)}{x^2} \, dx}{a}-\frac {b \int \frac {x \log (c+d x)}{a+b x^3} \, dx}{a}\\ &=-\frac {\log (c+d x)}{a x}-\frac {b \int \left (-\frac {\log (c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {(-1)^{2/3} \log (c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}+\frac {\sqrt [3]{-1} \log (c+d x)}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}\right ) \, dx}{a}+\frac {d \int \frac {1}{x (c+d x)} \, dx}{a}\\ &=-\frac {\log (c+d x)}{a x}+\frac {b^{2/3} \int \frac {\log (c+d x)}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^{4/3}}-\frac {\left (\sqrt [3]{-1} b^{2/3}\right ) \int \frac {\log (c+d x)}{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x} \, dx}{3 a^{4/3}}+\frac {\left ((-1)^{2/3} b^{2/3}\right ) \int \frac {\log (c+d x)}{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x} \, dx}{3 a^{4/3}}+\frac {d \int \frac {1}{x} \, dx}{a c}-\frac {d^2 \int \frac {1}{c+d x} \, dx}{a c}\\ &=\frac {d \log (x)}{a c}-\frac {d \log (c+d x)}{a c}-\frac {\log (c+d x)}{a x}+\frac {\sqrt [3]{b} \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 a^{4/3}}-\frac {\sqrt [3]{-1} \sqrt [3]{b} \log \left (\frac {d \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\sqrt [3]{-1} \sqrt [3]{b} c+\sqrt [3]{a} d}\right ) \log (c+d x)}{3 a^{4/3}}+\frac {(-1)^{2/3} \sqrt [3]{b} \log \left (-\frac {d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right ) \log (c+d x)}{3 a^{4/3}}-\frac {\left (\sqrt [3]{b} d\right ) \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 a^{4/3}}+\frac {\left (\sqrt [3]{-1} \sqrt [3]{b} d\right ) \int \frac {\log \left (\frac {d \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\sqrt [3]{-1} \sqrt [3]{b} c+\sqrt [3]{a} d}\right )}{c+d x} \, dx}{3 a^{4/3}}-\frac {\left ((-1)^{2/3} \sqrt [3]{b} d\right ) \int \frac {\log \left (\frac {d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{-(-1)^{2/3} \sqrt [3]{b} c+\sqrt [3]{a} d}\right )}{c+d x} \, dx}{3 a^{4/3}}\\ &=\frac {d \log (x)}{a c}-\frac {d \log (c+d x)}{a c}-\frac {\log (c+d x)}{a x}+\frac {\sqrt [3]{b} \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 a^{4/3}}-\frac {\sqrt [3]{-1} \sqrt [3]{b} \log \left (\frac {d \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\sqrt [3]{-1} \sqrt [3]{b} c+\sqrt [3]{a} d}\right ) \log (c+d x)}{3 a^{4/3}}+\frac {(-1)^{2/3} \sqrt [3]{b} \log \left (-\frac {d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right ) \log (c+d x)}{3 a^{4/3}}-\frac {\sqrt [3]{b} \operatorname {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 a^{4/3}}+\frac {\left (\sqrt [3]{-1} \sqrt [3]{b}\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt [3]{-1} \sqrt [3]{b} x}{\sqrt [3]{-1} \sqrt [3]{b} c+\sqrt [3]{a} d}\right )}{x} \, dx,x,c+d x\right )}{3 a^{4/3}}-\frac {\left ((-1)^{2/3} \sqrt [3]{b}\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {(-1)^{2/3} \sqrt [3]{b} x}{-(-1)^{2/3} \sqrt [3]{b} c+\sqrt [3]{a} d}\right )}{x} \, dx,x,c+d x\right )}{3 a^{4/3}}\\ &=\frac {d \log (x)}{a c}-\frac {d \log (c+d x)}{a c}-\frac {\log (c+d x)}{a x}+\frac {\sqrt [3]{b} \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 a^{4/3}}-\frac {\sqrt [3]{-1} \sqrt [3]{b} \log \left (\frac {d \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\sqrt [3]{-1} \sqrt [3]{b} c+\sqrt [3]{a} d}\right ) \log (c+d x)}{3 a^{4/3}}+\frac {(-1)^{2/3} \sqrt [3]{b} \log \left (-\frac {d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right ) \log (c+d x)}{3 a^{4/3}}+\frac {\sqrt [3]{b} \text {Li}_2\left (\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^{4/3}}+\frac {(-1)^{2/3} \sqrt [3]{b} \text {Li}_2\left (\frac {(-1)^{2/3} \sqrt [3]{b} (c+d x)}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^{4/3}}-\frac {\sqrt [3]{-1} \sqrt [3]{b} \text {Li}_2\left (\frac {\sqrt [3]{-1} \sqrt [3]{b} (c+d x)}{\sqrt [3]{-1} \sqrt [3]{b} c+\sqrt [3]{a} d}\right )}{3 a^{4/3}}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 378, normalized size = 0.95 \[ \frac {\sqrt [3]{b} c x \text {Li}_2\left (\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )+(-1)^{2/3} \sqrt [3]{b} c x \text {Li}_2\left (\frac {(-1)^{2/3} \sqrt [3]{b} (c+d x)}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right )-\sqrt [3]{-1} \sqrt [3]{b} c x \text {Li}_2\left (\frac {\sqrt [3]{-1} \sqrt [3]{b} (c+d x)}{\sqrt [3]{-1} \sqrt [3]{b} c+\sqrt [3]{a} d}\right )+\sqrt [3]{b} c x \log (c+d x) \log \left (\frac {d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a} d-\sqrt [3]{b} c}\right )-\sqrt [3]{-1} \sqrt [3]{b} c x \log (c+d x) \log \left (\frac {d \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} c}\right )+(-1)^{2/3} \sqrt [3]{b} c x \log (c+d x) \log \left (\frac {d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{a} d-(-1)^{2/3} \sqrt [3]{b} c}\right )-3 \sqrt [3]{a} c \log (c+d x)-3 \sqrt [3]{a} d x \log (c+d x)+3 \sqrt [3]{a} d x \log (x)}{3 a^{4/3} c x} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left (d x + c\right )}{b x^{5} + a x^{2}}, x\right ) \]
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 \[ \int \frac {\log \left (d x + c\right )}{{\left (b x^{3} + a\right )} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.28, size = 128, normalized size = 0.32 \[ -\frac {d \left (\ln \left (\frac {-d x +\RootOf \left (b \,\textit {\_Z}^{3}-3 b c \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )-c}{\RootOf \left (b \,\textit {\_Z}^{3}-3 b c \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}\right ) \ln \left (d x +c \right )+\dilog \left (\frac {-d x +\RootOf \left (b \,\textit {\_Z}^{3}-3 b c \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )-c}{\RootOf \left (b \,\textit {\_Z}^{3}-3 b c \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}\right )\right )}{3 a \left (\RootOf \left (b \,\textit {\_Z}^{3}-3 b c \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )-c \right )}+\frac {d \ln \left (d x \right )}{a c}-\frac {d \ln \left (d x +c \right )}{a c}-\frac {\ln \left (d x +c \right )}{a x} \]
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 \[ \int \frac {\log \left (d x + c\right )}{{\left (b x^{3} + a\right )} x^{2}}\,{d x} \]
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 \[ \int \frac {\ln \left (c+d\,x\right )}{x^2\,\left (b\,x^3+a\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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