Optimal. Leaf size=139 \[ -\frac{b^{2/3} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{4 a^{2/3}}+\frac{b^{2/3} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 a^{2/3}}-\frac{\sqrt{3} b^{2/3} p \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{2 a^{2/3}}-\frac{\log \left (c \left (a+b x^3\right )^p\right )}{2 x^2} \]
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Rubi [A] time = 0.0747968, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438, Rules used = {2455, 200, 31, 634, 617, 204, 628} \[ -\frac{b^{2/3} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{4 a^{2/3}}+\frac{b^{2/3} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 a^{2/3}}-\frac{\sqrt{3} b^{2/3} p \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{2 a^{2/3}}-\frac{\log \left (c \left (a+b x^3\right )^p\right )}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 2455
Rule 200
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{\log \left (c \left (a+b x^3\right )^p\right )}{x^3} \, dx &=-\frac{\log \left (c \left (a+b x^3\right )^p\right )}{2 x^2}+\frac{1}{2} (3 b p) \int \frac{1}{a+b x^3} \, dx\\ &=-\frac{\log \left (c \left (a+b x^3\right )^p\right )}{2 x^2}+\frac{(b p) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{2 a^{2/3}}+\frac{(b p) \int \frac{2 \sqrt [3]{a}-\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 a^{2/3}}\\ &=\frac{b^{2/3} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 a^{2/3}}-\frac{\log \left (c \left (a+b x^3\right )^p\right )}{2 x^2}-\frac{\left (b^{2/3} p\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}{4 a^{2/3}}+\frac{(3 b p) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{4 \sqrt [3]{a}}\\ &=\frac{b^{2/3} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 a^{2/3}}-\frac{b^{2/3} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{4 a^{2/3}}-\frac{\log \left (c \left (a+b x^3\right )^p\right )}{2 x^2}+\frac{\left (3 b^{2/3} p\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{2 a^{2/3}}\\ &=-\frac{\sqrt{3} b^{2/3} p \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{2 a^{2/3}}+\frac{b^{2/3} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{2 a^{2/3}}-\frac{b^{2/3} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{4 a^{2/3}}-\frac{\log \left (c \left (a+b x^3\right )^p\right )}{2 x^2}\\ \end{align*}
Mathematica [A] time = 0.0327894, size = 134, normalized size = 0.96 \[ -\frac{b^{2/3} p x^2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+2 a^{2/3} \log \left (c \left (a+b x^3\right )^p\right )-2 b^{2/3} p x^2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+2 \sqrt{3} b^{2/3} p x^2 \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{4 a^{2/3} x^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.322, size = 197, normalized size = 1.4 \begin{align*} -{\frac{\ln \left ( \left ( b{x}^{3}+a \right ) ^{p} \right ) }{2\,{x}^{2}}}-{\frac{i\pi \,{\it csgn} \left ( i \left ( b{x}^{3}+a \right ) ^{p} \right ) \left ({\it csgn} \left ( ic \left ( b{x}^{3}+a \right ) ^{p} \right ) \right ) ^{2}-i\pi \,{\it csgn} \left ( i \left ( b{x}^{3}+a \right ) ^{p} \right ){\it csgn} \left ( ic \left ( b{x}^{3}+a \right ) ^{p} \right ){\it csgn} \left ( ic \right ) -i\pi \, \left ({\it csgn} \left ( ic \left ( b{x}^{3}+a \right ) ^{p} \right ) \right ) ^{3}+i\pi \, \left ({\it csgn} \left ( ic \left ( b{x}^{3}+a \right ) ^{p} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) -2\,\sum _{{\it \_R}={\it RootOf} \left ({a}^{2}{{\it \_Z}}^{3}-{b}^{2}{p}^{3} \right ) }{\it \_R}\,\ln \left ( \left ( -4\,{a}^{2}{{\it \_R}}^{3}+3\,{b}^{2}{p}^{3} \right ) x-a{p}^{2}{\it \_R}\,b \right ){x}^{2}+2\,\ln \left ( c \right ) }{4\,{x}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.01815, size = 366, normalized size = 2.63 \begin{align*} \frac{2 \, \sqrt{3} p x^{2} \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} a x \left (\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}} - \sqrt{3} b}{3 \, b}\right ) - p x^{2} \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b^{2} x^{2} - a b x \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} + a^{2} \left (\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}}\right ) + 2 \, p x^{2} \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b x + a \left (\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}\right ) - 2 \, p \log \left (b x^{3} + a\right ) - 2 \, \log \left (c\right )}{4 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.30659, size = 186, normalized size = 1.34 \begin{align*} -\frac{1}{4} \, b p{\left (\frac{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{a} - \frac{2 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{1}{3}} \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 )}{a b} - \frac{\left (-a b^{2}\right )^{\frac{1}{3}} \log \left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{a b}\right )} - \frac{p \log \left (b x^{3} + a\right )}{2 \, x^{2}} - \frac{\log \left (c\right )}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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