Optimal. Leaf size=151 \[ \frac{b^{5/3} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{10 a^{5/3}}-\frac{b^{5/3} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{5 a^{5/3}}+\frac{\sqrt{3} b^{5/3} p \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{5 a^{5/3}}-\frac{\log \left (c \left (a+b x^3\right )^p\right )}{5 x^5}-\frac{3 b p}{10 a x^2} \]
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Rubi [A] time = 0.0911043, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {2455, 325, 200, 31, 634, 617, 204, 628} \[ \frac{b^{5/3} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{10 a^{5/3}}-\frac{b^{5/3} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{5 a^{5/3}}+\frac{\sqrt{3} b^{5/3} p \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{5 a^{5/3}}-\frac{\log \left (c \left (a+b x^3\right )^p\right )}{5 x^5}-\frac{3 b p}{10 a x^2} \]
Antiderivative was successfully verified.
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Rule 2455
Rule 325
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^6} \, dx &=-\frac{\log \left (c \left (a+b x^3\right )^p\right )}{5 x^5}+\frac{1}{5} (3 b p) \int \frac{1}{x^3 \left (a+b x^3\right )} \, dx\\ &=-\frac{3 b p}{10 a x^2}-\frac{\log \left (c \left (a+b x^3\right )^p\right )}{5 x^5}-\frac{\left (3 b^2 p\right ) \int \frac{1}{a+b x^3} \, dx}{5 a}\\ &=-\frac{3 b p}{10 a x^2}-\frac{\log \left (c \left (a+b x^3\right )^p\right )}{5 x^5}-\frac{\left (b^2 p\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{5 a^{5/3}}-\frac{\left (b^2 p\right ) \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}{5 a^{5/3}}\\ &=-\frac{3 b p}{10 a x^2}-\frac{b^{5/3} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{5 a^{5/3}}-\frac{\log \left (c \left (a+b x^3\right )^p\right )}{5 x^5}+\frac{\left (b^{5/3} 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}{10 a^{5/3}}-\frac{\left (3 b^2 p\right ) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{10 a^{4/3}}\\ &=-\frac{3 b p}{10 a x^2}-\frac{b^{5/3} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{5 a^{5/3}}+\frac{b^{5/3} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{10 a^{5/3}}-\frac{\log \left (c \left (a+b x^3\right )^p\right )}{5 x^5}-\frac{\left (3 b^{5/3} p\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{5 a^{5/3}}\\ &=-\frac{3 b p}{10 a x^2}+\frac{\sqrt{3} b^{5/3} p \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{5 a^{5/3}}-\frac{b^{5/3} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{5 a^{5/3}}+\frac{b^{5/3} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{10 a^{5/3}}-\frac{\log \left (c \left (a+b x^3\right )^p\right )}{5 x^5}\\ \end{align*}
Mathematica [C] time = 0.0026382, size = 49, normalized size = 0.32 \[ -\frac{\log \left (c \left (a+b x^3\right )^p\right )}{5 x^5}-\frac{3 b p \, _2F_1\left (-\frac{2}{3},1;\frac{1}{3};-\frac{b x^3}{a}\right )}{10 a x^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.322, size = 216, normalized size = 1.4 \begin{align*} -{\frac{\ln \left ( \left ( b{x}^{3}+a \right ) ^{p} \right ) }{5\,{x}^{5}}}-{\frac{-2\,\sum _{{\it \_R}={\it RootOf} \left ({a}^{5}{{\it \_Z}}^{3}+{b}^{5}{p}^{3} \right ) }{\it \_R}\,\ln \left ( \left ( -4\,{a}^{5}{{\it \_R}}^{3}-3\,{b}^{5}{p}^{3} \right ) x-{p}^{2}{\it \_R}\,{b}^{3}{a}^{2} \right ) a{x}^{5}+i\pi \,a{\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 \,a{\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 \,a \left ({\it csgn} \left ( ic \left ( b{x}^{3}+a \right ) ^{p} \right ) \right ) ^{3}+i\pi \,a \left ({\it csgn} \left ( ic \left ( b{x}^{3}+a \right ) ^{p} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +3\,bp{x}^{3}+2\,\ln \left ( c \right ) a}{10\,a{x}^{5}}} \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 = 2.37267, size = 412, normalized size = 2.73 \begin{align*} \frac{2 \, \sqrt{3} b p x^{5} \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 ) - b p x^{5} \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 \, b p x^{5} \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 ) - 3 \, b p x^{3} - 2 \, a p \log \left (b x^{3} + a\right ) - 2 \, a \log \left (c\right )}{10 \, a x^{5}} \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 = 1.17594, size = 208, normalized size = 1.38 \begin{align*} \frac{1}{10} \, b^{2} 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^{2}} - \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^{2} 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^{2} b}\right )} - \frac{p \log \left (b x^{3} + a\right )}{5 \, x^{5}} - \frac{3 \, b p x^{3} + 2 \, a \log \left (c\right )}{10 \, a x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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