Optimal. Leaf size=159 \[ -\frac{a^{5/3} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{10 b^{5/3}}+\frac{a^{5/3} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{5 b^{5/3}}+\frac{\sqrt{3} a^{5/3} p \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{5 b^{5/3}}+\frac{1}{5} x^5 \log \left (c \left (a+b x^3\right )^p\right )+\frac{3 a p x^2}{10 b}-\frac{3 p x^5}{25} \]
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Rubi [A] time = 0.129039, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {2455, 302, 292, 31, 634, 617, 204, 628} \[ -\frac{a^{5/3} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{10 b^{5/3}}+\frac{a^{5/3} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{5 b^{5/3}}+\frac{\sqrt{3} a^{5/3} p \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{5 b^{5/3}}+\frac{1}{5} x^5 \log \left (c \left (a+b x^3\right )^p\right )+\frac{3 a p x^2}{10 b}-\frac{3 p x^5}{25} \]
Antiderivative was successfully verified.
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Rule 2455
Rule 302
Rule 292
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int x^4 \log \left (c \left (a+b x^3\right )^p\right ) \, dx &=\frac{1}{5} x^5 \log \left (c \left (a+b x^3\right )^p\right )-\frac{1}{5} (3 b p) \int \frac{x^7}{a+b x^3} \, dx\\ &=\frac{1}{5} x^5 \log \left (c \left (a+b x^3\right )^p\right )-\frac{1}{5} (3 b p) \int \left (-\frac{a x}{b^2}+\frac{x^4}{b}+\frac{a^2 x}{b^2 \left (a+b x^3\right )}\right ) \, dx\\ &=\frac{3 a p x^2}{10 b}-\frac{3 p x^5}{25}+\frac{1}{5} x^5 \log \left (c \left (a+b x^3\right )^p\right )-\frac{\left (3 a^2 p\right ) \int \frac{x}{a+b x^3} \, dx}{5 b}\\ &=\frac{3 a p x^2}{10 b}-\frac{3 p x^5}{25}+\frac{1}{5} x^5 \log \left (c \left (a+b x^3\right )^p\right )+\frac{\left (a^{5/3} p\right ) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{5 b^{4/3}}-\frac{\left (a^{5/3} p\right ) \int \frac{\sqrt [3]{a}+\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{5 b^{4/3}}\\ &=\frac{3 a p x^2}{10 b}-\frac{3 p x^5}{25}+\frac{a^{5/3} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{5 b^{5/3}}+\frac{1}{5} x^5 \log \left (c \left (a+b x^3\right )^p\right )-\frac{\left (a^{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 b^{5/3}}-\frac{\left (3 a^2 p\right ) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{10 b^{4/3}}\\ &=\frac{3 a p x^2}{10 b}-\frac{3 p x^5}{25}+\frac{a^{5/3} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{5 b^{5/3}}-\frac{a^{5/3} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{10 b^{5/3}}+\frac{1}{5} x^5 \log \left (c \left (a+b x^3\right )^p\right )-\frac{\left (3 a^{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 b^{5/3}}\\ &=\frac{3 a p x^2}{10 b}-\frac{3 p x^5}{25}+\frac{\sqrt{3} a^{5/3} p \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{5 b^{5/3}}+\frac{a^{5/3} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{5 b^{5/3}}-\frac{a^{5/3} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{10 b^{5/3}}+\frac{1}{5} x^5 \log \left (c \left (a+b x^3\right )^p\right )\\ \end{align*}
Mathematica [C] time = 0.0032979, size = 69, normalized size = 0.43 \[ \frac{1}{5} x^5 \log \left (c \left (a+b x^3\right )^p\right )-\frac{3 a p x^2 \, _2F_1\left (\frac{2}{3},1;\frac{5}{3};-\frac{b x^3}{a}\right )}{10 b}+\frac{3 a p x^2}{10 b}-\frac{3 p x^5}{25} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.452, size = 196, normalized size = 1.2 \begin{align*}{\frac{{x}^{5}\ln \left ( \left ( b{x}^{3}+a \right ) ^{p} \right ) }{5}}-{\frac{i}{10}}\pi \,{x}^{5}{\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 ) +{\frac{i}{10}}\pi \,{x}^{5} \left ({\it csgn} \left ( ic \left ( b{x}^{3}+a \right ) ^{p} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +{\frac{i}{10}}\pi \,{x}^{5}{\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}-{\frac{i}{10}}\pi \,{x}^{5} \left ({\it csgn} \left ( ic \left ( b{x}^{3}+a \right ) ^{p} \right ) \right ) ^{3}+{\frac{\ln \left ( c \right ){x}^{5}}{5}}-{\frac{3\,p{x}^{5}}{25}}+{\frac{3\,ap{x}^{2}}{10\,b}}-{\frac{{a}^{2}p}{5\,{b}^{2}}\sum _{{\it \_R}={\it RootOf} \left ( b{{\it \_Z}}^{3}+a \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{{\it \_R}}}} \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 = 2.14814, size = 406, normalized size = 2.55 \begin{align*} \frac{10 \, b p x^{5} \log \left (b x^{3} + a\right ) - 6 \, b p x^{5} + 10 \, b x^{5} \log \left (c\right ) + 15 \, a p x^{2} - 10 \, \sqrt{3} a p \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} b x \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} - \sqrt{3} a}{3 \, a}\right ) - 5 \, a p \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \log \left (a x^{2} - b x \left (\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}} + a \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}}\right ) + 10 \, a p \left (\frac{a^{2}}{b^{2}}\right )^{\frac{1}{3}} \log \left (a x + b \left (\frac{a^{2}}{b^{2}}\right )^{\frac{2}{3}}\right )}{50 \, b} \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 = 1.27326, size = 219, normalized size = 1.38 \begin{align*} \frac{1}{10} \, a^{2} b^{4} p{\left (\frac{2 \, \left (-\frac{a}{b}\right )^{\frac{2}{3}} \log \left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{a b^{5}} + \frac{2 \, \sqrt{3} \left (-a b^{2}\right )^{\frac{2}{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^{7}} - \frac{\left (-a b^{2}\right )^{\frac{2}{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^{7}}\right )} + \frac{1}{5} \, p x^{5} \log \left (b x^{3} + a\right ) - \frac{1}{25} \,{\left (3 \, p - 5 \, \log \left (c\right )\right )} x^{5} + \frac{3 \, a p x^{2}}{10 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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