Optimal. Leaf size=133 \[ -\frac {\sqrt [3]{a} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{b}}+x \log \left (c \left (a+b x^3\right )^p\right )+\frac {\sqrt [3]{a} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}-\frac {\sqrt {3} \sqrt [3]{a} p \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt [3]{b}}-3 p x \]
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Rubi [A] time = 0.09, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {2448, 321, 200, 31, 634, 617, 204, 628} \[ -\frac {\sqrt [3]{a} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{b}}+x \log \left (c \left (a+b x^3\right )^p\right )+\frac {\sqrt [3]{a} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}-\frac {\sqrt {3} \sqrt [3]{a} p \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt [3]{b}}-3 p x \]
Antiderivative was successfully verified.
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Rule 31
Rule 200
Rule 204
Rule 321
Rule 617
Rule 628
Rule 634
Rule 2448
Rubi steps
\begin {align*} \int \log \left (c \left (a+b x^3\right )^p\right ) \, dx &=x \log \left (c \left (a+b x^3\right )^p\right )-(3 b p) \int \frac {x^3}{a+b x^3} \, dx\\ &=-3 p x+x \log \left (c \left (a+b x^3\right )^p\right )+(3 a p) \int \frac {1}{a+b x^3} \, dx\\ &=-3 p x+x \log \left (c \left (a+b x^3\right )^p\right )+\left (\sqrt [3]{a} p\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx+\left (\sqrt [3]{a} 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\\ &=-3 p x+\frac {\sqrt [3]{a} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}+x \log \left (c \left (a+b x^3\right )^p\right )+\frac {1}{2} \left (3 a^{2/3} p\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx-\frac {\left (\sqrt [3]{a} 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}{2 \sqrt [3]{b}}\\ &=-3 p x+\frac {\sqrt [3]{a} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}-\frac {\sqrt [3]{a} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{b}}+x \log \left (c \left (a+b x^3\right )^p\right )+\frac {\left (3 \sqrt [3]{a} p\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}\\ &=-3 p x-\frac {\sqrt {3} \sqrt [3]{a} p \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt [3]{b}}+\frac {\sqrt [3]{a} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}-\frac {\sqrt [3]{a} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{b}}+x \log \left (c \left (a+b x^3\right )^p\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 129, normalized size = 0.97 \[ -\frac {\sqrt [3]{a} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{b}}+x \log \left (c \left (a+b x^3\right )^p\right )+\frac {\sqrt [3]{a} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}-\frac {\sqrt {3} \sqrt [3]{a} p \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}-3 p x \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 110, normalized size = 0.83 \[ p x \log \left (b x^{3} + a\right ) + \sqrt {3} p \left (\frac {a}{b}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} b x \left (\frac {a}{b}\right )^{\frac {2}{3}} - \sqrt {3} a}{3 \, a}\right ) - \frac {1}{2} \, p \left (\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) + p \left (\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) - 3 \, p x + x \log \relax (c) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 143, normalized size = 1.08 \[ -\frac {1}{2} \, a b p {\left (\frac {2 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{a b} - \frac {2 \, \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{a b^{2}} - \frac {\left (-a b^{2}\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{a b^{2}}\right )} + p x \log \left (b x^{3} + a\right ) - {\left (3 \, p - \log \relax (c)\right )} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 113, normalized size = 0.85 \[ \frac {\sqrt {3}\, a p \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{\left (\frac {a}{b}\right )^{\frac {2}{3}} b}+\frac {a p \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{\left (\frac {a}{b}\right )^{\frac {2}{3}} b}-\frac {a p \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{2 \left (\frac {a}{b}\right )^{\frac {2}{3}} b}-3 p x +x \ln \left (c \left (b \,x^{3}+a \right )^{p}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.01, size = 125, normalized size = 0.94 \[ -\frac {1}{2} \, b p {\left (\frac {6 \, x}{b} - \frac {2 \, \sqrt {3} a \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {a \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {2 \, a \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )} + x \log \left ({\left (b x^{3} + a\right )}^{p} c\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.00, size = 134, normalized size = 1.01 \[ x\,\ln \left (c\,{\left (b\,x^3+a\right )}^p\right )-3\,p\,x-\frac {{\left (-a\right )}^{1/3}\,p\,\ln \left ({\left (-a\right )}^{4/3}+a\,b^{1/3}\,x\right )}{b^{1/3}}+\frac {{\left (-a\right )}^{1/3}\,p\,\ln \left (2\,a\,b^{1/3}\,x-{\left (-a\right )}^{4/3}-\sqrt {3}\,{\left (-a\right )}^{4/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{b^{1/3}}-\frac {{\left (-a\right )}^{1/3}\,p\,\ln \left (2\,a\,b^{1/3}\,x-{\left (-a\right )}^{4/3}+\sqrt {3}\,{\left (-a\right )}^{4/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{b^{1/3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 60.15, size = 231, normalized size = 1.74 \[ \begin {cases} x \log {\left (0^{p} c \right )} & \text {for}\: a = 0 \wedge b = 0 \\x \log {\left (a^{p} c \right )} & \text {for}\: b = 0 \\p x \log {\relax (b )} + 3 p x \log {\relax (x )} - 3 p x + x \log {\relax (c )} & \text {for}\: a = 0 \\- \sqrt [3]{-1} \sqrt [3]{a} b p \left (\frac {1}{b}\right )^{\frac {4}{3}} \log {\left (a + b x^{3} \right )} + \frac {3 \sqrt [3]{-1} \sqrt [3]{a} b p \left (\frac {1}{b}\right )^{\frac {4}{3}} \log {\left (4 \left (-1\right )^{\frac {2}{3}} a^{\frac {2}{3}} \left (\frac {1}{b}\right )^{\frac {2}{3}} + 4 \sqrt [3]{-1} \sqrt [3]{a} x \sqrt [3]{\frac {1}{b}} + 4 x^{2} \right )}}{2} + \sqrt [3]{-1} \sqrt {3} \sqrt [3]{a} b p \left (\frac {1}{b}\right )^{\frac {4}{3}} \operatorname {atan}{\left (\frac {\sqrt {3}}{3} - \frac {2 \left (-1\right )^{\frac {2}{3}} \sqrt {3} x}{3 \sqrt [3]{a} \sqrt [3]{\frac {1}{b}}} \right )} + p x \log {\left (a + b x^{3} \right )} - 3 p x + x \log {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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