Optimal. Leaf size=151 \[ -\frac {3 b p}{4 a x}+\frac {\sqrt {3} b^{4/3} p \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{4 a^{4/3}}+\frac {b^{4/3} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{4 a^{4/3}}-\frac {b^{4/3} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{8 a^{4/3}}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{4 x^4} \]
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Rubi [A]
time = 0.06, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2505, 331, 298,
31, 648, 631, 210, 642} \begin {gather*} \frac {\sqrt {3} b^{4/3} p \text {ArcTan}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{4 a^{4/3}}-\frac {b^{4/3} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{8 a^{4/3}}+\frac {b^{4/3} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{4 a^{4/3}}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{4 x^4}-\frac {3 b p}{4 a x} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 210
Rule 298
Rule 331
Rule 631
Rule 642
Rule 648
Rule 2505
Rubi steps
\begin {align*} \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x^5} \, dx &=-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{4 x^4}+\frac {1}{4} (3 b p) \int \frac {1}{x^2 \left (a+b x^3\right )} \, dx\\ &=-\frac {3 b p}{4 a x}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{4 x^4}-\frac {\left (3 b^2 p\right ) \int \frac {x}{a+b x^3} \, dx}{4 a}\\ &=-\frac {3 b p}{4 a x}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{4 x^4}+\frac {\left (b^{5/3} p\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{4 a^{4/3}}-\frac {\left (b^{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}{4 a^{4/3}}\\ &=-\frac {3 b p}{4 a x}+\frac {b^{4/3} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{4 a^{4/3}}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{4 x^4}-\frac {\left (b^{4/3} 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}{8 a^{4/3}}-\frac {\left (3 b^{5/3} p\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{8 a}\\ &=-\frac {3 b p}{4 a x}+\frac {b^{4/3} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{4 a^{4/3}}-\frac {b^{4/3} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{8 a^{4/3}}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{4 x^4}-\frac {\left (3 b^{4/3} p\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{4 a^{4/3}}\\ &=-\frac {3 b p}{4 a x}+\frac {\sqrt {3} b^{4/3} p \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{4 a^{4/3}}+\frac {b^{4/3} p \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{4 a^{4/3}}-\frac {b^{4/3} p \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{8 a^{4/3}}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{4 x^4}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.00, size = 49, normalized size = 0.32 \begin {gather*} -\frac {3 b p \, _2F_1\left (-\frac {1}{3},1;\frac {2}{3};-\frac {b x^3}{a}\right )}{4 a x}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{4 x^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.30, size = 215, normalized size = 1.42
method | result | size |
risch | \(-\frac {\ln \left (\left (x^{3} b +a \right )^{p}\right )}{4 x^{4}}-\frac {i \pi a \,\mathrm {csgn}\left (i \left (x^{3} b +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (x^{3} b +a \right )^{p}\right )^{2}-i \pi a \,\mathrm {csgn}\left (i \left (x^{3} b +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (x^{3} b +a \right )^{p}\right ) \mathrm {csgn}\left (i c \right )-i \pi a \mathrm {csgn}\left (i c \left (x^{3} b +a \right )^{p}\right )^{3}+i \pi a \mathrm {csgn}\left (i c \left (x^{3} b +a \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right )-2 \left (\munderset {\textit {\_R} =\RootOf \left (a^{4} \textit {\_Z}^{3}-b^{4} p^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 a^{4} \textit {\_R}^{3}+3 b^{4} p^{3}\right ) x -a^{3} b p \,\textit {\_R}^{2}\right )\right ) a \,x^{4}+6 b p \,x^{3}+2 \ln \left (c \right ) a}{8 a \,x^{4}}\) | \(215\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 127, normalized size = 0.84 \begin {gather*} -\frac {1}{8} \, b p {\left (\frac {2 \, \sqrt {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 \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {\log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{a \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {2 \, \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{a \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {6}{a x}\right )} - \frac {\log \left ({\left (b x^{3} + a\right )}^{p} c\right )}{4 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 138, normalized size = 0.91 \begin {gather*} -\frac {2 \, \sqrt {3} b p x^{4} \left (\frac {b}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {2}{3} \, \sqrt {3} x \left (\frac {b}{a}\right )^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + b p x^{4} \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x^{2} - a x \left (\frac {b}{a}\right )^{\frac {2}{3}} + a \left (\frac {b}{a}\right )^{\frac {1}{3}}\right ) - 2 \, b p x^{4} \left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x + a \left (\frac {b}{a}\right )^{\frac {2}{3}}\right ) + 6 \, b p x^{3} + 2 \, a p \log \left (b x^{3} + a\right ) + 2 \, a \log \left (c\right )}{8 \, a x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.74, size = 153, normalized size = 1.01 \begin {gather*} \frac {1}{8} \, b^{2} 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^{2}} + \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^{2} b^{2}} - \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^{2} b^{2}}\right )} - \frac {p \log \left (b x^{3} + a\right )}{4 \, x^{4}} - \frac {3 \, b p x^{3} + a \log \left (c\right )}{4 \, a x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.36, size = 125, normalized size = 0.83 \begin {gather*} \frac {b^{4/3}\,p\,\ln \left (b^{1/3}\,x+a^{1/3}\right )}{4\,a^{4/3}}-\frac {\ln \left (c\,{\left (b\,x^3+a\right )}^p\right )}{4\,x^4}-\frac {3\,b\,p}{4\,a\,x}+\frac {b^{4/3}\,p\,\ln \left (4\,b^{1/3}\,x-2\,a^{1/3}-\sqrt {3}\,a^{1/3}\,2{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{4\,a^{4/3}}-\frac {b^{4/3}\,p\,\ln \left (4\,b^{1/3}\,x-2\,a^{1/3}+\sqrt {3}\,a^{1/3}\,2{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{4\,a^{4/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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