Optimal. Leaf size=64 \[ -\frac {b p}{6 a x^3}-\frac {b^2 p \log (x)}{2 a^2}+\frac {b^2 p \log \left (a+b x^3\right )}{6 a^2}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{6 x^6} \]
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Rubi [A]
time = 0.04, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2504, 2442, 46}
\begin {gather*} \frac {b^2 p \log \left (a+b x^3\right )}{6 a^2}-\frac {b^2 p \log (x)}{2 a^2}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{6 x^6}-\frac {b p}{6 a x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 2442
Rule 2504
Rubi steps
\begin {align*} \int \frac {\log \left (c \left (a+b x^3\right )^p\right )}{x^7} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {\log \left (c (a+b x)^p\right )}{x^3} \, dx,x,x^3\right )\\ &=-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{6 x^6}+\frac {1}{6} (b p) \text {Subst}\left (\int \frac {1}{x^2 (a+b x)} \, dx,x,x^3\right )\\ &=-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{6 x^6}+\frac {1}{6} (b p) \text {Subst}\left (\int \left (\frac {1}{a x^2}-\frac {b}{a^2 x}+\frac {b^2}{a^2 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=-\frac {b p}{6 a x^3}-\frac {b^2 p \log (x)}{2 a^2}+\frac {b^2 p \log \left (a+b x^3\right )}{6 a^2}-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{6 x^6}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 56, normalized size = 0.88 \begin {gather*} \frac {1}{6} b p \left (-\frac {1}{a x^3}-\frac {3 b \log (x)}{a^2}+\frac {b \log \left (a+b x^3\right )}{a^2}\right )-\frac {\log \left (c \left (a+b x^3\right )^p\right )}{6 x^6} \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 = 0.26, size = 198, normalized size = 3.09
method | result | size |
risch | \(-\frac {\ln \left (\left (x^{3} b +a \right )^{p}\right )}{6 x^{6}}-\frac {6 b^{2} p \ln \left (x \right ) x^{6}-2 b^{2} p \ln \left (-x^{3} b -a \right ) x^{6}+i \pi \,a^{2} \mathrm {csgn}\left (i \left (x^{3} b +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (x^{3} b +a \right )^{p}\right )^{2}-i \pi \,a^{2} \mathrm {csgn}\left (i \left (x^{3} b +a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (x^{3} b +a \right )^{p}\right ) \mathrm {csgn}\left (i c \right )-i \pi \,a^{2} \mathrm {csgn}\left (i c \left (x^{3} b +a \right )^{p}\right )^{3}+i \pi \,a^{2} \mathrm {csgn}\left (i c \left (x^{3} b +a \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right )+2 a b p \,x^{3}+2 \ln \left (c \right ) a^{2}}{12 a^{2} x^{6}}\) | \(198\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 54, normalized size = 0.84 \begin {gather*} \frac {1}{6} \, b p {\left (\frac {b \log \left (b x^{3} + a\right )}{a^{2}} - \frac {b \log \left (x^{3}\right )}{a^{2}} - \frac {1}{a x^{3}}\right )} - \frac {\log \left ({\left (b x^{3} + a\right )}^{p} c\right )}{6 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 58, normalized size = 0.91 \begin {gather*} -\frac {3 \, b^{2} p x^{6} \log \left (x\right ) + a b p x^{3} + a^{2} \log \left (c\right ) - {\left (b^{2} p x^{6} - a^{2} p\right )} \log \left (b x^{3} + a\right )}{6 \, a^{2} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 9.27, size = 83, normalized size = 1.30 \begin {gather*} \begin {cases} - \frac {\log {\left (c \left (a + b x^{3}\right )^{p} \right )}}{6 x^{6}} - \frac {b p}{6 a x^{3}} - \frac {b^{2} p \log {\left (x \right )}}{2 a^{2}} + \frac {b^{2} \log {\left (c \left (a + b x^{3}\right )^{p} \right )}}{6 a^{2}} & \text {for}\: a \neq 0 \\- \frac {p}{12 x^{6}} - \frac {\log {\left (c \left (b x^{3}\right )^{p} \right )}}{6 x^{6}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 132 vs.
\(2 (56) = 112\).
time = 5.69, size = 132, normalized size = 2.06 \begin {gather*} -\frac {\frac {b^{3} p \log \left (b x^{3} + a\right )}{{\left (b x^{3} + a\right )}^{2} - 2 \, {\left (b x^{3} + a\right )} a + a^{2}} - \frac {b^{3} p \log \left (b x^{3} + a\right )}{a^{2}} + \frac {b^{3} p \log \left (b x^{3}\right )}{a^{2}} + \frac {{\left (b x^{3} + a\right )} b^{3} p - a b^{3} p + a b^{3} \log \left (c\right )}{{\left (b x^{3} + a\right )}^{2} a - 2 \, {\left (b x^{3} + a\right )} a^{2} + a^{3}}}{6 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.28, size = 56, normalized size = 0.88 \begin {gather*} \frac {b^2\,p\,\ln \left (b\,x^3+a\right )}{6\,a^2}-\frac {\ln \left (c\,{\left (b\,x^3+a\right )}^p\right )}{6\,x^6}-\frac {b^2\,p\,\ln \left (x\right )}{2\,a^2}-\frac {b\,p}{6\,a\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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