Optimal. Leaf size=73 \[ \frac {p}{9 x^3}-\frac {a p}{6 b x^2}+\frac {a^2 p}{3 b^2 x}-\frac {a^3 p \log \left (a+\frac {b}{x}\right )}{3 b^3}-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{3 x^3} \]
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Rubi [A]
time = 0.03, antiderivative size = 73, 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, 45}
\begin {gather*} -\frac {a^3 p \log \left (a+\frac {b}{x}\right )}{3 b^3}+\frac {a^2 p}{3 b^2 x}-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{3 x^3}-\frac {a p}{6 b x^2}+\frac {p}{9 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2442
Rule 2504
Rubi steps
\begin {align*} \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{x^4} \, dx &=-\text {Subst}\left (\int x^2 \log \left (c (a+b x)^p\right ) \, dx,x,\frac {1}{x}\right )\\ &=-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{3 x^3}+\frac {1}{3} (b p) \text {Subst}\left (\int \frac {x^3}{a+b x} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{3 x^3}+\frac {1}{3} (b p) \text {Subst}\left (\int \left (\frac {a^2}{b^3}-\frac {a x}{b^2}+\frac {x^2}{b}-\frac {a^3}{b^3 (a+b x)}\right ) \, dx,x,\frac {1}{x}\right )\\ &=\frac {p}{9 x^3}-\frac {a p}{6 b x^2}+\frac {a^2 p}{3 b^2 x}-\frac {a^3 p \log \left (a+\frac {b}{x}\right )}{3 b^3}-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{3 x^3}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 73, normalized size = 1.00 \begin {gather*} \frac {p}{9 x^3}-\frac {a p}{6 b x^2}+\frac {a^2 p}{3 b^2 x}-\frac {a^3 p \log \left (a+\frac {b}{x}\right )}{3 b^3}-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\ln \left (c \left (a +\frac {b}{x}\right )^{p}\right )}{x^{4}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 74, normalized size = 1.01 \begin {gather*} -\frac {1}{18} \, b p {\left (\frac {6 \, a^{3} \log \left (a x + b\right )}{b^{4}} - \frac {6 \, a^{3} \log \left (x\right )}{b^{4}} - \frac {6 \, a^{2} x^{2} - 3 \, a b x + 2 \, b^{2}}{b^{3} x^{3}}\right )} - \frac {\log \left ({\left (a + \frac {b}{x}\right )}^{p} c\right )}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 66, normalized size = 0.90 \begin {gather*} \frac {6 \, a^{2} b p x^{2} - 3 \, a b^{2} p x + 2 \, b^{3} p - 6 \, b^{3} \log \left (c\right ) - 6 \, {\left (a^{3} p x^{3} + b^{3} p\right )} \log \left (\frac {a x + b}{x}\right )}{18 \, b^{3} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.40, size = 75, normalized size = 1.03 \begin {gather*} \begin {cases} - \frac {a^{3} \log {\left (c \left (a + \frac {b}{x}\right )^{p} \right )}}{3 b^{3}} + \frac {a^{2} p}{3 b^{2} x} - \frac {a p}{6 b x^{2}} + \frac {p}{9 x^{3}} - \frac {\log {\left (c \left (a + \frac {b}{x}\right )^{p} \right )}}{3 x^{3}} & \text {for}\: b \neq 0 \\- \frac {\log {\left (a^{p} c \right )}}{3 x^{3}} & \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. 234 vs.
\(2 (63) = 126\).
time = 5.72, size = 234, normalized size = 3.21 \begin {gather*} -\frac {\frac {18 \, {\left (a x + b\right )} a^{2} p \log \left (-b {\left (\frac {a}{b} - \frac {a x + b}{b x}\right )} + a\right )}{b^{2} x} - \frac {18 \, {\left (a x + b\right )} a^{2} p}{b^{2} x} - \frac {18 \, {\left (a x + b\right )}^{2} a p \log \left (-b {\left (\frac {a}{b} - \frac {a x + b}{b x}\right )} + a\right )}{b^{2} x^{2}} + \frac {18 \, {\left (a x + b\right )} a^{2} \log \left (c\right )}{b^{2} x} + \frac {9 \, {\left (a x + b\right )}^{2} a p}{b^{2} x^{2}} + \frac {6 \, {\left (a x + b\right )}^{3} p \log \left (-b {\left (\frac {a}{b} - \frac {a x + b}{b x}\right )} + a\right )}{b^{2} x^{3}} - \frac {18 \, {\left (a x + b\right )}^{2} a \log \left (c\right )}{b^{2} x^{2}} - \frac {2 \, {\left (a x + b\right )}^{3} p}{b^{2} x^{3}} + \frac {6 \, {\left (a x + b\right )}^{3} \log \left (c\right )}{b^{2} x^{3}}}{18 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.32, size = 65, normalized size = 0.89 \begin {gather*} \frac {\frac {p}{3}+\frac {a^2\,p\,x^2}{b^2}-\frac {a\,p\,x}{2\,b}}{3\,x^3}-\frac {\ln \left (c\,{\left (a+\frac {b}{x}\right )}^p\right )}{3\,x^3}-\frac {2\,a^3\,p\,\mathrm {atanh}\left (\frac {2\,a\,x}{b}+1\right )}{3\,b^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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