Optimal. Leaf size=231 \[ -\frac {a^3 3^{-q} x^{3 m} \left (c x^n\right )^{-\frac {3 m}{n}} \log ^q\left (c x^n\right ) \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-q} \Gamma \left (q,-\frac {3 m \log \left (c x^n\right )}{n}\right )}{n}-\frac {3 a^2 b 4^{-q} x^{2 m} \left (c x^n\right )^{-\frac {2 m}{n}} \log ^{2 q}\left (c x^n\right ) \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-2 q} \Gamma \left (2 q,-\frac {2 m \log \left (c x^n\right )}{n}\right )}{n}-\frac {3 a b^2 x^m \left (c x^n\right )^{-\frac {m}{n}} \log ^{3 q}\left (c x^n\right ) \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-3 q} \Gamma \left (3 q,-\frac {m \log \left (c x^n\right )}{n}\right )}{n}+\frac {b^3 \log ^{4 q}\left (c x^n\right )}{4 n q} \]
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Rubi [A] time = 0.35, antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {2539, 2310, 2181, 2302, 30} \[ -\frac {3 a^2 b 4^{-q} x^{2 m} \left (c x^n\right )^{-\frac {2 m}{n}} \log ^{2 q}\left (c x^n\right ) \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-2 q} \text {Gamma}\left (2 q,-\frac {2 m \log \left (c x^n\right )}{n}\right )}{n}-\frac {a^3 3^{-q} x^{3 m} \left (c x^n\right )^{-\frac {3 m}{n}} \log ^q\left (c x^n\right ) \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-q} \text {Gamma}\left (q,-\frac {3 m \log \left (c x^n\right )}{n}\right )}{n}-\frac {3 a b^2 x^m \left (c x^n\right )^{-\frac {m}{n}} \log ^{3 q}\left (c x^n\right ) \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-3 q} \text {Gamma}\left (3 q,-\frac {m \log \left (c x^n\right )}{n}\right )}{n}+\frac {b^3 \log ^{4 q}\left (c x^n\right )}{4 n q} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2181
Rule 2302
Rule 2310
Rule 2539
Rubi steps
\begin {align*} \int \frac {\log ^{-1+q}\left (c x^n\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^3}{x} \, dx &=\int \left (a^3 x^{-1+3 m} \log ^{-1+q}\left (c x^n\right )+3 a^2 b x^{-1+2 m} \log ^{-1+2 q}\left (c x^n\right )+3 a b^2 x^{-1+m} \log ^{-1+3 q}\left (c x^n\right )+\frac {b^3 \log ^{-1+4 q}\left (c x^n\right )}{x}\right ) \, dx\\ &=a^3 \int x^{-1+3 m} \log ^{-1+q}\left (c x^n\right ) \, dx+\left (3 a^2 b\right ) \int x^{-1+2 m} \log ^{-1+2 q}\left (c x^n\right ) \, dx+\left (3 a b^2\right ) \int x^{-1+m} \log ^{-1+3 q}\left (c x^n\right ) \, dx+b^3 \int \frac {\log ^{-1+4 q}\left (c x^n\right )}{x} \, dx\\ &=\frac {b^3 \operatorname {Subst}\left (\int x^{-1+4 q} \, dx,x,\log \left (c x^n\right )\right )}{n}+\frac {\left (a^3 x^{3 m} \left (c x^n\right )^{-\frac {3 m}{n}}\right ) \operatorname {Subst}\left (\int e^{\frac {3 m x}{n}} x^{-1+q} \, dx,x,\log \left (c x^n\right )\right )}{n}+\frac {\left (3 a^2 b x^{2 m} \left (c x^n\right )^{-\frac {2 m}{n}}\right ) \operatorname {Subst}\left (\int e^{\frac {2 m x}{n}} x^{-1+2 q} \, dx,x,\log \left (c x^n\right )\right )}{n}+\frac {\left (3 a b^2 x^m \left (c x^n\right )^{-\frac {m}{n}}\right ) \operatorname {Subst}\left (\int e^{\frac {m x}{n}} x^{-1+3 q} \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac {b^3 \log ^{4 q}\left (c x^n\right )}{4 n q}-\frac {3 a b^2 x^m \left (c x^n\right )^{-\frac {m}{n}} \Gamma \left (3 q,-\frac {m \log \left (c x^n\right )}{n}\right ) \log ^{3 q}\left (c x^n\right ) \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-3 q}}{n}-\frac {3\ 4^{-q} a^2 b x^{2 m} \left (c x^n\right )^{-\frac {2 m}{n}} \Gamma \left (2 q,-\frac {2 m \log \left (c x^n\right )}{n}\right ) \log ^{2 q}\left (c x^n\right ) \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-2 q}}{n}-\frac {3^{-q} a^3 x^{3 m} \left (c x^n\right )^{-\frac {3 m}{n}} \Gamma \left (q,-\frac {3 m \log \left (c x^n\right )}{n}\right ) \log ^q\left (c x^n\right ) \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-q}}{n}\\ \end {align*}
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Mathematica [A] time = 0.77, size = 223, normalized size = 0.97 \[ \frac {\log ^q\left (c x^n\right ) \left (-4 a^3 3^{-q} x^{3 m} \left (c x^n\right )^{-\frac {3 m}{n}} \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-q} \Gamma \left (q,-\frac {3 m \log \left (c x^n\right )}{n}\right )-3 a^2 b 4^{1-q} x^{2 m} \left (c x^n\right )^{-\frac {2 m}{n}} \log ^q\left (c x^n\right ) \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-2 q} \Gamma \left (2 q,-\frac {2 m \log \left (c x^n\right )}{n}\right )-12 a b^2 x^m \left (c x^n\right )^{-\frac {m}{n}} \log ^{2 q}\left (c x^n\right ) \left (-\frac {m \log \left (c x^n\right )}{n}\right )^{-3 q} \Gamma \left (3 q,-\frac {m \log \left (c x^n\right )}{n}\right )+\frac {b^3 \log ^{3 q}\left (c x^n\right )}{q}\right )}{4 n} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {3 \, a b^{2} x^{m} \log \left (c x^{n}\right )^{2 \, q} \log \left (c x^{n}\right )^{q - 1} + 3 \, a^{2} b x^{2 \, m} \log \left (c x^{n}\right )^{q - 1} \log \left (c x^{n}\right )^{q} + a^{3} x^{3 \, m} \log \left (c x^{n}\right )^{q - 1} + b^{3} \log \left (c x^{n}\right )^{3 \, q} \log \left (c x^{n}\right )^{q - 1}}{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a x^{m} + b \log \left (c x^{n}\right )^{q}\right )}^{3} \log \left (c x^{n}\right )^{q - 1}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 20.38, size = 0, normalized size = 0.00 \[ \int \frac {\left (a \,x^{m}+b \ln \left (c \,x^{n}\right )^{q}\right )^{3} \ln \left (c \,x^{n}\right )^{q -1}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\ln \left (c\,x^n\right )}^{q-1}\,{\left (a\,x^m+b\,{\ln \left (c\,x^n\right )}^q\right )}^3}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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