Optimal. Leaf size=99 \[ \frac {(g+h x)^{m+1} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h (m+1)}+\frac {b f p q (g+h x)^{m+2} \, _2F_1\left (1,m+2;m+3;\frac {f (g+h x)}{f g-e h}\right )}{h (m+1) (m+2) (f g-e h)} \]
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Rubi [A] time = 0.11, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {2395, 68, 2445} \[ \frac {(g+h x)^{m+1} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h (m+1)}+\frac {b f p q (g+h x)^{m+2} \, _2F_1\left (1,m+2;m+3;\frac {f (g+h x)}{f g-e h}\right )}{h (m+1) (m+2) (f g-e h)} \]
Antiderivative was successfully verified.
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Rule 68
Rule 2395
Rule 2445
Rubi steps
\begin {align*} \int (g+h x)^m \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right ) \, dx &=\operatorname {Subst}\left (\int (g+h x)^m \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right ) \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {(g+h x)^{1+m} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h (1+m)}-\operatorname {Subst}\left (\frac {(b f p q) \int \frac {(g+h x)^{1+m}}{e+f x} \, dx}{h (1+m)},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {b f p q (g+h x)^{2+m} \, _2F_1\left (1,2+m;3+m;\frac {f (g+h x)}{f g-e h}\right )}{h (f g-e h) (1+m) (2+m)}+\frac {(g+h x)^{1+m} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 86, normalized size = 0.87 \[ \frac {(g+h x)^{m+1} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )+\frac {b f p q (g+h x) \, _2F_1\left (1,m+2;m+3;\frac {f (g+h x)}{f g-e h}\right )}{(m+2) (f g-e h)}\right )}{h (m+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (h x + g\right )}^{m} b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + {\left (h x + g\right )}^{m} a, x\right ) \]
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 \[ \int {\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )} {\left (h x + g\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.40, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )+a \right ) \left (h x +g \right )^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ b {\left (\frac {{\left (h x + g\right )} {\left (h x + g\right )}^{m} \log \left ({\left ({\left (f x + e\right )}^{p}\right )}^{q}\right )}{h {\left (m + 1\right )}} + \int -\frac {{\left (f g p q - e h {\left (m + 1\right )} \log \relax (c) - {\left (m q + q\right )} e h \log \relax (d) + {\left (f h p q - f h {\left (m + 1\right )} \log \relax (c) - {\left (m q + q\right )} f h \log \relax (d)\right )} x\right )} {\left (h x + g\right )}^{m}}{f h {\left (m + 1\right )} x + e h {\left (m + 1\right )}}\,{d x}\right )} + \frac {{\left (h x + g\right )}^{m + 1} a}{h {\left (m + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (g+h\,x\right )}^m\,\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: HeuristicGCDFailed} \]
Verification of antiderivative is not currently implemented for this CAS.
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