Optimal. Leaf size=76 \[ \frac {(f+g x) \log \left (c \left (d+e (f+g x)^p\right )^q\right )}{g}-\frac {e p q (f+g x)^{p+1} \, _2F_1\left (1,1+\frac {1}{p};2+\frac {1}{p};-\frac {e (f+g x)^p}{d}\right )}{d g (p+1)} \]
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Rubi [A] time = 0.04, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2483, 2448, 364} \[ \frac {(f+g x) \log \left (c \left (d+e (f+g x)^p\right )^q\right )}{g}-\frac {e p q (f+g x)^{p+1} \, _2F_1\left (1,1+\frac {1}{p};2+\frac {1}{p};-\frac {e (f+g x)^p}{d}\right )}{d g (p+1)} \]
Antiderivative was successfully verified.
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Rule 364
Rule 2448
Rule 2483
Rubi steps
\begin {align*} \int \log \left (c \left (d+e (f+g x)^p\right )^q\right ) \, dx &=\frac {\operatorname {Subst}\left (\int \log \left (c \left (d+e x^p\right )^q\right ) \, dx,x,f+g x\right )}{g}\\ &=\frac {(f+g x) \log \left (c \left (d+e (f+g x)^p\right )^q\right )}{g}-\frac {(e p q) \operatorname {Subst}\left (\int \frac {x^p}{d+e x^p} \, dx,x,f+g x\right )}{g}\\ &=-\frac {e p q (f+g x)^{1+p} \, _2F_1\left (1,1+\frac {1}{p};2+\frac {1}{p};-\frac {e (f+g x)^p}{d}\right )}{d g (1+p)}+\frac {(f+g x) \log \left (c \left (d+e (f+g x)^p\right )^q\right )}{g}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 65, normalized size = 0.86 \[ \frac {(f+g x) \log \left (c \left (d+e (f+g x)^p\right )^q\right )}{g}+\frac {p q (f+g x) \, _2F_1\left (1,\frac {1}{p};1+\frac {1}{p};-\frac {e (f+g x)^p}{d}\right )}{g}-p q x \]
Antiderivative was successfully verified.
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fricas [F] time = 0.87, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\log \left ({\left ({\left (g x + f\right )}^{p} e + d\right )}^{q} c\right ), x\right ) \]
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 \[ \int \log \left ({\left ({\left (g x + f\right )}^{p} e + d\right )}^{q} c\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.06, size = 0, normalized size = 0.00 \[ \int \ln \left (c \left (e \left (g x +f \right )^{p}+d \right )^{q}\right )\, dx \]
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 \[ d g p q \int \frac {x}{d g x + {\left (e g x + e f\right )} {\left (g x + f\right )}^{p} + d f}\,{d x} + \frac {f p q \log \left (g x + f\right ) + g x \log \left ({\left ({\left (g x + f\right )}^{p} e + d\right )}^{q}\right ) - {\left (g p q - g \log \relax (c)\right )} x}{g} \]
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 \[ \int \ln \left (c\,{\left (d+e\,{\left (f+g\,x\right )}^p\right )}^q\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \log {\left (c \left (d + e \left (f + g x\right )^{p}\right )^{q} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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