Optimal. Leaf size=92 \[ -\frac {b e m n x^{1+m} (f x)^q \, _2F_1\left (1,\frac {1+m+q}{m};\frac {1+2 m+q}{m};-\frac {e x^m}{d}\right )}{d (1+q) (1+m+q)}+\frac {(f x)^{1+q} \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{f (1+q)} \]
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Rubi [A]
time = 0.04, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2505, 20, 371}
\begin {gather*} \frac {(f x)^{q+1} \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{f (q+1)}-\frac {b e m n x^{m+1} (f x)^q \, _2F_1\left (1,\frac {m+q+1}{m};\frac {2 m+q+1}{m};-\frac {e x^m}{d}\right )}{d (q+1) (m+q+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 20
Rule 371
Rule 2505
Rubi steps
\begin {align*} \int (f x)^q \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right ) \, dx &=\frac {(f x)^{1+q} \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{f (1+q)}-\frac {(b e m n) \int \frac {x^{-1+m} (f x)^{1+q}}{d+e x^m} \, dx}{f (1+q)}\\ &=\frac {(f x)^{1+q} \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{f (1+q)}-\frac {\left (b e m n x^{-q} (f x)^q\right ) \int \frac {x^{m+q}}{d+e x^m} \, dx}{1+q}\\ &=-\frac {b e m n x^{1+m} (f x)^q \, _2F_1\left (1,\frac {1+m+q}{m};\frac {1+2 m+q}{m};-\frac {e x^m}{d}\right )}{d (1+q) (1+m+q)}+\frac {(f x)^{1+q} \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{f (1+q)}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 82, normalized size = 0.89 \begin {gather*} \frac {x (f x)^q \left (-b e m n x^m \, _2F_1\left (1,\frac {1+m+q}{m};\frac {1+2 m+q}{m};-\frac {e x^m}{d}\right )+d (1+m+q) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )\right )}{d (1+q) (1+m+q)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.08, size = 0, normalized size = 0.00 \[\int \left (f x \right )^{q} \left (a +b \ln \left (c \left (d +e \,x^{m}\right )^{n}\right )\right )\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \left (f x\right )^{q} \left (a + b \log {\left (c \left (d + e x^{m}\right )^{n} \right )}\right )\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int {\left (f\,x\right )}^q\,\left (a+b\,\ln \left (c\,{\left (d+e\,x^m\right )}^n\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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