Optimal. Leaf size=132 \[ -\frac {e f n p x^{1+n} \, _2F_1\left (1,1+\frac {1}{n};2+\frac {1}{n};-\frac {e x^n}{d}\right )}{d (1+n)}-\frac {e g n p x^{2+n} \, _2F_1\left (1,\frac {2+n}{n};2 \left (1+\frac {1}{n}\right );-\frac {e x^n}{d}\right )}{2 d (2+n)}-\frac {f^2 p \log \left (d+e x^n\right )}{2 g}+\frac {(f+g x)^2 \log \left (c \left (d+e x^n\right )^p\right )}{2 g} \]
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Rubi [A]
time = 0.09, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2513, 1858,
266, 371} \begin {gather*} \frac {(f+g x)^2 \log \left (c \left (d+e x^n\right )^p\right )}{2 g}-\frac {f^2 p \log \left (d+e x^n\right )}{2 g}-\frac {e f n p x^{n+1} \, _2F_1\left (1,1+\frac {1}{n};2+\frac {1}{n};-\frac {e x^n}{d}\right )}{d (n+1)}-\frac {e g n p x^{n+2} \, _2F_1\left (1,\frac {n+2}{n};2 \left (1+\frac {1}{n}\right );-\frac {e x^n}{d}\right )}{2 d (n+2)} \end {gather*}
Antiderivative was successfully verified.
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Rule 266
Rule 371
Rule 1858
Rule 2513
Rubi steps
\begin {align*} \int (f+g x) \log \left (c \left (d+e x^n\right )^p\right ) \, dx &=\frac {(f+g x)^2 \log \left (c \left (d+e x^n\right )^p\right )}{2 g}-\frac {(e n p) \int \frac {x^{-1+n} (f+g x)^2}{d+e x^n} \, dx}{2 g}\\ &=\frac {(f+g x)^2 \log \left (c \left (d+e x^n\right )^p\right )}{2 g}-\frac {(e n p) \int \left (\frac {f^2 x^{-1+n}}{d+e x^n}+\frac {2 f g x^n}{d+e x^n}+\frac {g^2 x^{1+n}}{d+e x^n}\right ) \, dx}{2 g}\\ &=\frac {(f+g x)^2 \log \left (c \left (d+e x^n\right )^p\right )}{2 g}-(e f n p) \int \frac {x^n}{d+e x^n} \, dx-\frac {\left (e f^2 n p\right ) \int \frac {x^{-1+n}}{d+e x^n} \, dx}{2 g}-\frac {1}{2} (e g n p) \int \frac {x^{1+n}}{d+e x^n} \, dx\\ &=-\frac {e f n p x^{1+n} \, _2F_1\left (1,1+\frac {1}{n};2+\frac {1}{n};-\frac {e x^n}{d}\right )}{d (1+n)}-\frac {e g n p x^{2+n} \, _2F_1\left (1,\frac {2+n}{n};2 \left (1+\frac {1}{n}\right );-\frac {e x^n}{d}\right )}{2 d (2+n)}-\frac {f^2 p \log \left (d+e x^n\right )}{2 g}+\frac {(f+g x)^2 \log \left (c \left (d+e x^n\right )^p\right )}{2 g}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 130, normalized size = 0.98 \begin {gather*} -\frac {e f n p x^{1+n} \, _2F_1\left (1,\frac {1+n}{n};1+\frac {1+n}{n};-\frac {e x^n}{d}\right )}{d (1+n)}-\frac {e g n p x^{2+n} \, _2F_1\left (1,\frac {2+n}{n};1+\frac {2+n}{n};-\frac {e x^n}{d}\right )}{2 d (2+n)}+f x \log \left (c \left (d+e x^n\right )^p\right )+\frac {1}{2} g x^2 \log \left (c \left (d+e x^n\right )^p\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \left (g x +f \right ) \ln \left (c \left (d +e \,x^{n}\right )^{p}\right )\, dx\]
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 [C] Result contains complex when optimal does not.
time = 6.30, size = 162, normalized size = 1.23 \begin {gather*} f x \log {\left (c \left (d + e x^{n}\right )^{p} \right )} + \frac {f p x \Phi \left (\frac {d x^{- n} e^{i \pi }}{e}, 1, \frac {e^{i \pi }}{n}\right ) \Gamma \left (\frac {1}{n}\right )}{n \Gamma \left (1 + \frac {1}{n}\right )} + \frac {g x^{2} \log {\left (c \left (d + e x^{n}\right )^{p} \right )}}{2} - \frac {e g p x^{2} x^{n} \Phi \left (\frac {e x^{n} e^{i \pi }}{d}, 1, 1 + \frac {2}{n}\right ) \Gamma \left (1 + \frac {2}{n}\right )}{2 d \Gamma \left (2 + \frac {2}{n}\right )} - \frac {e g p x^{2} x^{n} \Phi \left (\frac {e x^{n} e^{i \pi }}{d}, 1, 1 + \frac {2}{n}\right ) \Gamma \left (1 + \frac {2}{n}\right )}{d n \Gamma \left (2 + \frac {2}{n}\right )} \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 \ln \left (c\,{\left (d+e\,x^n\right )}^p\right )\,\left (f+g\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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