Optimal. Leaf size=84 \[ -\frac {p (b d-a e)^2 \log (a+b x)}{2 b^2 e}+\frac {(d+e x)^2 \log \left (c (a+b x)^p\right )}{2 e}-\frac {p x (b d-a e)}{2 b}-\frac {p (d+e x)^2}{4 e} \]
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Rubi [A] time = 0.04, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2395, 43} \[ -\frac {p (b d-a e)^2 \log (a+b x)}{2 b^2 e}+\frac {(d+e x)^2 \log \left (c (a+b x)^p\right )}{2 e}-\frac {p x (b d-a e)}{2 b}-\frac {p (d+e x)^2}{4 e} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2395
Rubi steps
\begin {align*} \int (d+e x) \log \left (c (a+b x)^p\right ) \, dx &=\frac {(d+e x)^2 \log \left (c (a+b x)^p\right )}{2 e}-\frac {(b p) \int \frac {(d+e x)^2}{a+b x} \, dx}{2 e}\\ &=\frac {(d+e x)^2 \log \left (c (a+b x)^p\right )}{2 e}-\frac {(b p) \int \left (\frac {e (b d-a e)}{b^2}+\frac {(b d-a e)^2}{b^2 (a+b x)}+\frac {e (d+e x)}{b}\right ) \, dx}{2 e}\\ &=-\frac {(b d-a e) p x}{2 b}-\frac {p (d+e x)^2}{4 e}-\frac {(b d-a e)^2 p \log (a+b x)}{2 b^2 e}+\frac {(d+e x)^2 \log \left (c (a+b x)^p\right )}{2 e}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 82, normalized size = 0.98 \[ -\frac {a^2 e p \log (a+b x)}{2 b^2}+\frac {d (a+b x) \log \left (c (a+b x)^p\right )}{b}+\frac {1}{2} e x^2 \log \left (c (a+b x)^p\right )+\frac {a e p x}{2 b}-d p x-\frac {1}{4} e p x^2 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 91, normalized size = 1.08 \[ -\frac {b^{2} e p x^{2} + 2 \, {\left (2 \, b^{2} d - a b e\right )} p x - 2 \, {\left (b^{2} e p x^{2} + 2 \, b^{2} d p x + {\left (2 \, a b d - a^{2} e\right )} p\right )} \log \left (b x + a\right ) - 2 \, {\left (b^{2} e x^{2} + 2 \, b^{2} d x\right )} \log \relax (c)}{4 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 142, normalized size = 1.69 \[ \frac {{\left (b x + a\right )} d p \log \left (b x + a\right )}{b} + \frac {{\left (b x + a\right )}^{2} p e \log \left (b x + a\right )}{2 \, b^{2}} - \frac {{\left (b x + a\right )} a p e \log \left (b x + a\right )}{b^{2}} - \frac {{\left (b x + a\right )} d p}{b} - \frac {{\left (b x + a\right )}^{2} p e}{4 \, b^{2}} + \frac {{\left (b x + a\right )} a p e}{b^{2}} + \frac {{\left (b x + a\right )} d \log \relax (c)}{b} + \frac {{\left (b x + a\right )}^{2} e \log \relax (c)}{2 \, b^{2}} - \frac {{\left (b x + a\right )} a e \log \relax (c)}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 83, normalized size = 0.99 \[ -\frac {e p \,x^{2}}{4}+\frac {e \,x^{2} \ln \left (c \,{\mathrm e}^{p \ln \left (b x +a \right )}\right )}{2}-\frac {a^{2} e p \ln \left (b x +a \right )}{2 b^{2}}+\frac {a d p \ln \left (b x +a \right )}{b}+\frac {a e p x}{2 b}-d p x +d x \ln \left (c \left (b x +a \right )^{p}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 74, normalized size = 0.88 \[ -\frac {1}{4} \, b p {\left (\frac {b e x^{2} + 2 \, {\left (2 \, b d - a e\right )} x}{b^{2}} - \frac {2 \, {\left (2 \, a b d - a^{2} e\right )} \log \left (b x + a\right )}{b^{3}}\right )} + \frac {1}{2} \, {\left (e x^{2} + 2 \, d x\right )} \log \left ({\left (b x + a\right )}^{p} c\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.25, size = 68, normalized size = 0.81 \[ \ln \left (c\,{\left (a+b\,x\right )}^p\right )\,\left (\frac {e\,x^2}{2}+d\,x\right )-x\,\left (d\,p-\frac {a\,e\,p}{2\,b}\right )-\frac {e\,p\,x^2}{4}-\frac {\ln \left (a+b\,x\right )\,\left (a^2\,e\,p-2\,a\,b\,d\,p\right )}{2\,b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.38, size = 116, normalized size = 1.38 \[ \begin {cases} - \frac {a^{2} e p \log {\left (a + b x \right )}}{2 b^{2}} + \frac {a d p \log {\left (a + b x \right )}}{b} + \frac {a e p x}{2 b} + d p x \log {\left (a + b x \right )} - d p x + d x \log {\relax (c )} + \frac {e p x^{2} \log {\left (a + b x \right )}}{2} - \frac {e p x^{2}}{4} + \frac {e x^{2} \log {\relax (c )}}{2} & \text {for}\: b \neq 0 \\\left (d x + \frac {e x^{2}}{2}\right ) \log {\left (a^{p} c \right )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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