Integrand size = 27, antiderivative size = 116 \[ \int (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=-\frac {1}{2} a p r x+\frac {(b c-a d) q r x}{2 d}-\frac {1}{4} b p r x^2-\frac {q r (a+b x)^2}{4 b}-\frac {(b c-a d)^2 q r \log (c+d x)}{2 b d^2}+\frac {(a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b} \]
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Time = 0.03 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2581, 45} \[ \int (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=-\frac {q r (b c-a d)^2 \log (c+d x)}{2 b d^2}+\frac {(a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}+\frac {q r x (b c-a d)}{2 d}-\frac {q r (a+b x)^2}{4 b}-\frac {1}{2} a p r x-\frac {1}{4} b p r x^2 \]
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Rule 45
Rule 2581
Rubi steps \begin{align*} \text {integral}& = \frac {(a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}-\frac {1}{2} (p r) \int (a+b x) \, dx-\frac {(d q r) \int \frac {(a+b x)^2}{c+d x} \, dx}{2 b} \\ & = -\frac {1}{2} a p r x-\frac {1}{4} b p r x^2+\frac {(a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b}-\frac {(d q r) \int \left (-\frac {b (b c-a d)}{d^2}+\frac {b (a+b x)}{d}+\frac {(-b c+a d)^2}{d^2 (c+d x)}\right ) \, dx}{2 b} \\ & = -\frac {1}{2} a p r x+\frac {(b c-a d) q r x}{2 d}-\frac {1}{4} b p r x^2-\frac {q r (a+b x)^2}{4 b}-\frac {(b c-a d)^2 q r \log (c+d x)}{2 b d^2}+\frac {(a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.91 \[ \int (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\frac {a^2 p r \log (a+b x)}{2 b}-\frac {2 c (b c-2 a d) q r \log (c+d x)+d x \left (r (-2 b c q+2 a d (p+2 q)+b d (p+q) x)-2 d (2 a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{4 d^2} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(304\) vs. \(2(104)=208\).
Time = 12.56 (sec) , antiderivative size = 305, normalized size of antiderivative = 2.63
method | result | size |
parallelrisch | \(\frac {-x^{2} b^{2} d^{2} p r -x^{2} b^{2} d^{2} q r +6 \ln \left (b x +a \right ) a^{2} d^{2} p r +6 \ln \left (b x +a \right ) a b c d p r +4 \ln \left (d x +c \right ) a^{2} d^{2} q r +10 \ln \left (d x +c \right ) a b c d q r -2 \ln \left (d x +c \right ) b^{2} c^{2} q r +2 x^{2} \ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right ) b^{2} d^{2}-2 x a b \,d^{2} p r -4 x a b \,d^{2} q r +2 x \,b^{2} c d q r +4 x \ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right ) a b \,d^{2}+2 a^{2} d^{2} p r +4 a^{2} q r \,d^{2}+3 a b c d p r +3 a b c d q r -2 b^{2} c^{2} q r -4 \ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right ) a^{2} d^{2}-6 \ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right ) a b c d}{4 b \,d^{2}}\) | \(305\) |
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Time = 0.32 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.70 \[ \int (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=-\frac {{\left (b^{2} d^{2} p + b^{2} d^{2} q\right )} r x^{2} + 2 \, {\left (a b d^{2} p - {\left (b^{2} c d - 2 \, a b d^{2}\right )} q\right )} r x - 2 \, {\left (b^{2} d^{2} p r x^{2} + 2 \, a b d^{2} p r x + a^{2} d^{2} p r\right )} \log \left (b x + a\right ) - 2 \, {\left (b^{2} d^{2} q r x^{2} + 2 \, a b d^{2} q r x - {\left (b^{2} c^{2} - 2 \, a b c d\right )} q r\right )} \log \left (d x + c\right ) - 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x\right )} \log \left (e\right ) - 2 \, {\left (b^{2} d^{2} r x^{2} + 2 \, a b d^{2} r x\right )} \log \left (f\right )}{4 \, b d^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 325 vs. \(2 (104) = 208\).
Time = 20.45 (sec) , antiderivative size = 325, normalized size of antiderivative = 2.80 \[ \int (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\begin {cases} a x \log {\left (e \left (a^{p} c^{q} f\right )^{r} \right )} & \text {for}\: b = 0 \wedge d = 0 \\a \left (\frac {c \log {\left (e \left (a^{p} f \left (c + d x\right )^{q}\right )^{r} \right )}}{d} - q r x + x \log {\left (e \left (a^{p} f \left (c + d x\right )^{q}\right )^{r} \right )}\right ) & \text {for}\: b = 0 \\\frac {a^{2} \log {\left (e \left (c^{q} f \left (a + b x\right )^{p}\right )^{r} \right )}}{2 b} - \frac {a p r x}{2} + a x \log {\left (e \left (c^{q} f \left (a + b x\right )^{p}\right )^{r} \right )} - \frac {b p r x^{2}}{4} + \frac {b x^{2} \log {\left (e \left (c^{q} f \left (a + b x\right )^{p}\right )^{r} \right )}}{2} & \text {for}\: d = 0 \\- \frac {a^{2} q r \log {\left (\frac {c}{d} + x \right )}}{2 b} + \frac {a^{2} \log {\left (e \left (f \left (a + b x\right )^{p} \left (c + d x\right )^{q}\right )^{r} \right )}}{2 b} + \frac {a c q r \log {\left (\frac {c}{d} + x \right )}}{d} - \frac {a p r x}{2} - a q r x + a x \log {\left (e \left (f \left (a + b x\right )^{p} \left (c + d x\right )^{q}\right )^{r} \right )} - \frac {b c^{2} q r \log {\left (\frac {c}{d} + x \right )}}{2 d^{2}} + \frac {b c q r x}{2 d} - \frac {b p r x^{2}}{4} - \frac {b q r x^{2}}{4} + \frac {b x^{2} \log {\left (e \left (f \left (a + b x\right )^{p} \left (c + d x\right )^{q}\right )^{r} \right )}}{2} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.02 \[ \int (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\frac {1}{2} \, {\left (b x^{2} + 2 \, a x\right )} \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right ) + \frac {{\left (\frac {2 \, a^{2} f p \log \left (b x + a\right )}{b} - \frac {b d f {\left (p + q\right )} x^{2} + 2 \, {\left (a d f {\left (p + 2 \, q\right )} - b c f q\right )} x}{d} - \frac {2 \, {\left (b c^{2} f q - 2 \, a c d f q\right )} \log \left (d x + c\right )}{d^{2}}\right )} r}{4 \, f} \]
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Time = 0.50 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.31 \[ \int (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\frac {a^{2} p r \log \left (b x + a\right )}{2 \, b} - \frac {1}{4} \, {\left (b p r + b q r - 2 \, b r \log \left (f\right ) - 2 \, b \log \left (e\right )\right )} x^{2} + \frac {1}{2} \, {\left (b p r x^{2} + 2 \, a p r x\right )} \log \left (b x + a\right ) + \frac {1}{2} \, {\left (b q r x^{2} + 2 \, a q r x\right )} \log \left (d x + c\right ) - \frac {{\left (a d p r - b c q r + 2 \, a d q r - 2 \, a d r \log \left (f\right ) - 2 \, a d \log \left (e\right )\right )} x}{2 \, d} - \frac {{\left (b c^{2} q r - 2 \, a c d q r\right )} \log \left (-d x - c\right )}{2 \, d^{2}} \]
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Time = 1.42 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.10 \[ \int (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\ln \left (e\,{\left (f\,{\left (a+b\,x\right )}^p\,{\left (c+d\,x\right )}^q\right )}^r\right )\,\left (\frac {b\,x^2}{2}+a\,x\right )-x\,\left (\frac {r\,\left (2\,a\,d\,p+b\,c\,p+3\,a\,d\,q\right )}{2\,d}-\frac {r\,\left (p+q\right )\,\left (2\,a\,d+2\,b\,c\right )}{4\,d}\right )-\frac {\ln \left (c+d\,x\right )\,\left (b\,c^2\,q\,r-2\,a\,c\,d\,q\,r\right )}{2\,d^2}-\frac {b\,r\,x^2\,\left (p+q\right )}{4}+\frac {a^2\,p\,r\,\ln \left (a+b\,x\right )}{2\,b} \]
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