Integrand size = 18, antiderivative size = 112 \[ \int (d+e x)^2 \log \left (c (a+b x)^p\right ) \, dx=-\frac {(b d-a e)^2 p x}{3 b^2}-\frac {(b d-a e) p (d+e x)^2}{6 b e}-\frac {p (d+e x)^3}{9 e}-\frac {(b d-a e)^3 p \log (a+b x)}{3 b^3 e}+\frac {(d+e x)^3 \log \left (c (a+b x)^p\right )}{3 e} \]
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Time = 0.04 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2442, 45} \[ \int (d+e x)^2 \log \left (c (a+b x)^p\right ) \, dx=-\frac {p (b d-a e)^3 \log (a+b x)}{3 b^3 e}-\frac {p x (b d-a e)^2}{3 b^2}+\frac {(d+e x)^3 \log \left (c (a+b x)^p\right )}{3 e}-\frac {p (d+e x)^2 (b d-a e)}{6 b e}-\frac {p (d+e x)^3}{9 e} \]
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Rule 45
Rule 2442
Rubi steps \begin{align*} \text {integral}& = \frac {(d+e x)^3 \log \left (c (a+b x)^p\right )}{3 e}-\frac {(b p) \int \frac {(d+e x)^3}{a+b x} \, dx}{3 e} \\ & = \frac {(d+e x)^3 \log \left (c (a+b x)^p\right )}{3 e}-\frac {(b p) \int \left (\frac {e (b d-a e)^2}{b^3}+\frac {(b d-a e)^3}{b^3 (a+b x)}+\frac {e (b d-a e) (d+e x)}{b^2}+\frac {e (d+e x)^2}{b}\right ) \, dx}{3 e} \\ & = -\frac {(b d-a e)^2 p x}{3 b^2}-\frac {(b d-a e) p (d+e x)^2}{6 b e}-\frac {p (d+e x)^3}{9 e}-\frac {(b d-a e)^3 p \log (a+b x)}{3 b^3 e}+\frac {(d+e x)^3 \log \left (c (a+b x)^p\right )}{3 e} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.08 \[ \int (d+e x)^2 \log \left (c (a+b x)^p\right ) \, dx=\frac {6 a^2 e (-3 b d+a e) p \log (a+b x)+b \left (-p x \left (6 a^2 e^2-3 a b e (6 d+e x)+b^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )\right )+6 b \left (3 a d^2+b x \left (3 d^2+3 d e x+e^2 x^2\right )\right ) \log \left (c (a+b x)^p\right )\right )}{18 b^3} \]
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Time = 0.94 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.64
method | result | size |
parts | \(\frac {\ln \left (c \left (b x +a \right )^{p}\right ) e^{2} x^{3}}{3}+\ln \left (c \left (b x +a \right )^{p}\right ) e d \,x^{2}+d^{2} \ln \left (c \left (b x +a \right )^{p}\right ) x +\frac {\ln \left (c \left (b x +a \right )^{p}\right ) d^{3}}{3 e}-\frac {p b \left (\frac {e \left (\frac {1}{3} x^{3} b^{2} e^{2}-\frac {1}{2} a b \,e^{2} x^{2}+\frac {3}{2} d e \,b^{2} x^{2}+x \,a^{2} e^{2}-3 a b d e x +3 x \,b^{2} d^{2}\right )}{b^{3}}+\frac {\left (-a^{3} e^{3}+3 a^{2} b d \,e^{2}-3 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) \ln \left (b x +a \right )}{b^{4}}\right )}{3 e}\) | \(184\) |
parallelrisch | \(\frac {6 x^{3} \ln \left (c \left (b x +a \right )^{p}\right ) b^{3} e^{2}-2 x^{3} b^{3} e^{2} p +18 x^{2} \ln \left (c \left (b x +a \right )^{p}\right ) b^{3} d e +3 x^{2} a \,b^{2} e^{2} p -9 x^{2} b^{3} d e p +6 \ln \left (b x +a \right ) a^{3} e^{2} p -18 \ln \left (b x +a \right ) a^{2} b d e p +36 \ln \left (b x +a \right ) a \,b^{2} d^{2} p +18 x \ln \left (c \left (b x +a \right )^{p}\right ) b^{3} d^{2}-6 x \,a^{2} b \,e^{2} p +18 x a \,b^{2} d e p -18 b^{3} d^{2} p x -18 \ln \left (c \left (b x +a \right )^{p}\right ) a \,b^{2} d^{2}+6 a^{3} e^{2} p -18 a^{2} b d e p +18 a \,b^{2} d^{2} p}{18 b^{3}}\) | \(227\) |
risch | \(-\frac {i \pi \,d^{2} x \,\operatorname {csgn}\left (i \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}+\frac {\left (e x +d \right )^{3} \ln \left (\left (b x +a \right )^{p}\right )}{3 e}+\frac {i e^{2} \pi \,x^{3} \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2} \operatorname {csgn}\left (i c \right )}{6}+\frac {i e^{2} \pi \,x^{3} \operatorname {csgn}\left (i \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2}}{6}-\frac {i e \pi d \,x^{2} \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{3}}{2}+e \ln \left (c \right ) d \,x^{2}-\frac {\ln \left (b x +a \right ) d^{3} p}{3 e}+\frac {d^{2} p a \ln \left (b x +a \right )}{b}+\frac {e^{2} a p \,x^{2}}{6 b}-\frac {e^{2} a^{2} p x}{3 b^{2}}+\frac {e^{2} \ln \left (b x +a \right ) a^{3} p}{3 b^{3}}+\frac {e a d p x}{b}-\frac {e \ln \left (b x +a \right ) a^{2} d p}{b^{2}}+\frac {i \pi \,d^{2} x \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}+\frac {i \pi \,d^{2} x \,\operatorname {csgn}\left (i \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2}}{2}+\frac {e^{2} \ln \left (c \right ) x^{3}}{3}+\ln \left (c \right ) d^{2} x -\frac {i \pi \,d^{2} x \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{3}}{2}-\frac {i e \pi d \,x^{2} \operatorname {csgn}\left (i \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i e^{2} \pi \,x^{3} \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{3}}{6}-\frac {i e^{2} \pi \,x^{3} \operatorname {csgn}\left (i \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{6}+\frac {i e \pi d \,x^{2} \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2} \operatorname {csgn}\left (i c \right )}{2}+\frac {i e \pi d \,x^{2} \operatorname {csgn}\left (i \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2}}{2}-\frac {d e p \,x^{2}}{2}-\frac {e^{2} p \,x^{3}}{9}-d^{2} p x\) | \(537\) |
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Time = 0.31 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.54 \[ \int (d+e x)^2 \log \left (c (a+b x)^p\right ) \, dx=-\frac {2 \, b^{3} e^{2} p x^{3} + 3 \, {\left (3 \, b^{3} d e - a b^{2} e^{2}\right )} p x^{2} + 6 \, {\left (3 \, b^{3} d^{2} - 3 \, a b^{2} d e + a^{2} b e^{2}\right )} p x - 6 \, {\left (b^{3} e^{2} p x^{3} + 3 \, b^{3} d e p x^{2} + 3 \, b^{3} d^{2} p x + {\left (3 \, a b^{2} d^{2} - 3 \, a^{2} b d e + a^{3} e^{2}\right )} p\right )} \log \left (b x + a\right ) - 6 \, {\left (b^{3} e^{2} x^{3} + 3 \, b^{3} d e x^{2} + 3 \, b^{3} d^{2} x\right )} \log \left (c\right )}{18 \, b^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 202 vs. \(2 (94) = 188\).
Time = 0.57 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.80 \[ \int (d+e x)^2 \log \left (c (a+b x)^p\right ) \, dx=\begin {cases} \frac {a^{3} e^{2} \log {\left (c \left (a + b x\right )^{p} \right )}}{3 b^{3}} - \frac {a^{2} d e \log {\left (c \left (a + b x\right )^{p} \right )}}{b^{2}} - \frac {a^{2} e^{2} p x}{3 b^{2}} + \frac {a d^{2} \log {\left (c \left (a + b x\right )^{p} \right )}}{b} + \frac {a d e p x}{b} + \frac {a e^{2} p x^{2}}{6 b} - d^{2} p x + d^{2} x \log {\left (c \left (a + b x\right )^{p} \right )} - \frac {d e p x^{2}}{2} + d e x^{2} \log {\left (c \left (a + b x\right )^{p} \right )} - \frac {e^{2} p x^{3}}{9} + \frac {e^{2} x^{3} \log {\left (c \left (a + b x\right )^{p} \right )}}{3} & \text {for}\: b \neq 0 \\\left (d^{2} x + d e x^{2} + \frac {e^{2} x^{3}}{3}\right ) \log {\left (a^{p} c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.21 \[ \int (d+e x)^2 \log \left (c (a+b x)^p\right ) \, dx=-\frac {1}{18} \, b p {\left (\frac {2 \, b^{2} e^{2} x^{3} + 3 \, {\left (3 \, b^{2} d e - a b e^{2}\right )} x^{2} + 6 \, {\left (3 \, b^{2} d^{2} - 3 \, a b d e + a^{2} e^{2}\right )} x}{b^{3}} - \frac {6 \, {\left (3 \, a b^{2} d^{2} - 3 \, a^{2} b d e + a^{3} e^{2}\right )} \log \left (b x + a\right )}{b^{4}}\right )} + \frac {1}{3} \, {\left (e^{2} x^{3} + 3 \, d e x^{2} + 3 \, d^{2} x\right )} \log \left ({\left (b x + a\right )}^{p} c\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 316 vs. \(2 (102) = 204\).
Time = 0.30 (sec) , antiderivative size = 316, normalized size of antiderivative = 2.82 \[ \int (d+e x)^2 \log \left (c (a+b x)^p\right ) \, dx=\frac {{\left (b x + a\right )} d^{2} p \log \left (b x + a\right )}{b} + \frac {{\left (b x + a\right )}^{2} d e p \log \left (b x + a\right )}{b^{2}} - \frac {2 \, {\left (b x + a\right )} a d e p \log \left (b x + a\right )}{b^{2}} + \frac {{\left (b x + a\right )}^{3} e^{2} p \log \left (b x + a\right )}{3 \, b^{3}} - \frac {{\left (b x + a\right )}^{2} a e^{2} p \log \left (b x + a\right )}{b^{3}} + \frac {{\left (b x + a\right )} a^{2} e^{2} p \log \left (b x + a\right )}{b^{3}} - \frac {{\left (b x + a\right )} d^{2} p}{b} - \frac {{\left (b x + a\right )}^{2} d e p}{2 \, b^{2}} + \frac {2 \, {\left (b x + a\right )} a d e p}{b^{2}} - \frac {{\left (b x + a\right )}^{3} e^{2} p}{9 \, b^{3}} + \frac {{\left (b x + a\right )}^{2} a e^{2} p}{2 \, b^{3}} - \frac {{\left (b x + a\right )} a^{2} e^{2} p}{b^{3}} + \frac {{\left (b x + a\right )} d^{2} \log \left (c\right )}{b} + \frac {{\left (b x + a\right )}^{2} d e \log \left (c\right )}{b^{2}} - \frac {2 \, {\left (b x + a\right )} a d e \log \left (c\right )}{b^{2}} + \frac {{\left (b x + a\right )}^{3} e^{2} \log \left (c\right )}{3 \, b^{3}} - \frac {{\left (b x + a\right )}^{2} a e^{2} \log \left (c\right )}{b^{3}} + \frac {{\left (b x + a\right )} a^{2} e^{2} \log \left (c\right )}{b^{3}} \]
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Time = 1.31 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.17 \[ \int (d+e x)^2 \log \left (c (a+b x)^p\right ) \, dx=\ln \left (c\,{\left (a+b\,x\right )}^p\right )\,\left (d^2\,x+d\,e\,x^2+\frac {e^2\,x^3}{3}\right )-x^2\,\left (\frac {d\,e\,p}{2}-\frac {a\,e^2\,p}{6\,b}\right )-x\,\left (d^2\,p-\frac {a\,\left (d\,e\,p-\frac {a\,e^2\,p}{3\,b}\right )}{b}\right )-\frac {e^2\,p\,x^3}{9}+\frac {\ln \left (a+b\,x\right )\,\left (p\,a^3\,e^2-3\,p\,a^2\,b\,d\,e+3\,p\,a\,b^2\,d^2\right )}{3\,b^3} \]
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