Integrand size = 16, antiderivative size = 285 \[ \int x^2 \log ^3\left (c (a+b x)^n\right ) \, dx=-\frac {6 a^2 n^3 x}{b^2}+\frac {3 a n^3 (a+b x)^2}{4 b^3}-\frac {2 n^3 (a+b x)^3}{27 b^3}+\frac {6 a^2 n^2 (a+b x) \log \left (c (a+b x)^n\right )}{b^3}-\frac {3 a n^2 (a+b x)^2 \log \left (c (a+b x)^n\right )}{2 b^3}+\frac {2 n^2 (a+b x)^3 \log \left (c (a+b x)^n\right )}{9 b^3}-\frac {3 a^2 n (a+b x) \log ^2\left (c (a+b x)^n\right )}{b^3}+\frac {3 a n (a+b x)^2 \log ^2\left (c (a+b x)^n\right )}{2 b^3}-\frac {n (a+b x)^3 \log ^2\left (c (a+b x)^n\right )}{3 b^3}+\frac {a^2 (a+b x) \log ^3\left (c (a+b x)^n\right )}{b^3}-\frac {a (a+b x)^2 \log ^3\left (c (a+b x)^n\right )}{b^3}+\frac {(a+b x)^3 \log ^3\left (c (a+b x)^n\right )}{3 b^3} \]
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Time = 0.14 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {2448, 2436, 2333, 2332, 2437, 2342, 2341} \[ \int x^2 \log ^3\left (c (a+b x)^n\right ) \, dx=\frac {6 a^2 n^2 (a+b x) \log \left (c (a+b x)^n\right )}{b^3}+\frac {a^2 (a+b x) \log ^3\left (c (a+b x)^n\right )}{b^3}-\frac {3 a^2 n (a+b x) \log ^2\left (c (a+b x)^n\right )}{b^3}-\frac {6 a^2 n^3 x}{b^2}+\frac {2 n^2 (a+b x)^3 \log \left (c (a+b x)^n\right )}{9 b^3}-\frac {3 a n^2 (a+b x)^2 \log \left (c (a+b x)^n\right )}{2 b^3}+\frac {(a+b x)^3 \log ^3\left (c (a+b x)^n\right )}{3 b^3}-\frac {a (a+b x)^2 \log ^3\left (c (a+b x)^n\right )}{b^3}-\frac {n (a+b x)^3 \log ^2\left (c (a+b x)^n\right )}{3 b^3}+\frac {3 a n (a+b x)^2 \log ^2\left (c (a+b x)^n\right )}{2 b^3}-\frac {2 n^3 (a+b x)^3}{27 b^3}+\frac {3 a n^3 (a+b x)^2}{4 b^3} \]
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Rule 2332
Rule 2333
Rule 2341
Rule 2342
Rule 2436
Rule 2437
Rule 2448
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^2 \log ^3\left (c (a+b x)^n\right )}{b^2}-\frac {2 a (a+b x) \log ^3\left (c (a+b x)^n\right )}{b^2}+\frac {(a+b x)^2 \log ^3\left (c (a+b x)^n\right )}{b^2}\right ) \, dx \\ & = \frac {\int (a+b x)^2 \log ^3\left (c (a+b x)^n\right ) \, dx}{b^2}-\frac {(2 a) \int (a+b x) \log ^3\left (c (a+b x)^n\right ) \, dx}{b^2}+\frac {a^2 \int \log ^3\left (c (a+b x)^n\right ) \, dx}{b^2} \\ & = \frac {\text {Subst}\left (\int x^2 \log ^3\left (c x^n\right ) \, dx,x,a+b x\right )}{b^3}-\frac {(2 a) \text {Subst}\left (\int x \log ^3\left (c x^n\right ) \, dx,x,a+b x\right )}{b^3}+\frac {a^2 \text {Subst}\left (\int \log ^3\left (c x^n\right ) \, dx,x,a+b x\right )}{b^3} \\ & = \frac {a^2 (a+b x) \log ^3\left (c (a+b x)^n\right )}{b^3}-\frac {a (a+b x)^2 \log ^3\left (c (a+b x)^n\right )}{b^3}+\frac {(a+b x)^3 \log ^3\left (c (a+b x)^n\right )}{3 b^3}-\frac {n \text {Subst}\left (\int x^2 \log ^2\left (c x^n\right ) \, dx,x,a+b x\right )}{b^3}+\frac {(3 a n) \text {Subst}\left (\int x \log ^2\left (c x^n\right ) \, dx,x,a+b x\right )}{b^3}-\frac {\left (3 a^2 n\right ) \text {Subst}\left (\int \log ^2\left (c x^n\right ) \, dx,x,a+b x\right )}{b^3} \\ & = -\frac {3 a^2 n (a+b x) \log ^2\left (c (a+b x)^n\right )}{b^3}+\frac {3 a n (a+b x)^2 \log ^2\left (c (a+b x)^n\right )}{2 b^3}-\frac {n (a+b x)^3 \log ^2\left (c (a+b x)^n\right )}{3 b^3}+\frac {a^2 (a+b x) \log ^3\left (c (a+b x)^n\right )}{b^3}-\frac {a (a+b x)^2 \log ^3\left (c (a+b x)^n\right )}{b^3}+\frac {(a+b x)^3 \log ^3\left (c (a+b x)^n\right )}{3 b^3}+\frac {\left (2 n^2\right ) \text {Subst}\left (\int x^2 \log \left (c x^n\right ) \, dx,x,a+b x\right )}{3 b^3}-\frac {\left (3 a n^2\right ) \text {Subst}\left (\int x \log \left (c x^n\right ) \, dx,x,a+b x\right )}{b^3}+\frac {\left (6 a^2 n^2\right ) \text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,a+b x\right )}{b^3} \\ & = -\frac {6 a^2 n^3 x}{b^2}+\frac {3 a n^3 (a+b x)^2}{4 b^3}-\frac {2 n^3 (a+b x)^3}{27 b^3}+\frac {6 a^2 n^2 (a+b x) \log \left (c (a+b x)^n\right )}{b^3}-\frac {3 a n^2 (a+b x)^2 \log \left (c (a+b x)^n\right )}{2 b^3}+\frac {2 n^2 (a+b x)^3 \log \left (c (a+b x)^n\right )}{9 b^3}-\frac {3 a^2 n (a+b x) \log ^2\left (c (a+b x)^n\right )}{b^3}+\frac {3 a n (a+b x)^2 \log ^2\left (c (a+b x)^n\right )}{2 b^3}-\frac {n (a+b x)^3 \log ^2\left (c (a+b x)^n\right )}{3 b^3}+\frac {a^2 (a+b x) \log ^3\left (c (a+b x)^n\right )}{b^3}-\frac {a (a+b x)^2 \log ^3\left (c (a+b x)^n\right )}{b^3}+\frac {(a+b x)^3 \log ^3\left (c (a+b x)^n\right )}{3 b^3} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 260, normalized size of antiderivative = 0.91 \[ \int x^2 \log ^3\left (c (a+b x)^n\right ) \, dx=-\frac {85 a^2 n^3 x}{18 b^2}+\frac {19 a n^3 x^2}{36 b}-\frac {2 n^3 x^3}{27}+\frac {85 a^3 n^2 \log \left (c (a+b x)^n\right )}{18 b^3}+\frac {11 a^2 n^2 x \log \left (c (a+b x)^n\right )}{3 b^2}-\frac {5 a n^2 x^2 \log \left (c (a+b x)^n\right )}{6 b}+\frac {2}{9} n^2 x^3 \log \left (c (a+b x)^n\right )-\frac {11 a^3 n \log ^2\left (c (a+b x)^n\right )}{6 b^3}-\frac {a^2 n x \log ^2\left (c (a+b x)^n\right )}{b^2}+\frac {a n x^2 \log ^2\left (c (a+b x)^n\right )}{2 b}-\frac {1}{3} n x^3 \log ^2\left (c (a+b x)^n\right )+\frac {a^3 \log ^3\left (c (a+b x)^n\right )}{3 b^3}+\frac {1}{3} x^3 \log ^3\left (c (a+b x)^n\right ) \]
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Time = 0.83 (sec) , antiderivative size = 261, normalized size of antiderivative = 0.92
method | result | size |
parallelrisch | \(\frac {36 x^{3} \ln \left (c \left (b x +a \right )^{n}\right )^{3} b^{3}-36 x^{3} \ln \left (c \left (b x +a \right )^{n}\right )^{2} b^{3} n +24 x^{3} \ln \left (c \left (b x +a \right )^{n}\right ) b^{3} n^{2}-8 b^{3} n^{3} x^{3}+54 x^{2} \ln \left (c \left (b x +a \right )^{n}\right )^{2} a \,b^{2} n -90 x^{2} \ln \left (c \left (b x +a \right )^{n}\right ) a \,b^{2} n^{2}+57 a \,b^{2} n^{3} x^{2}+906 \ln \left (b x +a \right ) a^{3} n^{3}-108 x \ln \left (c \left (b x +a \right )^{n}\right )^{2} a^{2} b n +396 x \ln \left (c \left (b x +a \right )^{n}\right ) a^{2} b \,n^{2}-510 a^{2} b \,n^{3} x +36 \ln \left (c \left (b x +a \right )^{n}\right )^{3} a^{3}-198 \ln \left (c \left (b x +a \right )^{n}\right )^{2} a^{3} n -396 \ln \left (c \left (b x +a \right )^{n}\right ) a^{3} n^{2}+510 a^{3} n^{3}}{108 b^{3}}\) | \(261\) |
risch | \(\text {Expression too large to display}\) | \(5345\) |
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Time = 0.31 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.20 \[ \int x^2 \log ^3\left (c (a+b x)^n\right ) \, dx=-\frac {8 \, b^{3} n^{3} x^{3} - 36 \, b^{3} x^{3} \log \left (c\right )^{3} - 57 \, a b^{2} n^{3} x^{2} + 510 \, a^{2} b n^{3} x - 36 \, {\left (b^{3} n^{3} x^{3} + a^{3} n^{3}\right )} \log \left (b x + a\right )^{3} + 18 \, {\left (2 \, b^{3} n^{3} x^{3} - 3 \, a b^{2} n^{3} x^{2} + 6 \, a^{2} b n^{3} x + 11 \, a^{3} n^{3} - 6 \, {\left (b^{3} n^{2} x^{3} + a^{3} n^{2}\right )} \log \left (c\right )\right )} \log \left (b x + a\right )^{2} + 18 \, {\left (2 \, b^{3} n x^{3} - 3 \, a b^{2} n x^{2} + 6 \, a^{2} b n x\right )} \log \left (c\right )^{2} - 6 \, {\left (4 \, b^{3} n^{3} x^{3} - 15 \, a b^{2} n^{3} x^{2} + 66 \, a^{2} b n^{3} x + 85 \, a^{3} n^{3} + 18 \, {\left (b^{3} n x^{3} + a^{3} n\right )} \log \left (c\right )^{2} - 6 \, {\left (2 \, b^{3} n^{2} x^{3} - 3 \, a b^{2} n^{2} x^{2} + 6 \, a^{2} b n^{2} x + 11 \, a^{3} n^{2}\right )} \log \left (c\right )\right )} \log \left (b x + a\right ) - 6 \, {\left (4 \, b^{3} n^{2} x^{3} - 15 \, a b^{2} n^{2} x^{2} + 66 \, a^{2} b n^{2} x\right )} \log \left (c\right )}{108 \, b^{3}} \]
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Time = 1.37 (sec) , antiderivative size = 269, normalized size of antiderivative = 0.94 \[ \int x^2 \log ^3\left (c (a+b x)^n\right ) \, dx=\begin {cases} \frac {85 a^{3} n^{2} \log {\left (c \left (a + b x\right )^{n} \right )}}{18 b^{3}} - \frac {11 a^{3} n \log {\left (c \left (a + b x\right )^{n} \right )}^{2}}{6 b^{3}} + \frac {a^{3} \log {\left (c \left (a + b x\right )^{n} \right )}^{3}}{3 b^{3}} - \frac {85 a^{2} n^{3} x}{18 b^{2}} + \frac {11 a^{2} n^{2} x \log {\left (c \left (a + b x\right )^{n} \right )}}{3 b^{2}} - \frac {a^{2} n x \log {\left (c \left (a + b x\right )^{n} \right )}^{2}}{b^{2}} + \frac {19 a n^{3} x^{2}}{36 b} - \frac {5 a n^{2} x^{2} \log {\left (c \left (a + b x\right )^{n} \right )}}{6 b} + \frac {a n x^{2} \log {\left (c \left (a + b x\right )^{n} \right )}^{2}}{2 b} - \frac {2 n^{3} x^{3}}{27} + \frac {2 n^{2} x^{3} \log {\left (c \left (a + b x\right )^{n} \right )}}{9} - \frac {n x^{3} \log {\left (c \left (a + b x\right )^{n} \right )}^{2}}{3} + \frac {x^{3} \log {\left (c \left (a + b x\right )^{n} \right )}^{3}}{3} & \text {for}\: b \neq 0 \\\frac {x^{3} \log {\left (a^{n} c \right )}^{3}}{3} & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.75 \[ \int x^2 \log ^3\left (c (a+b x)^n\right ) \, dx=\frac {1}{3} \, x^{3} \log \left ({\left (b x + a\right )}^{n} c\right )^{3} + \frac {1}{6} \, b n {\left (\frac {6 \, a^{3} \log \left (b x + a\right )}{b^{4}} - \frac {2 \, b^{2} x^{3} - 3 \, a b x^{2} + 6 \, a^{2} x}{b^{3}}\right )} \log \left ({\left (b x + a\right )}^{n} c\right )^{2} - \frac {1}{108} \, b n {\left (\frac {{\left (8 \, b^{3} x^{3} - 36 \, a^{3} \log \left (b x + a\right )^{3} - 57 \, a b^{2} x^{2} - 198 \, a^{3} \log \left (b x + a\right )^{2} + 510 \, a^{2} b x - 510 \, a^{3} \log \left (b x + a\right )\right )} n^{2}}{b^{4}} - \frac {6 \, {\left (4 \, b^{3} x^{3} - 15 \, a b^{2} x^{2} - 18 \, a^{3} \log \left (b x + a\right )^{2} + 66 \, a^{2} b x - 66 \, a^{3} \log \left (b x + a\right )\right )} n \log \left ({\left (b x + a\right )}^{n} c\right )}{b^{4}}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 626 vs. \(2 (271) = 542\).
Time = 0.32 (sec) , antiderivative size = 626, normalized size of antiderivative = 2.20 \[ \int x^2 \log ^3\left (c (a+b x)^n\right ) \, dx=\frac {{\left (b x + a\right )}^{3} n^{3} \log \left (b x + a\right )^{3}}{3 \, b^{3}} - \frac {{\left (b x + a\right )}^{2} a n^{3} \log \left (b x + a\right )^{3}}{b^{3}} + \frac {{\left (b x + a\right )} a^{2} n^{3} \log \left (b x + a\right )^{3}}{b^{3}} - \frac {{\left (b x + a\right )}^{3} n^{3} \log \left (b x + a\right )^{2}}{3 \, b^{3}} + \frac {3 \, {\left (b x + a\right )}^{2} a n^{3} \log \left (b x + a\right )^{2}}{2 \, b^{3}} - \frac {3 \, {\left (b x + a\right )} a^{2} n^{3} \log \left (b x + a\right )^{2}}{b^{3}} + \frac {{\left (b x + a\right )}^{3} n^{2} \log \left (b x + a\right )^{2} \log \left (c\right )}{b^{3}} - \frac {3 \, {\left (b x + a\right )}^{2} a n^{2} \log \left (b x + a\right )^{2} \log \left (c\right )}{b^{3}} + \frac {3 \, {\left (b x + a\right )} a^{2} n^{2} \log \left (b x + a\right )^{2} \log \left (c\right )}{b^{3}} + \frac {2 \, {\left (b x + a\right )}^{3} n^{3} \log \left (b x + a\right )}{9 \, b^{3}} - \frac {3 \, {\left (b x + a\right )}^{2} a n^{3} \log \left (b x + a\right )}{2 \, b^{3}} + \frac {6 \, {\left (b x + a\right )} a^{2} n^{3} \log \left (b x + a\right )}{b^{3}} - \frac {2 \, {\left (b x + a\right )}^{3} n^{2} \log \left (b x + a\right ) \log \left (c\right )}{3 \, b^{3}} + \frac {3 \, {\left (b x + a\right )}^{2} a n^{2} \log \left (b x + a\right ) \log \left (c\right )}{b^{3}} - \frac {6 \, {\left (b x + a\right )} a^{2} n^{2} \log \left (b x + a\right ) \log \left (c\right )}{b^{3}} + \frac {{\left (b x + a\right )}^{3} n \log \left (b x + a\right ) \log \left (c\right )^{2}}{b^{3}} - \frac {3 \, {\left (b x + a\right )}^{2} a n \log \left (b x + a\right ) \log \left (c\right )^{2}}{b^{3}} + \frac {3 \, {\left (b x + a\right )} a^{2} n \log \left (b x + a\right ) \log \left (c\right )^{2}}{b^{3}} - \frac {2 \, {\left (b x + a\right )}^{3} n^{3}}{27 \, b^{3}} + \frac {3 \, {\left (b x + a\right )}^{2} a n^{3}}{4 \, b^{3}} - \frac {6 \, {\left (b x + a\right )} a^{2} n^{3}}{b^{3}} + \frac {2 \, {\left (b x + a\right )}^{3} n^{2} \log \left (c\right )}{9 \, b^{3}} - \frac {3 \, {\left (b x + a\right )}^{2} a n^{2} \log \left (c\right )}{2 \, b^{3}} + \frac {6 \, {\left (b x + a\right )} a^{2} n^{2} \log \left (c\right )}{b^{3}} - \frac {{\left (b x + a\right )}^{3} n \log \left (c\right )^{2}}{3 \, b^{3}} + \frac {3 \, {\left (b x + a\right )}^{2} a n \log \left (c\right )^{2}}{2 \, b^{3}} - \frac {3 \, {\left (b x + a\right )} a^{2} n \log \left (c\right )^{2}}{b^{3}} + \frac {{\left (b x + a\right )}^{3} \log \left (c\right )^{3}}{3 \, b^{3}} - \frac {{\left (b x + a\right )}^{2} a \log \left (c\right )^{3}}{b^{3}} + \frac {{\left (b x + a\right )} a^{2} \log \left (c\right )^{3}}{b^{3}} \]
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Time = 1.36 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.60 \[ \int x^2 \log ^3\left (c (a+b x)^n\right ) \, dx={\ln \left (c\,{\left (a+b\,x\right )}^n\right )}^3\,\left (\frac {x^3}{3}+\frac {a^3}{3\,b^3}\right )-\frac {2\,n^3\,x^3}{27}-{\ln \left (c\,{\left (a+b\,x\right )}^n\right )}^2\,\left (\frac {n\,x^3}{3}+\frac {11\,a^3\,n}{6\,b^3}-\frac {a\,n\,x^2}{2\,b}+\frac {a^2\,n\,x}{b^2}\right )+\frac {\ln \left (c\,{\left (a+b\,x\right )}^n\right )\,\left (\frac {2\,b\,n^2\,x^3}{3}-\frac {5\,a\,n^2\,x^2}{2}+\frac {11\,a^2\,n^2\,x}{b}\right )}{3\,b}+\frac {85\,a^3\,n^3\,\ln \left (a+b\,x\right )}{18\,b^3}+\frac {19\,a\,n^3\,x^2}{36\,b}-\frac {85\,a^2\,n^3\,x}{18\,b^2} \]
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