Integrand size = 16, antiderivative size = 99 \[ \int \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx=6 a b^2 n^2 x-6 b^3 n^3 x+\frac {6 b^3 n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac {3 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2436, 2333, 2332} \[ \int \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx=6 a b^2 n^2 x-\frac {3 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac {6 b^3 n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e}-6 b^3 n^3 x \]
[In]
[Out]
Rule 2332
Rule 2333
Rule 2436
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e x\right )}{e} \\ & = \frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}-\frac {(3 b n) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{e} \\ & = -\frac {3 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac {\left (6 b^2 n^2\right ) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{e} \\ & = 6 a b^2 n^2 x-\frac {3 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e}+\frac {\left (6 b^3 n^2\right ) \text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e} \\ & = 6 a b^2 n^2 x-6 b^3 n^3 x+\frac {6 b^3 n^2 (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac {3 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.86 \[ \int \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx=\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3-3 b n \left ((d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2-2 b n \left (e (a-b n) x+b (d+e x) \log \left (c (d+e x)^n\right )\right )\right )}{e} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(321\) vs. \(2(99)=198\).
Time = 0.26 (sec) , antiderivative size = 322, normalized size of antiderivative = 3.25
method | result | size |
parallelrisch | \(\frac {x \ln \left (c \left (e x +d \right )^{n}\right )^{3} b^{3} e n -3 x \ln \left (c \left (e x +d \right )^{n}\right )^{2} b^{3} e \,n^{2}+6 x \ln \left (c \left (e x +d \right )^{n}\right ) b^{3} e \,n^{3}-6 x \,b^{3} e \,n^{4}+3 x \ln \left (c \left (e x +d \right )^{n}\right )^{2} a \,b^{2} e n -6 x \ln \left (c \left (e x +d \right )^{n}\right ) a \,b^{2} e \,n^{2}+6 x a \,b^{2} e \,n^{3}+\ln \left (c \left (e x +d \right )^{n}\right )^{3} b^{3} d n -3 \ln \left (c \left (e x +d \right )^{n}\right )^{2} b^{3} d \,n^{2}+6 \ln \left (c \left (e x +d \right )^{n}\right ) b^{3} d \,n^{3}+6 b^{3} d \,n^{4}+3 x \ln \left (c \left (e x +d \right )^{n}\right ) a^{2} b e n -3 x \,a^{2} b e \,n^{2}+3 \ln \left (c \left (e x +d \right )^{n}\right )^{2} a \,b^{2} d n -6 \ln \left (c \left (e x +d \right )^{n}\right ) a \,b^{2} d \,n^{2}-6 a \,b^{2} d \,n^{3}+x \,a^{3} e n +3 \ln \left (c \left (e x +d \right )^{n}\right ) a^{2} b d n +3 a^{2} b d \,n^{2}-a^{3} d n}{e n}\) | \(322\) |
risch | \(\text {Expression too large to display}\) | \(4872\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 324 vs. \(2 (99) = 198\).
Time = 0.31 (sec) , antiderivative size = 324, normalized size of antiderivative = 3.27 \[ \int \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx=\frac {b^{3} e x \log \left (c\right )^{3} + {\left (b^{3} e n^{3} x + b^{3} d n^{3}\right )} \log \left (e x + d\right )^{3} - 3 \, {\left (b^{3} e n - a b^{2} e\right )} x \log \left (c\right )^{2} - 3 \, {\left (b^{3} d n^{3} - a b^{2} d n^{2} + {\left (b^{3} e n^{3} - a b^{2} e n^{2}\right )} x - {\left (b^{3} e n^{2} x + b^{3} d n^{2}\right )} \log \left (c\right )\right )} \log \left (e x + d\right )^{2} + 3 \, {\left (2 \, b^{3} e n^{2} - 2 \, a b^{2} e n + a^{2} b e\right )} x \log \left (c\right ) - {\left (6 \, b^{3} e n^{3} - 6 \, a b^{2} e n^{2} + 3 \, a^{2} b e n - a^{3} e\right )} x + 3 \, {\left (2 \, b^{3} d n^{3} - 2 \, a b^{2} d n^{2} + a^{2} b d n + {\left (b^{3} e n x + b^{3} d n\right )} \log \left (c\right )^{2} + {\left (2 \, b^{3} e n^{3} - 2 \, a b^{2} e n^{2} + a^{2} b e n\right )} x - 2 \, {\left (b^{3} d n^{2} - a b^{2} d n + {\left (b^{3} e n^{2} - a b^{2} e n\right )} x\right )} \log \left (c\right )\right )} \log \left (e x + d\right )}{e} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 294 vs. \(2 (95) = 190\).
Time = 0.50 (sec) , antiderivative size = 294, normalized size of antiderivative = 2.97 \[ \int \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx=\begin {cases} a^{3} x + \frac {3 a^{2} b d \log {\left (c \left (d + e x\right )^{n} \right )}}{e} - 3 a^{2} b n x + 3 a^{2} b x \log {\left (c \left (d + e x\right )^{n} \right )} - \frac {6 a b^{2} d n \log {\left (c \left (d + e x\right )^{n} \right )}}{e} + \frac {3 a b^{2} d \log {\left (c \left (d + e x\right )^{n} \right )}^{2}}{e} + 6 a b^{2} n^{2} x - 6 a b^{2} n x \log {\left (c \left (d + e x\right )^{n} \right )} + 3 a b^{2} x \log {\left (c \left (d + e x\right )^{n} \right )}^{2} + \frac {6 b^{3} d n^{2} \log {\left (c \left (d + e x\right )^{n} \right )}}{e} - \frac {3 b^{3} d n \log {\left (c \left (d + e x\right )^{n} \right )}^{2}}{e} + \frac {b^{3} d \log {\left (c \left (d + e x\right )^{n} \right )}^{3}}{e} - 6 b^{3} n^{3} x + 6 b^{3} n^{2} x \log {\left (c \left (d + e x\right )^{n} \right )} - 3 b^{3} n x \log {\left (c \left (d + e x\right )^{n} \right )}^{2} + b^{3} x \log {\left (c \left (d + e x\right )^{n} \right )}^{3} & \text {for}\: e \neq 0 \\x \left (a + b \log {\left (c d^{n} \right )}\right )^{3} & \text {otherwise} \end {cases} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 282 vs. \(2 (99) = 198\).
Time = 0.20 (sec) , antiderivative size = 282, normalized size of antiderivative = 2.85 \[ \int \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx=b^{3} x \log \left ({\left (e x + d\right )}^{n} c\right )^{3} - 3 \, a^{2} b e n {\left (\frac {x}{e} - \frac {d \log \left (e x + d\right )}{e^{2}}\right )} + 3 \, a b^{2} x \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + 3 \, a^{2} b x \log \left ({\left (e x + d\right )}^{n} c\right ) - 3 \, {\left (2 \, e n {\left (\frac {x}{e} - \frac {d \log \left (e x + d\right )}{e^{2}}\right )} \log \left ({\left (e x + d\right )}^{n} c\right ) + \frac {{\left (d \log \left (e x + d\right )^{2} - 2 \, e x + 2 \, d \log \left (e x + d\right )\right )} n^{2}}{e}\right )} a b^{2} - {\left (3 \, e n {\left (\frac {x}{e} - \frac {d \log \left (e x + d\right )}{e^{2}}\right )} \log \left ({\left (e x + d\right )}^{n} c\right )^{2} - e n {\left (\frac {{\left (d \log \left (e x + d\right )^{3} + 3 \, d \log \left (e x + d\right )^{2} - 6 \, e x + 6 \, d \log \left (e x + d\right )\right )} n^{2}}{e^{2}} - \frac {3 \, {\left (d \log \left (e x + d\right )^{2} - 2 \, e x + 2 \, d \log \left (e x + d\right )\right )} n \log \left ({\left (e x + d\right )}^{n} c\right )}{e^{2}}\right )}\right )} b^{3} + a^{3} x \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 399 vs. \(2 (99) = 198\).
Time = 0.33 (sec) , antiderivative size = 399, normalized size of antiderivative = 4.03 \[ \int \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx=\frac {{\left (e x + d\right )} b^{3} n^{3} \log \left (e x + d\right )^{3}}{e} - \frac {3 \, {\left (e x + d\right )} b^{3} n^{3} \log \left (e x + d\right )^{2}}{e} + \frac {3 \, {\left (e x + d\right )} b^{3} n^{2} \log \left (e x + d\right )^{2} \log \left (c\right )}{e} + \frac {6 \, {\left (e x + d\right )} b^{3} n^{3} \log \left (e x + d\right )}{e} + \frac {3 \, {\left (e x + d\right )} a b^{2} n^{2} \log \left (e x + d\right )^{2}}{e} - \frac {6 \, {\left (e x + d\right )} b^{3} n^{2} \log \left (e x + d\right ) \log \left (c\right )}{e} + \frac {3 \, {\left (e x + d\right )} b^{3} n \log \left (e x + d\right ) \log \left (c\right )^{2}}{e} - \frac {6 \, {\left (e x + d\right )} b^{3} n^{3}}{e} - \frac {6 \, {\left (e x + d\right )} a b^{2} n^{2} \log \left (e x + d\right )}{e} + \frac {6 \, {\left (e x + d\right )} b^{3} n^{2} \log \left (c\right )}{e} + \frac {6 \, {\left (e x + d\right )} a b^{2} n \log \left (e x + d\right ) \log \left (c\right )}{e} - \frac {3 \, {\left (e x + d\right )} b^{3} n \log \left (c\right )^{2}}{e} + \frac {{\left (e x + d\right )} b^{3} \log \left (c\right )^{3}}{e} + \frac {6 \, {\left (e x + d\right )} a b^{2} n^{2}}{e} + \frac {3 \, {\left (e x + d\right )} a^{2} b n \log \left (e x + d\right )}{e} - \frac {6 \, {\left (e x + d\right )} a b^{2} n \log \left (c\right )}{e} + \frac {3 \, {\left (e x + d\right )} a b^{2} \log \left (c\right )^{2}}{e} - \frac {3 \, {\left (e x + d\right )} a^{2} b n}{e} + \frac {3 \, {\left (e x + d\right )} a^{2} b \log \left (c\right )}{e} + \frac {{\left (e x + d\right )} a^{3}}{e} \]
[In]
[Out]
Time = 0.00 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.74 \[ \int \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx=x\,\left (a^3-3\,a^2\,b\,n+6\,a\,b^2\,n^2-6\,b^3\,n^3\right )+{\ln \left (c\,{\left (d+e\,x\right )}^n\right )}^3\,\left (b^3\,x+\frac {b^3\,d}{e}\right )+{\ln \left (c\,{\left (d+e\,x\right )}^n\right )}^2\,\left (\frac {3\,\left (a\,b^2\,d-b^3\,d\,n\right )}{e}+3\,b^2\,x\,\left (a-b\,n\right )\right )+\frac {\ln \left (d+e\,x\right )\,\left (3\,d\,a^2\,b\,n-6\,d\,a\,b^2\,n^2+6\,d\,b^3\,n^3\right )}{e}+3\,b\,x\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\,\left (a^2-2\,a\,b\,n+2\,b^2\,n^2\right ) \]
[In]
[Out]