Integrand size = 20, antiderivative size = 195 \[ \int \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \, dx=\frac {b^2 n^2 \left (d+e \sqrt {x}\right )^2}{2 e^2}+\frac {4 a b d n \sqrt {x}}{e}-\frac {4 b^2 d n^2 \sqrt {x}}{e}+\frac {4 b^2 d n \left (d+e \sqrt {x}\right ) \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{e^2}-\frac {b n \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{e^2}-\frac {2 d \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{e^2}+\frac {\left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{e^2} \]
[Out]
Time = 0.12 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2501, 2448, 2436, 2333, 2332, 2437, 2342, 2341} \[ \int \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \, dx=-\frac {b n \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{e^2}+\frac {\left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{e^2}-\frac {2 d \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{e^2}+\frac {4 a b d n \sqrt {x}}{e}+\frac {4 b^2 d n \left (d+e \sqrt {x}\right ) \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{e^2}+\frac {b^2 n^2 \left (d+e \sqrt {x}\right )^2}{2 e^2}-\frac {4 b^2 d n^2 \sqrt {x}}{e} \]
[In]
[Out]
Rule 2332
Rule 2333
Rule 2341
Rule 2342
Rule 2436
Rule 2437
Rule 2448
Rule 2501
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (-\frac {d \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 )^2}{e}\right ) \, dx,x,\sqrt {x}\right ) \\ & = \frac {2 \text {Subst}\left (\int (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx,x,\sqrt {x}\right )}{e}-\frac {(2 d) \text {Subst}\left (\int \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx,x,\sqrt {x}\right )}{e} \\ & = \frac {2 \text {Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e \sqrt {x}\right )}{e^2}-\frac {(2 d) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e \sqrt {x}\right )}{e^2} \\ & = -\frac {2 d \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{e^2}+\frac {\left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{e^2}-\frac {(2 b n) \text {Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e \sqrt {x}\right )}{e^2}+\frac {(4 b d n) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e \sqrt {x}\right )}{e^2} \\ & = \frac {b^2 n^2 \left (d+e \sqrt {x}\right )^2}{2 e^2}+\frac {4 a b d n \sqrt {x}}{e}-\frac {b n \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{e^2}-\frac {2 d \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{e^2}+\frac {\left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{e^2}+\frac {\left (4 b^2 d n\right ) \text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e \sqrt {x}\right )}{e^2} \\ & = \frac {b^2 n^2 \left (d+e \sqrt {x}\right )^2}{2 e^2}+\frac {4 a b d n \sqrt {x}}{e}-\frac {4 b^2 d n^2 \sqrt {x}}{e}+\frac {4 b^2 d n \left (d+e \sqrt {x}\right ) \log \left (c \left (d+e \sqrt {x}\right )^n\right )}{e^2}-\frac {b n \left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )}{e^2}-\frac {2 d \left (d+e \sqrt {x}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{e^2}+\frac {\left (d+e \sqrt {x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2}{e^2} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.77 \[ \int \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \, dx=\frac {-2 a b n \left (d-e \sqrt {x}\right )^2+b^2 e n^2 \left (-6 d+e \sqrt {x}\right ) \sqrt {x}-2 a^2 \left (d^2-e^2 x\right )+2 b \left (d+e \sqrt {x}\right ) \left (-2 a d+3 b d n+2 a e \sqrt {x}-b e n \sqrt {x}\right ) \log \left (c \left (d+e \sqrt {x}\right )^n\right )-2 b^2 \left (d^2-e^2 x\right ) \log ^2\left (c \left (d+e \sqrt {x}\right )^n\right )}{2 e^2} \]
[In]
[Out]
\[\int {\left (a +b \ln \left (c \left (d +e \sqrt {x}\right )^{n}\right )\right )}^{2}d x\]
[In]
[Out]
none
Time = 0.33 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.15 \[ \int \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \, dx=\frac {2 \, b^{2} e^{2} x \log \left (c\right )^{2} + 2 \, {\left (b^{2} e^{2} n^{2} x - b^{2} d^{2} n^{2}\right )} \log \left (e \sqrt {x} + d\right )^{2} - 2 \, {\left (b^{2} e^{2} n - 2 \, a b e^{2}\right )} x \log \left (c\right ) + {\left (b^{2} e^{2} n^{2} - 2 \, a b e^{2} n + 2 \, a^{2} e^{2}\right )} x + 2 \, {\left (2 \, b^{2} d e n^{2} \sqrt {x} + 3 \, b^{2} d^{2} n^{2} - 2 \, a b d^{2} n - {\left (b^{2} e^{2} n^{2} - 2 \, a b e^{2} n\right )} x + 2 \, {\left (b^{2} e^{2} n x - b^{2} d^{2} n\right )} \log \left (c\right )\right )} \log \left (e \sqrt {x} + d\right ) - 2 \, {\left (3 \, b^{2} d e n^{2} - 2 \, b^{2} d e n \log \left (c\right ) - 2 \, a b d e n\right )} \sqrt {x}}{2 \, e^{2}} \]
[In]
[Out]
\[ \int \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \, dx=\int \left (a + b \log {\left (c \left (d + e \sqrt {x}\right )^{n} \right )}\right )^{2}\, dx \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.92 \[ \int \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \, dx=-{\left (e n {\left (\frac {2 \, d^{2} \log \left (e \sqrt {x} + d\right )}{e^{3}} + \frac {e x - 2 \, d \sqrt {x}}{e^{2}}\right )} - 2 \, x \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right )\right )} a b - \frac {1}{2} \, {\left (2 \, e n {\left (\frac {2 \, d^{2} \log \left (e \sqrt {x} + d\right )}{e^{3}} + \frac {e x - 2 \, d \sqrt {x}}{e^{2}}\right )} \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right ) - 2 \, x \log \left ({\left (e \sqrt {x} + d\right )}^{n} c\right )^{2} - \frac {{\left (2 \, d^{2} \log \left (e \sqrt {x} + d\right )^{2} + e^{2} x + 6 \, d^{2} \log \left (e \sqrt {x} + d\right ) - 6 \, d e \sqrt {x}\right )} n^{2}}{e^{2}}\right )} b^{2} + a^{2} x \]
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.74 \[ \int \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \, dx=\frac {\frac {{\left (2 \, {\left (e \sqrt {x} + d\right )}^{2} \log \left (e \sqrt {x} + d\right )^{2} - 4 \, {\left (e \sqrt {x} + d\right )} d \log \left (e \sqrt {x} + d\right )^{2} - 2 \, {\left (e \sqrt {x} + d\right )}^{2} \log \left (e \sqrt {x} + d\right ) + 8 \, {\left (e \sqrt {x} + d\right )} d \log \left (e \sqrt {x} + d\right ) + {\left (e \sqrt {x} + d\right )}^{2} - 8 \, {\left (e \sqrt {x} + d\right )} d\right )} b^{2} n^{2}}{e} + \frac {2 \, {\left (2 \, {\left (e \sqrt {x} + d\right )}^{2} \log \left (e \sqrt {x} + d\right ) - 4 \, {\left (e \sqrt {x} + d\right )} d \log \left (e \sqrt {x} + d\right ) - {\left (e \sqrt {x} + d\right )}^{2} + 4 \, {\left (e \sqrt {x} + d\right )} d\right )} b^{2} n \log \left (c\right )}{e} + \frac {2 \, {\left ({\left (e \sqrt {x} + d\right )}^{2} - 2 \, {\left (e \sqrt {x} + d\right )} d\right )} b^{2} \log \left (c\right )^{2}}{e} + \frac {2 \, {\left (2 \, {\left (e \sqrt {x} + d\right )}^{2} \log \left (e \sqrt {x} + d\right ) - 4 \, {\left (e \sqrt {x} + d\right )} d \log \left (e \sqrt {x} + d\right ) - {\left (e \sqrt {x} + d\right )}^{2} + 4 \, {\left (e \sqrt {x} + d\right )} d\right )} a b n}{e} + \frac {4 \, {\left ({\left (e \sqrt {x} + d\right )}^{2} - 2 \, {\left (e \sqrt {x} + d\right )} d\right )} a b \log \left (c\right )}{e} + \frac {2 \, {\left ({\left (e \sqrt {x} + d\right )}^{2} - 2 \, {\left (e \sqrt {x} + d\right )} d\right )} a^{2}}{e}}{2 \, e} \]
[In]
[Out]
Time = 1.63 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.95 \[ \int \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \, dx=x\,\left (a^2-a\,b\,n+\frac {b^2\,n^2}{2}\right )-\sqrt {x}\,\left (\frac {d\,\left (2\,a^2-2\,a\,b\,n+b^2\,n^2\right )}{e}-\frac {2\,d\,\left (a^2-b^2\,n^2\right )}{e}\right )+{\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )}^2\,\left (b^2\,x-\frac {b^2\,d^2}{e^2}\right )-\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )\,\left (\sqrt {x}\,\left (\frac {2\,b\,d\,\left (2\,a-b\,n\right )}{e}-\frac {4\,a\,b\,d}{e}\right )-b\,x\,\left (2\,a-b\,n\right )\right )+\frac {\ln \left (d+e\,\sqrt {x}\right )\,\left (3\,b^2\,d^2\,n^2-2\,a\,b\,d^2\,n\right )}{e^2} \]
[In]
[Out]