Optimal. Leaf size=920 \[ -\frac {a (b c-a d)^3 p q r^2 x}{5 d^3}+\frac {2 (b c-a d)^4 p q r^2 x}{25 d^4}+\frac {77 (b c-a d)^4 q^2 r^2 x}{150 d^4}+\frac {2 (b c-a d)^4 q (p+q) r^2 x}{5 d^4}-\frac {b (b c-a d)^3 p q r^2 x^2}{10 d^3}-\frac {(b c-a d)^3 p q r^2 (a+b x)^2}{25 b d^3}-\frac {77 (b c-a d)^3 q^2 r^2 (a+b x)^2}{300 b d^3}+\frac {16 (b c-a d)^2 p q r^2 (a+b x)^3}{225 b d^2}+\frac {47 (b c-a d)^2 q^2 r^2 (a+b x)^3}{450 b d^2}-\frac {9 (b c-a d) p q r^2 (a+b x)^4}{200 b d}-\frac {9 (b c-a d) q^2 r^2 (a+b x)^4}{200 b d}+\frac {2 p^2 r^2 (a+b x)^5}{125 b}+\frac {4 p q r^2 (a+b x)^5}{125 b}+\frac {2 q^2 r^2 (a+b x)^5}{125 b}-\frac {2 (b c-a d)^5 p q r^2 \log (c+d x)}{25 b d^5}-\frac {137 (b c-a d)^5 q^2 r^2 \log (c+d x)}{150 b d^5}-\frac {2 (b c-a d)^5 p q r^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{5 b d^5}-\frac {(b c-a d)^5 q^2 r^2 \log ^2(c+d x)}{5 b d^5}-\frac {2 (b c-a d)^4 q r (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b d^4}+\frac {(b c-a d)^3 q r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b d^3}-\frac {2 (b c-a d)^2 q r (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{15 b d^2}+\frac {(b c-a d) q r (a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{10 b d}-\frac {2 p r (a+b x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{25 b}-\frac {2 q r (a+b x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{25 b}+\frac {2 (b c-a d)^5 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b d^5}+\frac {(a+b x)^5 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b}-\frac {2 (b c-a d)^5 p q r^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{5 b d^5} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.61, antiderivative size = 920, normalized size of antiderivative = 1.00, number
of steps used = 32, number of rules used = 14, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.452, Rules
used = {2584, 2581, 32, 45, 2594, 2579, 31, 8, 2580, 2441, 2440, 2438, 2437, 2338}
\begin {gather*} -\frac {q^2 r^2 \log ^2(c+d x) (b c-a d)^5}{5 b d^5}-\frac {137 q^2 r^2 \log (c+d x) (b c-a d)^5}{150 b d^5}-\frac {2 p q r^2 \log (c+d x) (b c-a d)^5}{25 b d^5}-\frac {2 p q r^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) (b c-a d)^5}{5 b d^5}+\frac {2 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) (b c-a d)^5}{5 b d^5}-\frac {2 p q r^2 \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right ) (b c-a d)^5}{5 b d^5}+\frac {77 q^2 r^2 x (b c-a d)^4}{150 d^4}+\frac {2 p q r^2 x (b c-a d)^4}{25 d^4}+\frac {2 q (p+q) r^2 x (b c-a d)^4}{5 d^4}-\frac {2 q r (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) (b c-a d)^4}{5 b d^4}-\frac {b p q r^2 x^2 (b c-a d)^3}{10 d^3}-\frac {77 q^2 r^2 (a+b x)^2 (b c-a d)^3}{300 b d^3}-\frac {p q r^2 (a+b x)^2 (b c-a d)^3}{25 b d^3}-\frac {a p q r^2 x (b c-a d)^3}{5 d^3}+\frac {q r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) (b c-a d)^3}{5 b d^3}+\frac {47 q^2 r^2 (a+b x)^3 (b c-a d)^2}{450 b d^2}+\frac {16 p q r^2 (a+b x)^3 (b c-a d)^2}{225 b d^2}-\frac {2 q r (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) (b c-a d)^2}{15 b d^2}-\frac {9 q^2 r^2 (a+b x)^4 (b c-a d)}{200 b d}-\frac {9 p q r^2 (a+b x)^4 (b c-a d)}{200 b d}+\frac {q r (a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) (b c-a d)}{10 b d}+\frac {2 p^2 r^2 (a+b x)^5}{125 b}+\frac {2 q^2 r^2 (a+b x)^5}{125 b}+\frac {4 p q r^2 (a+b x)^5}{125 b}+\frac {(a+b x)^5 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b}-\frac {2 p r (a+b x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{25 b}-\frac {2 q r (a+b x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{25 b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 31
Rule 32
Rule 45
Rule 2338
Rule 2437
Rule 2438
Rule 2440
Rule 2441
Rule 2579
Rule 2580
Rule 2581
Rule 2584
Rule 2594
Rubi steps
\begin {align*} \int (a+b x)^4 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx &=\frac {(a+b x)^5 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b}-\frac {1}{5} (2 p r) \int (a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx-\frac {(2 d q r) \int \frac {(a+b x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{c+d x} \, dx}{5 b}\\ &=-\frac {2 p r (a+b x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{25 b}+\frac {(a+b x)^5 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b}-\frac {(2 d q r) \int \left (\frac {b (b c-a d)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d^5}-\frac {b (b c-a d)^3 (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d^4}+\frac {b (b c-a d)^2 (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d^3}-\frac {b (b c-a d) (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d^2}+\frac {b (a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d}+\frac {(-b c+a d)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d^5 (c+d x)}\right ) \, dx}{5 b}+\frac {1}{25} \left (2 p^2 r^2\right ) \int (a+b x)^4 \, dx+\frac {\left (2 d p q r^2\right ) \int \frac {(a+b x)^5}{c+d x} \, dx}{25 b}\\ &=\frac {2 p^2 r^2 (a+b x)^5}{125 b}-\frac {2 p r (a+b x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{25 b}+\frac {(a+b x)^5 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b}-\frac {1}{5} (2 q r) \int (a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx+\frac {(2 (b c-a d) q r) \int (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx}{5 d}-\frac {\left (2 (b c-a d)^2 q r\right ) \int (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx}{5 d^2}+\frac {\left (2 (b c-a d)^3 q r\right ) \int (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx}{5 d^3}-\frac {\left (2 (b c-a d)^4 q r\right ) \int \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx}{5 d^4}+\frac {\left (2 (b c-a d)^5 q r\right ) \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{c+d x} \, dx}{5 b d^4}+\frac {\left (2 d p q r^2\right ) \int \left (\frac {b (b c-a d)^4}{d^5}-\frac {b (b c-a d)^3 (a+b x)}{d^4}+\frac {b (b c-a d)^2 (a+b x)^2}{d^3}-\frac {b (b c-a d) (a+b x)^3}{d^2}+\frac {b (a+b x)^4}{d}+\frac {(-b c+a d)^5}{d^5 (c+d x)}\right ) \, dx}{25 b}\\ &=\frac {2 (b c-a d)^4 p q r^2 x}{25 d^4}-\frac {(b c-a d)^3 p q r^2 (a+b x)^2}{25 b d^3}+\frac {2 (b c-a d)^2 p q r^2 (a+b x)^3}{75 b d^2}-\frac {(b c-a d) p q r^2 (a+b x)^4}{50 b d}+\frac {2 p^2 r^2 (a+b x)^5}{125 b}+\frac {2 p q r^2 (a+b x)^5}{125 b}-\frac {2 (b c-a d)^5 p q r^2 \log (c+d x)}{25 b d^5}-\frac {2 (b c-a d)^4 q r (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b d^4}+\frac {(b c-a d)^3 q r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b d^3}-\frac {2 (b c-a d)^2 q r (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{15 b d^2}+\frac {(b c-a d) q r (a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{10 b d}-\frac {2 p r (a+b x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{25 b}-\frac {2 q r (a+b x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{25 b}+\frac {2 (b c-a d)^5 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b d^5}+\frac {(a+b x)^5 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b}+\frac {1}{25} \left (2 p q r^2\right ) \int (a+b x)^4 \, dx-\frac {\left ((b c-a d) p q r^2\right ) \int (a+b x)^3 \, dx}{10 d}+\frac {\left (2 (b c-a d)^2 p q r^2\right ) \int (a+b x)^2 \, dx}{15 d^2}-\frac {\left ((b c-a d)^3 p q r^2\right ) \int (a+b x) \, dx}{5 d^3}-\frac {\left (2 (b c-a d)^5 p q r^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{5 d^5}+\frac {\left (2 d q^2 r^2\right ) \int \frac {(a+b x)^5}{c+d x} \, dx}{25 b}-\frac {\left ((b c-a d) q^2 r^2\right ) \int \frac {(a+b x)^4}{c+d x} \, dx}{10 b}+\frac {\left (2 (b c-a d)^2 q^2 r^2\right ) \int \frac {(a+b x)^3}{c+d x} \, dx}{15 b d}-\frac {\left ((b c-a d)^3 q^2 r^2\right ) \int \frac {(a+b x)^2}{c+d x} \, dx}{5 b d^2}-\frac {\left (2 (b c-a d)^5 q^2 r^2\right ) \int \frac {1}{c+d x} \, dx}{5 b d^4}-\frac {\left (2 (b c-a d)^5 q^2 r^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{5 b d^4}+\frac {\left (2 (b c-a d)^4 q (p+q) r^2\right ) \int 1 \, dx}{5 d^4}\\ &=-\frac {a (b c-a d)^3 p q r^2 x}{5 d^3}+\frac {2 (b c-a d)^4 p q r^2 x}{25 d^4}+\frac {2 (b c-a d)^4 q (p+q) r^2 x}{5 d^4}-\frac {b (b c-a d)^3 p q r^2 x^2}{10 d^3}-\frac {(b c-a d)^3 p q r^2 (a+b x)^2}{25 b d^3}+\frac {16 (b c-a d)^2 p q r^2 (a+b x)^3}{225 b d^2}-\frac {9 (b c-a d) p q r^2 (a+b x)^4}{200 b d}+\frac {2 p^2 r^2 (a+b x)^5}{125 b}+\frac {4 p q r^2 (a+b x)^5}{125 b}-\frac {2 (b c-a d)^5 p q r^2 \log (c+d x)}{25 b d^5}-\frac {2 (b c-a d)^5 q^2 r^2 \log (c+d x)}{5 b d^5}-\frac {2 (b c-a d)^5 p q r^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{5 b d^5}-\frac {2 (b c-a d)^4 q r (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b d^4}+\frac {(b c-a d)^3 q r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b d^3}-\frac {2 (b c-a d)^2 q r (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{15 b d^2}+\frac {(b c-a d) q r (a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{10 b d}-\frac {2 p r (a+b x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{25 b}-\frac {2 q r (a+b x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{25 b}+\frac {2 (b c-a d)^5 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b d^5}+\frac {(a+b x)^5 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b}+\frac {\left (2 (b c-a d)^5 p q r^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{5 b d^4}+\frac {\left (2 d q^2 r^2\right ) \int \left (\frac {b (b c-a d)^4}{d^5}-\frac {b (b c-a d)^3 (a+b x)}{d^4}+\frac {b (b c-a d)^2 (a+b x)^2}{d^3}-\frac {b (b c-a d) (a+b x)^3}{d^2}+\frac {b (a+b x)^4}{d}+\frac {(-b c+a d)^5}{d^5 (c+d x)}\right ) \, dx}{25 b}-\frac {\left ((b c-a d) q^2 r^2\right ) \int \left (-\frac {b (b c-a d)^3}{d^4}+\frac {b (b c-a d)^2 (a+b x)}{d^3}-\frac {b (b c-a d) (a+b x)^2}{d^2}+\frac {b (a+b x)^3}{d}+\frac {(-b c+a d)^4}{d^4 (c+d x)}\right ) \, dx}{10 b}+\frac {\left (2 (b c-a d)^2 q^2 r^2\right ) \int \left (\frac {b (b c-a d)^2}{d^3}-\frac {b (b c-a d) (a+b x)}{d^2}+\frac {b (a+b x)^2}{d}+\frac {(-b c+a d)^3}{d^3 (c+d x)}\right ) \, dx}{15 b d}-\frac {\left ((b c-a d)^3 q^2 r^2\right ) \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}{5 b d^2}-\frac {\left (2 (b c-a d)^5 q^2 r^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{5 b d^5}\\ &=-\frac {a (b c-a d)^3 p q r^2 x}{5 d^3}+\frac {2 (b c-a d)^4 p q r^2 x}{25 d^4}+\frac {77 (b c-a d)^4 q^2 r^2 x}{150 d^4}+\frac {2 (b c-a d)^4 q (p+q) r^2 x}{5 d^4}-\frac {b (b c-a d)^3 p q r^2 x^2}{10 d^3}-\frac {(b c-a d)^3 p q r^2 (a+b x)^2}{25 b d^3}-\frac {77 (b c-a d)^3 q^2 r^2 (a+b x)^2}{300 b d^3}+\frac {16 (b c-a d)^2 p q r^2 (a+b x)^3}{225 b d^2}+\frac {47 (b c-a d)^2 q^2 r^2 (a+b x)^3}{450 b d^2}-\frac {9 (b c-a d) p q r^2 (a+b x)^4}{200 b d}-\frac {9 (b c-a d) q^2 r^2 (a+b x)^4}{200 b d}+\frac {2 p^2 r^2 (a+b x)^5}{125 b}+\frac {4 p q r^2 (a+b x)^5}{125 b}+\frac {2 q^2 r^2 (a+b x)^5}{125 b}-\frac {2 (b c-a d)^5 p q r^2 \log (c+d x)}{25 b d^5}-\frac {137 (b c-a d)^5 q^2 r^2 \log (c+d x)}{150 b d^5}-\frac {2 (b c-a d)^5 p q r^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{5 b d^5}-\frac {(b c-a d)^5 q^2 r^2 \log ^2(c+d x)}{5 b d^5}-\frac {2 (b c-a d)^4 q r (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b d^4}+\frac {(b c-a d)^3 q r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b d^3}-\frac {2 (b c-a d)^2 q r (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{15 b d^2}+\frac {(b c-a d) q r (a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{10 b d}-\frac {2 p r (a+b x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{25 b}-\frac {2 q r (a+b x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{25 b}+\frac {2 (b c-a d)^5 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b d^5}+\frac {(a+b x)^5 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b}+\frac {\left (2 (b c-a d)^5 p q r^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{5 b d^5}\\ &=-\frac {a (b c-a d)^3 p q r^2 x}{5 d^3}+\frac {2 (b c-a d)^4 p q r^2 x}{25 d^4}+\frac {77 (b c-a d)^4 q^2 r^2 x}{150 d^4}+\frac {2 (b c-a d)^4 q (p+q) r^2 x}{5 d^4}-\frac {b (b c-a d)^3 p q r^2 x^2}{10 d^3}-\frac {(b c-a d)^3 p q r^2 (a+b x)^2}{25 b d^3}-\frac {77 (b c-a d)^3 q^2 r^2 (a+b x)^2}{300 b d^3}+\frac {16 (b c-a d)^2 p q r^2 (a+b x)^3}{225 b d^2}+\frac {47 (b c-a d)^2 q^2 r^2 (a+b x)^3}{450 b d^2}-\frac {9 (b c-a d) p q r^2 (a+b x)^4}{200 b d}-\frac {9 (b c-a d) q^2 r^2 (a+b x)^4}{200 b d}+\frac {2 p^2 r^2 (a+b x)^5}{125 b}+\frac {4 p q r^2 (a+b x)^5}{125 b}+\frac {2 q^2 r^2 (a+b x)^5}{125 b}-\frac {2 (b c-a d)^5 p q r^2 \log (c+d x)}{25 b d^5}-\frac {137 (b c-a d)^5 q^2 r^2 \log (c+d x)}{150 b d^5}-\frac {2 (b c-a d)^5 p q r^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{5 b d^5}-\frac {(b c-a d)^5 q^2 r^2 \log ^2(c+d x)}{5 b d^5}-\frac {2 (b c-a d)^4 q r (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b d^4}+\frac {(b c-a d)^3 q r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b d^3}-\frac {2 (b c-a d)^2 q r (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{15 b d^2}+\frac {(b c-a d) q r (a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{10 b d}-\frac {2 p r (a+b x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{25 b}-\frac {2 q r (a+b x)^5 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{25 b}+\frac {2 (b c-a d)^5 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b d^5}+\frac {(a+b x)^5 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{5 b}-\frac {2 (b c-a d)^5 p q r^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{5 b d^5}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(2508\) vs. \(2(920)=1840\).
time = 1.73, size = 2508, normalized size = 2.73 \begin {gather*} \text {Result too large to show} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
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Maple [F]
time = 0.20, size = 0, normalized size = 0.00 \[\int \left (b x +a \right )^{4} \ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right )^{2}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.34, size = 1423, normalized size = 1.55 \begin {gather*} \frac {1}{5} \, {\left (b^{4} x^{5} + 5 \, a b^{3} x^{4} + 10 \, a^{2} b^{2} x^{3} + 10 \, a^{3} b x^{2} + 5 \, a^{4} x\right )} \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2} + \frac {{\left (\frac {60 \, a^{5} f p \log \left (b x + a\right )}{b} - \frac {12 \, b^{4} d^{4} f {\left (p + q\right )} x^{5} + 15 \, {\left (a b^{3} d^{4} f {\left (4 \, p + 5 \, q\right )} - b^{4} c d^{3} f q\right )} x^{4} + 20 \, {\left (2 \, a^{2} b^{2} d^{4} f {\left (3 \, p + 5 \, q\right )} + b^{4} c^{2} d^{2} f q - 5 \, a b^{3} c d^{3} f q\right )} x^{3} + 30 \, {\left (2 \, a^{3} b d^{4} f {\left (2 \, p + 5 \, q\right )} - b^{4} c^{3} d f q + 5 \, a b^{3} c^{2} d^{2} f q - 10 \, a^{2} b^{2} c d^{3} f q\right )} x^{2} + 60 \, {\left (a^{4} d^{4} f {\left (p + 5 \, q\right )} + b^{4} c^{4} f q - 5 \, a b^{3} c^{3} d f q + 10 \, a^{2} b^{2} c^{2} d^{2} f q - 10 \, a^{3} b c d^{3} f q\right )} x}{d^{4}} + \frac {60 \, {\left (b^{4} c^{5} f q - 5 \, a b^{3} c^{4} d f q + 10 \, a^{2} b^{2} c^{3} d^{2} f q - 10 \, a^{3} b c^{2} d^{3} f q + 5 \, a^{4} c d^{4} f q\right )} \log \left (d x + c\right )}{d^{5}}\right )} r \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )}{150 \, f} - \frac {r^{2} {\left (\frac {60 \, {\left ({\left (12 \, p q + 137 \, q^{2}\right )} b^{4} c^{5} f^{2} - 5 \, {\left (12 \, p q + 125 \, q^{2}\right )} a b^{3} c^{4} d f^{2} + 20 \, {\left (6 \, p q + 55 \, q^{2}\right )} a^{2} b^{2} c^{3} d^{2} f^{2} - 60 \, {\left (2 \, p q + 15 \, q^{2}\right )} a^{3} b c^{2} d^{3} f^{2} + 60 \, {\left (p q + 5 \, q^{2}\right )} a^{4} c d^{4} f^{2}\right )} \log \left (d x + c\right )}{d^{5}} - \frac {3600 \, {\left (b^{5} c^{5} f^{2} p q - 5 \, a b^{4} c^{4} d f^{2} p q + 10 \, a^{2} b^{3} c^{3} d^{2} f^{2} p q - 10 \, a^{3} b^{2} c^{2} d^{3} f^{2} p q + 5 \, a^{4} b c d^{4} f^{2} p q - a^{5} d^{5} f^{2} p q\right )} {\left (\log \left (b x + a\right ) \log \left (\frac {b d x + a d}{b c - a d} + 1\right ) + {\rm Li}_2\left (-\frac {b d x + a d}{b c - a d}\right )\right )}}{b d^{5}} - \frac {144 \, {\left (p^{2} + 2 \, p q + q^{2}\right )} b^{5} d^{5} f^{2} x^{5} - 1800 \, a^{5} d^{5} f^{2} p^{2} \log \left (b x + a\right )^{2} - 45 \, {\left (9 \, {\left (p q + q^{2}\right )} b^{5} c d^{4} f^{2} - {\left (16 \, p^{2} + 41 \, p q + 25 \, q^{2}\right )} a b^{4} d^{5} f^{2}\right )} x^{4} + 20 \, {\left ({\left (32 \, p q + 47 \, q^{2}\right )} b^{5} c^{2} d^{3} f^{2} - 5 \, {\left (29 \, p q + 35 \, q^{2}\right )} a b^{4} c d^{4} f^{2} + {\left (72 \, p^{2} + 257 \, p q + 200 \, q^{2}\right )} a^{2} b^{3} d^{5} f^{2}\right )} x^{3} - 30 \, {\left (7 \, {\left (6 \, p q + 11 \, q^{2}\right )} b^{5} c^{3} d^{2} f^{2} - 5 \, {\left (38 \, p q + 65 \, q^{2}\right )} a b^{4} c^{2} d^{3} f^{2} + 5 \, {\left (67 \, p q + 100 \, q^{2}\right )} a^{2} b^{3} c d^{4} f^{2} - {\left (48 \, p^{2} + 283 \, p q + 300 \, q^{2}\right )} a^{3} b^{2} d^{5} f^{2}\right )} x^{2} - 3600 \, {\left (b^{5} c^{5} f^{2} p q - 5 \, a b^{4} c^{4} d f^{2} p q + 10 \, a^{2} b^{3} c^{3} d^{2} f^{2} p q - 10 \, a^{3} b^{2} c^{2} d^{3} f^{2} p q + 5 \, a^{4} b c d^{4} f^{2} p q\right )} \log \left (b x + a\right ) \log \left (d x + c\right ) - 1800 \, {\left (b^{5} c^{5} f^{2} q^{2} - 5 \, a b^{4} c^{4} d f^{2} q^{2} + 10 \, a^{2} b^{3} c^{3} d^{2} f^{2} q^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} f^{2} q^{2} + 5 \, a^{4} b c d^{4} f^{2} q^{2}\right )} \log \left (d x + c\right )^{2} + 60 \, {\left ({\left (72 \, p q + 137 \, q^{2}\right )} b^{5} c^{4} d f^{2} - 5 \, {\left (66 \, p q + 125 \, q^{2}\right )} a b^{4} c^{3} d^{2} f^{2} + 10 \, {\left (59 \, p q + 110 \, q^{2}\right )} a^{2} b^{3} c^{2} d^{3} f^{2} - 5 \, {\left (101 \, p q + 180 \, q^{2}\right )} a^{3} b^{2} c d^{4} f^{2} + {\left (12 \, p^{2} + 197 \, p q + 300 \, q^{2}\right )} a^{4} b d^{5} f^{2}\right )} x - 60 \, {\left (60 \, a b^{4} c^{4} d f^{2} p q - 270 \, a^{2} b^{3} c^{3} d^{2} f^{2} p q + 470 \, a^{3} b^{2} c^{2} d^{3} f^{2} p q - 385 \, a^{4} b c d^{4} f^{2} p q + {\left (12 \, p^{2} + 137 \, p q\right )} a^{5} d^{5} f^{2}\right )} \log \left (b x + a\right )}{b d^{5}}\right )}}{9000 \, f^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\ln \left (e\,{\left (f\,{\left (a+b\,x\right )}^p\,{\left (c+d\,x\right )}^q\right )}^r\right )}^2\,{\left (a+b\,x\right )}^4 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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