Optimal. Leaf size=805 \[ \frac {q^2 r^2 \log ^2(c+d x) (b c-a d)^4}{4 b d^4}+\frac {25 q^2 r^2 \log (c+d x) (b c-a d)^4}{24 b d^4}+\frac {p q r^2 \log (c+d x) (b c-a d)^4}{8 b d^4}+\frac {p q r^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) (b c-a d)^4}{2 b d^4}-\frac {q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) (b c-a d)^4}{2 b d^4}+\frac {p q r^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right ) (b c-a d)^4}{2 b d^4}-\frac {13 q^2 r^2 x (b c-a d)^3}{24 d^3}-\frac {p q r^2 x (b c-a d)^3}{8 d^3}-\frac {q (p+q) r^2 x (b c-a d)^3}{2 d^3}+\frac {q r (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) (b c-a d)^3}{2 b d^3}+\frac {b p q r^2 x^2 (b c-a d)^2}{8 d^2}+\frac {13 q^2 r^2 (a+b x)^2 (b c-a d)^2}{48 b d^2}+\frac {p q r^2 (a+b x)^2 (b c-a d)^2}{16 b d^2}+\frac {a p q r^2 x (b c-a d)^2}{4 d^2}-\frac {q r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) (b c-a d)^2}{4 b d^2}-\frac {7 q^2 r^2 (a+b x)^3 (b c-a d)}{72 b d}-\frac {7 p q r^2 (a+b x)^3 (b c-a d)}{72 b d}+\frac {q r (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) (b c-a d)}{6 b d}+\frac {p^2 r^2 (a+b x)^4}{32 b}+\frac {q^2 r^2 (a+b x)^4}{32 b}+\frac {p q r^2 (a+b x)^4}{16 b}+\frac {(a+b x)^4 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b}-\frac {p r (a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{8 b}-\frac {q r (a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{8 b} \]
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Rubi [A] time = 0.66, antiderivative size = 805, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 14, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.452, Rules used = {2498, 2495, 32, 43, 2514, 2487, 31, 8, 2494, 2394, 2393, 2391, 2390, 2301} \[ \frac {q^2 r^2 \log ^2(c+d x) (b c-a d)^4}{4 b d^4}+\frac {25 q^2 r^2 \log (c+d x) (b c-a d)^4}{24 b d^4}+\frac {p q r^2 \log (c+d x) (b c-a d)^4}{8 b d^4}+\frac {p q r^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) (b c-a d)^4}{2 b d^4}-\frac {q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) (b c-a d)^4}{2 b d^4}+\frac {p q r^2 \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right ) (b c-a d)^4}{2 b d^4}-\frac {13 q^2 r^2 x (b c-a d)^3}{24 d^3}-\frac {p q r^2 x (b c-a d)^3}{8 d^3}-\frac {q (p+q) r^2 x (b c-a d)^3}{2 d^3}+\frac {q r (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) (b c-a d)^3}{2 b d^3}+\frac {b p q r^2 x^2 (b c-a d)^2}{8 d^2}+\frac {13 q^2 r^2 (a+b x)^2 (b c-a d)^2}{48 b d^2}+\frac {p q r^2 (a+b x)^2 (b c-a d)^2}{16 b d^2}+\frac {a p q r^2 x (b c-a d)^2}{4 d^2}-\frac {q r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) (b c-a d)^2}{4 b d^2}-\frac {7 q^2 r^2 (a+b x)^3 (b c-a d)}{72 b d}-\frac {7 p q r^2 (a+b x)^3 (b c-a d)}{72 b d}+\frac {q r (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) (b c-a d)}{6 b d}+\frac {p^2 r^2 (a+b x)^4}{32 b}+\frac {q^2 r^2 (a+b x)^4}{32 b}+\frac {p q r^2 (a+b x)^4}{16 b}+\frac {(a+b x)^4 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b}-\frac {p r (a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{8 b}-\frac {q r (a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{8 b} \]
Antiderivative was successfully verified.
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Rule 8
Rule 31
Rule 32
Rule 43
Rule 2301
Rule 2390
Rule 2391
Rule 2393
Rule 2394
Rule 2487
Rule 2494
Rule 2495
Rule 2498
Rule 2514
Rubi steps
\begin {align*} \int (a+b x)^3 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx &=\frac {(a+b x)^4 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b}-\frac {1}{2} (p r) \int (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx-\frac {(d q r) \int \frac {(a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{c+d x} \, dx}{2 b}\\ &=-\frac {p r (a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{8 b}+\frac {(a+b x)^4 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b}-\frac {(d q r) \int \left (-\frac {b (b c-a d)^3 \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) \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)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d^2}+\frac {b (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d}+\frac {(-b c+a d)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{d^4 (c+d x)}\right ) \, dx}{2 b}+\frac {1}{8} \left (p^2 r^2\right ) \int (a+b x)^3 \, dx+\frac {\left (d p q r^2\right ) \int \frac {(a+b x)^4}{c+d x} \, dx}{8 b}\\ &=\frac {p^2 r^2 (a+b x)^4}{32 b}-\frac {p r (a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{8 b}+\frac {(a+b x)^4 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b}-\frac {1}{2} (q r) \int (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx+\frac {((b c-a d) q r) \int (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx}{2 d}-\frac {\left ((b c-a d)^2 q r\right ) \int (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx}{2 d^2}+\frac {\left ((b c-a d)^3 q r\right ) \int \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx}{2 d^3}-\frac {\left ((b c-a d)^4 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}{2 b d^3}+\frac {\left (d p q 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}{8 b}\\ &=-\frac {(b c-a d)^3 p q r^2 x}{8 d^3}+\frac {(b c-a d)^2 p q r^2 (a+b x)^2}{16 b d^2}-\frac {(b c-a d) p q r^2 (a+b x)^3}{24 b d}+\frac {p^2 r^2 (a+b x)^4}{32 b}+\frac {p q r^2 (a+b x)^4}{32 b}+\frac {(b c-a d)^4 p q r^2 \log (c+d x)}{8 b d^4}+\frac {(b c-a d)^3 q r (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b d^3}-\frac {(b c-a d)^2 q r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b d^2}+\frac {(b c-a d) q r (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{6 b d}-\frac {p r (a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{8 b}-\frac {q r (a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{8 b}-\frac {(b c-a d)^4 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b d^4}+\frac {(a+b x)^4 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b}+\frac {1}{8} \left (p q r^2\right ) \int (a+b x)^3 \, dx-\frac {\left ((b c-a d) p q r^2\right ) \int (a+b x)^2 \, dx}{6 d}+\frac {\left ((b c-a d)^2 p q r^2\right ) \int (a+b x) \, dx}{4 d^2}+\frac {\left ((b c-a d)^4 p q r^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{2 d^4}+\frac {\left (d q^2 r^2\right ) \int \frac {(a+b x)^4}{c+d x} \, dx}{8 b}-\frac {\left ((b c-a d) q^2 r^2\right ) \int \frac {(a+b x)^3}{c+d x} \, dx}{6 b}+\frac {\left ((b c-a d)^2 q^2 r^2\right ) \int \frac {(a+b x)^2}{c+d x} \, dx}{4 b d}+\frac {\left ((b c-a d)^4 q^2 r^2\right ) \int \frac {1}{c+d x} \, dx}{2 b d^3}+\frac {\left ((b c-a d)^4 q^2 r^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{2 b d^3}-\frac {\left ((b c-a d)^3 q (p+q) r^2\right ) \int 1 \, dx}{2 d^3}\\ &=\frac {a (b c-a d)^2 p q r^2 x}{4 d^2}-\frac {(b c-a d)^3 p q r^2 x}{8 d^3}-\frac {(b c-a d)^3 q (p+q) r^2 x}{2 d^3}+\frac {b (b c-a d)^2 p q r^2 x^2}{8 d^2}+\frac {(b c-a d)^2 p q r^2 (a+b x)^2}{16 b d^2}-\frac {7 (b c-a d) p q r^2 (a+b x)^3}{72 b d}+\frac {p^2 r^2 (a+b x)^4}{32 b}+\frac {p q r^2 (a+b x)^4}{16 b}+\frac {(b c-a d)^4 p q r^2 \log (c+d x)}{8 b d^4}+\frac {(b c-a d)^4 q^2 r^2 \log (c+d x)}{2 b d^4}+\frac {(b c-a d)^4 p q r^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2 b d^4}+\frac {(b c-a d)^3 q r (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b d^3}-\frac {(b c-a d)^2 q r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b d^2}+\frac {(b c-a d) q r (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{6 b d}-\frac {p r (a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{8 b}-\frac {q r (a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{8 b}-\frac {(b c-a d)^4 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b d^4}+\frac {(a+b x)^4 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b}-\frac {\left ((b c-a d)^4 p q r^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{2 b d^3}+\frac {\left (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}{8 b}-\frac {\left ((b c-a d) 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}{6 b}+\frac {\left ((b c-a d)^2 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}{4 b d}+\frac {\left ((b c-a d)^4 q^2 r^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{2 b d^4}\\ &=\frac {a (b c-a d)^2 p q r^2 x}{4 d^2}-\frac {(b c-a d)^3 p q r^2 x}{8 d^3}-\frac {13 (b c-a d)^3 q^2 r^2 x}{24 d^3}-\frac {(b c-a d)^3 q (p+q) r^2 x}{2 d^3}+\frac {b (b c-a d)^2 p q r^2 x^2}{8 d^2}+\frac {(b c-a d)^2 p q r^2 (a+b x)^2}{16 b d^2}+\frac {13 (b c-a d)^2 q^2 r^2 (a+b x)^2}{48 b d^2}-\frac {7 (b c-a d) p q r^2 (a+b x)^3}{72 b d}-\frac {7 (b c-a d) q^2 r^2 (a+b x)^3}{72 b d}+\frac {p^2 r^2 (a+b x)^4}{32 b}+\frac {p q r^2 (a+b x)^4}{16 b}+\frac {q^2 r^2 (a+b x)^4}{32 b}+\frac {(b c-a d)^4 p q r^2 \log (c+d x)}{8 b d^4}+\frac {25 (b c-a d)^4 q^2 r^2 \log (c+d x)}{24 b d^4}+\frac {(b c-a d)^4 p q r^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2 b d^4}+\frac {(b c-a d)^4 q^2 r^2 \log ^2(c+d x)}{4 b d^4}+\frac {(b c-a d)^3 q r (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b d^3}-\frac {(b c-a d)^2 q r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b d^2}+\frac {(b c-a d) q r (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{6 b d}-\frac {p r (a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{8 b}-\frac {q r (a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{8 b}-\frac {(b c-a d)^4 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b d^4}+\frac {(a+b x)^4 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b}-\frac {\left ((b c-a d)^4 p q r^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{2 b d^4}\\ &=\frac {a (b c-a d)^2 p q r^2 x}{4 d^2}-\frac {(b c-a d)^3 p q r^2 x}{8 d^3}-\frac {13 (b c-a d)^3 q^2 r^2 x}{24 d^3}-\frac {(b c-a d)^3 q (p+q) r^2 x}{2 d^3}+\frac {b (b c-a d)^2 p q r^2 x^2}{8 d^2}+\frac {(b c-a d)^2 p q r^2 (a+b x)^2}{16 b d^2}+\frac {13 (b c-a d)^2 q^2 r^2 (a+b x)^2}{48 b d^2}-\frac {7 (b c-a d) p q r^2 (a+b x)^3}{72 b d}-\frac {7 (b c-a d) q^2 r^2 (a+b x)^3}{72 b d}+\frac {p^2 r^2 (a+b x)^4}{32 b}+\frac {p q r^2 (a+b x)^4}{16 b}+\frac {q^2 r^2 (a+b x)^4}{32 b}+\frac {(b c-a d)^4 p q r^2 \log (c+d x)}{8 b d^4}+\frac {25 (b c-a d)^4 q^2 r^2 \log (c+d x)}{24 b d^4}+\frac {(b c-a d)^4 p q r^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2 b d^4}+\frac {(b c-a d)^4 q^2 r^2 \log ^2(c+d x)}{4 b d^4}+\frac {(b c-a d)^3 q r (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b d^3}-\frac {(b c-a d)^2 q r (a+b x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b d^2}+\frac {(b c-a d) q r (a+b x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{6 b d}-\frac {p r (a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{8 b}-\frac {q r (a+b x)^4 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{8 b}-\frac {(b c-a d)^4 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 b d^4}+\frac {(a+b x)^4 \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b}+\frac {(b c-a d)^4 p q r^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{2 b d^4}\\ \end {align*}
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Mathematica [B] time = 2.09, size = 1853, normalized size = 2.30 \[ \text {result too large to display} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right )} \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x + a\right )}^{3} \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.31, size = 0, normalized size = 0.00 \[ \int \left (b x +a \right )^{3} \ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.19, size = 1071, normalized size = 1.33 \[ \text {result too large to display} \]
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 \[ \int {\ln \left (e\,{\left (f\,{\left (a+b\,x\right )}^p\,{\left (c+d\,x\right )}^q\right )}^r\right )}^2\,{\left (a+b\,x\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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