Integrand size = 25, antiderivative size = 251 \[ \int x^5 \left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx=-\frac {d^2 (e f-d g)^2 p x^2}{2 e^4}+\frac {d (e f-2 d g) (e f-d g) p \left (d+e x^2\right )^2}{4 e^5}-\frac {\left (e^2 f^2-6 d e f g+6 d^2 g^2\right ) p \left (d+e x^2\right )^3}{18 e^5}-\frac {g (e f-2 d g) p \left (d+e x^2\right )^4}{16 e^5}-\frac {g^2 p \left (d+e x^2\right )^5}{50 e^5}+\frac {d^3 \left (10 e^2 f^2-15 d e f g+6 d^2 g^2\right ) p \log \left (d+e x^2\right )}{60 e^5}+\frac {1}{6} f^2 x^6 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{4} f g x^8 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{10} g^2 x^{10} \log \left (c \left (d+e x^2\right )^p\right ) \]
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Time = 0.30 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2525, 45, 2461, 12, 907} \[ \int x^5 \left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx=\frac {1}{6} f^2 x^6 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{4} f g x^8 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{10} g^2 x^{10} \log \left (c \left (d+e x^2\right )^p\right )-\frac {d^2 p x^2 (e f-d g)^2}{2 e^4}-\frac {p \left (d+e x^2\right )^3 \left (6 d^2 g^2-6 d e f g+e^2 f^2\right )}{18 e^5}+\frac {d^3 p \left (6 d^2 g^2-15 d e f g+10 e^2 f^2\right ) \log \left (d+e x^2\right )}{60 e^5}-\frac {g p \left (d+e x^2\right )^4 (e f-2 d g)}{16 e^5}+\frac {d p \left (d+e x^2\right )^2 (e f-2 d g) (e f-d g)}{4 e^5}-\frac {g^2 p \left (d+e x^2\right )^5}{50 e^5} \]
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Rule 12
Rule 45
Rule 907
Rule 2461
Rule 2525
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int x^2 (f+g x)^2 \log \left (c (d+e x)^p\right ) \, dx,x,x^2\right ) \\ & = \frac {1}{6} f^2 x^6 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{4} f g x^8 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{10} g^2 x^{10} \log \left (c \left (d+e x^2\right )^p\right )-\frac {1}{2} (e p) \text {Subst}\left (\int \frac {x^3 \left (10 f^2+15 f g x+6 g^2 x^2\right )}{30 (d+e x)} \, dx,x,x^2\right ) \\ & = \frac {1}{6} f^2 x^6 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{4} f g x^8 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{10} g^2 x^{10} \log \left (c \left (d+e x^2\right )^p\right )-\frac {1}{60} (e p) \text {Subst}\left (\int \frac {x^3 \left (10 f^2+15 f g x+6 g^2 x^2\right )}{d+e x} \, dx,x,x^2\right ) \\ & = \frac {1}{6} f^2 x^6 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{4} f g x^8 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{10} g^2 x^{10} \log \left (c \left (d+e x^2\right )^p\right )-\frac {1}{60} (e p) \text {Subst}\left (\int \left (\frac {30 d^2 (-e f+d g)^2}{e^5}-\frac {d^3 \left (10 e^2 f^2-15 d e f g+6 d^2 g^2\right )}{e^5 (d+e x)}+\frac {30 d (e f-2 d g) (-e f+d g) (d+e x)}{e^5}+\frac {10 \left (e^2 f^2-6 d e f g+6 d^2 g^2\right ) (d+e x)^2}{e^5}+\frac {15 g (e f-2 d g) (d+e x)^3}{e^5}+\frac {6 g^2 (d+e x)^4}{e^5}\right ) \, dx,x,x^2\right ) \\ & = -\frac {d^2 (e f-d g)^2 p x^2}{2 e^4}+\frac {d (e f-2 d g) (e f-d g) p \left (d+e x^2\right )^2}{4 e^5}-\frac {\left (e^2 f^2-6 d e f g+6 d^2 g^2\right ) p \left (d+e x^2\right )^3}{18 e^5}-\frac {g (e f-2 d g) p \left (d+e x^2\right )^4}{16 e^5}-\frac {g^2 p \left (d+e x^2\right )^5}{50 e^5}+\frac {d^3 \left (10 e^2 f^2-15 d e f g+6 d^2 g^2\right ) p \log \left (d+e x^2\right )}{60 e^5}+\frac {1}{6} f^2 x^6 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{4} f g x^8 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{10} g^2 x^{10} \log \left (c \left (d+e x^2\right )^p\right ) \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.82 \[ \int x^5 \left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx=\frac {-e p x^2 \left (360 d^4 g^2-180 d^3 e g \left (5 f+g x^2\right )-30 d e^3 x^2 \left (10 f^2+10 f g x^2+3 g^2 x^4\right )+30 d^2 e^2 \left (20 f^2+15 f g x^2+4 g^2 x^4\right )+e^4 x^4 \left (200 f^2+225 f g x^2+72 g^2 x^4\right )\right )+60 d^3 \left (10 e^2 f^2-15 d e f g+6 d^2 g^2\right ) p \log \left (d+e x^2\right )+60 e^5 x^6 \left (10 f^2+15 f g x^2+6 g^2 x^4\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3600 e^5} \]
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Time = 3.72 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.01
method | result | size |
parts | \(\frac {g^{2} x^{10} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{10}+\frac {f g \,x^{8} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{4}+\frac {f^{2} x^{6} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{6}-\frac {p e \left (\frac {\frac {6}{5} e^{4} g^{2} x^{10}-\frac {3}{2} x^{8} d \,e^{3} g^{2}+\frac {15}{4} e^{4} f g \,x^{8}+2 x^{6} d^{2} e^{2} g^{2}-5 x^{6} d \,e^{3} f g +\frac {10}{3} e^{4} f^{2} x^{6}-3 x^{4} d^{3} e \,g^{2}+\frac {15}{2} x^{4} d^{2} e^{2} f g -5 x^{4} d \,e^{3} f^{2}+6 d^{4} g^{2} x^{2}-15 d^{3} e f g \,x^{2}+10 d^{2} e^{2} f^{2} x^{2}}{2 e^{5}}-\frac {d^{3} \left (6 g^{2} d^{2}-15 d e f g +10 e^{2} f^{2}\right ) \ln \left (e \,x^{2}+d \right )}{2 e^{6}}\right )}{30}\) | \(253\) |
parallelrisch | \(\frac {360 x^{10} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) e^{5} g^{2}-72 e^{5} g^{2} p \,x^{10}+900 x^{8} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) e^{5} f g +90 d \,e^{4} g^{2} p \,x^{8}-225 e^{5} f g p \,x^{8}+600 x^{6} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) e^{5} f^{2}-120 d^{2} e^{3} g^{2} p \,x^{6}+300 d \,e^{4} f g p \,x^{6}-200 e^{5} f^{2} p \,x^{6}+180 d^{3} e^{2} g^{2} p \,x^{4}-450 d^{2} e^{3} f g p \,x^{4}+300 d \,e^{4} f^{2} p \,x^{4}-360 d^{4} e \,g^{2} p \,x^{2}+900 d^{3} e^{2} f g p \,x^{2}-600 d^{2} e^{3} f^{2} p \,x^{2}+360 \ln \left (e \,x^{2}+d \right ) d^{5} g^{2} p -900 \ln \left (e \,x^{2}+d \right ) d^{4} e f g p +600 \ln \left (e \,x^{2}+d \right ) d^{3} e^{2} f^{2} p +360 d^{5} g^{2} p -900 d^{4} e f g p +600 d^{3} e^{2} f^{2} p}{3600 e^{5}}\) | \(318\) |
risch | \(\frac {d f g p \,x^{6}}{12 e}+\frac {\ln \left (c \right ) g^{2} x^{10}}{10}+\frac {\ln \left (c \right ) f^{2} x^{6}}{6}+\frac {i \pi \,g^{2} x^{10} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{20}+\frac {i \pi \,g^{2} x^{10} \operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2}}{20}-\frac {i \pi f g \,x^{8} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}}{8}+\frac {i \pi \,f^{2} x^{6} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{12}+\frac {\ln \left (c \right ) f g \,x^{8}}{4}-\frac {i \pi f g \,x^{8} \operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{8}-\frac {g^{2} p \,x^{10}}{50}-\frac {f^{2} p \,x^{6}}{18}-\frac {f g p \,x^{8}}{16}+\frac {d \,g^{2} p \,x^{8}}{40 e}-\frac {d^{2} g^{2} p \,x^{6}}{30 e^{2}}+\frac {d^{3} g^{2} p \,x^{4}}{20 e^{3}}+\frac {d \,f^{2} p \,x^{4}}{12 e}-\frac {d^{4} g^{2} p \,x^{2}}{10 e^{4}}-\frac {d^{2} f^{2} p \,x^{2}}{6 e^{2}}+\frac {\ln \left (e \,x^{2}+d \right ) d^{5} g^{2} p}{10 e^{5}}+\frac {\ln \left (e \,x^{2}+d \right ) d^{3} f^{2} p}{6 e^{3}}+\frac {i \pi \,f^{2} x^{6} \operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2}}{12}-\frac {\ln \left (e \,x^{2}+d \right ) d^{4} f g p}{4 e^{4}}-\frac {d^{2} f g p \,x^{4}}{8 e^{2}}+\frac {d^{3} f g p \,x^{2}}{4 e^{3}}-\frac {i \pi \,g^{2} x^{10} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}}{20}-\frac {i \pi \,f^{2} x^{6} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}}{12}+\left (\frac {1}{10} g^{2} x^{10}+\frac {1}{4} f g \,x^{8}+\frac {1}{6} f^{2} x^{6}\right ) \ln \left (\left (e \,x^{2}+d \right )^{p}\right )-\frac {i \pi \,f^{2} x^{6} \operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{12}-\frac {i \pi \,g^{2} x^{10} \operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{20}+\frac {i \pi f g \,x^{8} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{8}+\frac {i \pi f g \,x^{8} \operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2}}{8}\) | \(687\) |
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Time = 0.31 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.04 \[ \int x^5 \left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx=-\frac {72 \, e^{5} g^{2} p x^{10} + 45 \, {\left (5 \, e^{5} f g - 2 \, d e^{4} g^{2}\right )} p x^{8} + 20 \, {\left (10 \, e^{5} f^{2} - 15 \, d e^{4} f g + 6 \, d^{2} e^{3} g^{2}\right )} p x^{6} - 30 \, {\left (10 \, d e^{4} f^{2} - 15 \, d^{2} e^{3} f g + 6 \, d^{3} e^{2} g^{2}\right )} p x^{4} + 60 \, {\left (10 \, d^{2} e^{3} f^{2} - 15 \, d^{3} e^{2} f g + 6 \, d^{4} e g^{2}\right )} p x^{2} - 60 \, {\left (6 \, e^{5} g^{2} p x^{10} + 15 \, e^{5} f g p x^{8} + 10 \, e^{5} f^{2} p x^{6} + {\left (10 \, d^{3} e^{2} f^{2} - 15 \, d^{4} e f g + 6 \, d^{5} g^{2}\right )} p\right )} \log \left (e x^{2} + d\right ) - 60 \, {\left (6 \, e^{5} g^{2} x^{10} + 15 \, e^{5} f g x^{8} + 10 \, e^{5} f^{2} x^{6}\right )} \log \left (c\right )}{3600 \, e^{5}} \]
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Timed out. \[ \int x^5 \left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx=\text {Timed out} \]
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Time = 0.19 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.89 \[ \int x^5 \left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx=-\frac {1}{3600} \, e p {\left (\frac {72 \, e^{4} g^{2} x^{10} + 45 \, {\left (5 \, e^{4} f g - 2 \, d e^{3} g^{2}\right )} x^{8} + 20 \, {\left (10 \, e^{4} f^{2} - 15 \, d e^{3} f g + 6 \, d^{2} e^{2} g^{2}\right )} x^{6} - 30 \, {\left (10 \, d e^{3} f^{2} - 15 \, d^{2} e^{2} f g + 6 \, d^{3} e g^{2}\right )} x^{4} + 60 \, {\left (10 \, d^{2} e^{2} f^{2} - 15 \, d^{3} e f g + 6 \, d^{4} g^{2}\right )} x^{2}}{e^{5}} - \frac {60 \, {\left (10 \, d^{3} e^{2} f^{2} - 15 \, d^{4} e f g + 6 \, d^{5} g^{2}\right )} \log \left (e x^{2} + d\right )}{e^{6}}\right )} + \frac {1}{60} \, {\left (6 \, g^{2} x^{10} + 15 \, f g x^{8} + 10 \, f^{2} x^{6}\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 753 vs. \(2 (233) = 466\).
Time = 0.33 (sec) , antiderivative size = 753, normalized size of antiderivative = 3.00 \[ \int x^5 \left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx=\frac {{\left (e x^{2} + d\right )}^{3} f^{2} p \log \left (e x^{2} + d\right )}{6 \, e^{3}} - \frac {{\left (e x^{2} + d\right )}^{2} d f^{2} p \log \left (e x^{2} + d\right )}{2 \, e^{3}} + \frac {{\left (e x^{2} + d\right )}^{4} f g p \log \left (e x^{2} + d\right )}{4 \, e^{4}} - \frac {{\left (e x^{2} + d\right )}^{3} d f g p \log \left (e x^{2} + d\right )}{e^{4}} + \frac {3 \, {\left (e x^{2} + d\right )}^{2} d^{2} f g p \log \left (e x^{2} + d\right )}{2 \, e^{4}} + \frac {{\left (e x^{2} + d\right )}^{5} g^{2} p \log \left (e x^{2} + d\right )}{10 \, e^{5}} - \frac {{\left (e x^{2} + d\right )}^{4} d g^{2} p \log \left (e x^{2} + d\right )}{2 \, e^{5}} + \frac {{\left (e x^{2} + d\right )}^{3} d^{2} g^{2} p \log \left (e x^{2} + d\right )}{e^{5}} - \frac {{\left (e x^{2} + d\right )}^{2} d^{3} g^{2} p \log \left (e x^{2} + d\right )}{e^{5}} - \frac {{\left (e x^{2} + d\right )}^{3} f^{2} p}{18 \, e^{3}} + \frac {{\left (e x^{2} + d\right )}^{2} d f^{2} p}{4 \, e^{3}} - \frac {{\left (e x^{2} + d\right )}^{4} f g p}{16 \, e^{4}} + \frac {{\left (e x^{2} + d\right )}^{3} d f g p}{3 \, e^{4}} - \frac {3 \, {\left (e x^{2} + d\right )}^{2} d^{2} f g p}{4 \, e^{4}} - \frac {{\left (e x^{2} + d\right )}^{5} g^{2} p}{50 \, e^{5}} + \frac {{\left (e x^{2} + d\right )}^{4} d g^{2} p}{8 \, e^{5}} - \frac {{\left (e x^{2} + d\right )}^{3} d^{2} g^{2} p}{3 \, e^{5}} + \frac {{\left (e x^{2} + d\right )}^{2} d^{3} g^{2} p}{2 \, e^{5}} + \frac {{\left (e x^{2} + d\right )}^{3} f^{2} \log \left (c\right )}{6 \, e^{3}} - \frac {{\left (e x^{2} + d\right )}^{2} d f^{2} \log \left (c\right )}{2 \, e^{3}} + \frac {{\left (e x^{2} + d\right )}^{4} f g \log \left (c\right )}{4 \, e^{4}} - \frac {{\left (e x^{2} + d\right )}^{3} d f g \log \left (c\right )}{e^{4}} + \frac {3 \, {\left (e x^{2} + d\right )}^{2} d^{2} f g \log \left (c\right )}{2 \, e^{4}} + \frac {{\left (e x^{2} + d\right )}^{5} g^{2} \log \left (c\right )}{10 \, e^{5}} - \frac {{\left (e x^{2} + d\right )}^{4} d g^{2} \log \left (c\right )}{2 \, e^{5}} + \frac {{\left (e x^{2} + d\right )}^{3} d^{2} g^{2} \log \left (c\right )}{e^{5}} - \frac {{\left (e x^{2} + d\right )}^{2} d^{3} g^{2} \log \left (c\right )}{e^{5}} - \frac {{\left (e x^{2} - {\left (e x^{2} + d\right )} \log \left (e x^{2} + d\right ) + d\right )} d^{2} e^{2} f^{2} p - 2 \, {\left (e x^{2} - {\left (e x^{2} + d\right )} \log \left (e x^{2} + d\right ) + d\right )} d^{3} e f g p + {\left (e x^{2} - {\left (e x^{2} + d\right )} \log \left (e x^{2} + d\right ) + d\right )} d^{4} g^{2} p - {\left (e x^{2} + d\right )} d^{2} e^{2} f^{2} \log \left (c\right ) + 2 \, {\left (e x^{2} + d\right )} d^{3} e f g \log \left (c\right ) - {\left (e x^{2} + d\right )} d^{4} g^{2} \log \left (c\right )}{2 \, e^{5}} \]
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Time = 1.61 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.89 \[ \int x^5 \left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx=\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\,\left (\frac {f^2\,x^6}{6}+\frac {f\,g\,x^8}{4}+\frac {g^2\,x^{10}}{10}\right )-x^6\,\left (\frac {f^2\,p}{18}-\frac {d\,\left (\frac {f\,g\,p}{2}-\frac {d\,g^2\,p}{5\,e}\right )}{6\,e}\right )-x^8\,\left (\frac {f\,g\,p}{16}-\frac {d\,g^2\,p}{40\,e}\right )-\frac {g^2\,p\,x^{10}}{50}+\frac {\ln \left (e\,x^2+d\right )\,\left (6\,p\,d^5\,g^2-15\,p\,d^4\,e\,f\,g+10\,p\,d^3\,e^2\,f^2\right )}{60\,e^5}+\frac {d\,x^4\,\left (\frac {f^2\,p}{3}-\frac {d\,\left (\frac {f\,g\,p}{2}-\frac {d\,g^2\,p}{5\,e}\right )}{e}\right )}{4\,e}-\frac {d^2\,x^2\,\left (\frac {f^2\,p}{3}-\frac {d\,\left (\frac {f\,g\,p}{2}-\frac {d\,g^2\,p}{5\,e}\right )}{e}\right )}{2\,e^2} \]
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