Integrand size = 25, antiderivative size = 210 \[ \int x^3 \left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx=\frac {d (e f-d g)^2 p x^2}{2 e^3}-\frac {(e f-3 d g) (e f-d g) p \left (d+e x^2\right )^2}{8 e^4}-\frac {g (2 e f-3 d g) p \left (d+e x^2\right )^3}{18 e^4}-\frac {g^2 p \left (d+e x^2\right )^4}{32 e^4}-\frac {d^2 \left (6 e^2 f^2-8 d e f g+3 d^2 g^2\right ) p \log \left (d+e x^2\right )}{24 e^4}+\frac {1}{4} f^2 x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{3} f g x^6 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{8} g^2 x^8 \log \left (c \left (d+e x^2\right )^p\right ) \]
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Time = 0.24 (sec) , antiderivative size = 210, 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} \[ \int x^3 \left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx=\frac {1}{4} f^2 x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{3} f g x^6 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{8} g^2 x^8 \log \left (c \left (d+e x^2\right )^p\right )-\frac {d^2 p \left (3 d^2 g^2-8 d e f g+6 e^2 f^2\right ) \log \left (d+e x^2\right )}{24 e^4}-\frac {g p \left (d+e x^2\right )^3 (2 e f-3 d g)}{18 e^4}-\frac {p \left (d+e x^2\right )^2 (e f-3 d g) (e f-d g)}{8 e^4}-\frac {g^2 p \left (d+e x^2\right )^4}{32 e^4}+\frac {d p x^2 (e f-d g)^2}{2 e^3} \]
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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 (f+g x)^2 \log \left (c (d+e x)^p\right ) \, dx,x,x^2\right ) \\ & = \frac {1}{4} f^2 x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{3} f g x^6 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{8} g^2 x^8 \log \left (c \left (d+e x^2\right )^p\right )-\frac {1}{2} (e p) \text {Subst}\left (\int \frac {x^2 \left (6 f^2+8 f g x+3 g^2 x^2\right )}{12 (d+e x)} \, dx,x,x^2\right ) \\ & = \frac {1}{4} f^2 x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{3} f g x^6 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{8} g^2 x^8 \log \left (c \left (d+e x^2\right )^p\right )-\frac {1}{24} (e p) \text {Subst}\left (\int \frac {x^2 \left (6 f^2+8 f g x+3 g^2 x^2\right )}{d+e x} \, dx,x,x^2\right ) \\ & = \frac {1}{4} f^2 x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{3} f g x^6 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{8} g^2 x^8 \log \left (c \left (d+e x^2\right )^p\right )-\frac {1}{24} (e p) \text {Subst}\left (\int \left (-\frac {12 d (-e f+d g)^2}{e^4}+\frac {d^2 \left (6 e^2 f^2-8 d e f g+3 d^2 g^2\right )}{e^4 (d+e x)}+\frac {6 (e f-3 d g) (e f-d g) (d+e x)}{e^4}+\frac {4 g (2 e f-3 d g) (d+e x)^2}{e^4}+\frac {3 g^2 (d+e x)^3}{e^4}\right ) \, dx,x,x^2\right ) \\ & = \frac {d (e f-d g)^2 p x^2}{2 e^3}-\frac {(e f-3 d g) (e f-d g) p \left (d+e x^2\right )^2}{8 e^4}-\frac {g (2 e f-3 d g) p \left (d+e x^2\right )^3}{18 e^4}-\frac {g^2 p \left (d+e x^2\right )^4}{32 e^4}-\frac {d^2 \left (6 e^2 f^2-8 d e f g+3 d^2 g^2\right ) p \log \left (d+e x^2\right )}{24 e^4}+\frac {1}{4} f^2 x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{3} f g x^6 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{8} g^2 x^8 \log \left (c \left (d+e x^2\right )^p\right ) \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.22 \[ \int x^3 \left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx=\frac {d f^2 p x^2}{4 e}-\frac {d^2 f g p x^2}{3 e^2}+\frac {d^3 g^2 p x^2}{8 e^3}-\frac {1}{8} f^2 p x^4+\frac {d f g p x^4}{6 e}-\frac {d^2 g^2 p x^4}{16 e^2}-\frac {1}{9} f g p x^6+\frac {d g^2 p x^6}{24 e}-\frac {1}{32} g^2 p x^8-\frac {d^2 f^2 p \log \left (d+e x^2\right )}{4 e^2}+\frac {d^3 f g p \log \left (d+e x^2\right )}{3 e^3}-\frac {d^4 g^2 p \log \left (d+e x^2\right )}{8 e^4}+\frac {1}{4} f^2 x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{3} f g x^6 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{8} g^2 x^8 \log \left (c \left (d+e x^2\right )^p\right ) \]
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Time = 2.47 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.00
method | result | size |
parts | \(\frac {g^{2} x^{8} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{8}+\frac {f g \,x^{6} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{3}+\frac {f^{2} x^{4} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{4}-\frac {p e \left (-\frac {-\frac {3}{4} e^{3} g^{2} x^{8}+d \,e^{2} g^{2} x^{6}-\frac {8}{3} e^{3} f g \,x^{6}-\frac {3}{2} d^{2} e \,g^{2} x^{4}+4 d f g \,x^{4} e^{2}-3 x^{4} e^{3} f^{2}+3 x^{2} d^{3} g^{2}-8 d^{2} e f g \,x^{2}+6 d \,e^{2} f^{2} x^{2}}{2 e^{4}}+\frac {d^{2} \left (3 g^{2} d^{2}-8 d e f g +6 e^{2} f^{2}\right ) \ln \left (e \,x^{2}+d \right )}{2 e^{5}}\right )}{12}\) | \(211\) |
parallelrisch | \(-\frac {-36 x^{8} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) e^{4} g^{2}+9 x^{8} e^{4} g^{2} p -96 x^{6} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) e^{4} f g -12 x^{6} d \,e^{3} g^{2} p +32 x^{6} e^{4} f g p -72 x^{4} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) e^{4} f^{2}+18 x^{4} d^{2} e^{2} g^{2} p -48 x^{4} d \,e^{3} f g p +36 x^{4} e^{4} f^{2} p -36 x^{2} d^{3} e \,g^{2} p +96 x^{2} d^{2} e^{2} f g p -72 x^{2} d \,e^{3} f^{2} p +36 \ln \left (e \,x^{2}+d \right ) d^{4} g^{2} p -96 \ln \left (e \,x^{2}+d \right ) d^{3} e f g p +72 \ln \left (e \,x^{2}+d \right ) d^{2} e^{2} f^{2} p +36 d^{4} g^{2} p -96 d^{3} e f g p +72 d^{2} e^{2} f^{2} p}{288 e^{4}}\) | \(274\) |
risch | \(\frac {\ln \left (c \right ) g^{2} x^{8}}{8}+\frac {\ln \left (c \right ) f^{2} x^{4}}{4}+\frac {i \pi f g \,x^{6} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{6}-\frac {i \pi \,f^{2} x^{4} \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 {i \pi \,g^{2} 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 )}{16}+\frac {i \pi f g \,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}}{6}+\frac {x^{6} d \,g^{2} p}{24 e}-\frac {x^{4} d^{2} g^{2} p}{16 e^{2}}+\frac {x^{2} d^{3} g^{2} p}{8 e^{3}}+\frac {x^{2} d \,f^{2} p}{4 e}-\frac {\ln \left (e \,x^{2}+d \right ) d^{4} g^{2} p}{8 e^{4}}-\frac {\ln \left (e \,x^{2}+d \right ) d^{2} f^{2} p}{4 e^{2}}-\frac {i \pi \,g^{2} x^{8} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}}{16}-\frac {i \pi \,f^{2} x^{4} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}}{8}+\frac {x^{4} d f g p}{6 e}-\frac {x^{2} d^{2} f g p}{3 e^{2}}+\frac {\ln \left (c \right ) f g \,x^{6}}{3}+\frac {\ln \left (e \,x^{2}+d \right ) d^{3} f g p}{3 e^{3}}-\frac {i \pi f g \,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 )}{6}-\frac {x^{8} g^{2} p}{32}-\frac {x^{4} f^{2} p}{8}-\frac {x^{6} f g p}{9}+\left (\frac {1}{8} g^{2} x^{8}+\frac {1}{3} f g \,x^{6}+\frac {1}{4} x^{4} f^{2}\right ) \ln \left (\left (e \,x^{2}+d \right )^{p}\right )+\frac {i \pi \,g^{2} 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}}{16}+\frac {i \pi \,g^{2} x^{8} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{16}-\frac {i \pi f g \,x^{6} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}}{6}+\frac {i \pi \,f^{2} x^{4} \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}+\frac {i \pi \,f^{2} x^{4} {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{8}\) | \(643\) |
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Time = 0.30 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.07 \[ \int x^3 \left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx=-\frac {9 \, e^{4} g^{2} p x^{8} + 4 \, {\left (8 \, e^{4} f g - 3 \, d e^{3} g^{2}\right )} p x^{6} + 6 \, {\left (6 \, e^{4} f^{2} - 8 \, d e^{3} f g + 3 \, d^{2} e^{2} g^{2}\right )} p x^{4} - 12 \, {\left (6 \, d e^{3} f^{2} - 8 \, d^{2} e^{2} f g + 3 \, d^{3} e g^{2}\right )} p x^{2} - 12 \, {\left (3 \, e^{4} g^{2} p x^{8} + 8 \, e^{4} f g p x^{6} + 6 \, e^{4} f^{2} p x^{4} - {\left (6 \, d^{2} e^{2} f^{2} - 8 \, d^{3} e f g + 3 \, d^{4} g^{2}\right )} p\right )} \log \left (e x^{2} + d\right ) - 12 \, {\left (3 \, e^{4} g^{2} x^{8} + 8 \, e^{4} f g x^{6} + 6 \, e^{4} f^{2} x^{4}\right )} \log \left (c\right )}{288 \, e^{4}} \]
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Timed out. \[ \int x^3 \left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx=\text {Timed out} \]
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Time = 0.19 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.88 \[ \int x^3 \left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx=-\frac {1}{288} \, e p {\left (\frac {9 \, e^{3} g^{2} x^{8} + 4 \, {\left (8 \, e^{3} f g - 3 \, d e^{2} g^{2}\right )} x^{6} + 6 \, {\left (6 \, e^{3} f^{2} - 8 \, d e^{2} f g + 3 \, d^{2} e g^{2}\right )} x^{4} - 12 \, {\left (6 \, d e^{2} f^{2} - 8 \, d^{2} e f g + 3 \, d^{3} g^{2}\right )} x^{2}}{e^{4}} + \frac {12 \, {\left (6 \, d^{2} e^{2} f^{2} - 8 \, d^{3} e f g + 3 \, d^{4} g^{2}\right )} \log \left (e x^{2} + d\right )}{e^{5}}\right )} + \frac {1}{24} \, {\left (3 \, g^{2} x^{8} + 8 \, f g x^{6} + 6 \, f^{2} x^{4}\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 544 vs. \(2 (194) = 388\).
Time = 0.32 (sec) , antiderivative size = 544, normalized size of antiderivative = 2.59 \[ \int x^3 \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 )}^{2} f^{2} p \log \left (e x^{2} + d\right )}{4 \, e^{2}} + \frac {{\left (e x^{2} + d\right )}^{3} f g p \log \left (e x^{2} + d\right )}{3 \, e^{3}} - \frac {{\left (e x^{2} + d\right )}^{2} d f g p \log \left (e x^{2} + d\right )}{e^{3}} + \frac {{\left (e x^{2} + d\right )}^{4} g^{2} p \log \left (e x^{2} + d\right )}{8 \, e^{4}} - \frac {{\left (e x^{2} + d\right )}^{3} d g^{2} p \log \left (e x^{2} + d\right )}{2 \, e^{4}} + \frac {3 \, {\left (e x^{2} + d\right )}^{2} d^{2} g^{2} p \log \left (e x^{2} + d\right )}{4 \, e^{4}} - \frac {{\left (e x^{2} + d\right )}^{2} f^{2} p}{8 \, e^{2}} - \frac {{\left (e x^{2} + d\right )}^{3} f g p}{9 \, e^{3}} + \frac {{\left (e x^{2} + d\right )}^{2} d f g p}{2 \, e^{3}} - \frac {{\left (e x^{2} + d\right )}^{4} g^{2} p}{32 \, e^{4}} + \frac {{\left (e x^{2} + d\right )}^{3} d g^{2} p}{6 \, e^{4}} - \frac {3 \, {\left (e x^{2} + d\right )}^{2} d^{2} g^{2} p}{8 \, e^{4}} + \frac {{\left (e x^{2} + d\right )}^{2} f^{2} \log \left (c\right )}{4 \, e^{2}} + \frac {{\left (e x^{2} + d\right )}^{3} f g \log \left (c\right )}{3 \, e^{3}} - \frac {{\left (e x^{2} + d\right )}^{2} d f g \log \left (c\right )}{e^{3}} + \frac {{\left (e x^{2} + d\right )}^{4} g^{2} \log \left (c\right )}{8 \, e^{4}} - \frac {{\left (e x^{2} + d\right )}^{3} d g^{2} \log \left (c\right )}{2 \, e^{4}} + \frac {3 \, {\left (e x^{2} + d\right )}^{2} d^{2} g^{2} \log \left (c\right )}{4 \, e^{4}} + \frac {{\left (e x^{2} - {\left (e x^{2} + d\right )} \log \left (e x^{2} + d\right ) + d\right )} d 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^{2} e f g p + {\left (e x^{2} - {\left (e x^{2} + d\right )} \log \left (e x^{2} + d\right ) + d\right )} d^{3} g^{2} p - {\left (e x^{2} + d\right )} d e^{2} f^{2} \log \left (c\right ) + 2 \, {\left (e x^{2} + d\right )} d^{2} e f g \log \left (c\right ) - {\left (e x^{2} + d\right )} d^{3} g^{2} \log \left (c\right )}{2 \, e^{4}} \]
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Time = 1.65 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.88 \[ \int x^3 \left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx=\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\,\left (\frac {f^2\,x^4}{4}+\frac {f\,g\,x^6}{3}+\frac {g^2\,x^8}{8}\right )-x^4\,\left (\frac {f^2\,p}{8}-\frac {d\,\left (\frac {2\,f\,g\,p}{3}-\frac {d\,g^2\,p}{4\,e}\right )}{4\,e}\right )-x^6\,\left (\frac {f\,g\,p}{9}-\frac {d\,g^2\,p}{24\,e}\right )-\frac {g^2\,p\,x^8}{32}-\frac {\ln \left (e\,x^2+d\right )\,\left (3\,p\,d^4\,g^2-8\,p\,d^3\,e\,f\,g+6\,p\,d^2\,e^2\,f^2\right )}{24\,e^4}+\frac {d\,x^2\,\left (\frac {f^2\,p}{2}-\frac {d\,\left (\frac {2\,f\,g\,p}{3}-\frac {d\,g^2\,p}{4\,e}\right )}{e}\right )}{2\,e} \]
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