Optimal. Leaf size=210 \[ \frac {1}{4} f^2 x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{3} f g x^6 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{8} g^2 x^8 \log \left (c \left (d+e x^2\right )^p\right )-\frac {d^2 p \left (3 d^2 g^2-8 d e f g+6 e^2 f^2\right ) \log \left (d+e x^2\right )}{24 e^4}-\frac {g p \left (d+e x^2\right )^3 (2 e f-3 d g)}{18 e^4}-\frac {p \left (d+e x^2\right )^2 (e f-3 d g) (e f-d g)}{8 e^4}-\frac {g^2 p \left (d+e x^2\right )^4}{32 e^4}+\frac {d p x^2 (e f-d g)^2}{2 e^3} \]
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Rubi [A] time = 0.36, antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2475, 43, 2414, 12, 893} \[ \frac {1}{4} f^2 x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{3} f g x^6 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{8} g^2 x^8 \log \left (c \left (d+e x^2\right )^p\right )-\frac {d^2 p \left (3 d^2 g^2-8 d e f g+6 e^2 f^2\right ) \log \left (d+e x^2\right )}{24 e^4}+\frac {d p x^2 (e f-d g)^2}{2 e^3}-\frac {g p \left (d+e x^2\right )^3 (2 e f-3 d g)}{18 e^4}-\frac {p \left (d+e x^2\right )^2 (e f-3 d g) (e f-d g)}{8 e^4}-\frac {g^2 p \left (d+e x^2\right )^4}{32 e^4} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 893
Rule 2414
Rule 2475
Rubi steps
\begin {align*} \int x^3 \left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx &=\frac {1}{2} \operatorname {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) \operatorname {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) \operatorname {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) \operatorname {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*}
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Mathematica [A] time = 0.14, size = 173, normalized size = 0.82 \[ \frac {12 e^4 x^4 \left (6 f^2+8 f g x^2+3 g^2 x^4\right ) \log \left (c \left (d+e x^2\right )^p\right )-12 d^2 p \left (3 d^2 g^2-8 d e f g+6 e^2 f^2\right ) \log \left (d+e x^2\right )+e p x^2 \left (36 d^3 g^2-6 d^2 e g \left (16 f+3 g x^2\right )+12 d e^2 \left (6 f^2+4 f g x^2+g^2 x^4\right )-e^3 x^2 \left (36 f^2+32 f g x^2+9 g^2 x^4\right )\right )}{288 e^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 224, normalized size = 1.07 \[ -\frac {9 \, e^{4} g^{2} p x^{8} + 4 \, {\left (8 \, e^{4} f g - 3 \, d e^{3} g^{2}\right )} p x^{6} + 6 \, {\left (6 \, e^{4} f^{2} - 8 \, d e^{3} f g + 3 \, d^{2} e^{2} g^{2}\right )} p x^{4} - 12 \, {\left (6 \, d e^{3} f^{2} - 8 \, d^{2} e^{2} f g + 3 \, d^{3} e g^{2}\right )} p x^{2} - 12 \, {\left (3 \, e^{4} g^{2} p x^{8} + 8 \, e^{4} f g p x^{6} + 6 \, e^{4} f^{2} p x^{4} - {\left (6 \, d^{2} e^{2} f^{2} - 8 \, d^{3} e f g + 3 \, d^{4} g^{2}\right )} p\right )} \log \left (e x^{2} + d\right ) - 12 \, {\left (3 \, e^{4} g^{2} x^{8} + 8 \, e^{4} f g x^{6} + 6 \, e^{4} f^{2} x^{4}\right )} \log \relax (c)}{288 \, e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 418, normalized size = 1.99 \[ \frac {1}{288} \, {\left (36 \, g^{2} x^{8} e \log \relax (c) + 96 \, f g x^{6} e \log \relax (c) + 36 \, {\left (2 \, {\left (x^{2} e + d\right )}^{2} \log \left (x^{2} e + d\right ) - 4 \, {\left (x^{2} e + d\right )} d \log \left (x^{2} e + d\right ) - {\left (x^{2} e + d\right )}^{2} + 4 \, {\left (x^{2} e + d\right )} d\right )} f^{2} p e^{\left (-1\right )} + 72 \, {\left ({\left (x^{2} e + d\right )}^{2} - 2 \, {\left (x^{2} e + d\right )} d\right )} f^{2} e^{\left (-1\right )} \log \relax (c) + 16 \, {\left (6 \, {\left (x^{2} e + d\right )}^{3} e^{\left (-2\right )} \log \left (x^{2} e + d\right ) - 18 \, {\left (x^{2} e + d\right )}^{2} d e^{\left (-2\right )} \log \left (x^{2} e + d\right ) + 18 \, {\left (x^{2} e + d\right )} d^{2} e^{\left (-2\right )} \log \left (x^{2} e + d\right ) - 2 \, {\left (x^{2} e + d\right )}^{3} e^{\left (-2\right )} + 9 \, {\left (x^{2} e + d\right )}^{2} d e^{\left (-2\right )} - 18 \, {\left (x^{2} e + d\right )} d^{2} e^{\left (-2\right )}\right )} f g p + 3 \, {\left (12 \, {\left (x^{2} e + d\right )}^{4} e^{\left (-3\right )} \log \left (x^{2} e + d\right ) - 48 \, {\left (x^{2} e + d\right )}^{3} d e^{\left (-3\right )} \log \left (x^{2} e + d\right ) + 72 \, {\left (x^{2} e + d\right )}^{2} d^{2} e^{\left (-3\right )} \log \left (x^{2} e + d\right ) - 48 \, {\left (x^{2} e + d\right )} d^{3} e^{\left (-3\right )} \log \left (x^{2} e + d\right ) - 3 \, {\left (x^{2} e + d\right )}^{4} e^{\left (-3\right )} + 16 \, {\left (x^{2} e + d\right )}^{3} d e^{\left (-3\right )} - 36 \, {\left (x^{2} e + d\right )}^{2} d^{2} e^{\left (-3\right )} + 48 \, {\left (x^{2} e + d\right )} d^{3} e^{\left (-3\right )}\right )} g^{2} p\right )} e^{\left (-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.47, size = 643, normalized size = 3.06 \[ \frac {f g \,x^{6} \ln \relax (c )}{3}-\frac {f^{2} p \,x^{4}}{8}-\frac {g^{2} p \,x^{8}}{32}+\frac {g^{2} x^{8} \ln \relax (c )}{8}+\frac {f^{2} x^{4} \ln \relax (c )}{4}+\frac {d^{3} f g p \ln \left (e \,x^{2}+d \right )}{3 e^{3}}+\left (\frac {1}{8} g^{2} x^{8}+\frac {1}{3} f g \,x^{6}+\frac {1}{4} f^{2} x^{4}\right ) \ln \left (\left (e \,x^{2}+d \right )^{p}\right )+\frac {i \pi f g \,x^{6} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2}}{6}+\frac {i \pi f g \,x^{6} \mathrm {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2}}{6}-\frac {i \pi \,f^{2} x^{4} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}{8}-\frac {i \pi f g \,x^{6} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}{6}-\frac {f g p \,x^{6}}{9}-\frac {i \pi \,g^{2} x^{8} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}{16}-\frac {i \pi \,g^{2} x^{8} \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{3}}{16}-\frac {i \pi \,f^{2} x^{4} \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{3}}{8}+\frac {d \,g^{2} p \,x^{6}}{24 e}-\frac {d^{2} g^{2} p \,x^{4}}{16 e^{2}}+\frac {d^{3} g^{2} p \,x^{2}}{8 e^{3}}+\frac {d \,f^{2} p \,x^{2}}{4 e}-\frac {d^{4} g^{2} p \ln \left (e \,x^{2}+d \right )}{8 e^{4}}-\frac {d^{2} f^{2} p \ln \left (e \,x^{2}+d \right )}{4 e^{2}}+\frac {i \pi \,g^{2} x^{8} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2}}{16}+\frac {i \pi \,g^{2} x^{8} \mathrm {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2}}{16}-\frac {i \pi f g \,x^{6} \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{3}}{6}+\frac {i \pi \,f^{2} x^{4} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2}}{8}+\frac {i \pi \,f^{2} x^{4} \mathrm {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2}}{8}+\frac {d f g p \,x^{4}}{6 e}-\frac {d^{2} f g p \,x^{2}}{3 e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 185, normalized size = 0.88 \[ -\frac {1}{288} \, e p {\left (\frac {9 \, e^{3} g^{2} x^{8} + 4 \, {\left (8 \, e^{3} f g - 3 \, d e^{2} g^{2}\right )} x^{6} + 6 \, {\left (6 \, e^{3} f^{2} - 8 \, d e^{2} f g + 3 \, d^{2} e g^{2}\right )} x^{4} - 12 \, {\left (6 \, d e^{2} f^{2} - 8 \, d^{2} e f g + 3 \, d^{3} g^{2}\right )} x^{2}}{e^{4}} + \frac {12 \, {\left (6 \, d^{2} e^{2} f^{2} - 8 \, d^{3} e f g + 3 \, d^{4} g^{2}\right )} \log \left (e x^{2} + d\right )}{e^{5}}\right )} + \frac {1}{24} \, {\left (3 \, g^{2} x^{8} + 8 \, f g x^{6} + 6 \, f^{2} x^{4}\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.34, size = 184, normalized size = 0.88 \[ \ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\,\left (\frac {f^2\,x^4}{4}+\frac {f\,g\,x^6}{3}+\frac {g^2\,x^8}{8}\right )-x^4\,\left (\frac {f^2\,p}{8}-\frac {d\,\left (\frac {2\,f\,g\,p}{3}-\frac {d\,g^2\,p}{4\,e}\right )}{4\,e}\right )-x^6\,\left (\frac {f\,g\,p}{9}-\frac {d\,g^2\,p}{24\,e}\right )-\frac {g^2\,p\,x^8}{32}-\frac {\ln \left (e\,x^2+d\right )\,\left (3\,p\,d^4\,g^2-8\,p\,d^3\,e\,f\,g+6\,p\,d^2\,e^2\,f^2\right )}{24\,e^4}+\frac {d\,x^2\,\left (\frac {f^2\,p}{2}-\frac {d\,\left (\frac {2\,f\,g\,p}{3}-\frac {d\,g^2\,p}{4\,e}\right )}{e}\right )}{2\,e} \]
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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