Optimal. Leaf size=216 \[ -\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{8 x^8}-\frac {f g \log \left (c \left (d+e x^2\right )^p\right )}{3 x^6}-\frac {g^2 \log \left (c \left (d+e x^2\right )^p\right )}{4 x^4}+\frac {e f p (3 e f-8 d g)}{48 d^2 x^4}+\frac {e^2 p \left (6 d^2 g^2-8 d e f g+3 e^2 f^2\right ) \log \left (d+e x^2\right )}{24 d^4}-\frac {e^2 p \log (x) \left (6 d^2 g^2-8 d e f g+3 e^2 f^2\right )}{12 d^4}-\frac {e p \left (6 d^2 g^2-8 d e f g+3 e^2 f^2\right )}{24 d^3 x^2}-\frac {e f^2 p}{24 d x^6} \]
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Rubi [A] time = 0.29, antiderivative size = 216, 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} \[ -\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{8 x^8}-\frac {f g \log \left (c \left (d+e x^2\right )^p\right )}{3 x^6}-\frac {g^2 \log \left (c \left (d+e x^2\right )^p\right )}{4 x^4}-\frac {e p \left (6 d^2 g^2-8 d e f g+3 e^2 f^2\right )}{24 d^3 x^2}+\frac {e^2 p \left (6 d^2 g^2-8 d e f g+3 e^2 f^2\right ) \log \left (d+e x^2\right )}{24 d^4}-\frac {e^2 p \log (x) \left (6 d^2 g^2-8 d e f g+3 e^2 f^2\right )}{12 d^4}+\frac {e f p (3 e f-8 d g)}{48 d^2 x^4}-\frac {e f^2 p}{24 d x^6} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 893
Rule 2414
Rule 2475
Rubi steps
\begin {align*} \int \frac {\left (f+g x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{x^9} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(f+g x)^2 \log \left (c (d+e x)^p\right )}{x^5} \, dx,x,x^2\right )\\ &=-\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{8 x^8}-\frac {f g \log \left (c \left (d+e x^2\right )^p\right )}{3 x^6}-\frac {g^2 \log \left (c \left (d+e x^2\right )^p\right )}{4 x^4}-\frac {1}{2} (e p) \operatorname {Subst}\left (\int \frac {-3 f^2-8 f g x-6 g^2 x^2}{12 x^4 (d+e x)} \, dx,x,x^2\right )\\ &=-\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{8 x^8}-\frac {f g \log \left (c \left (d+e x^2\right )^p\right )}{3 x^6}-\frac {g^2 \log \left (c \left (d+e x^2\right )^p\right )}{4 x^4}-\frac {1}{24} (e p) \operatorname {Subst}\left (\int \frac {-3 f^2-8 f g x-6 g^2 x^2}{x^4 (d+e x)} \, dx,x,x^2\right )\\ &=-\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{8 x^8}-\frac {f g \log \left (c \left (d+e x^2\right )^p\right )}{3 x^6}-\frac {g^2 \log \left (c \left (d+e x^2\right )^p\right )}{4 x^4}-\frac {1}{24} (e p) \operatorname {Subst}\left (\int \left (-\frac {3 f^2}{d x^4}-\frac {f (-3 e f+8 d g)}{d^2 x^3}+\frac {-3 e^2 f^2+8 d e f g-6 d^2 g^2}{d^3 x^2}+\frac {e \left (3 e^2 f^2-8 d e f g+6 d^2 g^2\right )}{d^4 x}-\frac {e^2 \left (3 e^2 f^2-8 d e f g+6 d^2 g^2\right )}{d^4 (d+e x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {e f^2 p}{24 d x^6}+\frac {e f (3 e f-8 d g) p}{48 d^2 x^4}-\frac {e \left (3 e^2 f^2-8 d e f g+6 d^2 g^2\right ) p}{24 d^3 x^2}-\frac {e^2 \left (3 e^2 f^2-8 d e f g+6 d^2 g^2\right ) p \log (x)}{12 d^4}+\frac {e^2 \left (3 e^2 f^2-8 d e f g+6 d^2 g^2\right ) p \log \left (d+e x^2\right )}{24 d^4}-\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{8 x^8}-\frac {f g \log \left (c \left (d+e x^2\right )^p\right )}{3 x^6}-\frac {g^2 \log \left (c \left (d+e x^2\right )^p\right )}{4 x^4}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 184, normalized size = 0.85 \[ -\frac {2 d^4 \left (3 f^2+8 f g x^2+6 g^2 x^4\right ) \log \left (c \left (d+e x^2\right )^p\right )+4 e^2 p x^8 \log (x) \left (6 d^2 g^2-8 d e f g+3 e^2 f^2\right )-2 e^2 p x^8 \left (6 d^2 g^2-8 d e f g+3 e^2 f^2\right ) \log \left (d+e x^2\right )+d e p x^2 \left (2 d^2 \left (f^2+4 f g x^2+6 g^2 x^4\right )-d e f x^2 \left (3 f+16 g x^2\right )+6 e^2 f^2 x^4\right )}{48 d^4 x^8} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 230, normalized size = 1.06 \[ -\frac {4 \, {\left (3 \, e^{4} f^{2} - 8 \, d e^{3} f g + 6 \, d^{2} e^{2} g^{2}\right )} p x^{8} \log \relax (x) + 2 \, d^{3} e f^{2} p x^{2} + 2 \, {\left (3 \, d e^{3} f^{2} - 8 \, d^{2} e^{2} f g + 6 \, d^{3} e g^{2}\right )} p x^{6} - {\left (3 \, d^{2} e^{2} f^{2} - 8 \, d^{3} e f g\right )} p x^{4} + 2 \, {\left (6 \, d^{4} g^{2} p x^{4} - {\left (3 \, e^{4} f^{2} - 8 \, d e^{3} f g + 6 \, d^{2} e^{2} g^{2}\right )} p x^{8} + 8 \, d^{4} f g p x^{2} + 3 \, d^{4} f^{2} p\right )} \log \left (e x^{2} + d\right ) + 2 \, {\left (6 \, d^{4} g^{2} x^{4} + 8 \, d^{4} f g x^{2} + 3 \, d^{4} f^{2}\right )} \log \relax (c)}{48 \, d^{4} x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.24, size = 1089, normalized size = 5.04 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.51, size = 713, normalized size = 3.30 \[ -\frac {\left (6 g^{2} x^{4}+8 f g \,x^{2}+3 f^{2}\right ) \ln \left (\left (e \,x^{2}+d \right )^{p}\right )}{24 x^{8}}-\frac {6 d^{4} f^{2} \ln \relax (c )-8 i \pi \,d^{4} f g \,x^{2} \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{3}-3 i \pi \,d^{4} f^{2} \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 )+12 d^{4} g^{2} x^{4} \ln \relax (c )+3 i \pi \,d^{4} f^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2}+3 i \pi \,d^{4} f^{2} \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}+12 d^{3} e \,g^{2} p \,x^{6}+6 d \,e^{3} f^{2} p \,x^{6}-3 d^{2} e^{2} f^{2} p \,x^{4}+2 d^{3} e \,f^{2} p \,x^{2}-3 i \pi \,d^{4} f^{2} \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{3}-6 e^{4} f^{2} p \,x^{8} \ln \left (-e \,x^{2}-d \right )+12 e^{4} f^{2} p \,x^{8} \ln \relax (x )+16 d^{4} f g \,x^{2} \ln \relax (c )-16 d^{2} e^{2} f g p \,x^{6}+8 d^{3} e f g p \,x^{4}+6 i \pi \,d^{4} g^{2} x^{4} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2}+6 i \pi \,d^{4} g^{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 i \pi \,d^{4} f g \,x^{2} \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 i \pi \,d^{4} g^{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 i \pi \,d^{4} f g \,x^{2} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2}+8 i \pi \,d^{4} f g \,x^{2} \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 i \pi \,d^{4} g^{2} x^{4} \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{3}-12 d^{2} e^{2} g^{2} p \,x^{8} \ln \left (-e \,x^{2}-d \right )+24 d^{2} e^{2} g^{2} p \,x^{8} \ln \relax (x )+16 d \,e^{3} f g p \,x^{8} \ln \left (-e \,x^{2}-d \right )-32 d \,e^{3} f g p \,x^{8} \ln \relax (x )}{48 d^{4} x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 183, normalized size = 0.85 \[ \frac {1}{48} \, e p {\left (\frac {2 \, {\left (3 \, e^{3} f^{2} - 8 \, d e^{2} f g + 6 \, d^{2} e g^{2}\right )} \log \left (e x^{2} + d\right )}{d^{4}} - \frac {2 \, {\left (3 \, e^{3} f^{2} - 8 \, d e^{2} f g + 6 \, d^{2} e g^{2}\right )} \log \left (x^{2}\right )}{d^{4}} - \frac {2 \, {\left (3 \, e^{2} f^{2} - 8 \, d e f g + 6 \, d^{2} g^{2}\right )} x^{4} + 2 \, d^{2} f^{2} - {\left (3 \, d e f^{2} - 8 \, d^{2} f g\right )} x^{2}}{d^{3} x^{6}}\right )} - \frac {{\left (6 \, g^{2} x^{4} + 8 \, f g x^{2} + 3 \, f^{2}\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{24 \, x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.43, size = 190, normalized size = 0.88 \[ \frac {\ln \left (e\,x^2+d\right )\,\left (6\,p\,d^2\,e^2\,g^2-8\,p\,d\,e^3\,f\,g+3\,p\,e^4\,f^2\right )}{24\,d^4}-\frac {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )\,\left (\frac {f^2}{8}+\frac {f\,g\,x^2}{3}+\frac {g^2\,x^4}{4}\right )}{x^8}-\frac {\frac {e\,f^2\,p}{2\,d}+\frac {e\,p\,x^4\,\left (6\,d^2\,g^2-8\,d\,e\,f\,g+3\,e^2\,f^2\right )}{2\,d^3}+\frac {e\,f\,p\,x^2\,\left (8\,d\,g-3\,e\,f\right )}{4\,d^2}}{12\,x^6}-\frac {\ln \relax (x)\,\left (6\,p\,d^2\,e^2\,g^2-8\,p\,d\,e^3\,f\,g+3\,p\,e^4\,f^2\right )}{12\,d^4} \]
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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