Optimal. Leaf size=231 \[ f^2 x \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{2} f g x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{7} g^2 x^7 \log \left (c \left (d+e x^2\right )^p\right )-\frac {2 d^{7/2} g^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{7 e^{7/2}}+\frac {2 d^3 g^2 p x}{7 e^3}-\frac {d^2 f g p \log \left (d+e x^2\right )}{2 e^2}-\frac {2 d^2 g^2 p x^3}{21 e^2}+\frac {2 \sqrt {d} f^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {d f g p x^2}{2 e}+\frac {2 d g^2 p x^5}{35 e}-2 f^2 p x-\frac {1}{4} f g p x^4-\frac {2}{49} g^2 p x^7 \]
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Rubi [A] time = 0.18, antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {2471, 2448, 321, 205, 2454, 2395, 43, 2455, 302} \[ f^2 x \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{2} f g x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{7} g^2 x^7 \log \left (c \left (d+e x^2\right )^p\right )-\frac {d^2 f g p \log \left (d+e x^2\right )}{2 e^2}-\frac {2 d^2 g^2 p x^3}{21 e^2}+\frac {2 d^3 g^2 p x}{7 e^3}-\frac {2 d^{7/2} g^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{7 e^{7/2}}+\frac {2 \sqrt {d} f^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {d f g p x^2}{2 e}+\frac {2 d g^2 p x^5}{35 e}-2 f^2 p x-\frac {1}{4} f g p x^4-\frac {2}{49} g^2 p x^7 \]
Antiderivative was successfully verified.
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Rule 43
Rule 205
Rule 302
Rule 321
Rule 2395
Rule 2448
Rule 2454
Rule 2455
Rule 2471
Rubi steps
\begin {align*} \int \left (f+g x^3\right )^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx &=\int \left (f^2 \log \left (c \left (d+e x^2\right )^p\right )+2 f g x^3 \log \left (c \left (d+e x^2\right )^p\right )+g^2 x^6 \log \left (c \left (d+e x^2\right )^p\right )\right ) \, dx\\ &=f^2 \int \log \left (c \left (d+e x^2\right )^p\right ) \, dx+(2 f g) \int x^3 \log \left (c \left (d+e x^2\right )^p\right ) \, dx+g^2 \int x^6 \log \left (c \left (d+e x^2\right )^p\right ) \, dx\\ &=f^2 x \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{7} g^2 x^7 \log \left (c \left (d+e x^2\right )^p\right )+(f g) \operatorname {Subst}\left (\int x \log \left (c (d+e x)^p\right ) \, dx,x,x^2\right )-\left (2 e f^2 p\right ) \int \frac {x^2}{d+e x^2} \, dx-\frac {1}{7} \left (2 e g^2 p\right ) \int \frac {x^8}{d+e x^2} \, dx\\ &=-2 f^2 p x+f^2 x \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{2} f g x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{7} g^2 x^7 \log \left (c \left (d+e x^2\right )^p\right )+\left (2 d f^2 p\right ) \int \frac {1}{d+e x^2} \, dx-\frac {1}{2} (e f g p) \operatorname {Subst}\left (\int \frac {x^2}{d+e x} \, dx,x,x^2\right )-\frac {1}{7} \left (2 e g^2 p\right ) \int \left (-\frac {d^3}{e^4}+\frac {d^2 x^2}{e^3}-\frac {d x^4}{e^2}+\frac {x^6}{e}+\frac {d^4}{e^4 \left (d+e x^2\right )}\right ) \, dx\\ &=-2 f^2 p x+\frac {2 d^3 g^2 p x}{7 e^3}-\frac {2 d^2 g^2 p x^3}{21 e^2}+\frac {2 d g^2 p x^5}{35 e}-\frac {2}{49} g^2 p x^7+\frac {2 \sqrt {d} f^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+f^2 x \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{2} f g x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{7} g^2 x^7 \log \left (c \left (d+e x^2\right )^p\right )-\frac {1}{2} (e f g p) \operatorname {Subst}\left (\int \left (-\frac {d}{e^2}+\frac {x}{e}+\frac {d^2}{e^2 (d+e x)}\right ) \, dx,x,x^2\right )-\frac {\left (2 d^4 g^2 p\right ) \int \frac {1}{d+e x^2} \, dx}{7 e^3}\\ &=-2 f^2 p x+\frac {2 d^3 g^2 p x}{7 e^3}+\frac {d f g p x^2}{2 e}-\frac {2 d^2 g^2 p x^3}{21 e^2}-\frac {1}{4} f g p x^4+\frac {2 d g^2 p x^5}{35 e}-\frac {2}{49} g^2 p x^7+\frac {2 \sqrt {d} f^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}-\frac {2 d^{7/2} g^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{7 e^{7/2}}-\frac {d^2 f g p \log \left (d+e x^2\right )}{2 e^2}+f^2 x \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{2} f g x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac {1}{7} g^2 x^7 \log \left (c \left (d+e x^2\right )^p\right )\\ \end {align*}
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Mathematica [A] time = 0.21, size = 178, normalized size = 0.77 \[ \frac {1}{14} x \left (14 f^2+7 f g x^3+2 g^2 x^6\right ) \log \left (c \left (d+e x^2\right )^p\right )-\frac {2 \sqrt {d} p \left (d^3 g^2-7 e^3 f^2\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{7 e^{7/2}}-\frac {d^2 f g p \log \left (d+e x^2\right )}{2 e^2}+\frac {p x \left (840 d^3 g^2-280 d^2 e g^2 x^2+42 d e^2 g x \left (35 f+4 g x^3\right )-15 e^3 \left (392 f^2+49 f g x^3+8 g^2 x^6\right )\right )}{2940 e^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 454, normalized size = 1.97 \[ \left [-\frac {120 \, e^{3} g^{2} p x^{7} - 168 \, d e^{2} g^{2} p x^{5} + 735 \, e^{3} f g p x^{4} + 280 \, d^{2} e g^{2} p x^{3} - 1470 \, d e^{2} f g p x^{2} + 420 \, {\left (7 \, e^{3} f^{2} - d^{3} g^{2}\right )} p \sqrt {-\frac {d}{e}} \log \left (\frac {e x^{2} - 2 \, e x \sqrt {-\frac {d}{e}} - d}{e x^{2} + d}\right ) + 840 \, {\left (7 \, e^{3} f^{2} - d^{3} g^{2}\right )} p x - 210 \, {\left (2 \, e^{3} g^{2} p x^{7} + 7 \, e^{3} f g p x^{4} + 14 \, e^{3} f^{2} p x - 7 \, d^{2} e f g p\right )} \log \left (e x^{2} + d\right ) - 210 \, {\left (2 \, e^{3} g^{2} x^{7} + 7 \, e^{3} f g x^{4} + 14 \, e^{3} f^{2} x\right )} \log \relax (c)}{2940 \, e^{3}}, -\frac {120 \, e^{3} g^{2} p x^{7} - 168 \, d e^{2} g^{2} p x^{5} + 735 \, e^{3} f g p x^{4} + 280 \, d^{2} e g^{2} p x^{3} - 1470 \, d e^{2} f g p x^{2} - 840 \, {\left (7 \, e^{3} f^{2} - d^{3} g^{2}\right )} p \sqrt {\frac {d}{e}} \arctan \left (\frac {e x \sqrt {\frac {d}{e}}}{d}\right ) + 840 \, {\left (7 \, e^{3} f^{2} - d^{3} g^{2}\right )} p x - 210 \, {\left (2 \, e^{3} g^{2} p x^{7} + 7 \, e^{3} f g p x^{4} + 14 \, e^{3} f^{2} p x - 7 \, d^{2} e f g p\right )} \log \left (e x^{2} + d\right ) - 210 \, {\left (2 \, e^{3} g^{2} x^{7} + 7 \, e^{3} f g x^{4} + 14 \, e^{3} f^{2} x\right )} \log \relax (c)}{2940 \, e^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 225, normalized size = 0.97 \[ -\frac {1}{2} \, d^{2} f g p e^{\left (-2\right )} \log \left (x^{2} e + d\right ) - \frac {2 \, {\left (d^{4} g^{2} p - 7 \, d f^{2} p e^{3}\right )} \arctan \left (\frac {x e^{\frac {1}{2}}}{\sqrt {d}}\right ) e^{\left (-\frac {7}{2}\right )}}{7 \, \sqrt {d}} + \frac {1}{2940} \, {\left (420 \, g^{2} p x^{7} e^{3} \log \left (x^{2} e + d\right ) - 120 \, g^{2} p x^{7} e^{3} + 420 \, g^{2} x^{7} e^{3} \log \relax (c) + 168 \, d g^{2} p x^{5} e^{2} - 280 \, d^{2} g^{2} p x^{3} e + 1470 \, f g p x^{4} e^{3} \log \left (x^{2} e + d\right ) - 735 \, f g p x^{4} e^{3} + 1470 \, f g x^{4} e^{3} \log \relax (c) + 840 \, d^{3} g^{2} p x + 1470 \, d f g p x^{2} e^{2} + 2940 \, f^{2} p x e^{3} \log \left (x^{2} e + d\right ) - 5880 \, f^{2} p x e^{3} + 2940 \, f^{2} x e^{3} \log \relax (c)\right )} e^{\left (-3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.54, size = 869, normalized size = 3.76 \[ -\frac {i \pi f g \,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 )}{4}+\frac {g^{2} x^{7} \ln \relax (c )}{7}+f^{2} x \ln \relax (c )-\frac {d^{2} f g p \ln \left (-d^{4} g^{2}+7 d \,e^{3} f^{2}-\sqrt {-d^{7} e \,g^{4}+14 d^{4} e^{4} f^{2} g^{2}-49 d \,e^{7} f^{4}}\, x \right )}{2 e^{2}}-\frac {d^{2} f g p \ln \left (-d^{4} g^{2}+7 d \,e^{3} f^{2}+\sqrt {-d^{7} e \,g^{4}+14 d^{4} e^{4} f^{2} g^{2}-49 d \,e^{7} f^{4}}\, x \right )}{2 e^{2}}-\frac {2 g^{2} p \,x^{7}}{49}-2 f^{2} p x +\left (\frac {1}{7} g^{2} x^{7}+\frac {1}{2} f g \,x^{4}+f^{2} x \right ) \ln \left (\left (e \,x^{2}+d \right )^{p}\right )+\frac {i \pi \,g^{2} x^{7} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2}}{14}+\frac {i \pi \,g^{2} x^{7} \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}}{14}+\frac {\sqrt {-d^{7} e \,g^{4}+14 d^{4} e^{4} f^{2} g^{2}-49 d \,e^{7} f^{4}}\, p \ln \left (-d^{4} g^{2}+7 d \,e^{3} f^{2}-\sqrt {-d^{7} e \,g^{4}+14 d^{4} e^{4} f^{2} g^{2}-49 d \,e^{7} f^{4}}\, x \right )}{7 e^{4}}-\frac {\sqrt {-d^{7} e \,g^{4}+14 d^{4} e^{4} f^{2} g^{2}-49 d \,e^{7} f^{4}}\, p \ln \left (-d^{4} g^{2}+7 d \,e^{3} f^{2}+\sqrt {-d^{7} e \,g^{4}+14 d^{4} e^{4} f^{2} g^{2}-49 d \,e^{7} f^{4}}\, x \right )}{7 e^{4}}+\frac {f g \,x^{4} \ln \relax (c )}{2}+\frac {i \pi \,f^{2} x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2}}{2}+\frac {i \pi \,f^{2} x \,\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}}{2}-\frac {f g p \,x^{4}}{4}+\frac {2 d^{3} g^{2} p x}{7 e^{3}}-\frac {2 d^{2} g^{2} p \,x^{3}}{21 e^{2}}+\frac {2 d \,g^{2} p \,x^{5}}{35 e}-\frac {i \pi \,f^{2} x \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{3}}{2}-\frac {i \pi \,g^{2} x^{7} \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{3}}{14}-\frac {i \pi \,g^{2} x^{7} \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 )}{14}+\frac {i \pi f g \,x^{4} \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{2}}{4}+\frac {i \pi f g \,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}}{4}-\frac {i \pi \,f^{2} x \,\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 )}{2}-\frac {i \pi f g \,x^{4} \mathrm {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )^{3}}{4}+\frac {d f g p \,x^{2}}{2 e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.99, size = 178, normalized size = 0.77 \[ -\frac {1}{2940} \, {\left (\frac {1470 \, d^{2} f g \log \left (e x^{2} + d\right )}{e^{3}} - \frac {840 \, {\left (7 \, d e^{3} f^{2} - d^{4} g^{2}\right )} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e} e^{4}} + \frac {120 \, e^{3} g^{2} x^{7} - 168 \, d e^{2} g^{2} x^{5} + 735 \, e^{3} f g x^{4} + 280 \, d^{2} e g^{2} x^{3} - 1470 \, d e^{2} f g x^{2} + 840 \, {\left (7 \, e^{3} f^{2} - d^{3} g^{2}\right )} x}{e^{4}}\right )} e p + \frac {1}{14} \, {\left (2 \, g^{2} x^{7} + 7 \, f g x^{4} + 14 \, f^{2} x\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.77, size = 317, normalized size = 1.37 \[ \frac {g^2\,x^7\,\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}{7}-2\,f^2\,p\,x-\frac {2\,g^2\,p\,x^7}{49}+f^2\,x\,\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )+\frac {f\,g\,x^4\,\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}{2}-\frac {f\,g\,p\,x^4}{4}+\frac {2\,d\,g^2\,p\,x^5}{35\,e}+\frac {2\,d^3\,g^2\,p\,x}{7\,e^3}-\frac {2\,\sqrt {d}\,f^2\,p\,\mathrm {atan}\left (\frac {7\,\sqrt {d}\,e^{7/2}\,f^2\,p\,x}{d^4\,g^2\,p-7\,d\,e^3\,f^2\,p}-\frac {d^{7/2}\,\sqrt {e}\,g^2\,p\,x}{d^4\,g^2\,p-7\,d\,e^3\,f^2\,p}\right )}{\sqrt {e}}+\frac {2\,d^{7/2}\,g^2\,p\,\mathrm {atan}\left (\frac {7\,\sqrt {d}\,e^{7/2}\,f^2\,p\,x}{d^4\,g^2\,p-7\,d\,e^3\,f^2\,p}-\frac {d^{7/2}\,\sqrt {e}\,g^2\,p\,x}{d^4\,g^2\,p-7\,d\,e^3\,f^2\,p}\right )}{7\,e^{7/2}}-\frac {2\,d^2\,g^2\,p\,x^3}{21\,e^2}+\frac {d\,f\,g\,p\,x^2}{2\,e}-\frac {d^2\,f\,g\,p\,\ln \left (e\,x^2+d\right )}{2\,e^2} \]
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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