Optimal. Leaf size=167 \[ -\frac {n \left (2 a^2 c^2-4 a b^2 c+b^4\right ) \log \left (a+b x+c x^2\right )}{8 c^4}-\frac {b n \sqrt {b^2-4 a c} \left (b^2-2 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{4 c^4}+\frac {b n x \left (b^2-3 a c\right )}{4 c^3}-\frac {n x^2 \left (b^2-2 a c\right )}{8 c^2}+\frac {1}{4} x^4 \log \left (d \left (a+b x+c x^2\right )^n\right )+\frac {b n x^3}{12 c}-\frac {n x^4}{8} \]
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Rubi [A] time = 0.19, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {2525, 800, 634, 618, 206, 628} \[ -\frac {n \left (2 a^2 c^2-4 a b^2 c+b^4\right ) \log \left (a+b x+c x^2\right )}{8 c^4}-\frac {n x^2 \left (b^2-2 a c\right )}{8 c^2}+\frac {b n x \left (b^2-3 a c\right )}{4 c^3}-\frac {b n \sqrt {b^2-4 a c} \left (b^2-2 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{4 c^4}+\frac {1}{4} x^4 \log \left (d \left (a+b x+c x^2\right )^n\right )+\frac {b n x^3}{12 c}-\frac {n x^4}{8} \]
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 628
Rule 634
Rule 800
Rule 2525
Rubi steps
\begin {align*} \int x^3 \log \left (d \left (a+b x+c x^2\right )^n\right ) \, dx &=\frac {1}{4} x^4 \log \left (d \left (a+b x+c x^2\right )^n\right )-\frac {1}{4} n \int \frac {x^4 (b+2 c x)}{a+b x+c x^2} \, dx\\ &=\frac {1}{4} x^4 \log \left (d \left (a+b x+c x^2\right )^n\right )-\frac {1}{4} n \int \left (-\frac {b \left (b^2-3 a c\right )}{c^3}+\frac {\left (b^2-2 a c\right ) x}{c^2}-\frac {b x^2}{c}+2 x^3+\frac {a b \left (b^2-3 a c\right )+\left (b^4-4 a b^2 c+2 a^2 c^2\right ) x}{c^3 \left (a+b x+c x^2\right )}\right ) \, dx\\ &=\frac {b \left (b^2-3 a c\right ) n x}{4 c^3}-\frac {\left (b^2-2 a c\right ) n x^2}{8 c^2}+\frac {b n x^3}{12 c}-\frac {n x^4}{8}+\frac {1}{4} x^4 \log \left (d \left (a+b x+c x^2\right )^n\right )-\frac {n \int \frac {a b \left (b^2-3 a c\right )+\left (b^4-4 a b^2 c+2 a^2 c^2\right ) x}{a+b x+c x^2} \, dx}{4 c^3}\\ &=\frac {b \left (b^2-3 a c\right ) n x}{4 c^3}-\frac {\left (b^2-2 a c\right ) n x^2}{8 c^2}+\frac {b n x^3}{12 c}-\frac {n x^4}{8}+\frac {1}{4} x^4 \log \left (d \left (a+b x+c x^2\right )^n\right )+\frac {\left (b \left (b^2-4 a c\right ) \left (b^2-2 a c\right ) n\right ) \int \frac {1}{a+b x+c x^2} \, dx}{8 c^4}-\frac {\left (\left (b^4-4 a b^2 c+2 a^2 c^2\right ) n\right ) \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{8 c^4}\\ &=\frac {b \left (b^2-3 a c\right ) n x}{4 c^3}-\frac {\left (b^2-2 a c\right ) n x^2}{8 c^2}+\frac {b n x^3}{12 c}-\frac {n x^4}{8}-\frac {\left (b^4-4 a b^2 c+2 a^2 c^2\right ) n \log \left (a+b x+c x^2\right )}{8 c^4}+\frac {1}{4} x^4 \log \left (d \left (a+b x+c x^2\right )^n\right )-\frac {\left (b \left (b^2-4 a c\right ) \left (b^2-2 a c\right ) n\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{4 c^4}\\ &=\frac {b \left (b^2-3 a c\right ) n x}{4 c^3}-\frac {\left (b^2-2 a c\right ) n x^2}{8 c^2}+\frac {b n x^3}{12 c}-\frac {n x^4}{8}-\frac {b \sqrt {b^2-4 a c} \left (b^2-2 a c\right ) n \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{4 c^4}-\frac {\left (b^4-4 a b^2 c+2 a^2 c^2\right ) n \log \left (a+b x+c x^2\right )}{8 c^4}+\frac {1}{4} x^4 \log \left (d \left (a+b x+c x^2\right )^n\right )\\ \end {align*}
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Mathematica [A] time = 0.16, size = 151, normalized size = 0.90 \[ \frac {-3 n \left (2 a^2 c^2-4 a b^2 c+b^4\right ) \log (a+x (b+c x))-6 b n \sqrt {b^2-4 a c} \left (b^2-2 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )+c n x \left (2 b c \left (c x^2-9 a\right )-3 c^2 x \left (c x^2-2 a\right )+6 b^3-3 b^2 c x\right )+6 c^4 x^4 \log \left (d (a+x (b+c x))^n\right )}{24 c^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 364, normalized size = 2.18 \[ \left [-\frac {3 \, c^{4} n x^{4} - 6 \, c^{4} x^{4} \log \relax (d) - 2 \, b c^{3} n x^{3} + 3 \, {\left (b^{2} c^{2} - 2 \, a c^{3}\right )} n x^{2} + 3 \, {\left (b^{3} - 2 \, a b c\right )} \sqrt {b^{2} - 4 \, a c} n \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c + \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) - 6 \, {\left (b^{3} c - 3 \, a b c^{2}\right )} n x - 3 \, {\left (2 \, c^{4} n x^{4} - {\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} n\right )} \log \left (c x^{2} + b x + a\right )}{24 \, c^{4}}, -\frac {3 \, c^{4} n x^{4} - 6 \, c^{4} x^{4} \log \relax (d) - 2 \, b c^{3} n x^{3} + 3 \, {\left (b^{2} c^{2} - 2 \, a c^{3}\right )} n x^{2} + 6 \, {\left (b^{3} - 2 \, a b c\right )} \sqrt {-b^{2} + 4 \, a c} n \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) - 6 \, {\left (b^{3} c - 3 \, a b c^{2}\right )} n x - 3 \, {\left (2 \, c^{4} n x^{4} - {\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} n\right )} \log \left (c x^{2} + b x + a\right )}{24 \, c^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 176, normalized size = 1.05 \[ \frac {1}{4} \, n x^{4} \log \left (c x^{2} + b x + a\right ) - \frac {1}{8} \, {\left (n - 2 \, \log \relax (d)\right )} x^{4} + \frac {b n x^{3}}{12 \, c} - \frac {{\left (b^{2} n - 2 \, a c n\right )} x^{2}}{8 \, c^{2}} + \frac {{\left (b^{3} n - 3 \, a b c n\right )} x}{4 \, c^{3}} - \frac {{\left (b^{4} n - 4 \, a b^{2} c n + 2 \, a^{2} c^{2} n\right )} \log \left (c x^{2} + b x + a\right )}{8 \, c^{4}} + \frac {{\left (b^{5} n - 6 \, a b^{3} c n + 8 \, a^{2} b c^{2} n\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{4 \, \sqrt {-b^{2} + 4 \, a c} c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.61, size = 1146, normalized size = 6.86 \[ -\frac {i \pi \,x^{4} \mathrm {csgn}\left (i d \right ) \mathrm {csgn}\left (i \left (c \,x^{2}+b x +a \right )^{n}\right ) \mathrm {csgn}\left (i d \left (c \,x^{2}+b x +a \right )^{n}\right )}{8}+\frac {i \pi \,x^{4} \mathrm {csgn}\left (i d \right ) \mathrm {csgn}\left (i d \left (c \,x^{2}+b x +a \right )^{n}\right )^{2}}{8}+\frac {i \pi \,x^{4} \mathrm {csgn}\left (i \left (c \,x^{2}+b x +a \right )^{n}\right ) \mathrm {csgn}\left (i d \left (c \,x^{2}+b x +a \right )^{n}\right )^{2}}{8}-\frac {i \pi \,x^{4} \mathrm {csgn}\left (i d \left (c \,x^{2}+b x +a \right )^{n}\right )^{3}}{8}-\frac {n \,x^{4}}{8}+\frac {x^{4} \ln \relax (d )}{4}+\frac {x^{4} \ln \left (\left (c \,x^{2}+b x +a \right )^{n}\right )}{4}+\frac {b n \,x^{3}}{12 c}+\frac {a n \,x^{2}}{4 c}-\frac {b^{2} n \,x^{2}}{8 c^{2}}-\frac {a^{2} n \ln \left (8 a^{2} b \,c^{2}-6 a \,b^{3} c +b^{5}-2 \sqrt {-16 a^{3} b^{2} c^{3}+20 a^{2} b^{4} c^{2}-8 a \,b^{6} c +b^{8}}\, c x -\sqrt {-16 a^{3} b^{2} c^{3}+20 a^{2} b^{4} c^{2}-8 a \,b^{6} c +b^{8}}\, b \right )}{4 c^{2}}-\frac {a^{2} n \ln \left (8 a^{2} b \,c^{2}-6 a \,b^{3} c +b^{5}+2 \sqrt {-16 a^{3} b^{2} c^{3}+20 a^{2} b^{4} c^{2}-8 a \,b^{6} c +b^{8}}\, c x +\sqrt {-16 a^{3} b^{2} c^{3}+20 a^{2} b^{4} c^{2}-8 a \,b^{6} c +b^{8}}\, b \right )}{4 c^{2}}+\frac {a \,b^{2} n \ln \left (8 a^{2} b \,c^{2}-6 a \,b^{3} c +b^{5}-2 \sqrt {-16 a^{3} b^{2} c^{3}+20 a^{2} b^{4} c^{2}-8 a \,b^{6} c +b^{8}}\, c x -\sqrt {-16 a^{3} b^{2} c^{3}+20 a^{2} b^{4} c^{2}-8 a \,b^{6} c +b^{8}}\, b \right )}{2 c^{3}}+\frac {a \,b^{2} n \ln \left (8 a^{2} b \,c^{2}-6 a \,b^{3} c +b^{5}+2 \sqrt {-16 a^{3} b^{2} c^{3}+20 a^{2} b^{4} c^{2}-8 a \,b^{6} c +b^{8}}\, c x +\sqrt {-16 a^{3} b^{2} c^{3}+20 a^{2} b^{4} c^{2}-8 a \,b^{6} c +b^{8}}\, b \right )}{2 c^{3}}-\frac {3 a b n x}{4 c^{2}}-\frac {b^{4} n \ln \left (8 a^{2} b \,c^{2}-6 a \,b^{3} c +b^{5}-2 \sqrt {-16 a^{3} b^{2} c^{3}+20 a^{2} b^{4} c^{2}-8 a \,b^{6} c +b^{8}}\, c x -\sqrt {-16 a^{3} b^{2} c^{3}+20 a^{2} b^{4} c^{2}-8 a \,b^{6} c +b^{8}}\, b \right )}{8 c^{4}}-\frac {b^{4} n \ln \left (8 a^{2} b \,c^{2}-6 a \,b^{3} c +b^{5}+2 \sqrt {-16 a^{3} b^{2} c^{3}+20 a^{2} b^{4} c^{2}-8 a \,b^{6} c +b^{8}}\, c x +\sqrt {-16 a^{3} b^{2} c^{3}+20 a^{2} b^{4} c^{2}-8 a \,b^{6} c +b^{8}}\, b \right )}{8 c^{4}}+\frac {b^{3} n x}{4 c^{3}}+\frac {\sqrt {-16 a^{3} b^{2} c^{3}+20 a^{2} b^{4} c^{2}-8 a \,b^{6} c +b^{8}}\, n \ln \left (8 a^{2} b \,c^{2}-6 a \,b^{3} c +b^{5}-2 \sqrt {-16 a^{3} b^{2} c^{3}+20 a^{2} b^{4} c^{2}-8 a \,b^{6} c +b^{8}}\, c x -\sqrt {-16 a^{3} b^{2} c^{3}+20 a^{2} b^{4} c^{2}-8 a \,b^{6} c +b^{8}}\, b \right )}{8 c^{4}}-\frac {\sqrt {-16 a^{3} b^{2} c^{3}+20 a^{2} b^{4} c^{2}-8 a \,b^{6} c +b^{8}}\, n \ln \left (8 a^{2} b \,c^{2}-6 a \,b^{3} c +b^{5}+2 \sqrt {-16 a^{3} b^{2} c^{3}+20 a^{2} b^{4} c^{2}-8 a \,b^{6} c +b^{8}}\, c x +\sqrt {-16 a^{3} b^{2} c^{3}+20 a^{2} b^{4} c^{2}-8 a \,b^{6} c +b^{8}}\, b \right )}{8 c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.54, size = 288, normalized size = 1.72 \[ x\,\left (\frac {b\,\left (\frac {b^2\,n}{4\,c^2}-\frac {a\,n}{2\,c}\right )}{c}-\frac {a\,b\,n}{4\,c^2}\right )-\frac {n\,x^4}{8}+\frac {x^4\,\ln \left (d\,{\left (c\,x^2+b\,x+a\right )}^n\right )}{4}-x^2\,\left (\frac {b^2\,n}{8\,c^2}-\frac {a\,n}{4\,c}\right )+\frac {\ln \left (4\,a\,c+b\,\sqrt {b^2-4\,a\,c}-b^2+2\,c\,x\,\sqrt {b^2-4\,a\,c}\right )\,\left (c\,\left (\frac {a\,b^2\,n}{2}-\frac {a\,b\,n\,\sqrt {b^2-4\,a\,c}}{4}\right )-\frac {b^4\,n}{8}+\frac {b^3\,n\,\sqrt {b^2-4\,a\,c}}{8}-\frac {a^2\,c^2\,n}{4}\right )}{c^4}-\frac {\ln \left (b\,\sqrt {b^2-4\,a\,c}-4\,a\,c+b^2+2\,c\,x\,\sqrt {b^2-4\,a\,c}\right )\,\left (\frac {b^4\,n}{8}-c\,\left (\frac {a\,b^2\,n}{2}+\frac {a\,b\,n\,\sqrt {b^2-4\,a\,c}}{4}\right )+\frac {b^3\,n\,\sqrt {b^2-4\,a\,c}}{8}+\frac {a^2\,c^2\,n}{4}\right )}{c^4}+\frac {b\,n\,x^3}{12\,c} \]
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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