Optimal. Leaf size=136 \[ \frac {b n \left (b^2-3 a c\right ) \log \left (a+b x+c x^2\right )}{6 c^3}+\frac {n \sqrt {b^2-4 a c} \left (b^2-a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{3 c^3}-\frac {n x \left (b^2-2 a c\right )}{3 c^2}+\frac {1}{3} x^3 \log \left (d \left (a+b x+c x^2\right )^n\right )+\frac {b n x^2}{6 c}-\frac {2 n x^3}{9} \]
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Rubi [A] time = 0.15, antiderivative size = 136, 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} \[ \frac {b n \left (b^2-3 a c\right ) \log \left (a+b x+c x^2\right )}{6 c^3}-\frac {n x \left (b^2-2 a c\right )}{3 c^2}+\frac {n \sqrt {b^2-4 a c} \left (b^2-a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{3 c^3}+\frac {1}{3} x^3 \log \left (d \left (a+b x+c x^2\right )^n\right )+\frac {b n x^2}{6 c}-\frac {2 n x^3}{9} \]
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^2 \log \left (d \left (a+b x+c x^2\right )^n\right ) \, dx &=\frac {1}{3} x^3 \log \left (d \left (a+b x+c x^2\right )^n\right )-\frac {1}{3} n \int \frac {x^3 (b+2 c x)}{a+b x+c x^2} \, dx\\ &=\frac {1}{3} x^3 \log \left (d \left (a+b x+c x^2\right )^n\right )-\frac {1}{3} n \int \left (\frac {b^2-2 a c}{c^2}-\frac {b x}{c}+2 x^2-\frac {a \left (b^2-2 a c\right )+b \left (b^2-3 a c\right ) x}{c^2 \left (a+b x+c x^2\right )}\right ) \, dx\\ &=-\frac {\left (b^2-2 a c\right ) n x}{3 c^2}+\frac {b n x^2}{6 c}-\frac {2 n x^3}{9}+\frac {1}{3} x^3 \log \left (d \left (a+b x+c x^2\right )^n\right )+\frac {n \int \frac {a \left (b^2-2 a c\right )+b \left (b^2-3 a c\right ) x}{a+b x+c x^2} \, dx}{3 c^2}\\ &=-\frac {\left (b^2-2 a c\right ) n x}{3 c^2}+\frac {b n x^2}{6 c}-\frac {2 n x^3}{9}+\frac {1}{3} x^3 \log \left (d \left (a+b x+c x^2\right )^n\right )+\frac {\left (b \left (b^2-3 a c\right ) n\right ) \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{6 c^3}-\frac {\left (\left (b^4-5 a b^2 c+4 a^2 c^2\right ) n\right ) \int \frac {1}{a+b x+c x^2} \, dx}{6 c^3}\\ &=-\frac {\left (b^2-2 a c\right ) n x}{3 c^2}+\frac {b n x^2}{6 c}-\frac {2 n x^3}{9}+\frac {b \left (b^2-3 a c\right ) n \log \left (a+b x+c x^2\right )}{6 c^3}+\frac {1}{3} x^3 \log \left (d \left (a+b x+c x^2\right )^n\right )+\frac {\left (\left (b^4-5 a b^2 c+4 a^2 c^2\right ) n\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{3 c^3}\\ &=-\frac {\left (b^2-2 a c\right ) n x}{3 c^2}+\frac {b n x^2}{6 c}-\frac {2 n x^3}{9}+\frac {\sqrt {b^2-4 a c} \left (b^2-a c\right ) n \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{3 c^3}+\frac {b \left (b^2-3 a c\right ) n \log \left (a+b x+c x^2\right )}{6 c^3}+\frac {1}{3} x^3 \log \left (d \left (a+b x+c x^2\right )^n\right )\\ \end {align*}
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Mathematica [A] time = 0.14, size = 122, normalized size = 0.90 \[ \frac {c n x \left (-4 c \left (c x^2-3 a\right )-6 b^2+3 b c x\right )+3 b n \left (b^2-3 a c\right ) \log (a+x (b+c x))+6 n \sqrt {b^2-4 a c} \left (b^2-a c\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )+6 c^3 x^3 \log \left (d (a+x (b+c x))^n\right )}{18 c^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 299, normalized size = 2.20 \[ \left [-\frac {4 \, c^{3} n x^{3} - 6 \, c^{3} x^{3} \log \relax (d) - 3 \, b c^{2} n x^{2} + 3 \, {\left (b^{2} - a c\right )} \sqrt {b^{2} - 4 \, a c} n \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c - \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) + 6 \, {\left (b^{2} c - 2 \, a c^{2}\right )} n x - 3 \, {\left (2 \, c^{3} n x^{3} + {\left (b^{3} - 3 \, a b c\right )} n\right )} \log \left (c x^{2} + b x + a\right )}{18 \, c^{3}}, -\frac {4 \, c^{3} n x^{3} - 6 \, c^{3} x^{3} \log \relax (d) - 3 \, b c^{2} n x^{2} - 6 \, {\left (b^{2} - a c\right )} \sqrt {-b^{2} + 4 \, a c} n \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + 6 \, {\left (b^{2} c - 2 \, a c^{2}\right )} n x - 3 \, {\left (2 \, c^{3} n x^{3} + {\left (b^{3} - 3 \, a b c\right )} n\right )} \log \left (c x^{2} + b x + a\right )}{18 \, c^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 146, normalized size = 1.07 \[ \frac {1}{3} \, n x^{3} \log \left (c x^{2} + b x + a\right ) - \frac {1}{9} \, {\left (2 \, n - 3 \, \log \relax (d)\right )} x^{3} + \frac {b n x^{2}}{6 \, c} - \frac {{\left (b^{2} n - 2 \, a c n\right )} x}{3 \, c^{2}} + \frac {{\left (b^{3} n - 3 \, a b c n\right )} \log \left (c x^{2} + b x + a\right )}{6 \, c^{3}} - \frac {{\left (b^{4} n - 5 \, a b^{2} c n + 4 \, a^{2} c^{2} n\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{3 \, \sqrt {-b^{2} + 4 \, a c} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.64, size = 870, normalized size = 6.40 \[ -\frac {i \pi \,x^{3} \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 )}{6}+\frac {i \pi \,x^{3} \mathrm {csgn}\left (i d \right ) \mathrm {csgn}\left (i d \left (c \,x^{2}+b x +a \right )^{n}\right )^{2}}{6}+\frac {i \pi \,x^{3} \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}}{6}-\frac {i \pi \,x^{3} \mathrm {csgn}\left (i d \left (c \,x^{2}+b x +a \right )^{n}\right )^{3}}{6}-\frac {2 n \,x^{3}}{9}+\frac {x^{3} \ln \relax (d )}{3}+\frac {x^{3} \ln \left (\left (c \,x^{2}+b x +a \right )^{n}\right )}{3}+\frac {b n \,x^{2}}{6 c}-\frac {a b n \ln \left (-4 a^{2} c^{2}+5 a \,b^{2} c -b^{4}-2 \sqrt {-4 a^{3} c^{3}+9 a^{2} b^{2} c^{2}-6 a \,b^{4} c +b^{6}}\, c x -\sqrt {-4 a^{3} c^{3}+9 a^{2} b^{2} c^{2}-6 a \,b^{4} c +b^{6}}\, b \right )}{2 c^{2}}-\frac {a b n \ln \left (-4 a^{2} c^{2}+5 a \,b^{2} c -b^{4}+2 \sqrt {-4 a^{3} c^{3}+9 a^{2} b^{2} c^{2}-6 a \,b^{4} c +b^{6}}\, c x +\sqrt {-4 a^{3} c^{3}+9 a^{2} b^{2} c^{2}-6 a \,b^{4} c +b^{6}}\, b \right )}{2 c^{2}}+\frac {2 a n x}{3 c}+\frac {b^{3} n \ln \left (-4 a^{2} c^{2}+5 a \,b^{2} c -b^{4}-2 \sqrt {-4 a^{3} c^{3}+9 a^{2} b^{2} c^{2}-6 a \,b^{4} c +b^{6}}\, c x -\sqrt {-4 a^{3} c^{3}+9 a^{2} b^{2} c^{2}-6 a \,b^{4} c +b^{6}}\, b \right )}{6 c^{3}}+\frac {b^{3} n \ln \left (-4 a^{2} c^{2}+5 a \,b^{2} c -b^{4}+2 \sqrt {-4 a^{3} c^{3}+9 a^{2} b^{2} c^{2}-6 a \,b^{4} c +b^{6}}\, c x +\sqrt {-4 a^{3} c^{3}+9 a^{2} b^{2} c^{2}-6 a \,b^{4} c +b^{6}}\, b \right )}{6 c^{3}}-\frac {b^{2} n x}{3 c^{2}}+\frac {\sqrt {-4 a^{3} c^{3}+9 a^{2} b^{2} c^{2}-6 a \,b^{4} c +b^{6}}\, n \ln \left (-4 a^{2} c^{2}+5 a \,b^{2} c -b^{4}-2 \sqrt {-4 a^{3} c^{3}+9 a^{2} b^{2} c^{2}-6 a \,b^{4} c +b^{6}}\, c x -\sqrt {-4 a^{3} c^{3}+9 a^{2} b^{2} c^{2}-6 a \,b^{4} c +b^{6}}\, b \right )}{6 c^{3}}-\frac {\sqrt {-4 a^{3} c^{3}+9 a^{2} b^{2} c^{2}-6 a \,b^{4} c +b^{6}}\, n \ln \left (-4 a^{2} c^{2}+5 a \,b^{2} c -b^{4}+2 \sqrt {-4 a^{3} c^{3}+9 a^{2} b^{2} c^{2}-6 a \,b^{4} c +b^{6}}\, c x +\sqrt {-4 a^{3} c^{3}+9 a^{2} b^{2} c^{2}-6 a \,b^{4} c +b^{6}}\, b \right )}{6 c^{3}} \]
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 = 0.50, size = 229, normalized size = 1.68 \[ \frac {x^3\,\ln \left (d\,{\left (c\,x^2+b\,x+a\right )}^n\right )}{3}-\frac {2\,n\,x^3}{9}-x\,\left (\frac {b^2\,n}{3\,c^2}-\frac {2\,a\,n}{3\,c}\right )-\frac {\ln \left (4\,a\,c+b\,\sqrt {b^2-4\,a\,c}-b^2+2\,c\,x\,\sqrt {b^2-4\,a\,c}\right )\,\left (c\,\left (\frac {a\,b\,n}{2}-\frac {a\,n\,\sqrt {b^2-4\,a\,c}}{6}\right )-\frac {b^3\,n}{6}+\frac {b^2\,n\,\sqrt {b^2-4\,a\,c}}{6}\right )}{c^3}+\frac {\ln \left (b\,\sqrt {b^2-4\,a\,c}-4\,a\,c+b^2+2\,c\,x\,\sqrt {b^2-4\,a\,c}\right )\,\left (\frac {b^3\,n}{6}-c\,\left (\frac {a\,b\,n}{2}+\frac {a\,n\,\sqrt {b^2-4\,a\,c}}{6}\right )+\frac {b^2\,n\,\sqrt {b^2-4\,a\,c}}{6}\right )}{c^3}+\frac {b\,n\,x^2}{6\,c} \]
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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