Optimal. Leaf size=149 \[ -\frac{b n \left (b^2-3 a c\right ) \log \left (a+b x+c x^2\right )}{6 a^3}+\frac{n \left (b^2-2 a c\right )}{3 a^2 x}+\frac{b n \log (x) \left (b^2-3 a c\right )}{3 a^3}+\frac{n \sqrt{b^2-4 a c} \left (b^2-a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{3 a^3}-\frac{\log \left (d \left (a+b x+c x^2\right )^n\right )}{3 x^3}-\frac{b n}{6 a x^2} \]
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Rubi [A] time = 0.199732, antiderivative size = 149, normalized size of antiderivative = 1., 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} \[ -\frac{b n \left (b^2-3 a c\right ) \log \left (a+b x+c x^2\right )}{6 a^3}+\frac{n \left (b^2-2 a c\right )}{3 a^2 x}+\frac{b n \log (x) \left (b^2-3 a c\right )}{3 a^3}+\frac{n \sqrt{b^2-4 a c} \left (b^2-a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{3 a^3}-\frac{\log \left (d \left (a+b x+c x^2\right )^n\right )}{3 x^3}-\frac{b n}{6 a x^2} \]
Antiderivative was successfully verified.
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Rule 2525
Rule 800
Rule 634
Rule 618
Rule 206
Rule 628
Rubi steps
\begin{align*} \int \frac{\log \left (d \left (a+b x+c x^2\right )^n\right )}{x^4} \, dx &=-\frac{\log \left (d \left (a+b x+c x^2\right )^n\right )}{3 x^3}+\frac{1}{3} n \int \frac{b+2 c x}{x^3 \left (a+b x+c x^2\right )} \, dx\\ &=-\frac{\log \left (d \left (a+b x+c x^2\right )^n\right )}{3 x^3}+\frac{1}{3} n \int \left (\frac{b}{a x^3}+\frac{-b^2+2 a c}{a^2 x^2}+\frac{b^3-3 a b c}{a^3 x}+\frac{-b^4+4 a b^2 c-2 a^2 c^2-b c \left (b^2-3 a c\right ) x}{a^3 \left (a+b x+c x^2\right )}\right ) \, dx\\ &=-\frac{b n}{6 a x^2}+\frac{\left (b^2-2 a c\right ) n}{3 a^2 x}+\frac{b \left (b^2-3 a c\right ) n \log (x)}{3 a^3}-\frac{\log \left (d \left (a+b x+c x^2\right )^n\right )}{3 x^3}+\frac{n \int \frac{-b^4+4 a b^2 c-2 a^2 c^2-b c \left (b^2-3 a c\right ) x}{a+b x+c x^2} \, dx}{3 a^3}\\ &=-\frac{b n}{6 a x^2}+\frac{\left (b^2-2 a c\right ) n}{3 a^2 x}+\frac{b \left (b^2-3 a c\right ) n \log (x)}{3 a^3}-\frac{\log \left (d \left (a+b x+c x^2\right )^n\right )}{3 x^3}-\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 a^3}-\frac{\left (\left (b^2-4 a c\right ) \left (b^2-a c\right ) n\right ) \int \frac{1}{a+b x+c x^2} \, dx}{6 a^3}\\ &=-\frac{b n}{6 a x^2}+\frac{\left (b^2-2 a c\right ) n}{3 a^2 x}+\frac{b \left (b^2-3 a c\right ) n \log (x)}{3 a^3}-\frac{b \left (b^2-3 a c\right ) n \log \left (a+b x+c x^2\right )}{6 a^3}-\frac{\log \left (d \left (a+b x+c x^2\right )^n\right )}{3 x^3}+\frac{\left (\left (b^2-4 a c\right ) \left (b^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 )}{3 a^3}\\ &=-\frac{b n}{6 a x^2}+\frac{\left (b^2-2 a c\right ) n}{3 a^2 x}+\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 a^3}+\frac{b \left (b^2-3 a c\right ) n \log (x)}{3 a^3}-\frac{b \left (b^2-3 a c\right ) n \log \left (a+b x+c x^2\right )}{6 a^3}-\frac{\log \left (d \left (a+b x+c x^2\right )^n\right )}{3 x^3}\\ \end{align*}
Mathematica [A] time = 0.366032, size = 132, normalized size = 0.89 \[ -\frac{\frac{n x \left (a^2 b-2 b x^2 \log (x) \left (b^2-3 a c\right )+b x^2 \left (b^2-3 a c\right ) \log (a+x (b+c x))-2 x^2 \sqrt{b^2-4 a c} \left (b^2-a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )-2 a x \left (b^2-2 a c\right )\right )}{a^3}+2 \log \left (d (a+x (b+c x))^n\right )}{6 x^3} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.085, size = 423, normalized size = 2.8 \begin{align*} -{\frac{\ln \left ( \left ( c{x}^{2}+bx+a \right ) ^{n} \right ) }{3\,{x}^{3}}}-{\frac{-i\pi \,{a}^{3}{\it csgn} \left ( id \right ){\it csgn} \left ( i \left ( c{x}^{2}+bx+a \right ) ^{n} \right ){\it csgn} \left ( id \left ( c{x}^{2}+bx+a \right ) ^{n} \right ) +i\pi \,{a}^{3}{\it csgn} \left ( id \right ) \left ({\it csgn} \left ( id \left ( c{x}^{2}+bx+a \right ) ^{n} \right ) \right ) ^{2}+i\pi \,{a}^{3}{\it csgn} \left ( i \left ( c{x}^{2}+bx+a \right ) ^{n} \right ) \left ({\it csgn} \left ( id \left ( c{x}^{2}+bx+a \right ) ^{n} \right ) \right ) ^{2}-i\pi \,{a}^{3} \left ({\it csgn} \left ( id \left ( c{x}^{2}+bx+a \right ) ^{n} \right ) \right ) ^{3}+6\,\ln \left ( x \right ) abcn{x}^{3}-2\,\ln \left ( x \right ){b}^{3}n{x}^{3}-2\,\sum _{{\it \_R}={\it RootOf} \left ({a}^{3}{{\it \_Z}}^{2}+ \left ( -3\,cbna+{b}^{3}n \right ){\it \_Z}+{c}^{3}{n}^{2} \right ) }{\it \_R}\,\ln \left ( \left ( \left ( 6\,{a}^{5}c-2\,{a}^{4}{b}^{2} \right ){{\it \_R}}^{2}+ \left ( -7\,{a}^{3}b{c}^{2}n+2\,{a}^{2}{b}^{3}cn \right ){\it \_R}+4\,{a}^{2}{c}^{4}{n}^{2}-4\,a{b}^{2}{c}^{3}{n}^{2}+{b}^{4}{c}^{2}{n}^{2} \right ) x-{a}^{5}b{{\it \_R}}^{2}+ \left ( 2\,{a}^{4}{c}^{2}n-4\,{a}^{3}{b}^{2}cn+{a}^{2}{b}^{4}n \right ){\it \_R}+6\,{a}^{2}b{c}^{3}{n}^{2}-5\,a{b}^{3}{c}^{2}{n}^{2}+{b}^{5}c{n}^{2} \right ){a}^{3}{x}^{3}+4\,{a}^{2}cn{x}^{2}-2\,a{b}^{2}n{x}^{2}+{a}^{2}bnx+2\,\ln \left ( d \right ){a}^{3}}{6\,{a}^{3}{x}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.74687, size = 741, normalized size = 4.97 \begin{align*} \left [-\frac{{\left (b^{2} - a c\right )} \sqrt{b^{2} - 4 \, a c} n x^{3} \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 ) - 2 \,{\left (b^{3} - 3 \, a b c\right )} n x^{3} \log \left (x\right ) + a^{2} b n x - 2 \,{\left (a b^{2} - 2 \, a^{2} c\right )} n x^{2} + 2 \, a^{3} \log \left (d\right ) +{\left ({\left (b^{3} - 3 \, a b c\right )} n x^{3} + 2 \, a^{3} n\right )} \log \left (c x^{2} + b x + a\right )}{6 \, a^{3} x^{3}}, \frac{2 \,{\left (b^{2} - a c\right )} \sqrt{-b^{2} + 4 \, a c} n x^{3} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + 2 \,{\left (b^{3} - 3 \, a b c\right )} n x^{3} \log \left (x\right ) - a^{2} b n x + 2 \,{\left (a b^{2} - 2 \, a^{2} c\right )} n x^{2} - 2 \, a^{3} \log \left (d\right ) -{\left ({\left (b^{3} - 3 \, a b c\right )} n x^{3} + 2 \, a^{3} n\right )} \log \left (c x^{2} + b x + a\right )}{6 \, a^{3} x^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20834, size = 221, normalized size = 1.48 \begin{align*} -\frac{{\left (b^{3} n - 3 \, a b c n\right )} \log \left (c x^{2} + b x + a\right )}{6 \, a^{3}} - \frac{n \log \left (c x^{2} + b x + a\right )}{3 \, x^{3}} + \frac{{\left (b^{3} n - 3 \, a b c n\right )} \log \left (x\right )}{3 \, a^{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} a^{3}} + \frac{2 \, b^{2} n x^{2} - 4 \, a c n x^{2} - a b n x - 2 \, a^{2} \log \left (d\right )}{6 \, a^{2} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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