Optimal. Leaf size=97 \[ 48 c^2 d^7 \left (b^2-4 a c\right ) \log \left (a+b x+c x^2\right )-\frac{6 c d^7 (b+2 c x)^4}{a+b x+c x^2}-\frac{d^7 (b+2 c x)^6}{2 \left (a+b x+c x^2\right )^2}+48 c^2 d^7 (b+2 c x)^2 \]
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Rubi [A] time = 0.0604783, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {686, 692, 628} \[ 48 c^2 d^7 \left (b^2-4 a c\right ) \log \left (a+b x+c x^2\right )-\frac{6 c d^7 (b+2 c x)^4}{a+b x+c x^2}-\frac{d^7 (b+2 c x)^6}{2 \left (a+b x+c x^2\right )^2}+48 c^2 d^7 (b+2 c x)^2 \]
Antiderivative was successfully verified.
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Rule 686
Rule 692
Rule 628
Rubi steps
\begin{align*} \int \frac{(b d+2 c d x)^7}{\left (a+b x+c x^2\right )^3} \, dx &=-\frac{d^7 (b+2 c x)^6}{2 \left (a+b x+c x^2\right )^2}+\left (6 c d^2\right ) \int \frac{(b d+2 c d x)^5}{\left (a+b x+c x^2\right )^2} \, dx\\ &=-\frac{d^7 (b+2 c x)^6}{2 \left (a+b x+c x^2\right )^2}-\frac{6 c d^7 (b+2 c x)^4}{a+b x+c x^2}+\left (48 c^2 d^4\right ) \int \frac{(b d+2 c d x)^3}{a+b x+c x^2} \, dx\\ &=48 c^2 d^7 (b+2 c x)^2-\frac{d^7 (b+2 c x)^6}{2 \left (a+b x+c x^2\right )^2}-\frac{6 c d^7 (b+2 c x)^4}{a+b x+c x^2}+\left (48 c^2 \left (b^2-4 a c\right ) d^6\right ) \int \frac{b d+2 c d x}{a+b x+c x^2} \, dx\\ &=48 c^2 d^7 (b+2 c x)^2-\frac{d^7 (b+2 c x)^6}{2 \left (a+b x+c x^2\right )^2}-\frac{6 c d^7 (b+2 c x)^4}{a+b x+c x^2}+48 c^2 \left (b^2-4 a c\right ) d^7 \log \left (a+b x+c x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0376434, size = 92, normalized size = 0.95 \[ d^7 \left (48 c^2 \left (b^2-4 a c\right ) \log (a+x (b+c x))-\frac{12 c \left (b^2-4 a c\right )^2}{a+x (b+c x)}-\frac{\left (b^2-4 a c\right )^3}{2 (a+x (b+c x))^2}+64 b c^3 x+64 c^4 x^2\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.051, size = 307, normalized size = 3.2 \begin{align*} 64\,{d}^{7}{c}^{4}{x}^{2}+64\,{d}^{7}b{c}^{3}x-192\,{\frac{{d}^{7}{x}^{2}{a}^{2}{c}^{4}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}+96\,{\frac{{d}^{7}{x}^{2}a{b}^{2}{c}^{3}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-12\,{\frac{{d}^{7}{x}^{2}{b}^{4}{c}^{2}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-192\,{\frac{{d}^{7}b{a}^{2}{c}^{3}x}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}+96\,{\frac{{d}^{7}a{b}^{3}{c}^{2}x}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-12\,{\frac{{d}^{7}{b}^{5}cx}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-160\,{\frac{{d}^{7}{a}^{3}{c}^{3}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}+72\,{\frac{{d}^{7}{a}^{2}{b}^{2}{c}^{2}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-6\,{\frac{{d}^{7}a{b}^{4}c}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-{\frac{{d}^{7}{b}^{6}}{2\, \left ( c{x}^{2}+bx+a \right ) ^{2}}}-192\,{d}^{7}\ln \left ( c{x}^{2}+bx+a \right ) a{c}^{3}+48\,{d}^{7}\ln \left ( c{x}^{2}+bx+a \right ){b}^{2}{c}^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.12292, size = 255, normalized size = 2.63 \begin{align*} 64 \, c^{4} d^{7} x^{2} + 64 \, b c^{3} d^{7} x + 48 \,{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{7} \log \left (c x^{2} + b x + a\right ) - \frac{24 \,{\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d^{7} x^{2} + 24 \,{\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} d^{7} x +{\left (b^{6} + 12 \, a b^{4} c - 144 \, a^{2} b^{2} c^{2} + 320 \, a^{3} c^{3}\right )} d^{7}}{2 \,{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.11628, size = 724, normalized size = 7.46 \begin{align*} \frac{128 \, c^{6} d^{7} x^{6} + 384 \, b c^{5} d^{7} x^{5} + 128 \,{\left (3 \, b^{2} c^{4} + 2 \, a c^{5}\right )} d^{7} x^{4} + 128 \,{\left (b^{3} c^{3} + 4 \, a b c^{4}\right )} d^{7} x^{3} - 8 \,{\left (3 \, b^{4} c^{2} - 56 \, a b^{2} c^{3} + 32 \, a^{2} c^{4}\right )} d^{7} x^{2} - 8 \,{\left (3 \, b^{5} c - 24 \, a b^{3} c^{2} + 32 \, a^{2} b c^{3}\right )} d^{7} x -{\left (b^{6} + 12 \, a b^{4} c - 144 \, a^{2} b^{2} c^{2} + 320 \, a^{3} c^{3}\right )} d^{7} + 96 \,{\left ({\left (b^{2} c^{4} - 4 \, a c^{5}\right )} d^{7} x^{4} + 2 \,{\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} d^{7} x^{3} +{\left (b^{4} c^{2} - 2 \, a b^{2} c^{3} - 8 \, a^{2} c^{4}\right )} d^{7} x^{2} + 2 \,{\left (a b^{3} c^{2} - 4 \, a^{2} b c^{3}\right )} d^{7} x +{\left (a^{2} b^{2} c^{2} - 4 \, a^{3} c^{3}\right )} d^{7}\right )} \log \left (c x^{2} + b x + a\right )}{2 \,{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 13.4142, size = 219, normalized size = 2.26 \begin{align*} 64 b c^{3} d^{7} x + 64 c^{4} d^{7} x^{2} - 48 c^{2} d^{7} \left (4 a c - b^{2}\right ) \log{\left (a + b x + c x^{2} \right )} - \frac{320 a^{3} c^{3} d^{7} - 144 a^{2} b^{2} c^{2} d^{7} + 12 a b^{4} c d^{7} + b^{6} d^{7} + x^{2} \left (384 a^{2} c^{4} d^{7} - 192 a b^{2} c^{3} d^{7} + 24 b^{4} c^{2} d^{7}\right ) + x \left (384 a^{2} b c^{3} d^{7} - 192 a b^{3} c^{2} d^{7} + 24 b^{5} c d^{7}\right )}{2 a^{2} + 4 a b x + 4 b c x^{3} + 2 c^{2} x^{4} + x^{2} \left (4 a c + 2 b^{2}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.1791, size = 258, normalized size = 2.66 \begin{align*} 48 \,{\left (b^{2} c^{2} d^{7} - 4 \, a c^{3} d^{7}\right )} \log \left (c x^{2} + b x + a\right ) - \frac{b^{6} d^{7} + 12 \, a b^{4} c d^{7} - 144 \, a^{2} b^{2} c^{2} d^{7} + 320 \, a^{3} c^{3} d^{7} + 24 \,{\left (b^{4} c^{2} d^{7} - 8 \, a b^{2} c^{3} d^{7} + 16 \, a^{2} c^{4} d^{7}\right )} x^{2} + 24 \,{\left (b^{5} c d^{7} - 8 \, a b^{3} c^{2} d^{7} + 16 \, a^{2} b c^{3} d^{7}\right )} x}{2 \,{\left (c x^{2} + b x + a\right )}^{2}} + \frac{64 \,{\left (c^{10} d^{7} x^{2} + b c^{9} d^{7} x\right )}}{c^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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