Optimal. Leaf size=89 \[ 12 c d^7 \left (b^2-4 a c\right )^2 \log \left (a+b x+c x^2\right )+12 c d^7 \left (b^2-4 a c\right ) (b+2 c x)^2-\frac{d^7 (b+2 c x)^6}{a+b x+c x^2}+6 c d^7 (b+2 c x)^4 \]
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Rubi [A] time = 0.06212, antiderivative size = 89, 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} \[ 12 c d^7 \left (b^2-4 a c\right )^2 \log \left (a+b x+c x^2\right )+12 c d^7 \left (b^2-4 a c\right ) (b+2 c x)^2-\frac{d^7 (b+2 c x)^6}{a+b x+c x^2}+6 c d^7 (b+2 c x)^4 \]
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 )^2} \, dx &=-\frac{d^7 (b+2 c x)^6}{a+b x+c x^2}+\left (12 c d^2\right ) \int \frac{(b d+2 c d x)^5}{a+b x+c x^2} \, dx\\ &=6 c d^7 (b+2 c x)^4-\frac{d^7 (b+2 c x)^6}{a+b x+c x^2}+\left (12 c \left (b^2-4 a c\right ) d^4\right ) \int \frac{(b d+2 c d x)^3}{a+b x+c x^2} \, dx\\ &=12 c \left (b^2-4 a c\right ) d^7 (b+2 c x)^2+6 c d^7 (b+2 c x)^4-\frac{d^7 (b+2 c x)^6}{a+b x+c x^2}+\left (12 c \left (b^2-4 a c\right )^2 d^6\right ) \int \frac{b d+2 c d x}{a+b x+c x^2} \, dx\\ &=12 c \left (b^2-4 a c\right ) d^7 (b+2 c x)^2+6 c d^7 (b+2 c x)^4-\frac{d^7 (b+2 c x)^6}{a+b x+c x^2}+12 c \left (b^2-4 a c\right )^2 d^7 \log \left (a+b x+c x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0880245, size = 103, normalized size = 1.16 \[ d^7 \left (-16 c^3 x^2 \left (8 a c-5 b^2\right )+16 b c^2 x \left (3 b^2-8 a c\right )-\frac{\left (b^2-4 a c\right )^3}{a+x (b+c x)}+12 c \left (b^2-4 a c\right )^2 \log (a+x (b+c x))+64 b c^4 x^3+32 c^5 x^4\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.048, size = 230, normalized size = 2.6 \begin{align*} 32\,{d}^{7}{c}^{5}{x}^{4}+64\,{d}^{7}b{c}^{4}{x}^{3}-128\,{d}^{7}{x}^{2}a{c}^{4}+80\,{d}^{7}{x}^{2}{b}^{2}{c}^{3}-128\,{d}^{7}ab{c}^{3}x+48\,{d}^{7}{b}^{3}{c}^{2}x+64\,{\frac{{d}^{7}{a}^{3}{c}^{3}}{c{x}^{2}+bx+a}}-48\,{\frac{{d}^{7}{a}^{2}{b}^{2}{c}^{2}}{c{x}^{2}+bx+a}}+12\,{\frac{{d}^{7}a{b}^{4}c}{c{x}^{2}+bx+a}}-{\frac{{d}^{7}{b}^{6}}{c{x}^{2}+bx+a}}+192\,{d}^{7}\ln \left ( c{x}^{2}+bx+a \right ){a}^{2}{c}^{3}-96\,{d}^{7}\ln \left ( c{x}^{2}+bx+a \right ) a{b}^{2}{c}^{2}+12\,{d}^{7}\ln \left ( c{x}^{2}+bx+a \right ){b}^{4}c \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11781, size = 209, normalized size = 2.35 \begin{align*} 32 \, c^{5} d^{7} x^{4} + 64 \, b c^{4} d^{7} x^{3} + 16 \,{\left (5 \, b^{2} c^{3} - 8 \, a c^{4}\right )} d^{7} x^{2} + 16 \,{\left (3 \, b^{3} c^{2} - 8 \, a b c^{3}\right )} d^{7} x + 12 \,{\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} d^{7} \log \left (c x^{2} + b x + a\right ) - \frac{{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} d^{7}}{c x^{2} + b x + a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.9632, size = 587, normalized size = 6.6 \begin{align*} \frac{32 \, c^{6} d^{7} x^{6} + 96 \, b c^{5} d^{7} x^{5} + 48 \,{\left (3 \, b^{2} c^{4} - 2 \, a c^{5}\right )} d^{7} x^{4} + 64 \,{\left (2 \, b^{3} c^{3} - 3 \, a b c^{4}\right )} d^{7} x^{3} + 16 \,{\left (3 \, b^{4} c^{2} - 3 \, a b^{2} c^{3} - 8 \, a^{2} c^{4}\right )} d^{7} x^{2} + 16 \,{\left (3 \, a b^{3} c^{2} - 8 \, a^{2} b c^{3}\right )} d^{7} x -{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} d^{7} + 12 \,{\left ({\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d^{7} x^{2} +{\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} d^{7} x +{\left (a b^{4} c - 8 \, a^{2} b^{2} c^{2} + 16 \, a^{3} c^{3}\right )} d^{7}\right )} \log \left (c x^{2} + b x + a\right )}{c x^{2} + b x + a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.28406, size = 160, normalized size = 1.8 \begin{align*} 64 b c^{4} d^{7} x^{3} + 32 c^{5} d^{7} x^{4} + 12 c d^{7} \left (4 a c - b^{2}\right )^{2} \log{\left (a + b x + c x^{2} \right )} + x^{2} \left (- 128 a c^{4} d^{7} + 80 b^{2} c^{3} d^{7}\right ) + x \left (- 128 a b c^{3} d^{7} + 48 b^{3} c^{2} d^{7}\right ) + \frac{64 a^{3} c^{3} d^{7} - 48 a^{2} b^{2} c^{2} d^{7} + 12 a b^{4} c d^{7} - b^{6} d^{7}}{a + b x + c x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.15258, size = 244, normalized size = 2.74 \begin{align*} 12 \,{\left (b^{4} c d^{7} - 8 \, a b^{2} c^{2} d^{7} + 16 \, a^{2} 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} + 48 \, a^{2} b^{2} c^{2} d^{7} - 64 \, a^{3} c^{3} d^{7}}{c x^{2} + b x + a} + \frac{16 \,{\left (2 \, c^{13} d^{7} x^{4} + 4 \, b c^{12} d^{7} x^{3} + 5 \, b^{2} c^{11} d^{7} x^{2} - 8 \, a c^{12} d^{7} x^{2} + 3 \, b^{3} c^{10} d^{7} x - 8 \, a b c^{11} d^{7} x\right )}}{c^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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