Optimal. Leaf size=86 \[ d^7 \left (b^2-4 a c\right )^3 \log \left (a+b x+c x^2\right )+\frac{1}{2} d^7 \left (b^2-4 a c\right ) (b+2 c x)^4+d^7 \left (b^2-4 a c\right )^2 (b+2 c x)^2+\frac{1}{3} d^7 (b+2 c x)^6 \]
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Rubi [A] time = 0.0701957, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {692, 628} \[ d^7 \left (b^2-4 a c\right )^3 \log \left (a+b x+c x^2\right )+\frac{1}{2} d^7 \left (b^2-4 a c\right ) (b+2 c x)^4+d^7 \left (b^2-4 a c\right )^2 (b+2 c x)^2+\frac{1}{3} d^7 (b+2 c x)^6 \]
Antiderivative was successfully verified.
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Rule 692
Rule 628
Rubi steps
\begin{align*} \int \frac{(b d+2 c d x)^7}{a+b x+c x^2} \, dx &=\frac{1}{3} d^7 (b+2 c x)^6+\left (\left (b^2-4 a c\right ) d^2\right ) \int \frac{(b d+2 c d x)^5}{a+b x+c x^2} \, dx\\ &=\frac{1}{2} \left (b^2-4 a c\right ) d^7 (b+2 c x)^4+\frac{1}{3} d^7 (b+2 c x)^6+\left (\left (b^2-4 a c\right )^2 d^4\right ) \int \frac{(b d+2 c d x)^3}{a+b x+c x^2} \, dx\\ &=\left (b^2-4 a c\right )^2 d^7 (b+2 c x)^2+\frac{1}{2} \left (b^2-4 a c\right ) d^7 (b+2 c x)^4+\frac{1}{3} d^7 (b+2 c x)^6+\left (\left (b^2-4 a c\right )^3 d^6\right ) \int \frac{b d+2 c d x}{a+b x+c x^2} \, dx\\ &=\left (b^2-4 a c\right )^2 d^7 (b+2 c x)^2+\frac{1}{2} \left (b^2-4 a c\right ) d^7 (b+2 c x)^4+\frac{1}{3} d^7 (b+2 c x)^6+\left (b^2-4 a c\right )^3 d^7 \log \left (a+b x+c x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0552827, size = 110, normalized size = 1.28 \[ d^7 \left (\frac{4}{3} c x (b+c x) \left (8 c^2 \left (6 a^2-3 a c x^2+2 c^2 x^4\right )+b^2 \left (34 c^2 x^2-36 a c\right )+8 b c^2 x \left (4 c x^2-3 a\right )+18 b^3 c x+9 b^4\right )+\left (b^2-4 a c\right )^3 \log (a+x (b+c x))\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.042, size = 243, normalized size = 2.8 \begin{align*}{\frac{64\,{d}^{7}{c}^{6}{x}^{6}}{3}}+64\,{d}^{7}b{c}^{5}{x}^{5}-32\,{d}^{7}{x}^{4}a{c}^{5}+88\,{d}^{7}{x}^{4}{b}^{2}{c}^{4}-64\,{d}^{7}{x}^{3}ab{c}^{4}+{\frac{208\,{d}^{7}{x}^{3}{b}^{3}{c}^{3}}{3}}+64\,{d}^{7}{x}^{2}{a}^{2}{c}^{4}-80\,{d}^{7}{x}^{2}a{b}^{2}{c}^{3}+36\,{d}^{7}{x}^{2}{b}^{4}{c}^{2}+64\,{d}^{7}b{a}^{2}{c}^{3}x-48\,{d}^{7}a{b}^{3}{c}^{2}x+12\,{d}^{7}{b}^{5}cx-64\,{d}^{7}\ln \left ( c{x}^{2}+bx+a \right ){a}^{3}{c}^{3}+48\,{d}^{7}\ln \left ( c{x}^{2}+bx+a \right ){a}^{2}{b}^{2}{c}^{2}-12\,{d}^{7}\ln \left ( c{x}^{2}+bx+a \right ) a{b}^{4}c+{d}^{7}\ln \left ( c{x}^{2}+bx+a \right ){b}^{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.38226, size = 244, normalized size = 2.84 \begin{align*} \frac{64}{3} \, c^{6} d^{7} x^{6} + 64 \, b c^{5} d^{7} x^{5} + 8 \,{\left (11 \, b^{2} c^{4} - 4 \, a c^{5}\right )} d^{7} x^{4} + \frac{16}{3} \,{\left (13 \, b^{3} c^{3} - 12 \, a b c^{4}\right )} d^{7} x^{3} + 4 \,{\left (9 \, b^{4} c^{2} - 20 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d^{7} x^{2} + 4 \,{\left (3 \, b^{5} c - 12 \, a b^{3} c^{2} + 16 \, 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} \log \left (c x^{2} + b x + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.63438, size = 393, normalized size = 4.57 \begin{align*} \frac{64}{3} \, c^{6} d^{7} x^{6} + 64 \, b c^{5} d^{7} x^{5} + 8 \,{\left (11 \, b^{2} c^{4} - 4 \, a c^{5}\right )} d^{7} x^{4} + \frac{16}{3} \,{\left (13 \, b^{3} c^{3} - 12 \, a b c^{4}\right )} d^{7} x^{3} + 4 \,{\left (9 \, b^{4} c^{2} - 20 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d^{7} x^{2} + 4 \,{\left (3 \, b^{5} c - 12 \, a b^{3} c^{2} + 16 \, 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} \log \left (c x^{2} + b x + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.27699, size = 185, normalized size = 2.15 \begin{align*} 64 b c^{5} d^{7} x^{5} + \frac{64 c^{6} d^{7} x^{6}}{3} - d^{7} \left (4 a c - b^{2}\right )^{3} \log{\left (a + b x + c x^{2} \right )} + x^{4} \left (- 32 a c^{5} d^{7} + 88 b^{2} c^{4} d^{7}\right ) + x^{3} \left (- 64 a b c^{4} d^{7} + \frac{208 b^{3} c^{3} d^{7}}{3}\right ) + x^{2} \left (64 a^{2} c^{4} d^{7} - 80 a b^{2} c^{3} d^{7} + 36 b^{4} c^{2} d^{7}\right ) + x \left (64 a^{2} b c^{3} d^{7} - 48 a b^{3} c^{2} d^{7} + 12 b^{5} c d^{7}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.18118, size = 296, normalized size = 3.44 \begin{align*}{\left (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}\right )} \log \left (c x^{2} + b x + a\right ) + \frac{4 \,{\left (16 \, c^{12} d^{7} x^{6} + 48 \, b c^{11} d^{7} x^{5} + 66 \, b^{2} c^{10} d^{7} x^{4} - 24 \, a c^{11} d^{7} x^{4} + 52 \, b^{3} c^{9} d^{7} x^{3} - 48 \, a b c^{10} d^{7} x^{3} + 27 \, b^{4} c^{8} d^{7} x^{2} - 60 \, a b^{2} c^{9} d^{7} x^{2} + 48 \, a^{2} c^{10} d^{7} x^{2} + 9 \, b^{5} c^{7} d^{7} x - 36 \, a b^{3} c^{8} d^{7} x + 48 \, a^{2} b c^{9} d^{7} x\right )}}{3 \, c^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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