Optimal. Leaf size=126 \[ \frac{28}{3} c d^8 \left (b^2-4 a c\right ) (b+2 c x)^3+28 c d^8 \left (b^2-4 a c\right )^2 (b+2 c x)-28 c d^8 \left (b^2-4 a c\right )^{5/2} \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )-\frac{d^8 (b+2 c x)^7}{a+b x+c x^2}+\frac{28}{5} c d^8 (b+2 c x)^5 \]
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Rubi [A] time = 0.104732, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {686, 692, 618, 206} \[ \frac{28}{3} c d^8 \left (b^2-4 a c\right ) (b+2 c x)^3+28 c d^8 \left (b^2-4 a c\right )^2 (b+2 c x)-28 c d^8 \left (b^2-4 a c\right )^{5/2} \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )-\frac{d^8 (b+2 c x)^7}{a+b x+c x^2}+\frac{28}{5} c d^8 (b+2 c x)^5 \]
Antiderivative was successfully verified.
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Rule 686
Rule 692
Rule 618
Rule 206
Rubi steps
\begin{align*} \int \frac{(b d+2 c d x)^8}{\left (a+b x+c x^2\right )^2} \, dx &=-\frac{d^8 (b+2 c x)^7}{a+b x+c x^2}+\left (14 c d^2\right ) \int \frac{(b d+2 c d x)^6}{a+b x+c x^2} \, dx\\ &=\frac{28}{5} c d^8 (b+2 c x)^5-\frac{d^8 (b+2 c x)^7}{a+b x+c x^2}+\left (14 c \left (b^2-4 a c\right ) d^4\right ) \int \frac{(b d+2 c d x)^4}{a+b x+c x^2} \, dx\\ &=\frac{28}{3} c \left (b^2-4 a c\right ) d^8 (b+2 c x)^3+\frac{28}{5} c d^8 (b+2 c x)^5-\frac{d^8 (b+2 c x)^7}{a+b x+c x^2}+\left (14 c \left (b^2-4 a c\right )^2 d^6\right ) \int \frac{(b d+2 c d x)^2}{a+b x+c x^2} \, dx\\ &=28 c \left (b^2-4 a c\right )^2 d^8 (b+2 c x)+\frac{28}{3} c \left (b^2-4 a c\right ) d^8 (b+2 c x)^3+\frac{28}{5} c d^8 (b+2 c x)^5-\frac{d^8 (b+2 c x)^7}{a+b x+c x^2}+\left (14 c \left (b^2-4 a c\right )^3 d^8\right ) \int \frac{1}{a+b x+c x^2} \, dx\\ &=28 c \left (b^2-4 a c\right )^2 d^8 (b+2 c x)+\frac{28}{3} c \left (b^2-4 a c\right ) d^8 (b+2 c x)^3+\frac{28}{5} c d^8 (b+2 c x)^5-\frac{d^8 (b+2 c x)^7}{a+b x+c x^2}-\left (28 c \left (b^2-4 a c\right )^3 d^8\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )\\ &=28 c \left (b^2-4 a c\right )^2 d^8 (b+2 c x)+\frac{28}{3} c \left (b^2-4 a c\right ) d^8 (b+2 c x)^3+\frac{28}{5} c d^8 (b+2 c x)^5-\frac{d^8 (b+2 c x)^7}{a+b x+c x^2}-28 c \left (b^2-4 a c\right )^{5/2} d^8 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )\\ \end{align*}
Mathematica [A] time = 0.0819658, size = 155, normalized size = 1.23 \[ d^8 \left (32 c^2 x \left (24 a^2 c^2-16 a b^2 c+3 b^4\right )-\frac{512}{3} c^4 x^3 \left (a c-b^2\right )+128 b c^3 x^2 \left (b^2-2 a c\right )-\frac{\left (b^2-4 a c\right )^3 (b+2 c x)}{a+x (b+c x)}-28 c \left (4 a c-b^2\right )^{5/2} \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )+128 b c^5 x^4+\frac{256 c^6 x^5}{5}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.155, size = 479, normalized size = 3.8 \begin{align*}{\frac{256\,{d}^{8}{c}^{6}{x}^{5}}{5}}+128\,{d}^{8}b{c}^{5}{x}^{4}-{\frac{512\,{d}^{8}{x}^{3}a{c}^{5}}{3}}+{\frac{512\,{d}^{8}{x}^{3}{b}^{2}{c}^{4}}{3}}-256\,{d}^{8}{x}^{2}ab{c}^{4}+128\,{d}^{8}{x}^{2}{b}^{3}{c}^{3}+768\,{d}^{8}{a}^{2}{c}^{4}x-512\,{d}^{8}{b}^{2}a{c}^{3}x+96\,{d}^{8}{b}^{4}{c}^{2}x+128\,{\frac{{d}^{8}{a}^{3}{c}^{4}x}{c{x}^{2}+bx+a}}-96\,{\frac{{d}^{8}{b}^{2}{a}^{2}{c}^{3}x}{c{x}^{2}+bx+a}}+24\,{\frac{{d}^{8}a{b}^{4}{c}^{2}x}{c{x}^{2}+bx+a}}-2\,{\frac{{d}^{8}{b}^{6}cx}{c{x}^{2}+bx+a}}+64\,{\frac{{d}^{8}{a}^{3}b{c}^{3}}{c{x}^{2}+bx+a}}-48\,{\frac{{d}^{8}{a}^{2}{b}^{3}{c}^{2}}{c{x}^{2}+bx+a}}+12\,{\frac{{d}^{8}a{b}^{5}c}{c{x}^{2}+bx+a}}-{\frac{{d}^{8}{b}^{7}}{c{x}^{2}+bx+a}}-1792\,{\frac{{d}^{8}{a}^{3}{c}^{4}}{\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+1344\,{\frac{{d}^{8}{b}^{2}{a}^{2}{c}^{3}}{\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-336\,{\frac{{d}^{8}a{b}^{4}{c}^{2}}{\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+28\,{\frac{{d}^{8}{b}^{6}c}{\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \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 [B] time = 2.08082, size = 1634, normalized size = 12.97 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 3.36178, size = 476, normalized size = 3.78 \begin{align*} 128 b c^{5} d^{8} x^{4} + \frac{256 c^{6} d^{8} x^{5}}{5} + 14 c d^{8} \sqrt{- \left (4 a c - b^{2}\right )^{5}} \log{\left (x + \frac{224 a^{2} b c^{3} d^{8} - 112 a b^{3} c^{2} d^{8} + 14 b^{5} c d^{8} - 14 c d^{8} \sqrt{- \left (4 a c - b^{2}\right )^{5}}}{448 a^{2} c^{4} d^{8} - 224 a b^{2} c^{3} d^{8} + 28 b^{4} c^{2} d^{8}} \right )} - 14 c d^{8} \sqrt{- \left (4 a c - b^{2}\right )^{5}} \log{\left (x + \frac{224 a^{2} b c^{3} d^{8} - 112 a b^{3} c^{2} d^{8} + 14 b^{5} c d^{8} + 14 c d^{8} \sqrt{- \left (4 a c - b^{2}\right )^{5}}}{448 a^{2} c^{4} d^{8} - 224 a b^{2} c^{3} d^{8} + 28 b^{4} c^{2} d^{8}} \right )} + x^{3} \left (- \frac{512 a c^{5} d^{8}}{3} + \frac{512 b^{2} c^{4} d^{8}}{3}\right ) + x^{2} \left (- 256 a b c^{4} d^{8} + 128 b^{3} c^{3} d^{8}\right ) + x \left (768 a^{2} c^{4} d^{8} - 512 a b^{2} c^{3} d^{8} + 96 b^{4} c^{2} d^{8}\right ) + \frac{64 a^{3} b c^{3} d^{8} - 48 a^{2} b^{3} c^{2} d^{8} + 12 a b^{5} c d^{8} - b^{7} d^{8} + x \left (128 a^{3} c^{4} d^{8} - 96 a^{2} b^{2} c^{3} d^{8} + 24 a b^{4} c^{2} d^{8} - 2 b^{6} c d^{8}\right )}{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 = 1.1537, size = 416, normalized size = 3.3 \begin{align*} \frac{28 \,{\left (b^{6} c d^{8} - 12 \, a b^{4} c^{2} d^{8} + 48 \, a^{2} b^{2} c^{3} d^{8} - 64 \, a^{3} c^{4} d^{8}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c}} - \frac{2 \, b^{6} c d^{8} x - 24 \, a b^{4} c^{2} d^{8} x + 96 \, a^{2} b^{2} c^{3} d^{8} x - 128 \, a^{3} c^{4} d^{8} x + b^{7} d^{8} - 12 \, a b^{5} c d^{8} + 48 \, a^{2} b^{3} c^{2} d^{8} - 64 \, a^{3} b c^{3} d^{8}}{c x^{2} + b x + a} + \frac{32 \,{\left (24 \, c^{16} d^{8} x^{5} + 60 \, b c^{15} d^{8} x^{4} + 80 \, b^{2} c^{14} d^{8} x^{3} - 80 \, a c^{15} d^{8} x^{3} + 60 \, b^{3} c^{13} d^{8} x^{2} - 120 \, a b c^{14} d^{8} x^{2} + 45 \, b^{4} c^{12} d^{8} x - 240 \, a b^{2} c^{13} d^{8} x + 360 \, a^{2} c^{14} d^{8} x\right )}}{15 \, c^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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