Optimal. Leaf size=134 \[ 140 c^2 d^8 \left (b^2-4 a c\right ) (b+2 c x)-140 c^2 d^8 \left (b^2-4 a c\right )^{3/2} \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )-\frac{7 c d^8 (b+2 c x)^5}{a+b x+c x^2}-\frac{d^8 (b+2 c x)^7}{2 \left (a+b x+c x^2\right )^2}+\frac{140}{3} c^2 d^8 (b+2 c x)^3 \]
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Rubi [A] time = 0.101095, antiderivative size = 134, 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} \[ 140 c^2 d^8 \left (b^2-4 a c\right ) (b+2 c x)-140 c^2 d^8 \left (b^2-4 a c\right )^{3/2} \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )-\frac{7 c d^8 (b+2 c x)^5}{a+b x+c x^2}-\frac{d^8 (b+2 c x)^7}{2 \left (a+b x+c x^2\right )^2}+\frac{140}{3} c^2 d^8 (b+2 c x)^3 \]
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 )^3} \, dx &=-\frac{d^8 (b+2 c x)^7}{2 \left (a+b x+c x^2\right )^2}+\left (7 c d^2\right ) \int \frac{(b d+2 c d x)^6}{\left (a+b x+c x^2\right )^2} \, dx\\ &=-\frac{d^8 (b+2 c x)^7}{2 \left (a+b x+c x^2\right )^2}-\frac{7 c d^8 (b+2 c x)^5}{a+b x+c x^2}+\left (70 c^2 d^4\right ) \int \frac{(b d+2 c d x)^4}{a+b x+c x^2} \, dx\\ &=\frac{140}{3} c^2 d^8 (b+2 c x)^3-\frac{d^8 (b+2 c x)^7}{2 \left (a+b x+c x^2\right )^2}-\frac{7 c d^8 (b+2 c x)^5}{a+b x+c x^2}+\left (70 c^2 \left (b^2-4 a c\right ) d^6\right ) \int \frac{(b d+2 c d x)^2}{a+b x+c x^2} \, dx\\ &=140 c^2 \left (b^2-4 a c\right ) d^8 (b+2 c x)+\frac{140}{3} c^2 d^8 (b+2 c x)^3-\frac{d^8 (b+2 c x)^7}{2 \left (a+b x+c x^2\right )^2}-\frac{7 c d^8 (b+2 c x)^5}{a+b x+c x^2}+\left (70 c^2 \left (b^2-4 a c\right )^2 d^8\right ) \int \frac{1}{a+b x+c x^2} \, dx\\ &=140 c^2 \left (b^2-4 a c\right ) d^8 (b+2 c x)+\frac{140}{3} c^2 d^8 (b+2 c x)^3-\frac{d^8 (b+2 c x)^7}{2 \left (a+b x+c x^2\right )^2}-\frac{7 c d^8 (b+2 c x)^5}{a+b x+c x^2}-\left (140 c^2 \left (b^2-4 a c\right )^2 d^8\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )\\ &=140 c^2 \left (b^2-4 a c\right ) d^8 (b+2 c x)+\frac{140}{3} c^2 d^8 (b+2 c x)^3-\frac{d^8 (b+2 c x)^7}{2 \left (a+b x+c x^2\right )^2}-\frac{7 c d^8 (b+2 c x)^5}{a+b x+c x^2}-140 c^2 \left (b^2-4 a c\right )^{3/2} d^8 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )\\ \end{align*}
Mathematica [A] time = 0.0861327, size = 142, normalized size = 1.06 \[ d^8 \left (-256 c^3 x \left (3 a c-b^2\right )+140 c^2 \left (4 a c-b^2\right )^{3/2} \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )-\frac{13 c \left (b^2-4 a c\right )^2 (b+2 c x)}{a+x (b+c x)}-\frac{\left (b^2-4 a c\right )^3 (b+2 c x)}{2 (a+x (b+c x))^2}+128 b c^4 x^2+\frac{256 c^5 x^3}{3}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.162, size = 526, normalized size = 3.9 \begin{align*}{\frac{256\,{d}^{8}{c}^{5}{x}^{3}}{3}}+128\,{d}^{8}b{c}^{4}{x}^{2}-768\,{d}^{8}a{c}^{4}x+256\,{d}^{8}{b}^{2}{c}^{3}x-416\,{\frac{{d}^{8}{x}^{3}{a}^{2}{c}^{5}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}+208\,{\frac{{d}^{8}{x}^{3}a{b}^{2}{c}^{4}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-26\,{\frac{{d}^{8}{x}^{3}{b}^{4}{c}^{3}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-624\,{\frac{{d}^{8}{x}^{2}{a}^{2}b{c}^{4}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}+312\,{\frac{{d}^{8}{x}^{2}a{b}^{3}{c}^{3}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-39\,{\frac{{d}^{8}{x}^{2}{b}^{5}{c}^{2}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-352\,{\frac{{d}^{8}{a}^{3}{c}^{4}x}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-48\,{\frac{{d}^{8}{b}^{2}{a}^{2}{c}^{3}x}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}+90\,{\frac{{d}^{8}a{b}^{4}{c}^{2}x}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-14\,{\frac{{d}^{8}{b}^{6}cx}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-176\,{\frac{{d}^{8}{a}^{3}b{c}^{3}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}+80\,{\frac{{d}^{8}{a}^{2}{b}^{3}{c}^{2}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-7\,{\frac{{d}^{8}a{b}^{5}c}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-{\frac{{d}^{8}{b}^{7}}{2\, \left ( c{x}^{2}+bx+a \right ) ^{2}}}+2240\,{\frac{{c}^{4}{d}^{8}{a}^{2}}{\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-1120\,{\frac{{d}^{8}{c}^{3}a{b}^{2}}{\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+140\,{\frac{{d}^{8}{c}^{2}{b}^{4}}{\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 = 2.15014, size = 1879, normalized size = 14.02 \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 = 12.3837, size = 469, normalized size = 3.5 \begin{align*} 128 b c^{4} d^{8} x^{2} + \frac{256 c^{5} d^{8} x^{3}}{3} - 70 c^{2} d^{8} \sqrt{- \left (4 a c - b^{2}\right )^{3}} \log{\left (x + \frac{280 a b c^{3} d^{8} - 70 b^{3} c^{2} d^{8} - 70 c^{2} d^{8} \sqrt{- \left (4 a c - b^{2}\right )^{3}}}{560 a c^{4} d^{8} - 140 b^{2} c^{3} d^{8}} \right )} + 70 c^{2} d^{8} \sqrt{- \left (4 a c - b^{2}\right )^{3}} \log{\left (x + \frac{280 a b c^{3} d^{8} - 70 b^{3} c^{2} d^{8} + 70 c^{2} d^{8} \sqrt{- \left (4 a c - b^{2}\right )^{3}}}{560 a c^{4} d^{8} - 140 b^{2} c^{3} d^{8}} \right )} + x \left (- 768 a c^{4} d^{8} + 256 b^{2} c^{3} d^{8}\right ) - \frac{352 a^{3} b c^{3} d^{8} - 160 a^{2} b^{3} c^{2} d^{8} + 14 a b^{5} c d^{8} + b^{7} d^{8} + x^{3} \left (832 a^{2} c^{5} d^{8} - 416 a b^{2} c^{4} d^{8} + 52 b^{4} c^{3} d^{8}\right ) + x^{2} \left (1248 a^{2} b c^{4} d^{8} - 624 a b^{3} c^{3} d^{8} + 78 b^{5} c^{2} d^{8}\right ) + x \left (704 a^{3} c^{4} d^{8} + 96 a^{2} b^{2} c^{3} d^{8} - 180 a b^{4} c^{2} d^{8} + 28 b^{6} c d^{8}\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 = 1.19335, size = 425, normalized size = 3.17 \begin{align*} \frac{140 \,{\left (b^{4} c^{2} d^{8} - 8 \, a b^{2} c^{3} d^{8} + 16 \, a^{2} 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{52 \, b^{4} c^{3} d^{8} x^{3} - 416 \, a b^{2} c^{4} d^{8} x^{3} + 832 \, a^{2} c^{5} d^{8} x^{3} + 78 \, b^{5} c^{2} d^{8} x^{2} - 624 \, a b^{3} c^{3} d^{8} x^{2} + 1248 \, a^{2} b c^{4} d^{8} x^{2} + 28 \, b^{6} c d^{8} x - 180 \, a b^{4} c^{2} d^{8} x + 96 \, a^{2} b^{2} c^{3} d^{8} x + 704 \, a^{3} c^{4} d^{8} x + b^{7} d^{8} + 14 \, a b^{5} c d^{8} - 160 \, a^{2} b^{3} c^{2} d^{8} + 352 \, a^{3} b c^{3} d^{8}}{2 \,{\left (c x^{2} + b x + a\right )}^{2}} + \frac{128 \,{\left (2 \, c^{14} d^{8} x^{3} + 3 \, b c^{13} d^{8} x^{2} + 6 \, b^{2} c^{12} d^{8} x - 18 \, a c^{13} d^{8} x\right )}}{3 \, c^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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