Optimal. Leaf size=92 \[ -\frac{12 c^2 d^4 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c}}-\frac{3 c d^4 (b+2 c x)}{a+b x+c x^2}-\frac{d^4 (b+2 c x)^3}{2 \left (a+b x+c x^2\right )^2} \]
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Rubi [A] time = 0.0566576, antiderivative size = 92, 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, 618, 206} \[ -\frac{12 c^2 d^4 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c}}-\frac{3 c d^4 (b+2 c x)}{a+b x+c x^2}-\frac{d^4 (b+2 c x)^3}{2 \left (a+b x+c x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 686
Rule 618
Rule 206
Rubi steps
\begin{align*} \int \frac{(b d+2 c d x)^4}{\left (a+b x+c x^2\right )^3} \, dx &=-\frac{d^4 (b+2 c x)^3}{2 \left (a+b x+c x^2\right )^2}+\left (3 c d^2\right ) \int \frac{(b d+2 c d x)^2}{\left (a+b x+c x^2\right )^2} \, dx\\ &=-\frac{d^4 (b+2 c x)^3}{2 \left (a+b x+c x^2\right )^2}-\frac{3 c d^4 (b+2 c x)}{a+b x+c x^2}+\left (6 c^2 d^4\right ) \int \frac{1}{a+b x+c x^2} \, dx\\ &=-\frac{d^4 (b+2 c x)^3}{2 \left (a+b x+c x^2\right )^2}-\frac{3 c d^4 (b+2 c x)}{a+b x+c x^2}-\left (12 c^2 d^4\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )\\ &=-\frac{d^4 (b+2 c x)^3}{2 \left (a+b x+c x^2\right )^2}-\frac{3 c d^4 (b+2 c x)}{a+b x+c x^2}-\frac{12 c^2 d^4 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c}}\\ \end{align*}
Mathematica [A] time = 0.0647679, size = 89, normalized size = 0.97 \[ d^4 \left (\frac{12 c^2 \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}-\frac{(b+2 c x) \left (2 c \left (3 a+5 c x^2\right )+b^2+10 b c x\right )}{2 (a+x (b+c x))^2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.157, size = 173, normalized size = 1.9 \begin{align*} -10\,{\frac{{d}^{4}{c}^{3}{x}^{3}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-15\,{\frac{{d}^{4}b{c}^{2}{x}^{2}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-6\,{\frac{{d}^{4}a{c}^{2}x}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-6\,{\frac{{d}^{4}{b}^{2}cx}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-3\,{\frac{{d}^{4}abc}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-{\frac{{d}^{4}{b}^{3}}{2\, \left ( c{x}^{2}+bx+a \right ) ^{2}}}+12\,{\frac{{c}^{2}{d}^{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 = 2.26966, size = 1312, normalized size = 14.26 \begin{align*} \left [-\frac{20 \,{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d^{4} x^{3} + 30 \,{\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d^{4} x^{2} + 12 \,{\left (b^{4} c - 3 \, a b^{2} c^{2} - 4 \, a^{2} c^{3}\right )} d^{4} x +{\left (b^{5} + 2 \, a b^{3} c - 24 \, a^{2} b c^{2}\right )} d^{4} - 12 \,{\left (c^{4} d^{4} x^{4} + 2 \, b c^{3} d^{4} x^{3} + 2 \, a b c^{2} d^{4} x + a^{2} c^{2} d^{4} +{\left (b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} x^{2}\right )} \sqrt{b^{2} - 4 \, a c} \log \left (\frac{2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c - \sqrt{b^{2} - 4 \, a c}{\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right )}{2 \,{\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{4} + a^{2} b^{2} - 4 \, a^{3} c + 2 \,{\left (b^{3} c - 4 \, a b c^{2}\right )} x^{3} +{\left (b^{4} - 2 \, a b^{2} c - 8 \, a^{2} c^{2}\right )} x^{2} + 2 \,{\left (a b^{3} - 4 \, a^{2} b c\right )} x\right )}}, -\frac{20 \,{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d^{4} x^{3} + 30 \,{\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d^{4} x^{2} + 12 \,{\left (b^{4} c - 3 \, a b^{2} c^{2} - 4 \, a^{2} c^{3}\right )} d^{4} x +{\left (b^{5} + 2 \, a b^{3} c - 24 \, a^{2} b c^{2}\right )} d^{4} + 24 \,{\left (c^{4} d^{4} x^{4} + 2 \, b c^{3} d^{4} x^{3} + 2 \, a b c^{2} d^{4} x + a^{2} c^{2} d^{4} +{\left (b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} x^{2}\right )} \sqrt{-b^{2} + 4 \, a c} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right )}{2 \,{\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{4} + a^{2} b^{2} - 4 \, a^{3} c + 2 \,{\left (b^{3} c - 4 \, a b c^{2}\right )} x^{3} +{\left (b^{4} - 2 \, a b^{2} c - 8 \, a^{2} c^{2}\right )} x^{2} + 2 \,{\left (a b^{3} - 4 \, a^{2} b c\right )} x\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 3.96293, size = 303, normalized size = 3.29 \begin{align*} - 6 c^{2} d^{4} \sqrt{- \frac{1}{4 a c - b^{2}}} \log{\left (x + \frac{- 24 a c^{3} d^{4} \sqrt{- \frac{1}{4 a c - b^{2}}} + 6 b^{2} c^{2} d^{4} \sqrt{- \frac{1}{4 a c - b^{2}}} + 6 b c^{2} d^{4}}{12 c^{3} d^{4}} \right )} + 6 c^{2} d^{4} \sqrt{- \frac{1}{4 a c - b^{2}}} \log{\left (x + \frac{24 a c^{3} d^{4} \sqrt{- \frac{1}{4 a c - b^{2}}} - 6 b^{2} c^{2} d^{4} \sqrt{- \frac{1}{4 a c - b^{2}}} + 6 b c^{2} d^{4}}{12 c^{3} d^{4}} \right )} - \frac{6 a b c d^{4} + b^{3} d^{4} + 30 b c^{2} d^{4} x^{2} + 20 c^{3} d^{4} x^{3} + x \left (12 a c^{2} d^{4} + 12 b^{2} c d^{4}\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 [A] time = 1.26437, size = 154, normalized size = 1.67 \begin{align*} \frac{12 \, c^{2} d^{4} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c}} - \frac{20 \, c^{3} d^{4} x^{3} + 30 \, b c^{2} d^{4} x^{2} + 12 \, b^{2} c d^{4} x + 12 \, a c^{2} d^{4} x + b^{3} d^{4} + 6 \, a b c d^{4}}{2 \,{\left (c x^{2} + b x + a\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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