Optimal. Leaf size=72 \[ 2 d^4 \left (b^2-4 a c\right ) (b+2 c x)-2 d^4 \left (b^2-4 a c\right )^{3/2} \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )+\frac{2}{3} d^4 (b+2 c x)^3 \]
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Rubi [A] time = 0.070038, antiderivative size = 72, 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 = {692, 618, 206} \[ 2 d^4 \left (b^2-4 a c\right ) (b+2 c x)-2 d^4 \left (b^2-4 a c\right )^{3/2} \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )+\frac{2}{3} d^4 (b+2 c x)^3 \]
Antiderivative was successfully verified.
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Rule 692
Rule 618
Rule 206
Rubi steps
\begin{align*} \int \frac{(b d+2 c d x)^4}{a+b x+c x^2} \, dx &=\frac{2}{3} d^4 (b+2 c x)^3+\left (\left (b^2-4 a c\right ) d^2\right ) \int \frac{(b d+2 c d x)^2}{a+b x+c x^2} \, dx\\ &=2 \left (b^2-4 a c\right ) d^4 (b+2 c x)+\frac{2}{3} d^4 (b+2 c x)^3+\left (\left (b^2-4 a c\right )^2 d^4\right ) \int \frac{1}{a+b x+c x^2} \, dx\\ &=2 \left (b^2-4 a c\right ) d^4 (b+2 c x)+\frac{2}{3} d^4 (b+2 c x)^3-\left (2 \left (b^2-4 a c\right )^2 d^4\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )\\ &=2 \left (b^2-4 a c\right ) d^4 (b+2 c x)+\frac{2}{3} d^4 (b+2 c x)^3-2 \left (b^2-4 a c\right )^{3/2} d^4 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )\\ \end{align*}
Mathematica [A] time = 0.0418401, size = 72, normalized size = 1. \[ d^4 \left (\frac{8}{3} c x \left (2 c \left (c x^2-3 a\right )+3 b^2+3 b c x\right )+2 \left (4 a c-b^2\right )^{3/2} \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.147, size = 170, normalized size = 2.4 \begin{align*}{\frac{16\,{d}^{4}{c}^{3}{x}^{3}}{3}}+8\,{d}^{4}b{c}^{2}{x}^{2}-16\,{d}^{4}a{c}^{2}x+8\,{d}^{4}{b}^{2}cx+32\,{\frac{{d}^{4}{a}^{2}{c}^{2}}{\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-16\,{\frac{{d}^{4}ac{b}^{2}}{\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+2\,{\frac{{d}^{4}{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 [A] time = 1.72199, size = 464, normalized size = 6.44 \begin{align*} \left [\frac{16}{3} \, c^{3} d^{4} x^{3} + 8 \, b c^{2} d^{4} x^{2} -{\left (b^{2} - 4 \, a c\right )}^{\frac{3}{2}} d^{4} \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 ) + 8 \,{\left (b^{2} c - 2 \, a c^{2}\right )} d^{4} x, \frac{16}{3} \, c^{3} d^{4} x^{3} + 8 \, b c^{2} d^{4} x^{2} - 2 \,{\left (b^{2} - 4 \, a c\right )} \sqrt{-b^{2} + 4 \, a c} d^{4} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + 8 \,{\left (b^{2} c - 2 \, a c^{2}\right )} d^{4} x\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.745157, size = 204, normalized size = 2.83 \begin{align*} 8 b c^{2} d^{4} x^{2} + \frac{16 c^{3} d^{4} x^{3}}{3} - d^{4} \sqrt{- \left (4 a c - b^{2}\right )^{3}} \log{\left (x + \frac{4 a b c d^{4} - b^{3} d^{4} - d^{4} \sqrt{- \left (4 a c - b^{2}\right )^{3}}}{8 a c^{2} d^{4} - 2 b^{2} c d^{4}} \right )} + d^{4} \sqrt{- \left (4 a c - b^{2}\right )^{3}} \log{\left (x + \frac{4 a b c d^{4} - b^{3} d^{4} + d^{4} \sqrt{- \left (4 a c - b^{2}\right )^{3}}}{8 a c^{2} d^{4} - 2 b^{2} c d^{4}} \right )} + x \left (- 16 a c^{2} d^{4} + 8 b^{2} c d^{4}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11598, size = 155, normalized size = 2.15 \begin{align*} \frac{2 \,{\left (b^{4} d^{4} - 8 \, a b^{2} c d^{4} + 16 \, a^{2} c^{2} d^{4}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c}} + \frac{8 \,{\left (2 \, c^{6} d^{4} x^{3} + 3 \, b c^{5} d^{4} x^{2} + 3 \, b^{2} c^{4} d^{4} x - 6 \, a c^{5} d^{4} x\right )}}{3 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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