Optimal. Leaf size=131 \[ \frac{3 (b+2 c x) (2 c d-b e)}{2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{-2 a e+x (2 c d-b e)+b d}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac{6 c (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}} \]
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Rubi [A] time = 0.0498186, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {638, 614, 618, 206} \[ \frac{3 (b+2 c x) (2 c d-b e)}{2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{-2 a e+x (2 c d-b e)+b d}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac{6 c (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 638
Rule 614
Rule 618
Rule 206
Rubi steps
\begin{align*} \int \frac{d+e x}{\left (a+b x+c x^2\right )^3} \, dx &=-\frac{b d-2 a e+(2 c d-b e) x}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac{(3 (2 c d-b e)) \int \frac{1}{\left (a+b x+c x^2\right )^2} \, dx}{2 \left (b^2-4 a c\right )}\\ &=-\frac{b d-2 a e+(2 c d-b e) x}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac{3 (2 c d-b e) (b+2 c x)}{2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}+\frac{(3 c (2 c d-b e)) \int \frac{1}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right )^2}\\ &=-\frac{b d-2 a e+(2 c d-b e) x}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac{3 (2 c d-b e) (b+2 c x)}{2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{(6 c (2 c d-b e)) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{\left (b^2-4 a c\right )^2}\\ &=-\frac{b d-2 a e+(2 c d-b e) x}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac{3 (2 c d-b e) (b+2 c x)}{2 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac{6 c (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.138294, size = 128, normalized size = 0.98 \[ \frac{\frac{\left (b^2-4 a c\right ) (2 a e-b d+b e x-2 c d x)}{(a+x (b+c x))^2}-\frac{12 c (b e-2 c d) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}+\frac{3 (b+2 c x) (2 c d-b e)}{a+x (b+c x)}}{2 \left (b^2-4 a c\right )^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.152, size = 242, normalized size = 1.9 \begin{align*}{\frac{bd-2\,ae+ \left ( -be+2\,cd \right ) x}{ \left ( 8\,ac-2\,{b}^{2} \right ) \left ( c{x}^{2}+bx+a \right ) ^{2}}}-3\,{\frac{bcxe}{ \left ( 4\,ac-{b}^{2} \right ) ^{2} \left ( c{x}^{2}+bx+a \right ) }}+6\,{\frac{{c}^{2}xd}{ \left ( 4\,ac-{b}^{2} \right ) ^{2} \left ( c{x}^{2}+bx+a \right ) }}-{\frac{3\,{b}^{2}e}{2\, \left ( 4\,ac-{b}^{2} \right ) ^{2} \left ( c{x}^{2}+bx+a \right ) }}+3\,{\frac{bcd}{ \left ( 4\,ac-{b}^{2} \right ) ^{2} \left ( c{x}^{2}+bx+a \right ) }}-6\,{\frac{bce}{ \left ( 4\,ac-{b}^{2} \right ) ^{5/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+12\,{\frac{{c}^{2}d}{ \left ( 4\,ac-{b}^{2} \right ) ^{5/2}}\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 = 2.15295, size = 2340, normalized size = 17.86 \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 = 2.26577, size = 651, normalized size = 4.97 \begin{align*} 3 c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (b e - 2 c d\right ) \log{\left (x + \frac{- 192 a^{3} c^{4} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (b e - 2 c d\right ) + 144 a^{2} b^{2} c^{3} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (b e - 2 c d\right ) - 36 a b^{4} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (b e - 2 c d\right ) + 3 b^{6} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (b e - 2 c d\right ) + 3 b^{2} c e - 6 b c^{2} d}{6 b c^{2} e - 12 c^{3} d} \right )} - 3 c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (b e - 2 c d\right ) \log{\left (x + \frac{192 a^{3} c^{4} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (b e - 2 c d\right ) - 144 a^{2} b^{2} c^{3} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (b e - 2 c d\right ) + 36 a b^{4} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (b e - 2 c d\right ) - 3 b^{6} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (b e - 2 c d\right ) + 3 b^{2} c e - 6 b c^{2} d}{6 b c^{2} e - 12 c^{3} d} \right )} - \frac{8 a^{2} c e + a b^{2} e - 10 a b c d + b^{3} d + x^{3} \left (6 b c^{2} e - 12 c^{3} d\right ) + x^{2} \left (9 b^{2} c e - 18 b c^{2} d\right ) + x \left (10 a b c e - 20 a c^{2} d + 2 b^{3} e - 4 b^{2} c d\right )}{32 a^{4} c^{2} - 16 a^{3} b^{2} c + 2 a^{2} b^{4} + x^{4} \left (32 a^{2} c^{4} - 16 a b^{2} c^{3} + 2 b^{4} c^{2}\right ) + x^{3} \left (64 a^{2} b c^{3} - 32 a b^{3} c^{2} + 4 b^{5} c\right ) + x^{2} \left (64 a^{3} c^{3} - 12 a b^{4} c + 2 b^{6}\right ) + x \left (64 a^{3} b c^{2} - 32 a^{2} b^{3} c + 4 a b^{5}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12226, size = 278, normalized size = 2.12 \begin{align*} \frac{6 \,{\left (2 \, c^{2} d - b c e\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{12 \, c^{3} d x^{3} - 6 \, b c^{2} x^{3} e + 18 \, b c^{2} d x^{2} - 9 \, b^{2} c x^{2} e + 4 \, b^{2} c d x + 20 \, a c^{2} d x - 2 \, b^{3} x e - 10 \, a b c x e - b^{3} d + 10 \, a b c d - a b^{2} e - 8 \, a^{2} c e}{2 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )}{\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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