Optimal. Leaf size=126 \[ \frac{6 e \left (a e^2-b d e+c d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac{3 e (d+e x) (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac{(d+e x)^3}{2 \left (a+b x+c x^2\right )^2} \]
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Rubi [A] time = 0.0834652, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {768, 722, 618, 206} \[ \frac{6 e \left (a e^2-b d e+c d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac{3 e (d+e x) (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac{(d+e x)^3}{2 \left (a+b x+c x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 768
Rule 722
Rule 618
Rule 206
Rubi steps
\begin{align*} \int \frac{(b+2 c x) (d+e x)^3}{\left (a+b x+c x^2\right )^3} \, dx &=-\frac{(d+e x)^3}{2 \left (a+b x+c x^2\right )^2}+\frac{1}{2} (3 e) \int \frac{(d+e x)^2}{\left (a+b x+c x^2\right )^2} \, dx\\ &=-\frac{(d+e x)^3}{2 \left (a+b x+c x^2\right )^2}-\frac{3 e (d+e x) (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 )}-\frac{\left (3 e \left (c d^2-b d e+a e^2\right )\right ) \int \frac{1}{a+b x+c x^2} \, dx}{b^2-4 a c}\\ &=-\frac{(d+e x)^3}{2 \left (a+b x+c x^2\right )^2}-\frac{3 e (d+e x) (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 )}+\frac{\left (6 e \left (c d^2-b d e+a e^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{b^2-4 a c}\\ &=-\frac{(d+e x)^3}{2 \left (a+b x+c x^2\right )^2}-\frac{3 e (d+e x) (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 )}+\frac{6 e \left (c d^2-b d e+a e^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.308431, size = 216, normalized size = 1.71 \[ \frac{1}{2} \left (\frac{e \left (b c \left (7 a e^2+3 c d (d-2 e x)\right )+2 c^2 \left (3 c d^2 x-a e (12 d+5 e x)\right )+b^2 c e (3 d+4 e x)+b^3 \left (-e^2\right )\right )}{c^2 \left (4 a c-b^2\right ) (a+x (b+c x))}+\frac{12 e \left (e (a e-b d)+c d^2\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2}}+\frac{c e^2 (3 a d+a e x+3 b d x)-b e^3 (a+b x)-c^2 d^2 (d+3 e x)}{c^2 (a+x (b+c x))^2}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.012, size = 365, normalized size = 2.9 \begin{align*}{\frac{1}{ \left ( c{x}^{2}+bx+a \right ) ^{2}} \left ( -{\frac{e \left ( 5\,ac{e}^{2}-2\,{b}^{2}{e}^{2}+3\,bcde-3\,{c}^{2}{d}^{2} \right ){x}^{3}}{4\,ac-{b}^{2}}}-{\frac{3\,e \left ( c{e}^{2}ab+8\,a{c}^{2}de-{b}^{3}{e}^{2}+{b}^{2}cde-3\,b{d}^{2}{c}^{2} \right ){x}^{2}}{2\,c \left ( 4\,ac-{b}^{2} \right ) }}-3\,{\frac{e \left ( c{e}^{2}{a}^{2}-a{b}^{2}{e}^{2}+3\,abcde+a{c}^{2}{d}^{2}-{b}^{2}{d}^{2}c \right ) x}{c \left ( 4\,ac-{b}^{2} \right ) }}+{\frac{3\,{a}^{2}b{e}^{3}-12\,{a}^{2}cd{e}^{2}+3\,abc{d}^{2}e-4\,a{c}^{2}{d}^{3}+{b}^{2}{d}^{3}c}{2\,c \left ( 4\,ac-{b}^{2} \right ) }} \right ) }+6\,{\frac{{e}^{3}a}{ \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-6\,{\frac{bd{e}^{2}}{ \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+6\,{\frac{c{d}^{2}e}{ \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\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 = 2.16388, size = 2974, normalized size = 23.6 \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 = 48.6777, size = 762, normalized size = 6.05 \begin{align*} - 3 e \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (a e^{2} - b d e + c d^{2}\right ) \log{\left (x + \frac{- 48 a^{2} c^{2} e \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (a e^{2} - b d e + c d^{2}\right ) + 24 a b^{2} c e \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (a e^{2} - b d e + c d^{2}\right ) + 3 a b e^{3} - 3 b^{4} e \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (a e^{2} - b d e + c d^{2}\right ) - 3 b^{2} d e^{2} + 3 b c d^{2} e}{6 a c e^{3} - 6 b c d e^{2} + 6 c^{2} d^{2} e} \right )} + 3 e \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (a e^{2} - b d e + c d^{2}\right ) \log{\left (x + \frac{48 a^{2} c^{2} e \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (a e^{2} - b d e + c d^{2}\right ) - 24 a b^{2} c e \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (a e^{2} - b d e + c d^{2}\right ) + 3 a b e^{3} + 3 b^{4} e \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (a e^{2} - b d e + c d^{2}\right ) - 3 b^{2} d e^{2} + 3 b c d^{2} e}{6 a c e^{3} - 6 b c d e^{2} + 6 c^{2} d^{2} e} \right )} - \frac{- 3 a^{2} b e^{3} + 12 a^{2} c d e^{2} - 3 a b c d^{2} e + 4 a c^{2} d^{3} - b^{2} c d^{3} + x^{3} \left (10 a c^{2} e^{3} - 4 b^{2} c e^{3} + 6 b c^{2} d e^{2} - 6 c^{3} d^{2} e\right ) + x^{2} \left (3 a b c e^{3} + 24 a c^{2} d e^{2} - 3 b^{3} e^{3} + 3 b^{2} c d e^{2} - 9 b c^{2} d^{2} e\right ) + x \left (6 a^{2} c e^{3} - 6 a b^{2} e^{3} + 18 a b c d e^{2} + 6 a c^{2} d^{2} e - 6 b^{2} c d^{2} e\right )}{8 a^{3} c^{2} - 2 a^{2} b^{2} c + x^{4} \left (8 a c^{4} - 2 b^{2} c^{3}\right ) + x^{3} \left (16 a b c^{3} - 4 b^{3} c^{2}\right ) + x^{2} \left (16 a^{2} c^{3} + 4 a b^{2} c^{2} - 2 b^{4} c\right ) + x \left (16 a^{2} b c^{2} - 4 a b^{3} c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.20297, size = 394, normalized size = 3.13 \begin{align*} -\frac{6 \,{\left (c d^{2} e - b d e^{2} + a e^{3}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} - 4 \, a c\right )} \sqrt{-b^{2} + 4 \, a c}} - \frac{6 \, c^{3} d^{2} x^{3} e - 6 \, b c^{2} d x^{3} e^{2} + 9 \, b c^{2} d^{2} x^{2} e + 4 \, b^{2} c x^{3} e^{3} - 10 \, a c^{2} x^{3} e^{3} - 3 \, b^{2} c d x^{2} e^{2} - 24 \, a c^{2} d x^{2} e^{2} + 6 \, b^{2} c d^{2} x e - 6 \, a c^{2} d^{2} x e + b^{2} c d^{3} - 4 \, a c^{2} d^{3} + 3 \, b^{3} x^{2} e^{3} - 3 \, a b c x^{2} e^{3} - 18 \, a b c d x e^{2} + 3 \, a b c d^{2} e + 6 \, a b^{2} x e^{3} - 6 \, a^{2} c x e^{3} - 12 \, a^{2} c d e^{2} + 3 \, a^{2} b e^{3}}{2 \,{\left (b^{2} c - 4 \, a 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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