Optimal. Leaf size=91 \[ \frac{8 (b+2 c x) (2 c d-b e)}{3 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{2 (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.0213004, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {638, 613} \[ \frac{8 (b+2 c x) (2 c d-b e)}{3 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{2 (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 638
Rule 613
Rubi steps
\begin{align*} \int \frac{d+e x}{\left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac{2 (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac{(4 (2 c d-b e)) \int \frac{1}{\left (a+b x+c x^2\right )^{3/2}} \, dx}{3 \left (b^2-4 a c\right )}\\ &=-\frac{2 (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac{8 (2 c d-b e) (b+2 c x)}{3 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [A] time = 0.12243, size = 110, normalized size = 1.21 \[ -\frac{2 \left (8 c \left (a^2 e-3 a c d x-2 c^2 d x^3\right )+2 b^2 (a e+3 c x (2 e x-d))+4 b c \left (2 c x^2 (e x-3 d)-3 a (d-e x)\right )+b^3 (d+3 e x)\right )}{3 \left (b^2-4 a c\right )^2 (a+x (b+c x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 131, normalized size = 1.4 \begin{align*} -{\frac{16\,b{c}^{2}e{x}^{3}-32\,{c}^{3}d{x}^{3}+24\,{b}^{2}ce{x}^{2}-48\,b{c}^{2}d{x}^{2}+24\,abcex-48\,a{c}^{2}dx+6\,{b}^{3}ex-12\,{b}^{2}cdx+16\,{a}^{2}ce+4\,a{b}^{2}e-24\,cabd+2\,{b}^{3}d}{48\,{a}^{2}{c}^{2}-24\,ac{b}^{2}+3\,{b}^{4}} \left ( c{x}^{2}+bx+a \right ) ^{-{\frac{3}{2}}}} \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 = 8.44065, size = 521, normalized size = 5.73 \begin{align*} \frac{2 \,{\left (8 \,{\left (2 \, c^{3} d - b c^{2} e\right )} x^{3} + 12 \,{\left (2 \, b c^{2} d - b^{2} c e\right )} x^{2} -{\left (b^{3} - 12 \, a b c\right )} d - 2 \,{\left (a b^{2} + 4 \, a^{2} c\right )} e + 3 \,{\left (2 \,{\left (b^{2} c + 4 \, a c^{2}\right )} d -{\left (b^{3} + 4 \, a b c\right )} e\right )} x\right )} \sqrt{c x^{2} + b x + a}}{3 \,{\left (a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2} +{\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{4} + 2 \,{\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{3} +{\left (b^{6} - 6 \, a b^{4} c + 32 \, a^{3} c^{3}\right )} x^{2} + 2 \,{\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.12858, size = 306, normalized size = 3.36 \begin{align*} \frac{{\left (4 \,{\left (\frac{2 \,{\left (2 \, c^{3} d - b c^{2} e\right )} x}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}} + \frac{3 \,{\left (2 \, b c^{2} d - b^{2} c e\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x + \frac{3 \,{\left (2 \, b^{2} c d + 8 \, a c^{2} d - b^{3} e - 4 \, a b c e\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x - \frac{b^{3} d - 12 \, a b c d + 2 \, a b^{2} e + 8 \, a^{2} c e}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}}{3 \,{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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