Optimal. Leaf size=98 \[ \frac{8 (2 c d-b e) (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{2 (b+2 c x) (d+e x)^2}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.0374289, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {728, 636} \[ \frac{8 (2 c d-b e) (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{2 (b+2 c x) (d+e x)^2}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 728
Rule 636
Rubi steps
\begin{align*} \int \frac{(d+e x)^2}{\left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac{2 (b+2 c x) (d+e x)^2}{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{d+e x}{\left (a+b x+c x^2\right )^{3/2}} \, dx}{3 \left (b^2-4 a c\right )}\\ &=-\frac{2 (b+2 c x) (d+e x)^2}{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 d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [A] time = 0.730552, size = 167, normalized size = 1.7 \[ \frac{2 \left (4 b \left (2 a^2 e^2+3 a c (d-e x)^2+2 c^2 d x^2 (3 d-2 e x)\right )+8 c \left (-2 a^2 d e+a c x \left (3 d^2+e^2 x^2\right )+2 c^2 d^2 x^3\right )+b^2 \left (2 c x \left (3 d^2-12 d e x+e^2 x^2\right )-4 a e (d-3 e x)\right )+b^3 \left (-\left (d^2+6 d e x-3 e^2 x^2\right )\right )\right )}{3 \left (b^2-4 a c\right )^2 (a+x (b+c x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.047, size = 215, normalized size = 2.2 \begin{align*}{\frac{16\,a{c}^{2}{e}^{2}{x}^{3}+4\,{b}^{2}c{e}^{2}{x}^{3}-32\,b{c}^{2}de{x}^{3}+32\,{c}^{3}{d}^{2}{x}^{3}+24\,abc{e}^{2}{x}^{2}+6\,{b}^{3}{e}^{2}{x}^{2}-48\,{b}^{2}cde{x}^{2}+48\,b{c}^{2}{d}^{2}{x}^{2}+24\,a{b}^{2}{e}^{2}x-48\,abcdex+48\,a{c}^{2}{d}^{2}x-12\,{b}^{3}dex+12\,{b}^{2}c{d}^{2}x+16\,{a}^{2}b{e}^{2}-32\,{a}^{2}cde-8\,a{b}^{2}de+24\,abc{d}^{2}-2\,{b}^{3}{d}^{2}}{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 = 8.82085, size = 644, normalized size = 6.57 \begin{align*} \frac{2 \,{\left (8 \, a^{2} b e^{2} + 2 \,{\left (8 \, c^{3} d^{2} - 8 \, b c^{2} d e +{\left (b^{2} c + 4 \, a c^{2}\right )} e^{2}\right )} x^{3} -{\left (b^{3} - 12 \, a b c\right )} d^{2} - 4 \,{\left (a b^{2} + 4 \, a^{2} c\right )} d e + 3 \,{\left (8 \, b c^{2} d^{2} - 8 \, b^{2} c d e +{\left (b^{3} + 4 \, a b c\right )} e^{2}\right )} x^{2} + 6 \,{\left (2 \, a b^{2} e^{2} +{\left (b^{2} c + 4 \, a c^{2}\right )} d^{2} -{\left (b^{3} + 4 \, a b c\right )} d 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 = 1.11447, size = 387, normalized size = 3.95 \begin{align*} \frac{{\left ({\left (\frac{2 \,{\left (8 \, c^{3} d^{2} - 8 \, b c^{2} d e + b^{2} c e^{2} + 4 \, a c^{2} e^{2}\right )} x}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}} + \frac{3 \,{\left (8 \, b c^{2} d^{2} - 8 \, b^{2} c d e + b^{3} e^{2} + 4 \, a b c e^{2}\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x + \frac{6 \,{\left (b^{2} c d^{2} + 4 \, a c^{2} d^{2} - b^{3} d e - 4 \, a b c d e + 2 \, a b^{2} e^{2}\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x - \frac{b^{3} d^{2} - 12 \, a b c d^{2} + 4 \, a b^{2} d e + 16 \, a^{2} c d e - 8 \, a^{2} b e^{2}}{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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