Optimal. Leaf size=94 \[ \frac{8 (2 a+b x) (A b-2 a B)}{3 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{2 x^2 (-2 a B-x (b B-2 A c)+A b)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.0375209, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {804, 636} \[ \frac{8 (2 a+b x) (A b-2 a B)}{3 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{2 x^2 (-2 a B-x (b B-2 A c)+A b)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 804
Rule 636
Rubi steps
\begin{align*} \int \frac{x^2 (A+B x)}{\left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac{2 x^2 (A b-2 a B-(b B-2 A c) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac{(4 (A b-2 a B)) \int \frac{x}{\left (a+b x+c x^2\right )^{3/2}} \, dx}{3 \left (b^2-4 a c\right )}\\ &=-\frac{2 x^2 (A b-2 a B-(b B-2 A c) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac{8 (A b-2 a B) (2 a+b x)}{3 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [A] time = 0.289217, size = 110, normalized size = 1.17 \[ \frac{2 \left (8 a^2 (A b-3 B x (b+c x))-16 a^3 B+2 a x \left (A \left (6 b^2+6 b c x+4 c^2 x^2\right )-3 b B x (b+2 c x)\right )+b^2 x^2 (3 A b+2 A c x+b B x)\right )}{3 \left (b^2-4 a c\right )^2 (a+x (b+c x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 141, normalized size = 1.5 \begin{align*}{\frac{16\,aA{c}^{2}{x}^{3}+4\,A{x}^{3}{b}^{2}c-24\,B{x}^{3}abc+2\,{b}^{3}B{x}^{3}+24\,A{x}^{2}abc+6\,A{b}^{3}{x}^{2}-48\,{a}^{2}Bc{x}^{2}-12\,B{x}^{2}a{b}^{2}+24\,Aa{b}^{2}x-48\,B{a}^{2}bx+16\,Ab{a}^{2}-32\,B{a}^{3}}{48\,{a}^{2}{c}^{2}-24\,a{b}^{2}c+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 = 7.08546, size = 522, normalized size = 5.55 \begin{align*} -\frac{2 \,{\left (16 \, B a^{3} - 8 \, A a^{2} b -{\left (B b^{3} + 8 \, A a c^{2} - 2 \,{\left (6 \, B a b - A b^{2}\right )} c\right )} x^{3} + 3 \,{\left (2 \, B a b^{2} - A b^{3} + 4 \,{\left (2 \, B a^{2} - A a b\right )} c\right )} x^{2} + 12 \,{\left (2 \, B a^{2} b - A a b^{2}\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 = 1.26811, size = 296, normalized size = 3.15 \begin{align*} \frac{{\left ({\left (\frac{{\left (B b^{3} - 12 \, B a b c + 2 \, A b^{2} c + 8 \, A a c^{2}\right )} x}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}} - \frac{3 \,{\left (2 \, B a b^{2} - A b^{3} + 8 \, B a^{2} c - 4 \, A a b c\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x - \frac{12 \,{\left (2 \, B a^{2} b - A a b^{2}\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x - \frac{8 \,{\left (2 \, B a^{3} - A a^{2} b\right )}}{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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