Optimal. Leaf size=84 \[ \frac{256}{3} c^2 d^5 \sqrt{a+b x+c x^2}-\frac{32 c d^5 (b+2 c x)^2}{3 \sqrt{a+b x+c x^2}}-\frac{2 d^5 (b+2 c x)^4}{3 \left (a+b x+c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.0439912, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {686, 629} \[ \frac{256}{3} c^2 d^5 \sqrt{a+b x+c x^2}-\frac{32 c d^5 (b+2 c x)^2}{3 \sqrt{a+b x+c x^2}}-\frac{2 d^5 (b+2 c x)^4}{3 \left (a+b x+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 686
Rule 629
Rubi steps
\begin{align*} \int \frac{(b d+2 c d x)^5}{\left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac{2 d^5 (b+2 c x)^4}{3 \left (a+b x+c x^2\right )^{3/2}}+\frac{1}{3} \left (16 c d^2\right ) \int \frac{(b d+2 c d x)^3}{\left (a+b x+c x^2\right )^{3/2}} \, dx\\ &=-\frac{2 d^5 (b+2 c x)^4}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac{32 c d^5 (b+2 c x)^2}{3 \sqrt{a+b x+c x^2}}+\frac{1}{3} \left (128 c^2 d^4\right ) \int \frac{b d+2 c d x}{\sqrt{a+b x+c x^2}} \, dx\\ &=-\frac{2 d^5 (b+2 c x)^4}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac{32 c d^5 (b+2 c x)^2}{3 \sqrt{a+b x+c x^2}}+\frac{256}{3} c^2 d^5 \sqrt{a+b x+c x^2}\\ \end{align*}
Mathematica [A] time = 0.0532253, size = 91, normalized size = 1.08 \[ \frac{d^5 \left (32 c^2 \left (8 a^2+12 a c x^2+3 c^2 x^4\right )+16 b^2 c \left (3 c x^2-2 a\right )+192 b c^2 x \left (2 a+c x^2\right )-48 b^3 c x-2 b^4\right )}{3 (a+x (b+c x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 91, normalized size = 1.1 \begin{align*}{\frac{2\,{d}^{5} \left ( 48\,{c}^{4}{x}^{4}+96\,b{c}^{3}{x}^{3}+192\,a{c}^{3}{x}^{2}+24\,{b}^{2}{c}^{2}{x}^{2}+192\,ab{c}^{2}x-24\,{b}^{3}cx+128\,{a}^{2}{c}^{2}-16\,ac{b}^{2}-{b}^{4} \right ) }{3} \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 [A] time = 8.91066, size = 302, normalized size = 3.6 \begin{align*} \frac{2 \,{\left (48 \, c^{4} d^{5} x^{4} + 96 \, b c^{3} d^{5} x^{3} + 24 \,{\left (b^{2} c^{2} + 8 \, a c^{3}\right )} d^{5} x^{2} - 24 \,{\left (b^{3} c - 8 \, a b c^{2}\right )} d^{5} x -{\left (b^{4} + 16 \, a b^{2} c - 128 \, a^{2} c^{2}\right )} d^{5}\right )} \sqrt{c x^{2} + b x + a}}{3 \,{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.62557, size = 615, normalized size = 7.32 \begin{align*} \frac{256 a^{2} c^{2} d^{5}}{3 a \sqrt{a + b x + c x^{2}} + 3 b x \sqrt{a + b x + c x^{2}} + 3 c x^{2} \sqrt{a + b x + c x^{2}}} - \frac{32 a b^{2} c d^{5}}{3 a \sqrt{a + b x + c x^{2}} + 3 b x \sqrt{a + b x + c x^{2}} + 3 c x^{2} \sqrt{a + b x + c x^{2}}} + \frac{384 a b c^{2} d^{5} x}{3 a \sqrt{a + b x + c x^{2}} + 3 b x \sqrt{a + b x + c x^{2}} + 3 c x^{2} \sqrt{a + b x + c x^{2}}} + \frac{384 a c^{3} d^{5} x^{2}}{3 a \sqrt{a + b x + c x^{2}} + 3 b x \sqrt{a + b x + c x^{2}} + 3 c x^{2} \sqrt{a + b x + c x^{2}}} - \frac{2 b^{4} d^{5}}{3 a \sqrt{a + b x + c x^{2}} + 3 b x \sqrt{a + b x + c x^{2}} + 3 c x^{2} \sqrt{a + b x + c x^{2}}} - \frac{48 b^{3} c d^{5} x}{3 a \sqrt{a + b x + c x^{2}} + 3 b x \sqrt{a + b x + c x^{2}} + 3 c x^{2} \sqrt{a + b x + c x^{2}}} + \frac{48 b^{2} c^{2} d^{5} x^{2}}{3 a \sqrt{a + b x + c x^{2}} + 3 b x \sqrt{a + b x + c x^{2}} + 3 c x^{2} \sqrt{a + b x + c x^{2}}} + \frac{192 b c^{3} d^{5} x^{3}}{3 a \sqrt{a + b x + c x^{2}} + 3 b x \sqrt{a + b x + c x^{2}} + 3 c x^{2} \sqrt{a + b x + c x^{2}}} + \frac{96 c^{4} d^{5} x^{4}}{3 a \sqrt{a + b x + c x^{2}} + 3 b x \sqrt{a + b x + c x^{2}} + 3 c x^{2} \sqrt{a + b x + c x^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.15636, size = 521, normalized size = 6.2 \begin{align*} \frac{2 \,{\left (24 \,{\left ({\left (2 \,{\left (\frac{{\left (b^{4} c^{6} d^{5} - 8 \, a b^{2} c^{7} d^{5} + 16 \, a^{2} c^{8} d^{5}\right )} x}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}} + \frac{2 \,{\left (b^{5} c^{5} d^{5} - 8 \, a b^{3} c^{6} d^{5} + 16 \, a^{2} b c^{7} d^{5}\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x + \frac{b^{6} c^{4} d^{5} - 48 \, a^{2} b^{2} c^{6} d^{5} + 128 \, a^{3} c^{7} d^{5}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x - \frac{b^{7} c^{3} d^{5} - 16 \, a b^{5} c^{4} d^{5} + 80 \, a^{2} b^{3} c^{5} d^{5} - 128 \, a^{3} b c^{6} d^{5}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x - \frac{b^{8} c^{2} d^{5} + 8 \, a b^{6} c^{3} d^{5} - 240 \, a^{2} b^{4} c^{4} d^{5} + 1280 \, a^{3} b^{2} c^{5} d^{5} - 2048 \, a^{4} c^{6} d^{5}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )}}{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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