Optimal. Leaf size=128 \[ -\frac{2 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^4 \sqrt{d+e x}}+\frac{2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{3 e^4 (d+e x)^{3/2}}-\frac{6 c \sqrt{d+e x} (2 c d-b e)}{e^4}+\frac{4 c^2 (d+e x)^{3/2}}{3 e^4} \]
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Rubi [A] time = 0.0642856, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038, Rules used = {771} \[ -\frac{2 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^4 \sqrt{d+e x}}+\frac{2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{3 e^4 (d+e x)^{3/2}}-\frac{6 c \sqrt{d+e x} (2 c d-b e)}{e^4}+\frac{4 c^2 (d+e x)^{3/2}}{3 e^4} \]
Antiderivative was successfully verified.
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Rule 771
Rubi steps
\begin{align*} \int \frac{(b+2 c x) \left (a+b x+c x^2\right )}{(d+e x)^{5/2}} \, dx &=\int \left (\frac{(-2 c d+b e) \left (c d^2-b d e+a e^2\right )}{e^3 (d+e x)^{5/2}}+\frac{6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)}{e^3 (d+e x)^{3/2}}-\frac{3 c (2 c d-b e)}{e^3 \sqrt{d+e x}}+\frac{2 c^2 \sqrt{d+e x}}{e^3}\right ) \, dx\\ &=\frac{2 (2 c d-b e) \left (c d^2-b d e+a e^2\right )}{3 e^4 (d+e x)^{3/2}}-\frac{2 \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right )}{e^4 \sqrt{d+e x}}-\frac{6 c (2 c d-b e) \sqrt{d+e x}}{e^4}+\frac{4 c^2 (d+e x)^{3/2}}{3 e^4}\\ \end{align*}
Mathematica [A] time = 0.117407, size = 108, normalized size = 0.84 \[ -\frac{2 \left (c e \left (2 a e (2 d+3 e x)-3 b \left (8 d^2+12 d e x+3 e^2 x^2\right )\right )+b e^2 (a e+2 b d+3 b e x)+2 c^2 \left (24 d^2 e x+16 d^3+6 d e^2 x^2-e^3 x^3\right )\right )}{3 e^4 (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 122, normalized size = 1. \begin{align*} -{\frac{-4\,{c}^{2}{x}^{3}{e}^{3}-18\,bc{e}^{3}{x}^{2}+24\,{c}^{2}d{e}^{2}{x}^{2}+12\,ac{e}^{3}x+6\,{b}^{2}{e}^{3}x-72\,bcd{e}^{2}x+96\,{c}^{2}{d}^{2}ex+2\,ab{e}^{3}+8\,acd{e}^{2}+4\,{b}^{2}d{e}^{2}-48\,b{d}^{2}ce+64\,{c}^{2}{d}^{3}}{3\,{e}^{4}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.996618, size = 170, normalized size = 1.33 \begin{align*} \frac{2 \,{\left (\frac{2 \,{\left (e x + d\right )}^{\frac{3}{2}} c^{2} - 9 \,{\left (2 \, c^{2} d - b c e\right )} \sqrt{e x + d}}{e^{3}} + \frac{2 \, c^{2} d^{3} - 3 \, b c d^{2} e - a b e^{3} +{\left (b^{2} + 2 \, a c\right )} d e^{2} - 3 \,{\left (6 \, c^{2} d^{2} - 6 \, b c d e +{\left (b^{2} + 2 \, a c\right )} e^{2}\right )}{\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac{3}{2}} e^{3}}\right )}}{3 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.33419, size = 296, normalized size = 2.31 \begin{align*} \frac{2 \,{\left (2 \, c^{2} e^{3} x^{3} - 32 \, c^{2} d^{3} + 24 \, b c d^{2} e - a b e^{3} - 2 \,{\left (b^{2} + 2 \, a c\right )} d e^{2} - 3 \,{\left (4 \, c^{2} d e^{2} - 3 \, b c e^{3}\right )} x^{2} - 3 \,{\left (16 \, c^{2} d^{2} e - 12 \, b c d e^{2} +{\left (b^{2} + 2 \, a c\right )} e^{3}\right )} x\right )} \sqrt{e x + d}}{3 \,{\left (e^{6} x^{2} + 2 \, d e^{5} x + d^{2} e^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.55803, size = 536, normalized size = 4.19 \begin{align*} \begin{cases} - \frac{2 a b e^{3}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} - \frac{8 a c d e^{2}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} - \frac{12 a c e^{3} x}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} - \frac{4 b^{2} d e^{2}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} - \frac{6 b^{2} e^{3} x}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} + \frac{48 b c d^{2} e}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} + \frac{72 b c d e^{2} x}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} + \frac{18 b c e^{3} x^{2}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} - \frac{64 c^{2} d^{3}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} - \frac{96 c^{2} d^{2} e x}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} - \frac{24 c^{2} d e^{2} x^{2}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} + \frac{4 c^{2} e^{3} x^{3}}{3 d e^{4} \sqrt{d + e x} + 3 e^{5} x \sqrt{d + e x}} & \text{for}\: e \neq 0 \\\frac{a b x + a c x^{2} + \frac{b^{2} x^{2}}{2} + b c x^{3} + \frac{c^{2} x^{4}}{2}}{d^{\frac{5}{2}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.34233, size = 207, normalized size = 1.62 \begin{align*} \frac{2}{3} \,{\left (2 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{2} e^{8} - 18 \, \sqrt{x e + d} c^{2} d e^{8} + 9 \, \sqrt{x e + d} b c e^{9}\right )} e^{\left (-12\right )} - \frac{2 \,{\left (18 \,{\left (x e + d\right )} c^{2} d^{2} - 2 \, c^{2} d^{3} - 18 \,{\left (x e + d\right )} b c d e + 3 \, b c d^{2} e + 3 \,{\left (x e + d\right )} b^{2} e^{2} + 6 \,{\left (x e + d\right )} a c e^{2} - b^{2} d e^{2} - 2 \, a c d e^{2} + a b e^{3}\right )} e^{\left (-4\right )}}{3 \,{\left (x e + d\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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