Optimal. Leaf size=162 \[ \frac{2 \sqrt{d+e x} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^5}+\frac{4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{e^5 \sqrt{d+e x}}-\frac{2 \left (a e^2-b d e+c d^2\right )^2}{3 e^5 (d+e x)^{3/2}}-\frac{4 c (d+e x)^{3/2} (2 c d-b e)}{3 e^5}+\frac{2 c^2 (d+e x)^{5/2}}{5 e^5} \]
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Rubi [A] time = 0.0745995, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {698} \[ \frac{2 \sqrt{d+e x} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^5}+\frac{4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{e^5 \sqrt{d+e x}}-\frac{2 \left (a e^2-b d e+c d^2\right )^2}{3 e^5 (d+e x)^{3/2}}-\frac{4 c (d+e x)^{3/2} (2 c d-b e)}{3 e^5}+\frac{2 c^2 (d+e x)^{5/2}}{5 e^5} \]
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin{align*} \int \frac{\left (a+b x+c x^2\right )^2}{(d+e x)^{5/2}} \, dx &=\int \left (\frac{\left (c d^2-b d e+a e^2\right )^2}{e^4 (d+e x)^{5/2}}+\frac{2 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )}{e^4 (d+e x)^{3/2}}+\frac{6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)}{e^4 \sqrt{d+e x}}-\frac{2 c (2 c d-b e) \sqrt{d+e x}}{e^4}+\frac{c^2 (d+e x)^{3/2}}{e^4}\right ) \, dx\\ &=-\frac{2 \left (c d^2-b d e+a e^2\right )^2}{3 e^5 (d+e x)^{3/2}}+\frac{4 (2 c d-b e) \left (c d^2-b d e+a e^2\right )}{e^5 \sqrt{d+e x}}+\frac{2 \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) \sqrt{d+e x}}{e^5}-\frac{4 c (2 c d-b e) (d+e x)^{3/2}}{3 e^5}+\frac{2 c^2 (d+e x)^{5/2}}{5 e^5}\\ \end{align*}
Mathematica [A] time = 0.128175, size = 170, normalized size = 1.05 \[ \frac{2 \left (-5 e^2 \left (a^2 e^2+2 a b e (2 d+3 e x)+b^2 \left (-\left (8 d^2+12 d e x+3 e^2 x^2\right )\right )\right )+10 c e \left (a e \left (8 d^2+12 d e x+3 e^2 x^2\right )+b \left (-24 d^2 e x-16 d^3-6 d e^2 x^2+e^3 x^3\right )\right )+c^2 \left (48 d^2 e^2 x^2+192 d^3 e x+128 d^4-8 d e^3 x^3+3 e^4 x^4\right )\right )}{15 e^5 (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 194, normalized size = 1.2 \begin{align*} -{\frac{-6\,{c}^{2}{x}^{4}{e}^{4}-20\,bc{e}^{4}{x}^{3}+16\,{c}^{2}d{e}^{3}{x}^{3}-60\,ac{e}^{4}{x}^{2}-30\,{b}^{2}{e}^{4}{x}^{2}+120\,bcd{e}^{3}{x}^{2}-96\,{c}^{2}{d}^{2}{e}^{2}{x}^{2}+60\,ab{e}^{4}x-240\,acd{e}^{3}x-120\,{b}^{2}d{e}^{3}x+480\,bc{d}^{2}{e}^{2}x-384\,{c}^{2}{d}^{3}ex+10\,{a}^{2}{e}^{4}+40\,abd{e}^{3}-160\,ac{d}^{2}{e}^{2}-80\,{b}^{2}{d}^{2}{e}^{2}+320\,bc{d}^{3}e-256\,{c}^{2}{d}^{4}}{15\,{e}^{5}} \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 = 0.974549, size = 246, normalized size = 1.52 \begin{align*} \frac{2 \,{\left (\frac{3 \,{\left (e x + d\right )}^{\frac{5}{2}} c^{2} - 10 \,{\left (2 \, c^{2} d - b c e\right )}{\left (e x + d\right )}^{\frac{3}{2}} + 15 \,{\left (6 \, c^{2} d^{2} - 6 \, b c d e +{\left (b^{2} + 2 \, a c\right )} e^{2}\right )} \sqrt{e x + d}}{e^{4}} - \frac{5 \,{\left (c^{2} d^{4} - 2 \, b c d^{3} e - 2 \, a b d e^{3} + a^{2} e^{4} +{\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} - 6 \,{\left (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}\right )}{\left (e x + d\right )}\right )}}{{\left (e x + d\right )}^{\frac{3}{2}} e^{4}}\right )}}{15 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.00105, size = 436, normalized size = 2.69 \begin{align*} \frac{2 \,{\left (3 \, c^{2} e^{4} x^{4} + 128 \, c^{2} d^{4} - 160 \, b c d^{3} e - 20 \, a b d e^{3} - 5 \, a^{2} e^{4} + 40 \,{\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} - 2 \,{\left (4 \, c^{2} d e^{3} - 5 \, b c e^{4}\right )} x^{3} + 3 \,{\left (16 \, c^{2} d^{2} e^{2} - 20 \, b c d e^{3} + 5 \,{\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{2} + 6 \,{\left (32 \, c^{2} d^{3} e - 40 \, b c d^{2} e^{2} - 5 \, a b e^{4} + 10 \,{\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x\right )} \sqrt{e x + d}}{15 \,{\left (e^{7} x^{2} + 2 \, d e^{6} x + d^{2} e^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 43.371, size = 160, normalized size = 0.99 \begin{align*} \frac{2 c^{2} \left (d + e x\right )^{\frac{5}{2}}}{5 e^{5}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (4 b c e - 8 c^{2} d\right )}{3 e^{5}} + \frac{\sqrt{d + e x} \left (4 a c e^{2} + 2 b^{2} e^{2} - 12 b c d e + 12 c^{2} d^{2}\right )}{e^{5}} - \frac{4 \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )}{e^{5} \sqrt{d + e x}} - \frac{2 \left (a e^{2} - b d e + c d^{2}\right )^{2}}{3 e^{5} \left (d + e x\right )^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1425, size = 329, normalized size = 2.03 \begin{align*} \frac{2}{15} \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} c^{2} e^{20} - 20 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{2} d e^{20} + 90 \, \sqrt{x e + d} c^{2} d^{2} e^{20} + 10 \,{\left (x e + d\right )}^{\frac{3}{2}} b c e^{21} - 90 \, \sqrt{x e + d} b c d e^{21} + 15 \, \sqrt{x e + d} b^{2} e^{22} + 30 \, \sqrt{x e + d} a c e^{22}\right )} e^{\left (-25\right )} + \frac{2 \,{\left (12 \,{\left (x e + d\right )} c^{2} d^{3} - c^{2} d^{4} - 18 \,{\left (x e + d\right )} b c d^{2} e + 2 \, b c d^{3} e + 6 \,{\left (x e + d\right )} b^{2} d e^{2} + 12 \,{\left (x e + d\right )} a c d e^{2} - b^{2} d^{2} e^{2} - 2 \, a c d^{2} e^{2} - 6 \,{\left (x e + d\right )} a b e^{3} + 2 \, a b d e^{3} - a^{2} e^{4}\right )} e^{\left (-5\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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