Optimal. Leaf size=162 \[ -\frac{2 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^5 \sqrt{d+e x}}+\frac{4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{3 e^5 (d+e x)^{3/2}}-\frac{2 \left (a e^2-b d e+c d^2\right )^2}{5 e^5 (d+e x)^{5/2}}-\frac{4 c \sqrt{d+e x} (2 c d-b e)}{e^5}+\frac{2 c^2 (d+e x)^{3/2}}{3 e^5} \]
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Rubi [A] time = 0.0708952, 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} \[ -\frac{2 \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^5 \sqrt{d+e x}}+\frac{4 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{3 e^5 (d+e x)^{3/2}}-\frac{2 \left (a e^2-b d e+c d^2\right )^2}{5 e^5 (d+e x)^{5/2}}-\frac{4 c \sqrt{d+e x} (2 c d-b e)}{e^5}+\frac{2 c^2 (d+e x)^{3/2}}{3 e^5} \]
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)^{7/2}} \, dx &=\int \left (\frac{\left (c d^2-b d e+a e^2\right )^2}{e^4 (d+e x)^{7/2}}+\frac{2 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )}{e^4 (d+e x)^{5/2}}+\frac{6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)}{e^4 (d+e x)^{3/2}}-\frac{2 c (2 c d-b e)}{e^4 \sqrt{d+e x}}+\frac{c^2 \sqrt{d+e x}}{e^4}\right ) \, dx\\ &=-\frac{2 \left (c d^2-b d e+a e^2\right )^2}{5 e^5 (d+e x)^{5/2}}+\frac{4 (2 c d-b e) \left (c d^2-b d e+a e^2\right )}{3 e^5 (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^5 \sqrt{d+e x}}-\frac{4 c (2 c d-b e) \sqrt{d+e x}}{e^5}+\frac{2 c^2 (d+e x)^{3/2}}{3 e^5}\\ \end{align*}
Mathematica [A] time = 0.136638, size = 172, normalized size = 1.06 \[ -\frac{2 \left (3 a^2 e^4+2 a b e^3 (2 d+5 e x)+2 a c e^2 \left (8 d^2+20 d e x+15 e^2 x^2\right )+b^2 e^2 \left (8 d^2+20 d e x+15 e^2 x^2\right )-6 b c e \left (40 d^2 e x+16 d^3+30 d e^2 x^2+5 e^3 x^3\right )+c^2 \left (240 d^2 e^2 x^2+320 d^3 e x+128 d^4+40 d e^3 x^3-5 e^4 x^4\right )\right )}{15 e^5 (d+e x)^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 194, normalized size = 1.2 \begin{align*} -{\frac{-10\,{c}^{2}{x}^{4}{e}^{4}-60\,bc{e}^{4}{x}^{3}+80\,{c}^{2}d{e}^{3}{x}^{3}+60\,ac{e}^{4}{x}^{2}+30\,{b}^{2}{e}^{4}{x}^{2}-360\,bcd{e}^{3}{x}^{2}+480\,{c}^{2}{d}^{2}{e}^{2}{x}^{2}+20\,ab{e}^{4}x+80\,acd{e}^{3}x+40\,{b}^{2}d{e}^{3}x-480\,bc{d}^{2}{e}^{2}x+640\,{c}^{2}{d}^{3}ex+6\,{a}^{2}{e}^{4}+8\,abd{e}^{3}+32\,ac{d}^{2}{e}^{2}+16\,{b}^{2}{d}^{2}{e}^{2}-192\,bc{d}^{3}e+256\,{c}^{2}{d}^{4}}{15\,{e}^{5}} \left ( ex+d \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.964951, size = 250, normalized size = 1.54 \begin{align*} \frac{2 \,{\left (\frac{5 \,{\left ({\left (e x + d\right )}^{\frac{3}{2}} c^{2} - 6 \,{\left (2 \, c^{2} d - b c e\right )} \sqrt{e x + d}\right )}}{e^{4}} - \frac{3 \, c^{2} d^{4} - 6 \, b c d^{3} e - 6 \, a b d e^{3} + 3 \, a^{2} e^{4} + 3 \,{\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + 15 \,{\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 )}^{2} - 10 \,{\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 )}}{{\left (e x + d\right )}^{\frac{5}{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 = 1.99409, size = 451, normalized size = 2.78 \begin{align*} \frac{2 \,{\left (5 \, c^{2} e^{4} x^{4} - 128 \, c^{2} d^{4} + 96 \, b c d^{3} e - 4 \, a b d e^{3} - 3 \, a^{2} e^{4} - 8 \,{\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} - 10 \,{\left (4 \, c^{2} d e^{3} - 3 \, b c e^{4}\right )} x^{3} - 15 \,{\left (16 \, c^{2} d^{2} e^{2} - 12 \, b c d e^{3} +{\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{2} - 10 \,{\left (32 \, c^{2} d^{3} e - 24 \, b c d^{2} e^{2} + a b e^{4} + 2 \,{\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x\right )} \sqrt{e x + d}}{15 \,{\left (e^{8} x^{3} + 3 \, d e^{7} x^{2} + 3 \, d^{2} e^{6} x + d^{3} e^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.81245, size = 1180, normalized size = 7.28 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13767, size = 325, normalized size = 2.01 \begin{align*} \frac{2}{3} \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} c^{2} e^{10} - 12 \, \sqrt{x e + d} c^{2} d e^{10} + 6 \, \sqrt{x e + d} b c e^{11}\right )} e^{\left (-15\right )} - \frac{2 \,{\left (90 \,{\left (x e + d\right )}^{2} c^{2} d^{2} - 20 \,{\left (x e + d\right )} c^{2} d^{3} + 3 \, c^{2} d^{4} - 90 \,{\left (x e + d\right )}^{2} b c d e + 30 \,{\left (x e + d\right )} b c d^{2} e - 6 \, b c d^{3} e + 15 \,{\left (x e + d\right )}^{2} b^{2} e^{2} + 30 \,{\left (x e + d\right )}^{2} a c e^{2} - 10 \,{\left (x e + d\right )} b^{2} d e^{2} - 20 \,{\left (x e + d\right )} a c d e^{2} + 3 \, b^{2} d^{2} e^{2} + 6 \, a c d^{2} e^{2} + 10 \,{\left (x e + d\right )} a b e^{3} - 6 \, a b d e^{3} + 3 \, a^{2} e^{4}\right )} e^{\left (-5\right )}}{15 \,{\left (x e + d\right )}^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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