Optimal. Leaf size=143 \[ \frac{2 (d+e x)^{3/2} \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{3 e^5}-\frac{2 d^2 (c d-b e)^2}{e^5 \sqrt{d+e x}}-\frac{4 c (d+e x)^{5/2} (2 c d-b e)}{5 e^5}-\frac{4 d \sqrt{d+e x} (c d-b e) (2 c d-b e)}{e^5}+\frac{2 c^2 (d+e x)^{7/2}}{7 e^5} \]
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Rubi [A] time = 0.0582186, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {698} \[ \frac{2 (d+e x)^{3/2} \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{3 e^5}-\frac{2 d^2 (c d-b e)^2}{e^5 \sqrt{d+e x}}-\frac{4 c (d+e x)^{5/2} (2 c d-b e)}{5 e^5}-\frac{4 d \sqrt{d+e x} (c d-b e) (2 c d-b e)}{e^5}+\frac{2 c^2 (d+e x)^{7/2}}{7 e^5} \]
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin{align*} \int \frac{\left (b x+c x^2\right )^2}{(d+e x)^{3/2}} \, dx &=\int \left (\frac{d^2 (c d-b e)^2}{e^4 (d+e x)^{3/2}}+\frac{2 d (c d-b e) (-2 c d+b e)}{e^4 \sqrt{d+e x}}+\frac{\left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) \sqrt{d+e x}}{e^4}-\frac{2 c (2 c d-b e) (d+e x)^{3/2}}{e^4}+\frac{c^2 (d+e x)^{5/2}}{e^4}\right ) \, dx\\ &=-\frac{2 d^2 (c d-b e)^2}{e^5 \sqrt{d+e x}}-\frac{4 d (c d-b e) (2 c d-b e) \sqrt{d+e x}}{e^5}+\frac{2 \left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) (d+e x)^{3/2}}{3 e^5}-\frac{4 c (2 c d-b e) (d+e x)^{5/2}}{5 e^5}+\frac{2 c^2 (d+e x)^{7/2}}{7 e^5}\\ \end{align*}
Mathematica [A] time = 0.0710242, size = 123, normalized size = 0.86 \[ \frac{70 b^2 e^2 \left (-8 d^2-4 d e x+e^2 x^2\right )+84 b c e \left (8 d^2 e x+16 d^3-2 d e^2 x^2+e^3 x^3\right )-6 c^2 \left (-16 d^2 e^2 x^2+64 d^3 e x+128 d^4+8 d e^3 x^3-5 e^4 x^4\right )}{105 e^5 \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 141, normalized size = 1. \begin{align*} -{\frac{-30\,{c}^{2}{x}^{4}{e}^{4}-84\,bc{e}^{4}{x}^{3}+48\,{c}^{2}d{e}^{3}{x}^{3}-70\,{b}^{2}{e}^{4}{x}^{2}+168\,bcd{e}^{3}{x}^{2}-96\,{c}^{2}{d}^{2}{e}^{2}{x}^{2}+280\,{b}^{2}d{e}^{3}x-672\,bc{d}^{2}{e}^{2}x+384\,{c}^{2}{d}^{3}ex+560\,{b}^{2}{d}^{2}{e}^{2}-1344\,bc{d}^{3}e+768\,{c}^{2}{d}^{4}}{105\,{e}^{5}}{\frac{1}{\sqrt{ex+d}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.19355, size = 198, normalized size = 1.38 \begin{align*} \frac{2 \,{\left (\frac{15 \,{\left (e x + d\right )}^{\frac{7}{2}} c^{2} - 42 \,{\left (2 \, c^{2} d - b c e\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 35 \,{\left (6 \, c^{2} d^{2} - 6 \, b c d e + b^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{3}{2}} - 210 \,{\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e + b^{2} d e^{2}\right )} \sqrt{e x + d}}{e^{4}} - \frac{105 \,{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )}}{\sqrt{e x + d} e^{4}}\right )}}{105 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.81059, size = 324, normalized size = 2.27 \begin{align*} \frac{2 \,{\left (15 \, c^{2} e^{4} x^{4} - 384 \, c^{2} d^{4} + 672 \, b c d^{3} e - 280 \, b^{2} d^{2} e^{2} - 6 \,{\left (4 \, c^{2} d e^{3} - 7 \, b c e^{4}\right )} x^{3} +{\left (48 \, c^{2} d^{2} e^{2} - 84 \, b c d e^{3} + 35 \, b^{2} e^{4}\right )} x^{2} - 4 \,{\left (48 \, c^{2} d^{3} e - 84 \, b c d^{2} e^{2} + 35 \, b^{2} d e^{3}\right )} x\right )} \sqrt{e x + d}}{105 \,{\left (e^{6} x + d e^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 24.154, size = 150, normalized size = 1.05 \begin{align*} \frac{2 c^{2} \left (d + e x\right )^{\frac{7}{2}}}{7 e^{5}} - \frac{2 d^{2} \left (b e - c d\right )^{2}}{e^{5} \sqrt{d + e x}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (4 b c e - 8 c^{2} d\right )}{5 e^{5}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (2 b^{2} e^{2} - 12 b c d e + 12 c^{2} d^{2}\right )}{3 e^{5}} + \frac{\sqrt{d + e x} \left (- 4 b^{2} d e^{2} + 12 b c d^{2} e - 8 c^{2} d^{3}\right )}{e^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28524, size = 254, normalized size = 1.78 \begin{align*} \frac{2}{105} \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} c^{2} e^{30} - 84 \,{\left (x e + d\right )}^{\frac{5}{2}} c^{2} d e^{30} + 210 \,{\left (x e + d\right )}^{\frac{3}{2}} c^{2} d^{2} e^{30} - 420 \, \sqrt{x e + d} c^{2} d^{3} e^{30} + 42 \,{\left (x e + d\right )}^{\frac{5}{2}} b c e^{31} - 210 \,{\left (x e + d\right )}^{\frac{3}{2}} b c d e^{31} + 630 \, \sqrt{x e + d} b c d^{2} e^{31} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{2} e^{32} - 210 \, \sqrt{x e + d} b^{2} d e^{32}\right )} e^{\left (-35\right )} - \frac{2 \,{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} e^{\left (-5\right )}}{\sqrt{x e + d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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