Optimal. Leaf size=71 \[ -\frac{4 b (d+e x)^{7/2} (b d-a e)}{7 e^3}+\frac{2 (d+e x)^{5/2} (b d-a e)^2}{5 e^3}+\frac{2 b^2 (d+e x)^{9/2}}{9 e^3} \]
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Rubi [A] time = 0.0228057, antiderivative size = 71, 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 = {27, 43} \[ -\frac{4 b (d+e x)^{7/2} (b d-a e)}{7 e^3}+\frac{2 (d+e x)^{5/2} (b d-a e)^2}{5 e^3}+\frac{2 b^2 (d+e x)^{9/2}}{9 e^3} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin{align*} \int (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right ) \, dx &=\int (a+b x)^2 (d+e x)^{3/2} \, dx\\ &=\int \left (\frac{(-b d+a e)^2 (d+e x)^{3/2}}{e^2}-\frac{2 b (b d-a e) (d+e x)^{5/2}}{e^2}+\frac{b^2 (d+e x)^{7/2}}{e^2}\right ) \, dx\\ &=\frac{2 (b d-a e)^2 (d+e x)^{5/2}}{5 e^3}-\frac{4 b (b d-a e) (d+e x)^{7/2}}{7 e^3}+\frac{2 b^2 (d+e x)^{9/2}}{9 e^3}\\ \end{align*}
Mathematica [A] time = 0.0368846, size = 61, normalized size = 0.86 \[ \frac{2 (d+e x)^{5/2} \left (63 a^2 e^2+18 a b e (5 e x-2 d)+b^2 \left (8 d^2-20 d e x+35 e^2 x^2\right )\right )}{315 e^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 63, normalized size = 0.9 \begin{align*}{\frac{70\,{b}^{2}{x}^{2}{e}^{2}+180\,xab{e}^{2}-40\,x{b}^{2}de+126\,{a}^{2}{e}^{2}-72\,abde+16\,{b}^{2}{d}^{2}}{315\,{e}^{3}} \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 = 1.03479, size = 92, normalized size = 1.3 \begin{align*} \frac{2 \,{\left (35 \,{\left (e x + d\right )}^{\frac{9}{2}} b^{2} - 90 \,{\left (b^{2} d - a b e\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 63 \,{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{5}{2}}\right )}}{315 \, e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.53832, size = 300, normalized size = 4.23 \begin{align*} \frac{2 \,{\left (35 \, b^{2} e^{4} x^{4} + 8 \, b^{2} d^{4} - 36 \, a b d^{3} e + 63 \, a^{2} d^{2} e^{2} + 10 \,{\left (5 \, b^{2} d e^{3} + 9 \, a b e^{4}\right )} x^{3} + 3 \,{\left (b^{2} d^{2} e^{2} + 48 \, a b d e^{3} + 21 \, a^{2} e^{4}\right )} x^{2} - 2 \,{\left (2 \, b^{2} d^{3} e - 9 \, a b d^{2} e^{2} - 63 \, a^{2} d e^{3}\right )} x\right )} \sqrt{e x + d}}{315 \, e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 9.39779, size = 240, normalized size = 3.38 \begin{align*} a^{2} d \left (\begin{cases} \sqrt{d} x & \text{for}\: e = 0 \\\frac{2 \left (d + e x\right )^{\frac{3}{2}}}{3 e} & \text{otherwise} \end{cases}\right ) + \frac{2 a^{2} \left (- \frac{d \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e} + \frac{4 a b d \left (- \frac{d \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{2}} + \frac{4 a b \left (\frac{d^{2} \left (d + e x\right )^{\frac{3}{2}}}{3} - \frac{2 d \left (d + e x\right )^{\frac{5}{2}}}{5} + \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right )}{e^{2}} + \frac{2 b^{2} d \left (\frac{d^{2} \left (d + e x\right )^{\frac{3}{2}}}{3} - \frac{2 d \left (d + e x\right )^{\frac{5}{2}}}{5} + \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right )}{e^{3}} + \frac{2 b^{2} \left (- \frac{d^{3} \left (d + e x\right )^{\frac{3}{2}}}{3} + \frac{3 d^{2} \left (d + e x\right )^{\frac{5}{2}}}{5} - \frac{3 d \left (d + e x\right )^{\frac{7}{2}}}{7} + \frac{\left (d + e x\right )^{\frac{9}{2}}}{9}\right )}{e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.21849, size = 288, normalized size = 4.06 \begin{align*} \frac{2}{315} \,{\left (42 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} a b d e^{\left (-1\right )} + 3 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2}\right )} b^{2} d e^{\left (-2\right )} + 105 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2} d + 6 \,{\left (15 \,{\left (x e + d\right )}^{\frac{7}{2}} - 42 \,{\left (x e + d\right )}^{\frac{5}{2}} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2}\right )} a b e^{\left (-1\right )} +{\left (35 \,{\left (x e + d\right )}^{\frac{9}{2}} - 135 \,{\left (x e + d\right )}^{\frac{7}{2}} d + 189 \,{\left (x e + d\right )}^{\frac{5}{2}} d^{2} - 105 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{3}\right )} b^{2} e^{\left (-2\right )} + 21 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} a^{2}\right )} e^{\left (-1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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