Optimal. Leaf size=63 \[ \frac{2 (d+e x)^{5/2} \left (a e^2+c d^2\right )}{5 e^3}+\frac{2 c (d+e x)^{9/2}}{9 e^3}-\frac{4 c d (d+e x)^{7/2}}{7 e^3} \]
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Rubi [A] time = 0.0210846, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {697} \[ \frac{2 (d+e x)^{5/2} \left (a e^2+c d^2\right )}{5 e^3}+\frac{2 c (d+e x)^{9/2}}{9 e^3}-\frac{4 c d (d+e x)^{7/2}}{7 e^3} \]
Antiderivative was successfully verified.
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Rule 697
Rubi steps
\begin{align*} \int (d+e x)^{3/2} \left (a+c x^2\right ) \, dx &=\int \left (\frac{\left (c d^2+a e^2\right ) (d+e x)^{3/2}}{e^2}-\frac{2 c d (d+e x)^{5/2}}{e^2}+\frac{c (d+e x)^{7/2}}{e^2}\right ) \, dx\\ &=\frac{2 \left (c d^2+a e^2\right ) (d+e x)^{5/2}}{5 e^3}-\frac{4 c d (d+e x)^{7/2}}{7 e^3}+\frac{2 c (d+e x)^{9/2}}{9 e^3}\\ \end{align*}
Mathematica [A] time = 0.0370441, size = 44, normalized size = 0.7 \[ \frac{2 (d+e x)^{5/2} \left (63 a e^2+c \left (8 d^2-20 d e x+35 e^2 x^2\right )\right )}{315 e^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 41, normalized size = 0.7 \begin{align*}{\frac{70\,c{e}^{2}{x}^{2}-40\,cdex+126\,a{e}^{2}+16\,c{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 = 1.19216, size = 63, normalized size = 1. \begin{align*} \frac{2 \,{\left (35 \,{\left (e x + d\right )}^{\frac{9}{2}} c - 90 \,{\left (e x + d\right )}^{\frac{7}{2}} c d + 63 \,{\left (c d^{2} + a 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 [A] time = 1.71276, size = 194, normalized size = 3.08 \begin{align*} \frac{2 \,{\left (35 \, c e^{4} x^{4} + 50 \, c d e^{3} x^{3} + 8 \, c d^{4} + 63 \, a d^{2} e^{2} + 3 \,{\left (c d^{2} e^{2} + 21 \, a e^{4}\right )} x^{2} - 2 \,{\left (2 \, c d^{3} e - 63 \, a 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 = 5.60879, size = 155, normalized size = 2.46 \begin{align*} a 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 \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{2 c 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 c \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 = 1.41944, size = 182, normalized size = 2.89 \begin{align*} \frac{2}{315} \,{\left (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 )} c d e^{\left (-2\right )} + 105 \,{\left (x e + d\right )}^{\frac{3}{2}} a d +{\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 )} c 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\right )} e^{\left (-1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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