Optimal. Leaf size=71 \[ -\frac{4 b (d+e x)^{5/2} (b d-a e)}{5 e^3}+\frac{2 (d+e x)^{3/2} (b d-a e)^2}{3 e^3}+\frac{2 b^2 (d+e x)^{7/2}}{7 e^3} \]
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Rubi [A] time = 0.0225009, 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} \[ -\frac{4 b (d+e x)^{5/2} (b d-a e)}{5 e^3}+\frac{2 (d+e x)^{3/2} (b d-a e)^2}{3 e^3}+\frac{2 b^2 (d+e x)^{7/2}}{7 e^3} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin{align*} \int \sqrt{d+e x} \left (a^2+2 a b x+b^2 x^2\right ) \, dx &=\int (a+b x)^2 \sqrt{d+e x} \, dx\\ &=\int \left (\frac{(-b d+a e)^2 \sqrt{d+e x}}{e^2}-\frac{2 b (b d-a e) (d+e x)^{3/2}}{e^2}+\frac{b^2 (d+e x)^{5/2}}{e^2}\right ) \, dx\\ &=\frac{2 (b d-a e)^2 (d+e x)^{3/2}}{3 e^3}-\frac{4 b (b d-a e) (d+e x)^{5/2}}{5 e^3}+\frac{2 b^2 (d+e x)^{7/2}}{7 e^3}\\ \end{align*}
Mathematica [A] time = 0.0343695, size = 61, normalized size = 0.86 \[ \frac{2 (d+e x)^{3/2} \left (35 a^2 e^2+14 a b e (3 e x-2 d)+b^2 \left (8 d^2-12 d e x+15 e^2 x^2\right )\right )}{105 e^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 63, normalized size = 0.9 \begin{align*}{\frac{30\,{b}^{2}{x}^{2}{e}^{2}+84\,xab{e}^{2}-24\,x{b}^{2}de+70\,{a}^{2}{e}^{2}-56\,abde+16\,{b}^{2}{d}^{2}}{105\,{e}^{3}} \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 = 1.08315, size = 92, normalized size = 1.3 \begin{align*} \frac{2 \,{\left (15 \,{\left (e x + d\right )}^{\frac{7}{2}} b^{2} - 42 \,{\left (b^{2} d - a b e\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 35 \,{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )}{\left (e x + d\right )}^{\frac{3}{2}}\right )}}{105 \, e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56796, size = 220, normalized size = 3.1 \begin{align*} \frac{2 \,{\left (15 \, b^{2} e^{3} x^{3} + 8 \, b^{2} d^{3} - 28 \, a b d^{2} e + 35 \, a^{2} d e^{2} + 3 \,{\left (b^{2} d e^{2} + 14 \, a b e^{3}\right )} x^{2} -{\left (4 \, b^{2} d^{2} e - 14 \, a b d e^{2} - 35 \, a^{2} e^{3}\right )} x\right )} \sqrt{e x + d}}{105 \, e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.83358, size = 85, normalized size = 1.2 \begin{align*} \frac{2 \left (\frac{b^{2} \left (d + e x\right )^{\frac{7}{2}}}{7 e^{2}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (2 a b e - 2 b^{2} d\right )}{5 e^{2}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (a^{2} e^{2} - 2 a b d e + b^{2} d^{2}\right )}{3 e^{2}}\right )}{e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18315, size = 117, normalized size = 1.65 \begin{align*} \frac{2}{105} \,{\left (14 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} a b e^{\left (-1\right )} +{\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} e^{\left (-2\right )} + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{2}\right )} e^{\left (-1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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