Optimal. Leaf size=69 \[ -\frac{4 b (d+e x)^{3/2} (b d-a e)}{3 e^3}+\frac{2 \sqrt{d+e x} (b d-a e)^2}{e^3}+\frac{2 b^2 (d+e x)^{5/2}}{5 e^3} \]
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Rubi [A] time = 0.022423, antiderivative size = 69, 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} \[ -\frac{4 b (d+e x)^{3/2} (b d-a e)}{3 e^3}+\frac{2 \sqrt{d+e x} (b d-a e)^2}{e^3}+\frac{2 b^2 (d+e x)^{5/2}}{5 e^3} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin{align*} \int \frac{a^2+2 a b x+b^2 x^2}{\sqrt{d+e x}} \, dx &=\int \frac{(a+b x)^2}{\sqrt{d+e x}} \, dx\\ &=\int \left (\frac{(-b d+a e)^2}{e^2 \sqrt{d+e x}}-\frac{2 b (b d-a e) \sqrt{d+e x}}{e^2}+\frac{b^2 (d+e x)^{3/2}}{e^2}\right ) \, dx\\ &=\frac{2 (b d-a e)^2 \sqrt{d+e x}}{e^3}-\frac{4 b (b d-a e) (d+e x)^{3/2}}{3 e^3}+\frac{2 b^2 (d+e x)^{5/2}}{5 e^3}\\ \end{align*}
Mathematica [A] time = 0.0325503, size = 60, normalized size = 0.87 \[ \frac{2 \sqrt{d+e x} \left (15 a^2 e^2+10 a b e (e x-2 d)+b^2 \left (8 d^2-4 d e x+3 e^2 x^2\right )\right )}{15 e^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 63, normalized size = 0.9 \begin{align*}{\frac{6\,{b}^{2}{x}^{2}{e}^{2}+20\,xab{e}^{2}-8\,x{b}^{2}de+30\,{a}^{2}{e}^{2}-40\,abde+16\,{b}^{2}{d}^{2}}{15\,{e}^{3}}\sqrt{ex+d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05642, size = 111, normalized size = 1.61 \begin{align*} \frac{2 \,{\left (15 \, \sqrt{e x + d} a^{2} + \frac{10 \,{\left ({\left (e x + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{e x + d} d\right )} a b}{e} + \frac{{\left (3 \,{\left (e x + d\right )}^{\frac{5}{2}} - 10 \,{\left (e x + d\right )}^{\frac{3}{2}} d + 15 \, \sqrt{e x + d} d^{2}\right )} b^{2}}{e^{2}}\right )}}{15 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.5299, size = 146, normalized size = 2.12 \begin{align*} \frac{2 \,{\left (3 \, b^{2} e^{2} x^{2} + 8 \, b^{2} d^{2} - 20 \, a b d e + 15 \, a^{2} e^{2} - 2 \,{\left (2 \, b^{2} d e - 5 \, a b e^{2}\right )} x\right )} \sqrt{e x + d}}{15 \, e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 9.33131, size = 236, normalized size = 3.42 \begin{align*} \begin{cases} - \frac{\frac{2 a^{2} d}{\sqrt{d + e x}} + 2 a^{2} \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right ) + \frac{4 a b d \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right )}{e} + \frac{4 a b \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right )}{e} + \frac{2 b^{2} d \left (\frac{d^{2}}{\sqrt{d + e x}} + 2 d \sqrt{d + e x} - \frac{\left (d + e x\right )^{\frac{3}{2}}}{3}\right )}{e^{2}} + \frac{2 b^{2} \left (- \frac{d^{3}}{\sqrt{d + e x}} - 3 d^{2} \sqrt{d + e x} + d \left (d + e x\right )^{\frac{3}{2}} - \frac{\left (d + e x\right )^{\frac{5}{2}}}{5}\right )}{e^{2}}}{e} & \text{for}\: e \neq 0 \\\frac{a^{2} x + a b x^{2} + \frac{b^{2} x^{3}}{3}}{\sqrt{d}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15855, size = 115, normalized size = 1.67 \begin{align*} \frac{2}{15} \,{\left (10 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} a b e^{\left (-1\right )} +{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 10 \,{\left (x e + d\right )}^{\frac{3}{2}} d + 15 \, \sqrt{x e + d} d^{2}\right )} b^{2} e^{\left (-2\right )} + 15 \, \sqrt{x e + d} a^{2}\right )} e^{\left (-1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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