Optimal. Leaf size=96 \[ -\frac{6 b^2 (d+e x)^{5/2} (b d-a e)}{5 e^4}+\frac{2 b (d+e x)^{3/2} (b d-a e)^2}{e^4}-\frac{2 \sqrt{d+e x} (b d-a e)^3}{e^4}+\frac{2 b^3 (d+e x)^{7/2}}{7 e^4} \]
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Rubi [A] time = 0.0341398, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {27, 43} \[ -\frac{6 b^2 (d+e x)^{5/2} (b d-a e)}{5 e^4}+\frac{2 b (d+e x)^{3/2} (b d-a e)^2}{e^4}-\frac{2 \sqrt{d+e x} (b d-a e)^3}{e^4}+\frac{2 b^3 (d+e x)^{7/2}}{7 e^4} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )}{\sqrt{d+e x}} \, dx &=\int \frac{(a+b x)^3}{\sqrt{d+e x}} \, dx\\ &=\int \left (\frac{(-b d+a e)^3}{e^3 \sqrt{d+e x}}+\frac{3 b (b d-a e)^2 \sqrt{d+e x}}{e^3}-\frac{3 b^2 (b d-a e) (d+e x)^{3/2}}{e^3}+\frac{b^3 (d+e x)^{5/2}}{e^3}\right ) \, dx\\ &=-\frac{2 (b d-a e)^3 \sqrt{d+e x}}{e^4}+\frac{2 b (b d-a e)^2 (d+e x)^{3/2}}{e^4}-\frac{6 b^2 (b d-a e) (d+e x)^{5/2}}{5 e^4}+\frac{2 b^3 (d+e x)^{7/2}}{7 e^4}\\ \end{align*}
Mathematica [A] time = 0.048051, size = 79, normalized size = 0.82 \[ \frac{2 \sqrt{d+e x} \left (-21 b^2 (d+e x)^2 (b d-a e)+35 b (d+e x) (b d-a e)^2-35 (b d-a e)^3+5 b^3 (d+e x)^3\right )}{35 e^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 116, normalized size = 1.2 \begin{align*}{\frac{10\,{x}^{3}{b}^{3}{e}^{3}+42\,{x}^{2}a{b}^{2}{e}^{3}-12\,{x}^{2}{b}^{3}d{e}^{2}+70\,x{a}^{2}b{e}^{3}-56\,xa{b}^{2}d{e}^{2}+16\,x{b}^{3}{d}^{2}e+70\,{e}^{3}{a}^{3}-140\,d{e}^{2}{a}^{2}b+112\,a{d}^{2}e{b}^{2}-32\,{d}^{3}{b}^{3}}{35\,{e}^{4}}\sqrt{ex+d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.975474, size = 159, normalized size = 1.66 \begin{align*} \frac{2 \,{\left (5 \,{\left (e x + d\right )}^{\frac{7}{2}} b^{3} - 21 \,{\left (b^{3} d - a b^{2} e\right )}{\left (e x + d\right )}^{\frac{5}{2}} + 35 \,{\left (b^{3} d^{2} - 2 \, a b^{2} d e + a^{2} b e^{2}\right )}{\left (e x + d\right )}^{\frac{3}{2}} - 35 \,{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \sqrt{e x + d}\right )}}{35 \, e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.32685, size = 251, normalized size = 2.61 \begin{align*} \frac{2 \,{\left (5 \, b^{3} e^{3} x^{3} - 16 \, b^{3} d^{3} + 56 \, a b^{2} d^{2} e - 70 \, a^{2} b d e^{2} + 35 \, a^{3} e^{3} - 3 \,{\left (2 \, b^{3} d e^{2} - 7 \, a b^{2} e^{3}\right )} x^{2} +{\left (8 \, b^{3} d^{2} e - 28 \, a b^{2} d e^{2} + 35 \, a^{2} b e^{3}\right )} x\right )} \sqrt{e x + d}}{35 \, e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 29.2301, size = 394, normalized size = 4.1 \begin{align*} \begin{cases} - \frac{\frac{2 a^{3} d}{\sqrt{d + e x}} + 2 a^{3} \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right ) + \frac{6 a^{2} b d \left (- \frac{d}{\sqrt{d + e x}} - \sqrt{d + e x}\right )}{e} + \frac{6 a^{2} 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{6 a 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{6 a 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}} + \frac{2 b^{3} d \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^{3}} + \frac{2 b^{3} \left (\frac{d^{4}}{\sqrt{d + e x}} + 4 d^{3} \sqrt{d + e x} - 2 d^{2} \left (d + e x\right )^{\frac{3}{2}} + \frac{4 d \left (d + e x\right )^{\frac{5}{2}}}{5} - \frac{\left (d + e x\right )^{\frac{7}{2}}}{7}\right )}{e^{3}}}{e} & \text{for}\: e \neq 0 \\\frac{\begin{cases} a^{3} x & \text{for}\: b = 0 \\\frac{a^{3} b x + \frac{3 a^{2} b^{2} x^{2}}{2} + a b^{3} x^{3} + \frac{b^{4} x^{4}}{4}}{b} & \text{otherwise} \end{cases}}{\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 = 1.12404, size = 193, normalized size = 2.01 \begin{align*} \frac{2}{35} \,{\left (35 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} - 3 \, \sqrt{x e + d} d\right )} a^{2} b e^{\left (-1\right )} + 7 \,{\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 )} a b^{2} e^{\left (-2\right )} +{\left (5 \,{\left (x e + d\right )}^{\frac{7}{2}} - 21 \,{\left (x e + d\right )}^{\frac{5}{2}} d + 35 \,{\left (x e + d\right )}^{\frac{3}{2}} d^{2} - 35 \, \sqrt{x e + d} d^{3}\right )} b^{3} e^{\left (-3\right )} + 35 \, \sqrt{x e + d} a^{3}\right )} e^{\left (-1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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