Optimal. Leaf size=94 \[ -\frac{2 b^2 (d+e x)^{3/2} (b d-a e)}{e^4}+\frac{6 b \sqrt{d+e x} (b d-a e)^2}{e^4}+\frac{2 (b d-a e)^3}{e^4 \sqrt{d+e x}}+\frac{2 b^3 (d+e x)^{5/2}}{5 e^4} \]
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Rubi [A] time = 0.0331072, antiderivative size = 94, 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} \[ -\frac{2 b^2 (d+e x)^{3/2} (b d-a e)}{e^4}+\frac{6 b \sqrt{d+e x} (b d-a e)^2}{e^4}+\frac{2 (b d-a e)^3}{e^4 \sqrt{d+e x}}+\frac{2 b^3 (d+e x)^{5/2}}{5 e^4} \]
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 )}{(d+e x)^{3/2}} \, dx &=\int \frac{(a+b x)^3}{(d+e x)^{3/2}} \, dx\\ &=\int \left (\frac{(-b d+a e)^3}{e^3 (d+e x)^{3/2}}+\frac{3 b (b d-a e)^2}{e^3 \sqrt{d+e x}}-\frac{3 b^2 (b d-a e) \sqrt{d+e x}}{e^3}+\frac{b^3 (d+e x)^{3/2}}{e^3}\right ) \, dx\\ &=\frac{2 (b d-a e)^3}{e^4 \sqrt{d+e x}}+\frac{6 b (b d-a e)^2 \sqrt{d+e x}}{e^4}-\frac{2 b^2 (b d-a e) (d+e x)^{3/2}}{e^4}+\frac{2 b^3 (d+e x)^{5/2}}{5 e^4}\\ \end{align*}
Mathematica [A] time = 0.0499887, size = 78, normalized size = 0.83 \[ \frac{2 \left (-5 b^2 (d+e x)^2 (b d-a e)+15 b (d+e x) (b d-a e)^2+5 (b d-a e)^3+b^3 (d+e x)^3\right )}{5 e^4 \sqrt{d+e x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 116, normalized size = 1.2 \begin{align*} -{\frac{-2\,{x}^{3}{b}^{3}{e}^{3}-10\,{x}^{2}a{b}^{2}{e}^{3}+4\,{x}^{2}{b}^{3}d{e}^{2}-30\,x{a}^{2}b{e}^{3}+40\,xa{b}^{2}d{e}^{2}-16\,x{b}^{3}{d}^{2}e+10\,{e}^{3}{a}^{3}-60\,d{e}^{2}{a}^{2}b+80\,a{d}^{2}e{b}^{2}-32\,{d}^{3}{b}^{3}}{5\,{e}^{4}}{\frac{1}{\sqrt{ex+d}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.975297, size = 169, normalized size = 1.8 \begin{align*} \frac{2 \,{\left (\frac{{\left (e x + d\right )}^{\frac{5}{2}} b^{3} - 5 \,{\left (b^{3} d - a b^{2} e\right )}{\left (e x + d\right )}^{\frac{3}{2}} + 15 \,{\left (b^{3} d^{2} - 2 \, a b^{2} d e + a^{2} b e^{2}\right )} \sqrt{e x + d}}{e^{3}} + \frac{5 \,{\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} e^{3}}\right )}}{5 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.30665, size = 259, normalized size = 2.76 \begin{align*} \frac{2 \,{\left (b^{3} e^{3} x^{3} + 16 \, b^{3} d^{3} - 40 \, a b^{2} d^{2} e + 30 \, a^{2} b d e^{2} - 5 \, a^{3} e^{3} -{\left (2 \, b^{3} d e^{2} - 5 \, a b^{2} e^{3}\right )} x^{2} +{\left (8 \, b^{3} d^{2} e - 20 \, a b^{2} d e^{2} + 15 \, a^{2} b e^{3}\right )} x\right )} \sqrt{e x + d}}{5 \,{\left (e^{5} x + d e^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 15.3016, size = 109, normalized size = 1.16 \begin{align*} \frac{2 b^{3} \left (d + e x\right )^{\frac{5}{2}}}{5 e^{4}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (6 a b^{2} e - 6 b^{3} d\right )}{3 e^{4}} + \frac{\sqrt{d + e x} \left (6 a^{2} b e^{2} - 12 a b^{2} d e + 6 b^{3} d^{2}\right )}{e^{4}} - \frac{2 \left (a e - b d\right )^{3}}{e^{4} \sqrt{d + e x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1434, size = 203, normalized size = 2.16 \begin{align*} \frac{2}{5} \,{\left ({\left (x e + d\right )}^{\frac{5}{2}} b^{3} e^{16} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} b^{3} d e^{16} + 15 \, \sqrt{x e + d} b^{3} d^{2} e^{16} + 5 \,{\left (x e + d\right )}^{\frac{3}{2}} a b^{2} e^{17} - 30 \, \sqrt{x e + d} a b^{2} d e^{17} + 15 \, \sqrt{x e + d} a^{2} b e^{18}\right )} e^{\left (-20\right )} + \frac{2 \,{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} e^{\left (-4\right )}}{\sqrt{x e + d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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