Optimal. Leaf size=100 \[ -\frac{6 b^2 (d+e x)^{7/2} (b d-a e)}{7 e^4}+\frac{6 b (d+e x)^{5/2} (b d-a e)^2}{5 e^4}-\frac{2 (d+e x)^{3/2} (b d-a e)^3}{3 e^4}+\frac{2 b^3 (d+e x)^{9/2}}{9 e^4} \]
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Rubi [A] time = 0.032099, antiderivative size = 100, 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} \[ -\frac{6 b^2 (d+e x)^{7/2} (b d-a e)}{7 e^4}+\frac{6 b (d+e x)^{5/2} (b d-a e)^2}{5 e^4}-\frac{2 (d+e x)^{3/2} (b d-a e)^3}{3 e^4}+\frac{2 b^3 (d+e x)^{9/2}}{9 e^4} \]
Antiderivative was successfully verified.
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Rule 27
Rule 43
Rubi steps
\begin{align*} \int (a+b x) \sqrt{d+e x} \left (a^2+2 a b x+b^2 x^2\right ) \, dx &=\int (a+b x)^3 \sqrt{d+e x} \, dx\\ &=\int \left (\frac{(-b d+a e)^3 \sqrt{d+e x}}{e^3}+\frac{3 b (b d-a e)^2 (d+e x)^{3/2}}{e^3}-\frac{3 b^2 (b d-a e) (d+e x)^{5/2}}{e^3}+\frac{b^3 (d+e x)^{7/2}}{e^3}\right ) \, dx\\ &=-\frac{2 (b d-a e)^3 (d+e x)^{3/2}}{3 e^4}+\frac{6 b (b d-a e)^2 (d+e x)^{5/2}}{5 e^4}-\frac{6 b^2 (b d-a e) (d+e x)^{7/2}}{7 e^4}+\frac{2 b^3 (d+e x)^{9/2}}{9 e^4}\\ \end{align*}
Mathematica [A] time = 0.0568004, size = 79, normalized size = 0.79 \[ \frac{2 (d+e x)^{3/2} \left (-135 b^2 (d+e x)^2 (b d-a e)+189 b (d+e x) (b d-a e)^2-105 (b d-a e)^3+35 b^3 (d+e x)^3\right )}{315 e^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 116, normalized size = 1.2 \begin{align*}{\frac{70\,{x}^{3}{b}^{3}{e}^{3}+270\,{x}^{2}a{b}^{2}{e}^{3}-60\,{x}^{2}{b}^{3}d{e}^{2}+378\,x{a}^{2}b{e}^{3}-216\,xa{b}^{2}d{e}^{2}+48\,x{b}^{3}{d}^{2}e+210\,{e}^{3}{a}^{3}-252\,d{e}^{2}{a}^{2}b+144\,a{d}^{2}e{b}^{2}-32\,{d}^{3}{b}^{3}}{315\,{e}^{4}} \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 = 0.962159, size = 159, normalized size = 1.59 \begin{align*} \frac{2 \,{\left (35 \,{\left (e x + d\right )}^{\frac{9}{2}} b^{3} - 135 \,{\left (b^{3} d - a b^{2} e\right )}{\left (e x + d\right )}^{\frac{7}{2}} + 189 \,{\left (b^{3} d^{2} - 2 \, a b^{2} d e + a^{2} b e^{2}\right )}{\left (e x + d\right )}^{\frac{5}{2}} - 105 \,{\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )}{\left (e x + d\right )}^{\frac{3}{2}}\right )}}{315 \, e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.42544, size = 359, normalized size = 3.59 \begin{align*} \frac{2 \,{\left (35 \, b^{3} e^{4} x^{4} - 16 \, b^{3} d^{4} + 72 \, a b^{2} d^{3} e - 126 \, a^{2} b d^{2} e^{2} + 105 \, a^{3} d e^{3} + 5 \,{\left (b^{3} d e^{3} + 27 \, a b^{2} e^{4}\right )} x^{3} - 3 \,{\left (2 \, b^{3} d^{2} e^{2} - 9 \, a b^{2} d e^{3} - 63 \, a^{2} b e^{4}\right )} x^{2} +{\left (8 \, b^{3} d^{3} e - 36 \, a b^{2} d^{2} e^{2} + 63 \, a^{2} b d e^{3} + 105 \, a^{3} e^{4}\right )} x\right )} \sqrt{e x + d}}{315 \, e^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.33208, size = 146, normalized size = 1.46 \begin{align*} \frac{2 \left (\frac{b^{3} \left (d + e x\right )^{\frac{9}{2}}}{9 e^{3}} + \frac{\left (d + e x\right )^{\frac{7}{2}} \left (3 a b^{2} e - 3 b^{3} d\right )}{7 e^{3}} + \frac{\left (d + e x\right )^{\frac{5}{2}} \left (3 a^{2} b e^{2} - 6 a b^{2} d e + 3 b^{3} d^{2}\right )}{5 e^{3}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \left (a^{3} e^{3} - 3 a^{2} b d e^{2} + 3 a b^{2} d^{2} e - b^{3} d^{3}\right )}{3 e^{3}}\right )}{e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11792, size = 196, normalized size = 1.96 \begin{align*} \frac{2}{315} \,{\left (63 \,{\left (3 \,{\left (x e + d\right )}^{\frac{5}{2}} - 5 \,{\left (x e + d\right )}^{\frac{3}{2}} d\right )} a^{2} b e^{\left (-1\right )} + 9 \,{\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 )} a b^{2} e^{\left (-2\right )} +{\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 )} b^{3} e^{\left (-3\right )} + 105 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{3}\right )} e^{\left (-1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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