Optimal. Leaf size=208 \[ \frac{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2}}{9 e^4 (a+b x)}-\frac{6 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)}{7 e^4 (a+b x)}+\frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^2}{5 e^4 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^3}{3 e^4 (a+b x)} \]
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Rubi [A] time = 0.0664567, antiderivative size = 208, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {646, 43} \[ \frac{2 b^3 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{9/2}}{9 e^4 (a+b x)}-\frac{6 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)}{7 e^4 (a+b x)}+\frac{6 b \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^2}{5 e^4 (a+b x)}-\frac{2 \sqrt{a^2+2 a b x+b^2 x^2} (d+e x)^{3/2} (b d-a e)^3}{3 e^4 (a+b x)} \]
Antiderivative was successfully verified.
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Rule 646
Rule 43
Rubi steps
\begin{align*} \int \sqrt{d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (a b+b^2 x\right )^3 \sqrt{d+e x} \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac{\sqrt{a^2+2 a b x+b^2 x^2} \int \left (-\frac{b^3 (b d-a e)^3 \sqrt{d+e x}}{e^3}+\frac{3 b^4 (b d-a e)^2 (d+e x)^{3/2}}{e^3}-\frac{3 b^5 (b d-a e) (d+e x)^{5/2}}{e^3}+\frac{b^6 (d+e x)^{7/2}}{e^3}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=-\frac{2 (b d-a e)^3 (d+e x)^{3/2} \sqrt{a^2+2 a b x+b^2 x^2}}{3 e^4 (a+b x)}+\frac{6 b (b d-a e)^2 (d+e x)^{5/2} \sqrt{a^2+2 a b x+b^2 x^2}}{5 e^4 (a+b x)}-\frac{6 b^2 (b d-a e) (d+e x)^{7/2} \sqrt{a^2+2 a b x+b^2 x^2}}{7 e^4 (a+b x)}+\frac{2 b^3 (d+e x)^{9/2} \sqrt{a^2+2 a b x+b^2 x^2}}{9 e^4 (a+b x)}\\ \end{align*}
Mathematica [A] time = 0.0685035, size = 120, normalized size = 0.58 \[ \frac{2 \sqrt{(a+b x)^2} (d+e x)^{3/2} \left (63 a^2 b e^2 (3 e x-2 d)+105 a^3 e^3+9 a b^2 e \left (8 d^2-12 d e x+15 e^2 x^2\right )+b^3 \left (24 d^2 e x-16 d^3-30 d e^2 x^2+35 e^3 x^3\right )\right )}{315 e^4 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.184, size = 132, normalized size = 0.6 \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\,{a}^{3}{e}^{3}-252\,d{e}^{2}{a}^{2}b+144\,a{b}^{2}{d}^{2}e-32\,{b}^{3}{d}^{3}}{315\,{e}^{4} \left ( bx+a \right ) ^{3}} \left ( ex+d \right ) ^{{\frac{3}{2}}} \left ( \left ( bx+a \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11669, size = 221, normalized size = 1.06 \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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Fricas [A] time = 1.55473, size = 359, normalized size = 1.73 \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{d + e x} \left (\left (a + b x\right )^{2}\right )^{\frac{3}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20514, size = 228, normalized size = 1.1 \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 )} \mathrm{sgn}\left (b x + a\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 )} \mathrm{sgn}\left (b x + a\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 )} \mathrm{sgn}\left (b x + a\right ) + 105 \,{\left (x e + d\right )}^{\frac{3}{2}} a^{3} \mathrm{sgn}\left (b x + a\right )\right )} e^{\left (-1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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