Optimal. Leaf size=69 \[ \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2} (b d-a e)}{4 b^2}+\frac{e \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{5 b^2} \]
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Rubi [A] time = 0.0211858, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {640, 609} \[ \frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2} (b d-a e)}{4 b^2}+\frac{e \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{5 b^2} \]
Antiderivative was successfully verified.
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Rule 640
Rule 609
Rubi steps
\begin{align*} \int (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx &=\frac{e \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{5 b^2}+\frac{\left (2 b^2 d-2 a b e\right ) \int \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx}{2 b^2}\\ &=\frac{(b d-a e) (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{4 b^2}+\frac{e \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{5 b^2}\\ \end{align*}
Mathematica [A] time = 0.032854, size = 83, normalized size = 1.2 \[ \frac{x \sqrt{(a+b x)^2} \left (10 a^2 b x (3 d+2 e x)+10 a^3 (2 d+e x)+5 a b^2 x^2 (4 d+3 e x)+b^3 x^3 (5 d+4 e x)\right )}{20 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.178, size = 90, normalized size = 1.3 \begin{align*}{\frac{x \left ( 4\,e{b}^{3}{x}^{4}+15\,{x}^{3}e{b}^{2}a+5\,{x}^{3}d{b}^{3}+20\,{a}^{2}be{x}^{2}+20\,a{b}^{2}d{x}^{2}+10\,{a}^{3}ex+30\,xdb{a}^{2}+20\,d{a}^{3} \right ) }{20\, \left ( bx+a \right ) ^{3}} \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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52932, size = 150, normalized size = 2.17 \begin{align*} \frac{1}{5} \, b^{3} e x^{5} + a^{3} d x + \frac{1}{4} \,{\left (b^{3} d + 3 \, a b^{2} e\right )} x^{4} +{\left (a b^{2} d + a^{2} b e\right )} x^{3} + \frac{1}{2} \,{\left (3 \, a^{2} b d + a^{3} e\right )} x^{2} \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 \left (d + e x\right ) \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 [B] time = 1.13498, size = 167, normalized size = 2.42 \begin{align*} \frac{1}{5} \, b^{3} x^{5} e \mathrm{sgn}\left (b x + a\right ) + \frac{1}{4} \, b^{3} d x^{4} \mathrm{sgn}\left (b x + a\right ) + \frac{3}{4} \, a b^{2} x^{4} e \mathrm{sgn}\left (b x + a\right ) + a b^{2} d x^{3} \mathrm{sgn}\left (b x + a\right ) + a^{2} b x^{3} e \mathrm{sgn}\left (b x + a\right ) + \frac{3}{2} \, a^{2} b d x^{2} \mathrm{sgn}\left (b x + a\right ) + \frac{1}{2} \, a^{3} x^{2} e \mathrm{sgn}\left (b x + a\right ) + a^{3} d x \mathrm{sgn}\left (b x + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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