Optimal. Leaf size=41 \[ a^2 x+\frac{4 a b (c+d x)^{3/2}}{3 d}+\frac{b^2 (c+d x)^2}{2 d} \]
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Rubi [A] time = 0.0303472, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {247, 190, 43} \[ a^2 x+\frac{4 a b (c+d x)^{3/2}}{3 d}+\frac{b^2 (c+d x)^2}{2 d} \]
Antiderivative was successfully verified.
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Rule 247
Rule 190
Rule 43
Rubi steps
\begin{align*} \int \left (a+b \sqrt{c+d x}\right )^2 \, dx &=\frac{\operatorname{Subst}\left (\int \left (a+b \sqrt{x}\right )^2 \, dx,x,c+d x\right )}{d}\\ &=\frac{2 \operatorname{Subst}\left (\int x (a+b x)^2 \, dx,x,\sqrt{c+d x}\right )}{d}\\ &=\frac{2 \operatorname{Subst}\left (\int \left (a^2 x+2 a b x^2+b^2 x^3\right ) \, dx,x,\sqrt{c+d x}\right )}{d}\\ &=a^2 x+\frac{4 a b (c+d x)^{3/2}}{3 d}+\frac{b^2 (c+d x)^2}{2 d}\\ \end{align*}
Mathematica [A] time = 0.0270579, size = 40, normalized size = 0.98 \[ \frac{6 a^2 d x+8 a b (c+d x)^{3/2}+3 b^2 (c+d x)^2}{6 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 35, normalized size = 0.9 \begin{align*}{b}^{2} \left ({\frac{d{x}^{2}}{2}}+cx \right ) +{\frac{4\,ab}{3\,d} \left ( dx+c \right ) ^{{\frac{3}{2}}}}+x{a}^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.1046, size = 47, normalized size = 1.15 \begin{align*} \frac{1}{2} \,{\left (d x^{2} + 2 \, c x\right )} b^{2} + a^{2} x + \frac{4 \,{\left (d x + c\right )}^{\frac{3}{2}} a b}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.93548, size = 109, normalized size = 2.66 \begin{align*} \frac{3 \, b^{2} d^{2} x^{2} + 6 \,{\left (b^{2} c + a^{2}\right )} d x + 8 \,{\left (a b d x + a b c\right )} \sqrt{d x + c}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.190655, size = 68, normalized size = 1.66 \begin{align*} \begin{cases} a^{2} x + \frac{4 a b c \sqrt{c + d x}}{3 d} + \frac{4 a b x \sqrt{c + d x}}{3} + b^{2} c x + \frac{b^{2} d x^{2}}{2} & \text{for}\: d \neq 0 \\x \left (a + b \sqrt{c}\right )^{2} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23808, size = 53, normalized size = 1.29 \begin{align*} \frac{3 \,{\left (d x + c\right )}^{2} b^{2} + 8 \,{\left (d x + c\right )}^{\frac{3}{2}} a b + 6 \,{\left (d x + c\right )} a^{2}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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