Optimal. Leaf size=23 \[ \frac{2 (d (a+b x)+c)^{3/2}}{3 b d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0108698, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {33, 32} \[ \frac{2 (d (a+b x)+c)^{3/2}}{3 b d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 33
Rule 32
Rubi steps
\begin{align*} \int \sqrt{c+d (a+b x)} \, dx &=\frac{\operatorname{Subst}\left (\int \sqrt{c+d x} \, dx,x,a+b x\right )}{b}\\ &=\frac{2 (c+d (a+b x))^{3/2}}{3 b d}\\ \end{align*}
Mathematica [A] time = 0.0154463, size = 23, normalized size = 1. \[ \frac{2 (d (a+b x)+c)^{3/2}}{3 b d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.003, size = 20, normalized size = 0.9 \begin{align*}{\frac{2}{3\,bd} \left ( bdx+ad+c \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.03498, size = 26, normalized size = 1.13 \begin{align*} \frac{2 \,{\left ({\left (b x + a\right )} d + c\right )}^{\frac{3}{2}}}{3 \, b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.57262, size = 47, normalized size = 2.04 \begin{align*} \frac{2 \,{\left (b d x + a d + c\right )}^{\frac{3}{2}}}{3 \, b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.383804, size = 78, normalized size = 3.39 \begin{align*} \begin{cases} \sqrt{c} x & \text{for}\: d = 0 \wedge \left (b = 0 \vee d = 0\right ) \\x \sqrt{a d + c} & \text{for}\: b = 0 \\\frac{2 a \sqrt{a d + b d x + c}}{3 b} + \frac{2 x \sqrt{a d + b d x + c}}{3} + \frac{2 c \sqrt{a d + b d x + c}}{3 b d} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.15791, size = 26, normalized size = 1.13 \begin{align*} \frac{2 \,{\left (b d x + a d + c\right )}^{\frac{3}{2}}}{3 \, b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]