3.1380 \(\int \sqrt{c+d x} \, dx\)

Optimal. Leaf size=16 \[ \frac{2 (c+d x)^{3/2}}{3 d} \]

[Out]

(2*(c + d*x)^(3/2))/(3*d)

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Rubi [A]  time = 0.0015467, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {32} \[ \frac{2 (c+d x)^{3/2}}{3 d} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[c + d*x],x]

[Out]

(2*(c + d*x)^(3/2))/(3*d)

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin{align*} \int \sqrt{c+d x} \, dx &=\frac{2 (c+d x)^{3/2}}{3 d}\\ \end{align*}

Mathematica [A]  time = 0.0036568, size = 16, normalized size = 1. \[ \frac{2 (c+d x)^{3/2}}{3 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[c + d*x],x]

[Out]

(2*(c + d*x)^(3/2))/(3*d)

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Maple [A]  time = 0.001, size = 13, normalized size = 0.8 \begin{align*}{\frac{2}{3\,d} \left ( dx+c \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^(1/2),x)

[Out]

2/3*(d*x+c)^(3/2)/d

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Maxima [A]  time = 0.960383, size = 16, normalized size = 1. \begin{align*} \frac{2 \,{\left (d x + c\right )}^{\frac{3}{2}}}{3 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(1/2),x, algorithm="maxima")

[Out]

2/3*(d*x + c)^(3/2)/d

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Fricas [A]  time = 1.94815, size = 31, normalized size = 1.94 \begin{align*} \frac{2 \,{\left (d x + c\right )}^{\frac{3}{2}}}{3 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(1/2),x, algorithm="fricas")

[Out]

2/3*(d*x + c)^(3/2)/d

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Sympy [A]  time = 0.052876, size = 12, normalized size = 0.75 \begin{align*} \frac{2 \left (c + d x\right )^{\frac{3}{2}}}{3 d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**(1/2),x)

[Out]

2*(c + d*x)**(3/2)/(3*d)

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Giac [A]  time = 1.09003, size = 16, normalized size = 1. \begin{align*} \frac{2 \,{\left (d x + c\right )}^{\frac{3}{2}}}{3 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(1/2),x, algorithm="giac")

[Out]

2/3*(d*x + c)^(3/2)/d