\(\int \sqrt {c+d x} \, dx\) [1380]

Optimal result

Integrand size = 9, antiderivative size = 16 \[ \int \sqrt {c+d x} \, dx=\frac {2 (c+d x)^{3/2}}{3 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {32} \[ \int \sqrt {c+d x} \, dx=\frac {2 (c+d x)^{3/2}}{3 d} \]

[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*} \text {integral}& = \frac {2 (c+d x)^{3/2}}{3 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \sqrt {c+d x} \, dx=\frac {2 (c+d x)^{3/2}}{3 d} \]

[In]

Integrate[Sqrt[c + d*x],x]

[Out]

(2*(c + d*x)^(3/2))/(3*d)

Maple [A] (verified)

Time = 0.22 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81

method	result	size
gosper	\(\frac {2 \left (d x +c \right )^{\frac {3}{2}}}{3 d}\)	\(13\)
derivativedivides	\(\frac {2 \left (d x +c \right )^{\frac {3}{2}}}{3 d}\)	\(13\)
default	\(\frac {2 \left (d x +c \right )^{\frac {3}{2}}}{3 d}\)	\(13\)
trager	\(\frac {2 \left (d x +c \right )^{\frac {3}{2}}}{3 d}\)	\(13\)
risch	\(\frac {2 \left (d x +c \right )^{\frac {3}{2}}}{3 d}\)	\(13\)
pseudoelliptic	\(\frac {2 \left (d x +c \right )^{\frac {3}{2}}}{3 d}\)	\(13\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \sqrt {c+d x} \, dx=\frac {2 \, {\left (d x + c\right )}^{\frac {3}{2}}}{3 \, d} \]

[In]

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

[Out]

2/3*(d*x + c)^(3/2)/d

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \sqrt {c+d x} \, dx=\frac {2 \left (c + d x\right )^{\frac {3}{2}}}{3 d} \]

[In]

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

[Out]

2*(c + d*x)**(3/2)/(3*d)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \sqrt {c+d x} \, dx=\frac {2 \, {\left (d x + c\right )}^{\frac {3}{2}}}{3 \, d} \]

[In]

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

[Out]

2/3*(d*x + c)^(3/2)/d

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \sqrt {c+d x} \, dx=\frac {2 \, {\left (d x + c\right )}^{\frac {3}{2}}}{3 \, d} \]

[In]

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

[Out]

2/3*(d*x + c)^(3/2)/d

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \sqrt {c+d x} \, dx=\frac {2\,{\left (c+d\,x\right )}^{3/2}}{3\,d} \]

[In]

int((c + d*x)^(1/2),x)

[Out]

(2*(c + d*x)^(3/2))/(3*d)