\(\int \sqrt {a+b x} \, dx\) [287]

Optimal result

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

[Out]

2/3*(b*x+a)^(3/2)/b

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} \[ \int \sqrt {a+b x} \, dx=\frac {2 (a+b x)^{3/2}}{3 b} \]

[In]

Int[Sqrt[a + b*x],x]

[Out]

(2*(a + b*x)^(3/2))/(3*b)

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 (a+b x)^{3/2}}{3 b} \\ \end{align*}

Mathematica [A] (verified)

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

[In]

Integrate[Sqrt[a + b*x],x]

[Out]

(2*(a + b*x)^(3/2))/(3*b)

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81

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

[In]

int((b*x+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/3*(b*x+a)^(3/2)/b

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \sqrt {a+b x} \, dx=\frac {2 \, {\left (b x + a\right )}^{\frac {3}{2}}}{3 \, b} \]

[In]

integrate((b*x+a)^(1/2),x, algorithm="fricas")

[Out]

2/3*(b*x + a)^(3/2)/b

Sympy [A] (verification not implemented)

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

[In]

integrate((b*x+a)**(1/2),x)

[Out]

2*(a + b*x)**(3/2)/(3*b)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \sqrt {a+b x} \, dx=\frac {2 \, {\left (b x + a\right )}^{\frac {3}{2}}}{3 \, b} \]

[In]

integrate((b*x+a)^(1/2),x, algorithm="maxima")

[Out]

2/3*(b*x + a)^(3/2)/b

Giac [A] (verification not implemented)

none

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

[In]

integrate((b*x+a)^(1/2),x, algorithm="giac")

[Out]

2/3*(b*x + a)^(3/2)/b

Mupad [B] (verification not implemented)

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

[In]

int((a + b*x)^(1/2),x)

[Out]

(2*(a + b*x)^(3/2))/(3*b)