\(\int \sqrt {x} (a+b x) \, dx\) [431]

Optimal result

Integrand size = 11, antiderivative size = 21 \[ \int \sqrt {x} (a+b x) \, dx=\frac {2}{3} a x^{3/2}+\frac {2}{5} b x^{5/2} \]

[Out]

2/3*a*x^(3/2)+2/5*b*x^(5/2)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45} \[ \int \sqrt {x} (a+b x) \, dx=\frac {2}{3} a x^{3/2}+\frac {2}{5} b x^{5/2} \]

[In]

Int[Sqrt[x]*(a + b*x),x]

[Out]

(2*a*x^(3/2))/3 + (2*b*x^(5/2))/5

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps \begin{align*} \text {integral}& = \int \left (a \sqrt {x}+b x^{3/2}\right ) \, dx \\ & = \frac {2}{3} a x^{3/2}+\frac {2}{5} b x^{5/2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \sqrt {x} (a+b x) \, dx=\frac {2}{15} x^{3/2} (5 a+3 b x) \]

[In]

Integrate[Sqrt[x]*(a + b*x),x]

[Out]

(2*x^(3/2)*(5*a + 3*b*x))/15

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.67

method	result	size
gosper	\(\frac {2 x^{\frac {3}{2}} \left (3 b x +5 a \right )}{15}\)	\(14\)
derivativedivides	\(\frac {2 a \,x^{\frac {3}{2}}}{3}+\frac {2 b \,x^{\frac {5}{2}}}{5}\)	\(14\)
default	\(\frac {2 a \,x^{\frac {3}{2}}}{3}+\frac {2 b \,x^{\frac {5}{2}}}{5}\)	\(14\)
trager	\(\frac {2 x^{\frac {3}{2}} \left (3 b x +5 a \right )}{15}\)	\(14\)
risch	\(\frac {2 x^{\frac {3}{2}} \left (3 b x +5 a \right )}{15}\)	\(14\)

[In]

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

[Out]

2/15*x^(3/2)*(3*b*x+5*a)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.76 \[ \int \sqrt {x} (a+b x) \, dx=\frac {2}{15} \, {\left (3 \, b x^{2} + 5 \, a x\right )} \sqrt {x} \]

[In]

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

[Out]

2/15*(3*b*x^2 + 5*a*x)*sqrt(x)

Sympy [A] (verification not implemented)

Time = 0.87 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \sqrt {x} (a+b x) \, dx=\frac {2 a x^{\frac {3}{2}}}{3} + \frac {2 b x^{\frac {5}{2}}}{5} \]

[In]

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

[Out]

2*a*x**(3/2)/3 + 2*b*x**(5/2)/5

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.62 \[ \int \sqrt {x} (a+b x) \, dx=\frac {2}{5} \, b x^{\frac {5}{2}} + \frac {2}{3} \, a x^{\frac {3}{2}} \]

[In]

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

[Out]

2/5*b*x^(5/2) + 2/3*a*x^(3/2)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.62 \[ \int \sqrt {x} (a+b x) \, dx=\frac {2}{5} \, b x^{\frac {5}{2}} + \frac {2}{3} \, a x^{\frac {3}{2}} \]

[In]

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

[Out]

2/5*b*x^(5/2) + 2/3*a*x^(3/2)

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.62 \[ \int \sqrt {x} (a+b x) \, dx=\frac {2\,x^{3/2}\,\left (5\,a+3\,b\,x\right )}{15} \]

[In]

int(x^(1/2)*(a + b*x),x)

[Out]

(2*x^(3/2)*(5*a + 3*b*x))/15