3.2114 \(\int (a+b \sqrt {x}) x \, dx\)

Optimal. Leaf size=19 \[ \frac {a x^2}{2}+\frac {2}{5} b x^{5/2} \]

[Out]

1/2*a*x^2+2/5*b*x^(5/2)

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {14} \[ \frac {a x^2}{2}+\frac {2}{5} b x^{5/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])*x,x]

[Out]

(a*x^2)/2 + (2*b*x^(5/2))/5

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 19, normalized size = 1.00 \[ \frac {a x^2}{2}+\frac {2}{5} b x^{5/2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])*x,x]

[Out]

(a*x^2)/2 + (2*b*x^(5/2))/5

________________________________________________________________________________________

fricas [A]  time = 1.08, size = 13, normalized size = 0.68 \[ \frac {2}{5} \, b x^{\frac {5}{2}} + \frac {1}{2} \, a x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*b*x^(5/2) + 1/2*a*x^2

________________________________________________________________________________________

giac [A]  time = 0.15, size = 13, normalized size = 0.68 \[ \frac {2}{5} \, b x^{\frac {5}{2}} + \frac {1}{2} \, a x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*b*x^(5/2) + 1/2*a*x^2

________________________________________________________________________________________

maple [A]  time = 0.00, size = 14, normalized size = 0.74 \[ \frac {2 b \,x^{\frac {5}{2}}}{5}+\frac {a \,x^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a*x^2+2/5*b*x^(5/2)

________________________________________________________________________________________

maxima [B]  time = 0.92, size = 64, normalized size = 3.37 \[ \frac {2 \, {\left (b \sqrt {x} + a\right )}^{5}}{5 \, b^{4}} - \frac {3 \, {\left (b \sqrt {x} + a\right )}^{4} a}{2 \, b^{4}} + \frac {2 \, {\left (b \sqrt {x} + a\right )}^{3} a^{2}}{b^{4}} - \frac {{\left (b \sqrt {x} + a\right )}^{2} a^{3}}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*(b*sqrt(x) + a)^5/b^4 - 3/2*(b*sqrt(x) + a)^4*a/b^4 + 2*(b*sqrt(x) + a)^3*a^2/b^4 - (b*sqrt(x) + a)^2*a^3/
b^4

________________________________________________________________________________________

mupad [B]  time = 0.02, size = 13, normalized size = 0.68 \[ \frac {a\,x^2}{2}+\frac {2\,b\,x^{5/2}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b*x^(1/2)),x)

[Out]

(a*x^2)/2 + (2*b*x^(5/2))/5

________________________________________________________________________________________

sympy [A]  time = 1.19, size = 15, normalized size = 0.79 \[ \frac {a x^{2}}{2} + \frac {2 b x^{\frac {5}{2}}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x**2/2 + 2*b*x**(5/2)/5

________________________________________________________________________________________