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

Optimal. Leaf size=32 \[ \frac{2 (a+b x)^{3/2}}{3 b^2}-\frac{2 a \sqrt{a+b x}}{b^2} \]

[Out]

(-2*a*Sqrt[a + b*x])/b^2 + (2*(a + b*x)^(3/2))/(3*b^2)

________________________________________________________________________________________

Rubi [A]  time = 0.0080337, antiderivative size = 32, normalized size of antiderivative = 1., 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 = {43} \[ \frac{2 (a+b x)^{3/2}}{3 b^2}-\frac{2 a \sqrt{a+b x}}{b^2} \]

Antiderivative was successfully verified.

[In]

Int[x/Sqrt[a + b*x],x]

[Out]

(-2*a*Sqrt[a + b*x])/b^2 + (2*(a + b*x)^(3/2))/(3*b^2)

Rule 43

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

Mathematica [A]  time = 0.0099597, size = 23, normalized size = 0.72 \[ \frac{2 (b x-2 a) \sqrt{a+b x}}{3 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x/Sqrt[a + b*x],x]

[Out]

(2*(-2*a + b*x)*Sqrt[a + b*x])/(3*b^2)

________________________________________________________________________________________

Maple [A]  time = 0.002, size = 21, normalized size = 0.7 \begin{align*} -{\frac{-2\,bx+4\,a}{3\,{b}^{2}}\sqrt{bx+a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(b*x+a)^(1/2)*(-b*x+2*a)/b^2

________________________________________________________________________________________

Maxima [A]  time = 0.932934, size = 35, normalized size = 1.09 \begin{align*} \frac{2 \,{\left (b x + a\right )}^{\frac{3}{2}}}{3 \, b^{2}} - \frac{2 \, \sqrt{b x + a} a}{b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.50508, size = 47, normalized size = 1.47 \begin{align*} \frac{2 \, \sqrt{b x + a}{\left (b x - 2 \, a\right )}}{3 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*sqrt(b*x + a)*(b*x - 2*a)/b^2

________________________________________________________________________________________

Sympy [B]  time = 1.03835, size = 162, normalized size = 5.06 \begin{align*} - \frac{4 a^{\frac{7}{2}} \sqrt{1 + \frac{b x}{a}}}{3 a^{2} b^{2} + 3 a b^{3} x} + \frac{4 a^{\frac{7}{2}}}{3 a^{2} b^{2} + 3 a b^{3} x} - \frac{2 a^{\frac{5}{2}} b x \sqrt{1 + \frac{b x}{a}}}{3 a^{2} b^{2} + 3 a b^{3} x} + \frac{4 a^{\frac{5}{2}} b x}{3 a^{2} b^{2} + 3 a b^{3} x} + \frac{2 a^{\frac{3}{2}} b^{2} x^{2} \sqrt{1 + \frac{b x}{a}}}{3 a^{2} b^{2} + 3 a b^{3} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*a**(7/2)*sqrt(1 + b*x/a)/(3*a**2*b**2 + 3*a*b**3*x) + 4*a**(7/2)/(3*a**2*b**2 + 3*a*b**3*x) - 2*a**(5/2)*b*
x*sqrt(1 + b*x/a)/(3*a**2*b**2 + 3*a*b**3*x) + 4*a**(5/2)*b*x/(3*a**2*b**2 + 3*a*b**3*x) + 2*a**(3/2)*b**2*x**
2*sqrt(1 + b*x/a)/(3*a**2*b**2 + 3*a*b**3*x)

________________________________________________________________________________________

Giac [A]  time = 1.07206, size = 31, normalized size = 0.97 \begin{align*} \frac{2 \,{\left ({\left (b x + a\right )}^{\frac{3}{2}} - 3 \, \sqrt{b x + a} a\right )}}{3 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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