\(\int \frac {x}{\sqrt {a+b x}} \, dx\) [337]

Optimal result

Integrand size = 11, antiderivative size = 32 \[ \int \frac {x}{\sqrt {a+b x}} \, dx=-\frac {2 a \sqrt {a+b x}}{b^2}+\frac {2 (a+b x)^{3/2}}{3 b^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 32, 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} \[ \int \frac {x}{\sqrt {a+b x}} \, dx=\frac {2 (a+b x)^{3/2}}{3 b^2}-\frac {2 a \sqrt {a+b x}}{b^2} \]

[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 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 (-\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] (verified)

Time = 0.00 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.72 \[ \int \frac {x}{\sqrt {a+b x}} \, dx=\frac {2 (-2 a+b x) \sqrt {a+b x}}{3 b^2} \]

[In]

Integrate[x/Sqrt[a + b*x],x]

[Out]

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

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.66

method	result	size
gosper	\(-\frac {2 \sqrt {b x +a}\, \left (-b x +2 a \right )}{3 b^{2}}\)	\(21\)
trager	\(-\frac {2 \sqrt {b x +a}\, \left (-b x +2 a \right )}{3 b^{2}}\)	\(21\)
risch	\(-\frac {2 \sqrt {b x +a}\, \left (-b x +2 a \right )}{3 b^{2}}\)	\(21\)
pseudoelliptic	\(-\frac {2 \sqrt {b x +a}\, \left (-b x +2 a \right )}{3 b^{2}}\)	\(21\)
derivativedivides	\(\frac {\frac {2 \left (b x +a \right )^{\frac {3}{2}}}{3}-2 a \sqrt {b x +a}}{b^{2}}\)	\(26\)
default	\(\frac {\frac {2 \left (b x +a \right )^{\frac {3}{2}}}{3}-2 a \sqrt {b x +a}}{b^{2}}\)	\(26\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.59 \[ \int \frac {x}{\sqrt {a+b x}} \, dx=\frac {2 \, \sqrt {b x + a} {\left (b x - 2 \, a\right )}}{3 \, b^{2}} \]

[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] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 162 vs. \(2 (29) = 58\).

Time = 0.88 (sec) , antiderivative size = 162, normalized size of antiderivative = 5.06 \[ \int \frac {x}{\sqrt {a+b x}} \, dx=- \frac {4 a^{\frac {7}{2}} \sqrt {1 + \frac {b x}{a}}}{3 a^{2} b^{2} + 3 a b^{3} x} + \frac {4 a^{\frac {7}{2}}}{3 a^{2} b^{2} + 3 a b^{3} x} - \frac {2 a^{\frac {5}{2}} b x \sqrt {1 + \frac {b x}{a}}}{3 a^{2} b^{2} + 3 a b^{3} x} + \frac {4 a^{\frac {5}{2}} b x}{3 a^{2} b^{2} + 3 a b^{3} x} + \frac {2 a^{\frac {3}{2}} b^{2} x^{2} \sqrt {1 + \frac {b x}{a}}}{3 a^{2} b^{2} + 3 a b^{3} x} \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int \frac {x}{\sqrt {a+b x}} \, dx=\frac {2 \, {\left (b x + a\right )}^{\frac {3}{2}}}{3 \, b^{2}} - \frac {2 \, \sqrt {b x + a} a}{b^{2}} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.72 \[ \int \frac {x}{\sqrt {a+b x}} \, dx=\frac {2 \, {\left ({\left (b x + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {b x + a} a\right )}}{3 \, b^{2}} \]

[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

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.78 \[ \int \frac {x}{\sqrt {a+b x}} \, dx=-\frac {6\,a\,\sqrt {a+b\,x}-2\,{\left (a+b\,x\right )}^{3/2}}{3\,b^2} \]

[In]

int(x/(a + b*x)^(1/2),x)

[Out]

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