\(\int \frac {x}{a+b \sqrt {x}} \, dx\) [2193]

Optimal result

Integrand size = 13, antiderivative size = 51 \[ \int \frac {x}{a+b \sqrt {x}} \, dx=\frac {2 a^2 \sqrt {x}}{b^3}-\frac {a x}{b^2}+\frac {2 x^{3/2}}{3 b}-\frac {2 a^3 \log \left (a+b \sqrt {x}\right )}{b^4} \]

[Out]

-a*x/b^2+2/3*x^(3/2)/b-2*a^3*ln(a+b*x^(1/2))/b^4+2*a^2*x^(1/2)/b^3

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 45} \[ \int \frac {x}{a+b \sqrt {x}} \, dx=-\frac {2 a^3 \log \left (a+b \sqrt {x}\right )}{b^4}+\frac {2 a^2 \sqrt {x}}{b^3}-\frac {a x}{b^2}+\frac {2 x^{3/2}}{3 b} \]

[In]

Int[x/(a + b*Sqrt[x]),x]

[Out]

(2*a^2*Sqrt[x])/b^3 - (a*x)/b^2 + (2*x^(3/2))/(3*b) - (2*a^3*Log[a + b*Sqrt[x]])/b^4

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])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {x^3}{a+b x} \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (\frac {a^2}{b^3}-\frac {a x}{b^2}+\frac {x^2}{b}-\frac {a^3}{b^3 (a+b x)}\right ) \, dx,x,\sqrt {x}\right ) \\ & = \frac {2 a^2 \sqrt {x}}{b^3}-\frac {a x}{b^2}+\frac {2 x^{3/2}}{3 b}-\frac {2 a^3 \log \left (a+b \sqrt {x}\right )}{b^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.02 \[ \int \frac {x}{a+b \sqrt {x}} \, dx=\frac {\sqrt {x} \left (6 a^2-3 a b \sqrt {x}+2 b^2 x\right )}{3 b^3}-\frac {2 a^3 \log \left (a+b \sqrt {x}\right )}{b^4} \]

[In]

Integrate[x/(a + b*Sqrt[x]),x]

[Out]

(Sqrt[x]*(6*a^2 - 3*a*b*Sqrt[x] + 2*b^2*x))/(3*b^3) - (2*a^3*Log[a + b*Sqrt[x]])/b^4

Maple [A] (verified)

Time = 5.83 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.86

method	result	size
derivativedivides	\(\frac {\frac {2 b^{2} x^{\frac {3}{2}}}{3}-a b x +2 a^{2} \sqrt {x}}{b^{3}}-\frac {2 a^{3} \ln \left (a +b \sqrt {x}\right )}{b^{4}}\)	\(44\)
default	\(\frac {\frac {2 b^{2} x^{\frac {3}{2}}}{3}-a b x +2 a^{2} \sqrt {x}}{b^{3}}-\frac {2 a^{3} \ln \left (a +b \sqrt {x}\right )}{b^{4}}\)	\(44\)

[In]

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

[Out]

2/b^3*(1/3*b^2*x^(3/2)-1/2*a*b*x+a^2*x^(1/2))-2*a^3*ln(a+b*x^(1/2))/b^4

Fricas [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.84 \[ \int \frac {x}{a+b \sqrt {x}} \, dx=-\frac {3 \, a b^{2} x + 6 \, a^{3} \log \left (b \sqrt {x} + a\right ) - 2 \, {\left (b^{3} x + 3 \, a^{2} b\right )} \sqrt {x}}{3 \, b^{4}} \]

[In]

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

[Out]

-1/3*(3*a*b^2*x + 6*a^3*log(b*sqrt(x) + a) - 2*(b^3*x + 3*a^2*b)*sqrt(x))/b^4

Sympy [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.06 \[ \int \frac {x}{a+b \sqrt {x}} \, dx=\begin {cases} - \frac {2 a^{3} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{b^{4}} + \frac {2 a^{2} \sqrt {x}}{b^{3}} - \frac {a x}{b^{2}} + \frac {2 x^{\frac {3}{2}}}{3 b} & \text {for}\: b \neq 0 \\\frac {x^{2}}{2 a} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((-2*a**3*log(a/b + sqrt(x))/b**4 + 2*a**2*sqrt(x)/b**3 - a*x/b**2 + 2*x**(3/2)/(3*b), Ne(b, 0)), (x*
*2/(2*a), True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.20 \[ \int \frac {x}{a+b \sqrt {x}} \, dx=-\frac {2 \, a^{3} \log \left (b \sqrt {x} + a\right )}{b^{4}} + \frac {2 \, {\left (b \sqrt {x} + a\right )}^{3}}{3 \, b^{4}} - \frac {3 \, {\left (b \sqrt {x} + a\right )}^{2} a}{b^{4}} + \frac {6 \, {\left (b \sqrt {x} + a\right )} a^{2}}{b^{4}} \]

[In]

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

[Out]

-2*a^3*log(b*sqrt(x) + a)/b^4 + 2/3*(b*sqrt(x) + a)^3/b^4 - 3*(b*sqrt(x) + a)^2*a/b^4 + 6*(b*sqrt(x) + a)*a^2/
b^4

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.88 \[ \int \frac {x}{a+b \sqrt {x}} \, dx=-\frac {2 \, a^{3} \log \left ({\left | b \sqrt {x} + a \right |}\right )}{b^{4}} + \frac {2 \, b^{2} x^{\frac {3}{2}} - 3 \, a b x + 6 \, a^{2} \sqrt {x}}{3 \, b^{3}} \]

[In]

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

[Out]

-2*a^3*log(abs(b*sqrt(x) + a))/b^4 + 1/3*(2*b^2*x^(3/2) - 3*a*b*x + 6*a^2*sqrt(x))/b^3

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.84 \[ \int \frac {x}{a+b \sqrt {x}} \, dx=\frac {2\,x^{3/2}}{3\,b}-\frac {2\,a^3\,\ln \left (a+b\,\sqrt {x}\right )}{b^4}+\frac {2\,a^2\,\sqrt {x}}{b^3}-\frac {a\,x}{b^2} \]

[In]

int(x/(a + b*x^(1/2)),x)

[Out]

(2*x^(3/2))/(3*b) - (2*a^3*log(a + b*x^(1/2)))/b^4 + (2*a^2*x^(1/2))/b^3 - (a*x)/b^2