3.2193 \(\int \frac {x}{a+b \sqrt {x}} \, dx\)

Optimal. Leaf size=51 \[ -\frac {2 a^3 \log \left (a+b \sqrt {x}\right )}{b^4}+\frac {2 a^2 \sqrt {x}}{b^3}-\frac {a x}{b^2}+\frac {2 x^{3/2}}{3 b} \]

[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

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Rubi [A]  time = 0.03, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac {2 a^2 \sqrt {x}}{b^3}-\frac {2 a^3 \log \left (a+b \sqrt {x}\right )}{b^4}-\frac {a x}{b^2}+\frac {2 x^{3/2}}{3 b} \]

Antiderivative was successfully verified.

[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 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])

Rule 266

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*} \int \frac {x}{a+b \sqrt {x}} \, dx &=2 \operatorname {Subst}\left (\int \frac {x^3}{a+b x} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {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*}

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Mathematica [A]  time = 0.03, size = 51, normalized size = 1.00 \[ -\frac {2 a^3 \log \left (a+b \sqrt {x}\right )}{b^4}+\frac {2 a^2 \sqrt {x}}{b^3}-\frac {a x}{b^2}+\frac {2 x^{3/2}}{3 b} \]

Antiderivative was successfully verified.

[In]

Integrate[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

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fricas [A]  time = 0.74, size = 43, normalized size = 0.84 \[ -\frac {3 \, a b^{2} x + 6 \, a^{3} \log \left (b \sqrt {x} + a\right ) - 2 \, {\left (b^{3} x + 3 \, a^{2} b\right )} \sqrt {x}}{3 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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giac [A]  time = 0.17, size = 45, normalized size = 0.88 \[ -\frac {2 \, a^{3} \log \left ({\left | b \sqrt {x} + a \right |}\right )}{b^{4}} + \frac {2 \, b^{2} x^{\frac {3}{2}} - 3 \, a b x + 6 \, a^{2} \sqrt {x}}{3 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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maple [A]  time = 0.00, size = 44, normalized size = 0.86 \[ \frac {2 x^{\frac {3}{2}}}{3 b}-\frac {2 a^{3} \ln \left (b \sqrt {x}+a \right )}{b^{4}}-\frac {a x}{b^{2}}+\frac {2 a^{2} \sqrt {x}}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.77, size = 61, normalized size = 1.20 \[ -\frac {2 \, a^{3} \log \left (b \sqrt {x} + a\right )}{b^{4}} + \frac {2 \, {\left (b \sqrt {x} + a\right )}^{3}}{3 \, b^{4}} - \frac {3 \, {\left (b \sqrt {x} + a\right )}^{2} a}{b^{4}} + \frac {6 \, {\left (b \sqrt {x} + a\right )} a^{2}}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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mupad [B]  time = 0.04, size = 43, normalized size = 0.84 \[ \frac {2\,x^{3/2}}{3\,b}-\frac {2\,a^3\,\ln \left (a+b\,\sqrt {x}\right )}{b^4}+\frac {2\,a^2\,\sqrt {x}}{b^3}-\frac {a\,x}{b^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [A]  time = 0.31, size = 54, normalized size = 1.06 \[ \begin {cases} - \frac {2 a^{3} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{b^{4}} + \frac {2 a^{2} \sqrt {x}}{b^{3}} - \frac {a x}{b^{2}} + \frac {2 x^{\frac {3}{2}}}{3 b} & \text {for}\: b \neq 0 \\\frac {x^{2}}{2 a} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[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))

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