3.2135 \(\int \frac{(a+b \sqrt{x})^3}{x} \, dx\)

Optimal. Leaf size=37 \[ 6 a^2 b \sqrt{x}+a^3 \log (x)+3 a b^2 x+\frac{2}{3} b^3 x^{3/2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0175741, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ 6 a^2 b \sqrt{x}+a^3 \log (x)+3 a b^2 x+\frac{2}{3} b^3 x^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^3/x,x]

[Out]

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

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]]

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{\left (a+b \sqrt{x}\right )^3}{x} \, dx &=2 \operatorname{Subst}\left (\int \frac{(a+b x)^3}{x} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (3 a^2 b+\frac{a^3}{x}+3 a b^2 x+b^3 x^2\right ) \, dx,x,\sqrt{x}\right )\\ &=6 a^2 b \sqrt{x}+3 a b^2 x+\frac{2}{3} b^3 x^{3/2}+a^3 \log (x)\\ \end{align*}

Mathematica [A]  time = 0.012699, size = 37, normalized size = 1. \[ 6 a^2 b \sqrt{x}+a^3 \log (x)+3 a b^2 x+\frac{2}{3} b^3 x^{3/2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^3/x,x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.002, size = 32, normalized size = 0.9 \begin{align*} 3\,xa{b}^{2}+{\frac{2\,{b}^{3}}{3}{x}^{{\frac{3}{2}}}}+{a}^{3}\ln \left ( x \right ) +6\,{a}^{2}b\sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.961255, size = 42, normalized size = 1.14 \begin{align*} \frac{2}{3} \, b^{3} x^{\frac{3}{2}} + 3 \, a b^{2} x + a^{3} \log \left (x\right ) + 6 \, a^{2} b \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^3/x,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.47456, size = 86, normalized size = 2.32 \begin{align*} 3 \, a b^{2} x + 2 \, a^{3} \log \left (\sqrt{x}\right ) + \frac{2}{3} \,{\left (b^{3} x + 9 \, a^{2} b\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^3/x,x, algorithm="fricas")

[Out]

3*a*b^2*x + 2*a^3*log(sqrt(x)) + 2/3*(b^3*x + 9*a^2*b)*sqrt(x)

________________________________________________________________________________________

Sympy [A]  time = 0.305087, size = 37, normalized size = 1. \begin{align*} a^{3} \log{\left (x \right )} + 6 a^{2} b \sqrt{x} + 3 a b^{2} x + \frac{2 b^{3} x^{\frac{3}{2}}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x**(1/2))**3/x,x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.1018, size = 43, normalized size = 1.16 \begin{align*} \frac{2}{3} \, b^{3} x^{\frac{3}{2}} + 3 \, a b^{2} x + a^{3} \log \left ({\left | x \right |}\right ) + 6 \, a^{2} b \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^(1/2))^3/x,x, algorithm="giac")

[Out]

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