3.2137 \(\int \frac{(a+b \sqrt{x})^3}{x^3} \, dx\)

Optimal. Leaf size=21 \[ -\frac{\left (a+b \sqrt{x}\right )^4}{2 a x^2} \]

[Out]

-(a + b*Sqrt[x])^4/(2*a*x^2)

________________________________________________________________________________________

Rubi [A]  time = 0.0033667, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {264} \[ -\frac{\left (a+b \sqrt{x}\right )^4}{2 a x^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^3/x^3,x]

[Out]

-(a + b*Sqrt[x])^4/(2*a*x^2)

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{\left (a+b \sqrt{x}\right )^3}{x^3} \, dx &=-\frac{\left (a+b \sqrt{x}\right )^4}{2 a x^2}\\ \end{align*}

Mathematica [A]  time = 0.0032108, size = 21, normalized size = 1. \[ -\frac{\left (a+b \sqrt{x}\right )^4}{2 a x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^3/x^3,x]

[Out]

-(a + b*Sqrt[x])^4/(2*a*x^2)

________________________________________________________________________________________

Maple [B]  time = 0.002, size = 36, normalized size = 1.7 \begin{align*} -2\,{\frac{{b}^{3}}{\sqrt{x}}}-3\,{\frac{{b}^{2}a}{x}}-2\,{\frac{b{a}^{2}}{{x}^{3/2}}}-{\frac{{a}^{3}}{2\,{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.963568, size = 45, normalized size = 2.14 \begin{align*} -\frac{4 \, b^{3} x^{\frac{3}{2}} + 6 \, a b^{2} x + 4 \, a^{2} b \sqrt{x} + a^{3}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.48042, size = 76, normalized size = 3.62 \begin{align*} -\frac{6 \, a b^{2} x + a^{3} + 4 \,{\left (b^{3} x + a^{2} b\right )} \sqrt{x}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [B]  time = 0.928327, size = 39, normalized size = 1.86 \begin{align*} - \frac{a^{3}}{2 x^{2}} - \frac{2 a^{2} b}{x^{\frac{3}{2}}} - \frac{3 a b^{2}}{x} - \frac{2 b^{3}}{\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.11296, size = 45, normalized size = 2.14 \begin{align*} -\frac{4 \, b^{3} x^{\frac{3}{2}} + 6 \, a b^{2} x + 4 \, a^{2} b \sqrt{x} + a^{3}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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