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]

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

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Rubi [A]  time = 0.00, antiderivative size = 21, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 21, normalized size = 1.00 \[ -\frac {\left (a+b \sqrt {x}\right )^4}{2 a x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^3/x^3,x]

[Out]

-1/2*(a + b*Sqrt[x])^4/(a*x^2)

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fricas [A]  time = 1.08, size = 32, normalized size = 1.52 \[ -\frac {6 \, a b^{2} x + a^{3} + 4 \, {\left (b^{3} x + a^{2} b\right )} \sqrt {x}}{2 \, x^{2}} \]

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

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giac [A]  time = 0.15, size = 33, normalized size = 1.57 \[ -\frac {4 \, b^{3} x^{\frac {3}{2}} + 6 \, a b^{2} x + 4 \, a^{2} b \sqrt {x} + a^{3}}{2 \, x^{2}} \]

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

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maple [B]  time = 0.00, size = 36, normalized size = 1.71 \[ -\frac {2 b^{3}}{\sqrt {x}}-\frac {3 a \,b^{2}}{x}-\frac {2 a^{2} b}{x^{\frac {3}{2}}}-\frac {a^{3}}{2 x^{2}} \]

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

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maxima [A]  time = 0.81, size = 33, normalized size = 1.57 \[ -\frac {4 \, b^{3} x^{\frac {3}{2}} + 6 \, a b^{2} x + 4 \, a^{2} b \sqrt {x} + a^{3}}{2 \, x^{2}} \]

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

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mupad [B]  time = 0.03, size = 33, normalized size = 1.57 \[ -\frac {a^3+4\,b^3\,x^{3/2}+4\,a^2\,b\,\sqrt {x}+6\,a\,b^2\,x}{2\,x^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^(1/2))^3/x^3,x)

[Out]

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

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sympy [B]  time = 0.69, size = 39, normalized size = 1.86 \[ - \frac {a^{3}}{2 x^{2}} - \frac {2 a^{2} b}{x^{\frac {3}{2}}} - \frac {3 a b^{2}}{x} - \frac {2 b^{3}}{\sqrt {x}} \]

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)

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