3.2136 \(\int \frac {(a+b \sqrt {x})^3}{x^2} \, dx\)

Optimal. Leaf size=38 \[ -\frac {a^3}{x}-\frac {6 a^2 b}{\sqrt {x}}+3 a b^2 \log (x)+2 b^3 \sqrt {x} \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, 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} \[ -\frac {6 a^2 b}{\sqrt {x}}-\frac {a^3}{x}+3 a b^2 \log (x)+2 b^3 \sqrt {x} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sqrt[x])^3/x^2,x]

[Out]

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

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

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Mathematica [A]  time = 0.02, size = 38, normalized size = 1.00 \[ -\frac {a^3}{x}-\frac {6 a^2 b}{\sqrt {x}}+3 a b^2 \log (x)+2 b^3 \sqrt {x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sqrt[x])^3/x^2,x]

[Out]

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

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fricas [A]  time = 0.94, size = 38, normalized size = 1.00 \[ \frac {6 \, a b^{2} x \log \left (\sqrt {x}\right ) - a^{3} + 2 \, {\left (b^{3} x - 3 \, a^{2} b\right )} \sqrt {x}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.15, size = 36, normalized size = 0.95 \[ 3 \, a b^{2} \log \left ({\left | x \right |}\right ) + 2 \, b^{3} \sqrt {x} - \frac {6 \, a^{2} b \sqrt {x} + a^{3}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 35, normalized size = 0.92 \[ 3 a \,b^{2} \ln \relax (x )+2 b^{3} \sqrt {x}-\frac {6 a^{2} b}{\sqrt {x}}-\frac {a^{3}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.89, size = 35, normalized size = 0.92 \[ 3 \, a b^{2} \log \relax (x) + 2 \, b^{3} \sqrt {x} - \frac {6 \, a^{2} b \sqrt {x} + a^{3}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.10, size = 37, normalized size = 0.97 \[ 2\,b^3\,\sqrt {x}-\frac {a^3+6\,a^2\,b\,\sqrt {x}}{x}+6\,a\,b^2\,\ln \left (\sqrt {x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^(1/2))^3/x^2,x)

[Out]

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

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sympy [A]  time = 0.49, size = 36, normalized size = 0.95 \[ - \frac {a^{3}}{x} - \frac {6 a^{2} b}{\sqrt {x}} + 3 a b^{2} \log {\relax (x )} + 2 b^{3} \sqrt {x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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