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

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

[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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Rubi [A]  time = 0.0195794, antiderivative size = 38, 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} \[ -\frac{6 a^2 b}{\sqrt{x}}-\frac{a^3}{x}+3 a b^2 \log (x)+2 b^3 \sqrt{x} \]

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 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^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*}

Mathematica [A]  time = 0.0226885, size = 38, normalized size = 1. \[ -\frac{6 a^2 b}{\sqrt{x}}-\frac{a^3}{x}+3 a b^2 \log (x)+2 b^3 \sqrt{x} \]

Antiderivative was successfully verified.

[In]

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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Maple [A]  time = 0.003, size = 35, normalized size = 0.9 \begin{align*} -{\frac{{a}^{3}}{x}}+3\,a{b}^{2}\ln \left ( x \right ) -6\,{\frac{b{a}^{2}}{\sqrt{x}}}+2\,{b}^{3}\sqrt{x} \end{align*}

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 = 0.98843, size = 47, normalized size = 1.24 \begin{align*} 3 \, a b^{2} \log \left (x\right ) + 2 \, b^{3} \sqrt{x} - \frac{6 \, a^{2} b \sqrt{x} + a^{3}}{x} \end{align*}

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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Fricas [A]  time = 1.48879, size = 86, normalized size = 2.26 \begin{align*} \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} \end{align*}

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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Sympy [A]  time = 0.575448, size = 36, normalized size = 0.95 \begin{align*} - \frac{a^{3}}{x} - \frac{6 a^{2} b}{\sqrt{x}} + 3 a b^{2} \log{\left (x \right )} + 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**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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Giac [A]  time = 1.11535, size = 49, normalized size = 1.29 \begin{align*} 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} \end{align*}

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