3.1703 \(\int (a+\frac{b}{x})^{3/2} x \, dx\)

Optimal. Leaf size=66 \[ \frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{4 \sqrt{a}}+\frac{1}{2} x^2 \left (a+\frac{b}{x}\right )^{3/2}+\frac{3}{4} b x \sqrt{a+\frac{b}{x}} \]

[Out]

(3*b*Sqrt[a + b/x]*x)/4 + ((a + b/x)^(3/2)*x^2)/2 + (3*b^2*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/(4*Sqrt[a])

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Rubi [A]  time = 0.0276017, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {266, 47, 63, 208} \[ \frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{4 \sqrt{a}}+\frac{1}{2} x^2 \left (a+\frac{b}{x}\right )^{3/2}+\frac{3}{4} b x \sqrt{a+\frac{b}{x}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^(3/2)*x,x]

[Out]

(3*b*Sqrt[a + b/x]*x)/4 + ((a + b/x)^(3/2)*x^2)/2 + (3*b^2*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/(4*Sqrt[a])

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 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \left (a+\frac{b}{x}\right )^{3/2} x \, dx &=-\operatorname{Subst}\left (\int \frac{(a+b x)^{3/2}}{x^3} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{2} \left (a+\frac{b}{x}\right )^{3/2} x^2-\frac{1}{4} (3 b) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x^2} \, dx,x,\frac{1}{x}\right )\\ &=\frac{3}{4} b \sqrt{a+\frac{b}{x}} x+\frac{1}{2} \left (a+\frac{b}{x}\right )^{3/2} x^2-\frac{1}{8} \left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{3}{4} b \sqrt{a+\frac{b}{x}} x+\frac{1}{2} \left (a+\frac{b}{x}\right )^{3/2} x^2-\frac{1}{4} (3 b) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+\frac{b}{x}}\right )\\ &=\frac{3}{4} b \sqrt{a+\frac{b}{x}} x+\frac{1}{2} \left (a+\frac{b}{x}\right )^{3/2} x^2+\frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{4 \sqrt{a}}\\ \end{align*}

Mathematica [A]  time = 0.036563, size = 76, normalized size = 1.15 \[ \frac{x \sqrt{a+\frac{b}{x}} \left (2 a^2 x^2+3 b^2 \sqrt{\frac{b}{a x}+1} \tanh ^{-1}\left (\sqrt{\frac{b}{a x}+1}\right )+7 a b x+5 b^2\right )}{4 (a x+b)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^(3/2)*x,x]

[Out]

(Sqrt[a + b/x]*x*(5*b^2 + 7*a*b*x + 2*a^2*x^2 + 3*b^2*Sqrt[1 + b/(a*x)]*ArcTanh[Sqrt[1 + b/(a*x)]]))/(4*(b + a
*x))

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Maple [A]  time = 0.006, size = 96, normalized size = 1.5 \begin{align*}{\frac{x}{8}\sqrt{{\frac{ax+b}{x}}} \left ( 4\,\sqrt{a{x}^{2}+bx}{a}^{5/2}x+10\,\sqrt{a{x}^{2}+bx}{a}^{3/2}b+3\,{b}^{2}\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) a \right ){\frac{1}{\sqrt{ \left ( ax+b \right ) x}}}{a}^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*((a*x+b)/x)^(1/2)*x*(4*(a*x^2+b*x)^(1/2)*a^(5/2)*x+10*(a*x^2+b*x)^(1/2)*a^(3/2)*b+3*b^2*ln(1/2*(2*(a*x^2+b
*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a)/((a*x+b)*x)^(1/2)/a^(3/2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.78003, size = 300, normalized size = 4.55 \begin{align*} \left [\frac{3 \, \sqrt{a} b^{2} \log \left (2 \, a x + 2 \, \sqrt{a} x \sqrt{\frac{a x + b}{x}} + b\right ) + 2 \,{\left (2 \, a^{2} x^{2} + 5 \, a b x\right )} \sqrt{\frac{a x + b}{x}}}{8 \, a}, -\frac{3 \, \sqrt{-a} b^{2} \arctan \left (\frac{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}{a}\right ) -{\left (2 \, a^{2} x^{2} + 5 \, a b x\right )} \sqrt{\frac{a x + b}{x}}}{4 \, a}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/8*(3*sqrt(a)*b^2*log(2*a*x + 2*sqrt(a)*x*sqrt((a*x + b)/x) + b) + 2*(2*a^2*x^2 + 5*a*b*x)*sqrt((a*x + b)/x)
)/a, -1/4*(3*sqrt(-a)*b^2*arctan(sqrt(-a)*sqrt((a*x + b)/x)/a) - (2*a^2*x^2 + 5*a*b*x)*sqrt((a*x + b)/x))/a]

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Sympy [A]  time = 3.07049, size = 75, normalized size = 1.14 \begin{align*} \frac{a \sqrt{b} x^{\frac{3}{2}} \sqrt{\frac{a x}{b} + 1}}{2} + \frac{5 b^{\frac{3}{2}} \sqrt{x} \sqrt{\frac{a x}{b} + 1}}{4} + \frac{3 b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{4 \sqrt{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*sqrt(b)*x**(3/2)*sqrt(a*x/b + 1)/2 + 5*b**(3/2)*sqrt(x)*sqrt(a*x/b + 1)/4 + 3*b**2*asinh(sqrt(a)*sqrt(x)/sqr
t(b))/(4*sqrt(a))

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Giac [A]  time = 1.14985, size = 107, normalized size = 1.62 \begin{align*} -\frac{3 \, b^{2} \log \left ({\left | -2 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )} \sqrt{a} - b \right |}\right ) \mathrm{sgn}\left (x\right )}{8 \, \sqrt{a}} + \frac{3 \, b^{2} \log \left ({\left | b \right |}\right ) \mathrm{sgn}\left (x\right )}{8 \, \sqrt{a}} + \frac{1}{4} \, \sqrt{a x^{2} + b x}{\left (2 \, a x \mathrm{sgn}\left (x\right ) + 5 \, b \mathrm{sgn}\left (x\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/8*b^2*log(abs(-2*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a) - b))*sgn(x)/sqrt(a) + 3/8*b^2*log(abs(b))*sgn(x)/
sqrt(a) + 1/4*sqrt(a*x^2 + b*x)*(2*a*x*sgn(x) + 5*b*sgn(x))