3.1732 \(\int \frac{x^2}{(a+\frac{b}{x})^{3/2}} \, dx\)

Optimal. Leaf size=117 \[ \frac{35 b^3}{8 a^4 \sqrt{a+\frac{b}{x}}}+\frac{35 b^2 x}{24 a^3 \sqrt{a+\frac{b}{x}}}-\frac{35 b^3 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{8 a^{9/2}}-\frac{7 b x^2}{12 a^2 \sqrt{a+\frac{b}{x}}}+\frac{x^3}{3 a \sqrt{a+\frac{b}{x}}} \]

[Out]

(35*b^3)/(8*a^4*Sqrt[a + b/x]) + (35*b^2*x)/(24*a^3*Sqrt[a + b/x]) - (7*b*x^2)/(12*a^2*Sqrt[a + b/x]) + x^3/(3
*a*Sqrt[a + b/x]) - (35*b^3*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/(8*a^(9/2))

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Rubi [A]  time = 0.0529987, antiderivative size = 115, normalized size of antiderivative = 0.98, number of steps used = 7, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {266, 51, 63, 208} \[ \frac{35 b^2 x \sqrt{a+\frac{b}{x}}}{8 a^4}-\frac{35 b^3 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{8 a^{9/2}}-\frac{35 b x^2 \sqrt{a+\frac{b}{x}}}{12 a^3}+\frac{7 x^3 \sqrt{a+\frac{b}{x}}}{3 a^2}-\frac{2 x^3}{a \sqrt{a+\frac{b}{x}}} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b/x)^(3/2),x]

[Out]

(35*b^2*Sqrt[a + b/x]*x)/(8*a^4) - (35*b*Sqrt[a + b/x]*x^2)/(12*a^3) - (2*x^3)/(a*Sqrt[a + b/x]) + (7*Sqrt[a +
b/x]*x^3)/(3*a^2) - (35*b^3*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/(8*a^(9/2))

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 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && 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 \frac{x^2}{\left (a+\frac{b}{x}\right )^{3/2}} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{x^4 (a+b x)^{3/2}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{2 x^3}{a \sqrt{a+\frac{b}{x}}}-\frac{7 \operatorname{Subst}\left (\int \frac{1}{x^4 \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )}{a}\\ &=-\frac{2 x^3}{a \sqrt{a+\frac{b}{x}}}+\frac{7 \sqrt{a+\frac{b}{x}} x^3}{3 a^2}+\frac{(35 b) \operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )}{6 a^2}\\ &=-\frac{35 b \sqrt{a+\frac{b}{x}} x^2}{12 a^3}-\frac{2 x^3}{a \sqrt{a+\frac{b}{x}}}+\frac{7 \sqrt{a+\frac{b}{x}} x^3}{3 a^2}-\frac{\left (35 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )}{8 a^3}\\ &=\frac{35 b^2 \sqrt{a+\frac{b}{x}} x}{8 a^4}-\frac{35 b \sqrt{a+\frac{b}{x}} x^2}{12 a^3}-\frac{2 x^3}{a \sqrt{a+\frac{b}{x}}}+\frac{7 \sqrt{a+\frac{b}{x}} x^3}{3 a^2}+\frac{\left (35 b^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )}{16 a^4}\\ &=\frac{35 b^2 \sqrt{a+\frac{b}{x}} x}{8 a^4}-\frac{35 b \sqrt{a+\frac{b}{x}} x^2}{12 a^3}-\frac{2 x^3}{a \sqrt{a+\frac{b}{x}}}+\frac{7 \sqrt{a+\frac{b}{x}} x^3}{3 a^2}+\frac{\left (35 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+\frac{b}{x}}\right )}{8 a^4}\\ &=\frac{35 b^2 \sqrt{a+\frac{b}{x}} x}{8 a^4}-\frac{35 b \sqrt{a+\frac{b}{x}} x^2}{12 a^3}-\frac{2 x^3}{a \sqrt{a+\frac{b}{x}}}+\frac{7 \sqrt{a+\frac{b}{x}} x^3}{3 a^2}-\frac{35 b^3 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{8 a^{9/2}}\\ \end{align*}

Mathematica [C]  time = 0.0128803, size = 37, normalized size = 0.32 \[ \frac{2 b^3 \, _2F_1\left (-\frac{1}{2},4;\frac{1}{2};\frac{b}{a x}+1\right )}{a^4 \sqrt{a+\frac{b}{x}}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a + b/x)^(3/2),x]

[Out]

(2*b^3*Hypergeometric2F1[-1/2, 4, 1/2, 1 + b/(a*x)])/(a^4*Sqrt[a + b/x])

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Maple [B]  time = 0.015, size = 458, normalized size = 3.9 \begin{align*} -{\frac{x}{48\, \left ( ax+b \right ) ^{2}}\sqrt{{\frac{ax+b}{x}}} \left ( -16\, \left ( a{x}^{2}+bx \right ) ^{3/2}{a}^{9/2}{x}^{2}+60\,\sqrt{a{x}^{2}+bx}{a}^{9/2}{x}^{3}b-240\,\sqrt{ \left ( ax+b \right ) x}{a}^{7/2}{x}^{2}{b}^{2}-32\, \left ( a{x}^{2}+bx \right ) ^{3/2}{a}^{7/2}xb+150\,\sqrt{a{x}^{2}+bx}{a}^{7/2}{x}^{2}{b}^{2}+120\,{a}^{3}\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+b \right ) x}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{2}{b}^{3}+96\, \left ( \left ( ax+b \right ) x \right ) ^{3/2}{a}^{5/2}{b}^{2}-480\,\sqrt{ \left ( ax+b \right ) x}{a}^{5/2}x{b}^{3}-16\, \left ( a{x}^{2}+bx \right ) ^{3/2}{a}^{5/2}{b}^{2}+120\,\sqrt{a{x}^{2}+bx}{a}^{5/2}x{b}^{3}+240\,{a}^{2}\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+b \right ) x}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) x{b}^{4}-15\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{2}{a}^{3}{b}^{3}-240\,\sqrt{ \left ( ax+b \right ) x}{a}^{3/2}{b}^{4}+30\,\sqrt{a{x}^{2}+bx}{a}^{3/2}{b}^{4}+120\,a\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+b \right ) x}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){b}^{5}-30\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) x{a}^{2}{b}^{4}-15\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) a{b}^{5} \right ){a}^{-{\frac{11}{2}}}{\frac{1}{\sqrt{ \left ( ax+b \right ) x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/48*((a*x+b)/x)^(1/2)*x/a^(11/2)*(-16*(a*x^2+b*x)^(3/2)*a^(9/2)*x^2+60*(a*x^2+b*x)^(1/2)*a^(9/2)*x^3*b-240*(
(a*x+b)*x)^(1/2)*a^(7/2)*x^2*b^2-32*(a*x^2+b*x)^(3/2)*a^(7/2)*x*b+150*(a*x^2+b*x)^(1/2)*a^(7/2)*x^2*b^2+120*a^
3*ln(1/2*(2*((a*x+b)*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*x^2*b^3+96*((a*x+b)*x)^(3/2)*a^(5/2)*b^2-480*((a*x+b)*
x)^(1/2)*a^(5/2)*x*b^3-16*(a*x^2+b*x)^(3/2)*a^(5/2)*b^2+120*(a*x^2+b*x)^(1/2)*a^(5/2)*x*b^3+240*a^2*ln(1/2*(2*
((a*x+b)*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*x*b^4-15*ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*x^2
*a^3*b^3-240*((a*x+b)*x)^(1/2)*a^(3/2)*b^4+30*(a*x^2+b*x)^(1/2)*a^(3/2)*b^4+120*a*ln(1/2*(2*((a*x+b)*x)^(1/2)*
a^(1/2)+2*a*x+b)/a^(1/2))*b^5-30*ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*x*a^2*b^4-15*ln(1/2*(2*
(a*x^2+b*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a*b^5)/((a*x+b)*x)^(1/2)/(a*x+b)^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(x^2/(a+b/x)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.5206, size = 474, normalized size = 4.05 \begin{align*} \left [\frac{105 \,{\left (a b^{3} x + b^{4}\right )} \sqrt{a} \log \left (2 \, a x - 2 \, \sqrt{a} x \sqrt{\frac{a x + b}{x}} + b\right ) + 2 \,{\left (8 \, a^{4} x^{4} - 14 \, a^{3} b x^{3} + 35 \, a^{2} b^{2} x^{2} + 105 \, a b^{3} x\right )} \sqrt{\frac{a x + b}{x}}}{48 \,{\left (a^{6} x + a^{5} b\right )}}, \frac{105 \,{\left (a b^{3} x + b^{4}\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}{a}\right ) +{\left (8 \, a^{4} x^{4} - 14 \, a^{3} b x^{3} + 35 \, a^{2} b^{2} x^{2} + 105 \, a b^{3} x\right )} \sqrt{\frac{a x + b}{x}}}{24 \,{\left (a^{6} x + a^{5} b\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/48*(105*(a*b^3*x + b^4)*sqrt(a)*log(2*a*x - 2*sqrt(a)*x*sqrt((a*x + b)/x) + b) + 2*(8*a^4*x^4 - 14*a^3*b*x^
3 + 35*a^2*b^2*x^2 + 105*a*b^3*x)*sqrt((a*x + b)/x))/(a^6*x + a^5*b), 1/24*(105*(a*b^3*x + b^4)*sqrt(-a)*arcta
n(sqrt(-a)*sqrt((a*x + b)/x)/a) + (8*a^4*x^4 - 14*a^3*b*x^3 + 35*a^2*b^2*x^2 + 105*a*b^3*x)*sqrt((a*x + b)/x))
/(a^6*x + a^5*b)]

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Sympy [A]  time = 7.56225, size = 133, normalized size = 1.14 \begin{align*} \frac{x^{\frac{7}{2}}}{3 a \sqrt{b} \sqrt{\frac{a x}{b} + 1}} - \frac{7 \sqrt{b} x^{\frac{5}{2}}}{12 a^{2} \sqrt{\frac{a x}{b} + 1}} + \frac{35 b^{\frac{3}{2}} x^{\frac{3}{2}}}{24 a^{3} \sqrt{\frac{a x}{b} + 1}} + \frac{35 b^{\frac{5}{2}} \sqrt{x}}{8 a^{4} \sqrt{\frac{a x}{b} + 1}} - \frac{35 b^{3} \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{8 a^{\frac{9}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**(7/2)/(3*a*sqrt(b)*sqrt(a*x/b + 1)) - 7*sqrt(b)*x**(5/2)/(12*a**2*sqrt(a*x/b + 1)) + 35*b**(3/2)*x**(3/2)/(
24*a**3*sqrt(a*x/b + 1)) + 35*b**(5/2)*sqrt(x)/(8*a**4*sqrt(a*x/b + 1)) - 35*b**3*asinh(sqrt(a)*sqrt(x)/sqrt(b
))/(8*a**(9/2))

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Giac [A]  time = 1.2066, size = 194, normalized size = 1.66 \begin{align*} \frac{1}{24} \, b{\left (\frac{105 \, b^{2} \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{4}} + \frac{48 \, b^{2}}{a^{4} \sqrt{\frac{a x + b}{x}}} - \frac{87 \, a^{2} b^{2} \sqrt{\frac{a x + b}{x}} - \frac{136 \,{\left (a x + b\right )} a b^{2} \sqrt{\frac{a x + b}{x}}}{x} + \frac{57 \,{\left (a x + b\right )}^{2} b^{2} \sqrt{\frac{a x + b}{x}}}{x^{2}}}{{\left (a - \frac{a x + b}{x}\right )}^{3} a^{4}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*b*(105*b^2*arctan(sqrt((a*x + b)/x)/sqrt(-a))/(sqrt(-a)*a^4) + 48*b^2/(a^4*sqrt((a*x + b)/x)) - (87*a^2*b
^2*sqrt((a*x + b)/x) - 136*(a*x + b)*a*b^2*sqrt((a*x + b)/x)/x + 57*(a*x + b)^2*b^2*sqrt((a*x + b)/x)/x^2)/((a
- (a*x + b)/x)^3*a^4))