3.1743 \(\int \frac{x}{(a+\frac{b}{x})^{5/2}} \, dx\)

Optimal. Leaf size=114 \[ -\frac{35 b^2}{4 a^4 \sqrt{a+\frac{b}{x}}}-\frac{35 b^2}{12 a^3 \left (a+\frac{b}{x}\right )^{3/2}}+\frac{35 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{4 a^{9/2}}-\frac{7 b x}{4 a^2 \left (a+\frac{b}{x}\right )^{3/2}}+\frac{x^2}{2 a \left (a+\frac{b}{x}\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0502586, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {266, 51, 63, 208} \[ \frac{35 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{4 a^{9/2}}+\frac{35 x^2 \sqrt{a+\frac{b}{x}}}{6 a^3}-\frac{14 x^2}{3 a^2 \sqrt{a+\frac{b}{x}}}-\frac{35 b x \sqrt{a+\frac{b}{x}}}{4 a^4}-\frac{2 x^2}{3 a \left (a+\frac{b}{x}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[x/(a + b/x)^(5/2),x]

[Out]

(-35*b*Sqrt[a + b/x]*x)/(4*a^4) - (2*x^2)/(3*a*(a + b/x)^(3/2)) - (14*x^2)/(3*a^2*Sqrt[a + b/x]) + (35*Sqrt[a
+ b/x]*x^2)/(6*a^3) + (35*b^2*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/(4*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}{\left (a+\frac{b}{x}\right )^{5/2}} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{x^3 (a+b x)^{5/2}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{2 x^2}{3 a \left (a+\frac{b}{x}\right )^{3/2}}-\frac{7 \operatorname{Subst}\left (\int \frac{1}{x^3 (a+b x)^{3/2}} \, dx,x,\frac{1}{x}\right )}{3 a}\\ &=-\frac{2 x^2}{3 a \left (a+\frac{b}{x}\right )^{3/2}}-\frac{14 x^2}{3 a^2 \sqrt{a+\frac{b}{x}}}-\frac{35 \operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )}{3 a^2}\\ &=-\frac{2 x^2}{3 a \left (a+\frac{b}{x}\right )^{3/2}}-\frac{14 x^2}{3 a^2 \sqrt{a+\frac{b}{x}}}+\frac{35 \sqrt{a+\frac{b}{x}} x^2}{6 a^3}+\frac{(35 b) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )}{4 a^3}\\ &=-\frac{35 b \sqrt{a+\frac{b}{x}} x}{4 a^4}-\frac{2 x^2}{3 a \left (a+\frac{b}{x}\right )^{3/2}}-\frac{14 x^2}{3 a^2 \sqrt{a+\frac{b}{x}}}+\frac{35 \sqrt{a+\frac{b}{x}} x^2}{6 a^3}-\frac{\left (35 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )}{8 a^4}\\ &=-\frac{35 b \sqrt{a+\frac{b}{x}} x}{4 a^4}-\frac{2 x^2}{3 a \left (a+\frac{b}{x}\right )^{3/2}}-\frac{14 x^2}{3 a^2 \sqrt{a+\frac{b}{x}}}+\frac{35 \sqrt{a+\frac{b}{x}} x^2}{6 a^3}-\frac{(35 b) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+\frac{b}{x}}\right )}{4 a^4}\\ &=-\frac{35 b \sqrt{a+\frac{b}{x}} x}{4 a^4}-\frac{2 x^2}{3 a \left (a+\frac{b}{x}\right )^{3/2}}-\frac{14 x^2}{3 a^2 \sqrt{a+\frac{b}{x}}}+\frac{35 \sqrt{a+\frac{b}{x}} x^2}{6 a^3}+\frac{35 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{4 a^{9/2}}\\ \end{align*}

Mathematica [C]  time = 0.0106617, size = 39, normalized size = 0.34 \[ -\frac{2 b^2 \, _2F_1\left (-\frac{3}{2},3;-\frac{1}{2};\frac{b}{a x}+1\right )}{3 a^3 \left (a+\frac{b}{x}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x/(a + b/x)^(5/2),x]

[Out]

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

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Maple [B]  time = 0.009, size = 531, normalized size = 4.7 \begin{align*}{\frac{x}{24\, \left ( ax+b \right ) ^{3}}\sqrt{{\frac{ax+b}{x}}} \left ( 12\,\sqrt{a{x}^{2}+bx}{a}^{11/2}{x}^{4}-216\,\sqrt{ \left ( ax+b \right ) x}{a}^{9/2}{x}^{3}b+42\,\sqrt{a{x}^{2}+bx}{a}^{9/2}{x}^{3}b+108\,{a}^{4}\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+b \right ) x}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{3}{b}^{2}+144\, \left ( \left ( ax+b \right ) x \right ) ^{3/2}{a}^{7/2}xb-648\,\sqrt{ \left ( ax+b \right ) x}{a}^{7/2}{x}^{2}{b}^{2}+54\,\sqrt{a{x}^{2}+bx}{a}^{7/2}{x}^{2}{b}^{2}+324\,{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}-3\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{3}{a}^{4}{b}^{2}+128\, \left ( \left ( ax+b \right ) x \right ) ^{3/2}{a}^{5/2}{b}^{2}-648\,\sqrt{ \left ( ax+b \right ) x}{a}^{5/2}x{b}^{3}+30\,\sqrt{a{x}^{2}+bx}{a}^{5/2}x{b}^{3}+324\,{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}-9\,\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}-216\,\sqrt{ \left ( ax+b \right ) x}{a}^{3/2}{b}^{4}+6\,\sqrt{a{x}^{2}+bx}{a}^{3/2}{b}^{4}+108\,a\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+b \right ) x}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){b}^{5}-9\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) x{a}^{2}{b}^{4}-3\,\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/(a+b/x)^(5/2),x)

[Out]

1/24*((a*x+b)/x)^(1/2)*x/a^(11/2)*(12*(a*x^2+b*x)^(1/2)*a^(11/2)*x^4-216*((a*x+b)*x)^(1/2)*a^(9/2)*x^3*b+42*(a
*x^2+b*x)^(1/2)*a^(9/2)*x^3*b+108*a^4*ln(1/2*(2*((a*x+b)*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*x^3*b^2+144*((a*x+
b)*x)^(3/2)*a^(7/2)*x*b-648*((a*x+b)*x)^(1/2)*a^(7/2)*x^2*b^2+54*(a*x^2+b*x)^(1/2)*a^(7/2)*x^2*b^2+324*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^3-3*ln(1/2*(2*(a*x^2+b*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1
/2))*x^3*a^4*b^2+128*((a*x+b)*x)^(3/2)*a^(5/2)*b^2-648*((a*x+b)*x)^(1/2)*a^(5/2)*x*b^3+30*(a*x^2+b*x)^(1/2)*a^
(5/2)*x*b^3+324*a^2*ln(1/2*(2*((a*x+b)*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*x*b^4-9*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^3-216*((a*x+b)*x)^(1/2)*a^(3/2)*b^4+6*(a*x^2+b*x)^(1/2)*a^(3/2)*b^4+108*a*
ln(1/2*(2*((a*x+b)*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*b^5-9*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^4-3*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)^3

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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/(a+b/x)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.76382, size = 564, normalized size = 4.95 \begin{align*} \left [\frac{105 \,{\left (a^{2} b^{2} x^{2} + 2 \, 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 (6 \, a^{4} x^{4} - 21 \, a^{3} b x^{3} - 140 \, a^{2} b^{2} x^{2} - 105 \, a b^{3} x\right )} \sqrt{\frac{a x + b}{x}}}{24 \,{\left (a^{7} x^{2} + 2 \, a^{6} b x + a^{5} b^{2}\right )}}, -\frac{105 \,{\left (a^{2} b^{2} x^{2} + 2 \, a b^{3} x + b^{4}\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}{a}\right ) -{\left (6 \, a^{4} x^{4} - 21 \, a^{3} b x^{3} - 140 \, a^{2} b^{2} x^{2} - 105 \, a b^{3} x\right )} \sqrt{\frac{a x + b}{x}}}{12 \,{\left (a^{7} x^{2} + 2 \, a^{6} b x + a^{5} b^{2}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a+b/x)^(5/2),x, algorithm="fricas")

[Out]

[1/24*(105*(a^2*b^2*x^2 + 2*a*b^3*x + b^4)*sqrt(a)*log(2*a*x + 2*sqrt(a)*x*sqrt((a*x + b)/x) + b) + 2*(6*a^4*x
^4 - 21*a^3*b*x^3 - 140*a^2*b^2*x^2 - 105*a*b^3*x)*sqrt((a*x + b)/x))/(a^7*x^2 + 2*a^6*b*x + a^5*b^2), -1/12*(
105*(a^2*b^2*x^2 + 2*a*b^3*x + b^4)*sqrt(-a)*arctan(sqrt(-a)*sqrt((a*x + b)/x)/a) - (6*a^4*x^4 - 21*a^3*b*x^3
- 140*a^2*b^2*x^2 - 105*a*b^3*x)*sqrt((a*x + b)/x))/(a^7*x^2 + 2*a^6*b*x + a^5*b^2)]

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Sympy [B]  time = 7.36674, size = 464, normalized size = 4.07 \begin{align*} \frac{6 a^{\frac{89}{2}} b^{75} x^{49}}{12 a^{\frac{93}{2}} b^{\frac{151}{2}} x^{\frac{93}{2}} \sqrt{\frac{a x}{b} + 1} + 12 a^{\frac{91}{2}} b^{\frac{153}{2}} x^{\frac{91}{2}} \sqrt{\frac{a x}{b} + 1}} - \frac{21 a^{\frac{87}{2}} b^{76} x^{48}}{12 a^{\frac{93}{2}} b^{\frac{151}{2}} x^{\frac{93}{2}} \sqrt{\frac{a x}{b} + 1} + 12 a^{\frac{91}{2}} b^{\frac{153}{2}} x^{\frac{91}{2}} \sqrt{\frac{a x}{b} + 1}} - \frac{140 a^{\frac{85}{2}} b^{77} x^{47}}{12 a^{\frac{93}{2}} b^{\frac{151}{2}} x^{\frac{93}{2}} \sqrt{\frac{a x}{b} + 1} + 12 a^{\frac{91}{2}} b^{\frac{153}{2}} x^{\frac{91}{2}} \sqrt{\frac{a x}{b} + 1}} - \frac{105 a^{\frac{83}{2}} b^{78} x^{46}}{12 a^{\frac{93}{2}} b^{\frac{151}{2}} x^{\frac{93}{2}} \sqrt{\frac{a x}{b} + 1} + 12 a^{\frac{91}{2}} b^{\frac{153}{2}} x^{\frac{91}{2}} \sqrt{\frac{a x}{b} + 1}} + \frac{105 a^{42} b^{\frac{155}{2}} x^{\frac{93}{2}} \sqrt{\frac{a x}{b} + 1} \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{12 a^{\frac{93}{2}} b^{\frac{151}{2}} x^{\frac{93}{2}} \sqrt{\frac{a x}{b} + 1} + 12 a^{\frac{91}{2}} b^{\frac{153}{2}} x^{\frac{91}{2}} \sqrt{\frac{a x}{b} + 1}} + \frac{105 a^{41} b^{\frac{157}{2}} x^{\frac{91}{2}} \sqrt{\frac{a x}{b} + 1} \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{12 a^{\frac{93}{2}} b^{\frac{151}{2}} x^{\frac{93}{2}} \sqrt{\frac{a x}{b} + 1} + 12 a^{\frac{91}{2}} b^{\frac{153}{2}} x^{\frac{91}{2}} \sqrt{\frac{a x}{b} + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a+b/x)**(5/2),x)

[Out]

6*a**(89/2)*b**75*x**49/(12*a**(93/2)*b**(151/2)*x**(93/2)*sqrt(a*x/b + 1) + 12*a**(91/2)*b**(153/2)*x**(91/2)
*sqrt(a*x/b + 1)) - 21*a**(87/2)*b**76*x**48/(12*a**(93/2)*b**(151/2)*x**(93/2)*sqrt(a*x/b + 1) + 12*a**(91/2)
*b**(153/2)*x**(91/2)*sqrt(a*x/b + 1)) - 140*a**(85/2)*b**77*x**47/(12*a**(93/2)*b**(151/2)*x**(93/2)*sqrt(a*x
/b + 1) + 12*a**(91/2)*b**(153/2)*x**(91/2)*sqrt(a*x/b + 1)) - 105*a**(83/2)*b**78*x**46/(12*a**(93/2)*b**(151
/2)*x**(93/2)*sqrt(a*x/b + 1) + 12*a**(91/2)*b**(153/2)*x**(91/2)*sqrt(a*x/b + 1)) + 105*a**42*b**(155/2)*x**(
93/2)*sqrt(a*x/b + 1)*asinh(sqrt(a)*sqrt(x)/sqrt(b))/(12*a**(93/2)*b**(151/2)*x**(93/2)*sqrt(a*x/b + 1) + 12*a
**(91/2)*b**(153/2)*x**(91/2)*sqrt(a*x/b + 1)) + 105*a**41*b**(157/2)*x**(91/2)*sqrt(a*x/b + 1)*asinh(sqrt(a)*
sqrt(x)/sqrt(b))/(12*a**(93/2)*b**(151/2)*x**(93/2)*sqrt(a*x/b + 1) + 12*a**(91/2)*b**(153/2)*x**(91/2)*sqrt(a
*x/b + 1))

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Giac [A]  time = 1.27404, size = 169, normalized size = 1.48 \begin{align*} -\frac{1}{12} \, b^{2}{\left (\frac{8 \,{\left (a + \frac{9 \,{\left (a x + b\right )}}{x}\right )} x}{{\left (a x + b\right )} a^{4} \sqrt{\frac{a x + b}{x}}} + \frac{105 \, \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{4}} - \frac{3 \,{\left (13 \, a \sqrt{\frac{a x + b}{x}} - \frac{11 \,{\left (a x + b\right )} \sqrt{\frac{a x + b}{x}}}{x}\right )}}{{\left (a - \frac{a x + b}{x}\right )}^{2} a^{4}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(a+b/x)^(5/2),x, algorithm="giac")

[Out]

-1/12*b^2*(8*(a + 9*(a*x + b)/x)*x/((a*x + b)*a^4*sqrt((a*x + b)/x)) + 105*arctan(sqrt((a*x + b)/x)/sqrt(-a))/
(sqrt(-a)*a^4) - 3*(13*a*sqrt((a*x + b)/x) - 11*(a*x + b)*sqrt((a*x + b)/x)/x)/((a - (a*x + b)/x)^2*a^4))