3.268 \(\int \frac{\sqrt{a+\frac{b}{x}}}{(c+\frac{d}{x})^{3/2}} \, dx\)

Optimal. Leaf size=122 \[ -\frac{\sqrt{a+\frac{b}{x}} (b c-3 a d)}{a c^2 \sqrt{c+\frac{d}{x}}}+\frac{(b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+\frac{b}{x}}}{\sqrt{a} \sqrt{c+\frac{d}{x}}}\right )}{\sqrt{a} c^{5/2}}+\frac{x \left (a+\frac{b}{x}\right )^{3/2}}{a c \sqrt{c+\frac{d}{x}}} \]

[Out]

-(((b*c - 3*a*d)*Sqrt[a + b/x])/(a*c^2*Sqrt[c + d/x])) + ((a + b/x)^(3/2)*x)/(a*c*Sqrt[c + d/x]) + ((b*c - 3*a
*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b/x])/(Sqrt[a]*Sqrt[c + d/x])])/(Sqrt[a]*c^(5/2))

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Rubi [A]  time = 0.0803529, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {375, 96, 94, 93, 208} \[ -\frac{\sqrt{a+\frac{b}{x}} (b c-3 a d)}{a c^2 \sqrt{c+\frac{d}{x}}}+\frac{(b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+\frac{b}{x}}}{\sqrt{a} \sqrt{c+\frac{d}{x}}}\right )}{\sqrt{a} c^{5/2}}+\frac{x \left (a+\frac{b}{x}\right )^{3/2}}{a c \sqrt{c+\frac{d}{x}}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b/x]/(c + d/x)^(3/2),x]

[Out]

-(((b*c - 3*a*d)*Sqrt[a + b/x])/(a*c^2*Sqrt[c + d/x])) + ((a + b/x)^(3/2)*x)/(a*c*Sqrt[c + d/x]) + ((b*c - 3*a
*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b/x])/(Sqrt[a]*Sqrt[c + d/x])])/(Sqrt[a]*c^(5/2))

Rule 375

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> -Subst[Int[((a + b/x^n)^p*(c +
d/x^n)^q)/x^2, x], x, 1/x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && ILtQ[n, 0]

Rule 96

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[(a*d*f*(m + 1)
+ b*c*f*(n + 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*
x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || Sum
SimplerQ[m, 1])

Rule 94

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a + b
*x)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1))/((m + 1)*(b*e - a*f)), x] - Dist[(n*(d*e - c*f))/((m + 1)*(b*e - a*
f)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[
m + n + p + 2, 0] && GtQ[n, 0] &&  !(SumSimplerQ[p, 1] &&  !SumSimplerQ[m, 1])

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*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{\sqrt{a+\frac{b}{x}}}{\left (c+\frac{d}{x}\right )^{3/2}} \, dx &=-\operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x^2 (c+d x)^{3/2}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{\left (a+\frac{b}{x}\right )^{3/2} x}{a c \sqrt{c+\frac{d}{x}}}+\frac{\left (-\frac{b c}{2}+\frac{3 a d}{2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x (c+d x)^{3/2}} \, dx,x,\frac{1}{x}\right )}{a c}\\ &=-\frac{(b c-3 a d) \sqrt{a+\frac{b}{x}}}{a c^2 \sqrt{c+\frac{d}{x}}}+\frac{\left (a+\frac{b}{x}\right )^{3/2} x}{a c \sqrt{c+\frac{d}{x}}}-\frac{(b c-3 a d) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx,x,\frac{1}{x}\right )}{2 c^2}\\ &=-\frac{(b c-3 a d) \sqrt{a+\frac{b}{x}}}{a c^2 \sqrt{c+\frac{d}{x}}}+\frac{\left (a+\frac{b}{x}\right )^{3/2} x}{a c \sqrt{c+\frac{d}{x}}}-\frac{(b c-3 a d) \operatorname{Subst}\left (\int \frac{1}{-a+c x^2} \, dx,x,\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{c+\frac{d}{x}}}\right )}{c^2}\\ &=-\frac{(b c-3 a d) \sqrt{a+\frac{b}{x}}}{a c^2 \sqrt{c+\frac{d}{x}}}+\frac{\left (a+\frac{b}{x}\right )^{3/2} x}{a c \sqrt{c+\frac{d}{x}}}+\frac{(b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+\frac{b}{x}}}{\sqrt{a} \sqrt{c+\frac{d}{x}}}\right )}{\sqrt{a} c^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.0948413, size = 87, normalized size = 0.71 \[ \frac{\sqrt{a+\frac{b}{x}} (c x+3 d)}{c^2 \sqrt{c+\frac{d}{x}}}+\frac{(b c-3 a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+\frac{b}{x}}}{\sqrt{a} \sqrt{c+\frac{d}{x}}}\right )}{\sqrt{a} c^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b/x]/(c + d/x)^(3/2),x]

[Out]

(Sqrt[a + b/x]*(3*d + c*x))/(c^2*Sqrt[c + d/x]) + ((b*c - 3*a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b/x])/(Sqrt[a]*Sqrt
[c + d/x])])/(Sqrt[a]*c^(5/2))

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Maple [B]  time = 0.032, size = 280, normalized size = 2.3 \begin{align*}{\frac{x}{ \left ( 2\,cx+2\,d \right ){c}^{2}}\sqrt{{\frac{ax+b}{x}}}\sqrt{{\frac{cx+d}{x}}} \left ( -3\,\ln \left ( 1/2\,{\frac{2\,acx+2\,\sqrt{ \left ( cx+d \right ) \left ( ax+b \right ) }\sqrt{ac}+ad+bc}{\sqrt{ac}}} \right ) xacd+\ln \left ({\frac{1}{2} \left ( 2\,acx+2\,\sqrt{ \left ( cx+d \right ) \left ( ax+b \right ) }\sqrt{ac}+ad+bc \right ){\frac{1}{\sqrt{ac}}}} \right ) xb{c}^{2}+2\,xc\sqrt{ \left ( cx+d \right ) \left ( ax+b \right ) }\sqrt{ac}-3\,\ln \left ( 1/2\,{\frac{2\,acx+2\,\sqrt{ \left ( cx+d \right ) \left ( ax+b \right ) }\sqrt{ac}+ad+bc}{\sqrt{ac}}} \right ) a{d}^{2}+\ln \left ({\frac{1}{2} \left ( 2\,acx+2\,\sqrt{ \left ( cx+d \right ) \left ( ax+b \right ) }\sqrt{ac}+ad+bc \right ){\frac{1}{\sqrt{ac}}}} \right ) bcd+6\,d\sqrt{ \left ( cx+d \right ) \left ( ax+b \right ) }\sqrt{ac} \right ){\frac{1}{\sqrt{ \left ( cx+d \right ) \left ( ax+b \right ) }}}{\frac{1}{\sqrt{ac}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a + \frac{b}{x}}}{{\left (c + \frac{d}{x}\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(a + b/x)/(c + d/x)^(3/2), x)

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Fricas [A]  time = 2.08185, size = 710, normalized size = 5.82 \begin{align*} \left [-\frac{{\left (b c d - 3 \, a d^{2} +{\left (b c^{2} - 3 \, a c d\right )} x\right )} \sqrt{a c} \log \left (-8 \, a^{2} c^{2} x^{2} - b^{2} c^{2} - 6 \, a b c d - a^{2} d^{2} + 4 \,{\left (2 \, a c x^{2} +{\left (b c + a d\right )} x\right )} \sqrt{a c} \sqrt{\frac{a x + b}{x}} \sqrt{\frac{c x + d}{x}} - 8 \,{\left (a b c^{2} + a^{2} c d\right )} x\right ) - 4 \,{\left (a c^{2} x^{2} + 3 \, a c d x\right )} \sqrt{\frac{a x + b}{x}} \sqrt{\frac{c x + d}{x}}}{4 \,{\left (a c^{4} x + a c^{3} d\right )}}, -\frac{{\left (b c d - 3 \, a d^{2} +{\left (b c^{2} - 3 \, a c d\right )} x\right )} \sqrt{-a c} \arctan \left (\frac{2 \, \sqrt{-a c} x \sqrt{\frac{a x + b}{x}} \sqrt{\frac{c x + d}{x}}}{2 \, a c x + b c + a d}\right ) - 2 \,{\left (a c^{2} x^{2} + 3 \, a c d x\right )} \sqrt{\frac{a x + b}{x}} \sqrt{\frac{c x + d}{x}}}{2 \,{\left (a c^{4} x + a c^{3} d\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/4*((b*c*d - 3*a*d^2 + (b*c^2 - 3*a*c*d)*x)*sqrt(a*c)*log(-8*a^2*c^2*x^2 - b^2*c^2 - 6*a*b*c*d - a^2*d^2 +
4*(2*a*c*x^2 + (b*c + a*d)*x)*sqrt(a*c)*sqrt((a*x + b)/x)*sqrt((c*x + d)/x) - 8*(a*b*c^2 + a^2*c*d)*x) - 4*(a*
c^2*x^2 + 3*a*c*d*x)*sqrt((a*x + b)/x)*sqrt((c*x + d)/x))/(a*c^4*x + a*c^3*d), -1/2*((b*c*d - 3*a*d^2 + (b*c^2
- 3*a*c*d)*x)*sqrt(-a*c)*arctan(2*sqrt(-a*c)*x*sqrt((a*x + b)/x)*sqrt((c*x + d)/x)/(2*a*c*x + b*c + a*d)) - 2
*(a*c^2*x^2 + 3*a*c*d*x)*sqrt((a*x + b)/x)*sqrt((c*x + d)/x))/(a*c^4*x + a*c^3*d)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a + \frac{b}{x}}}{\left (c + \frac{d}{x}\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a + b/x)/(c + d/x)**(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError