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

Optimal. Leaf size=76 \[ \frac{3 b c-2 a d}{a^2 \sqrt{a+\frac{b}{x}}}-\frac{(3 b c-2 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{5/2}}+\frac{c x}{a \sqrt{a+\frac{b}{x}}} \]

[Out]

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

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Rubi [A]  time = 0.050474, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {375, 78, 51, 63, 208} \[ \frac{3 b c-2 a d}{a^2 \sqrt{a+\frac{b}{x}}}-\frac{(3 b c-2 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{5/2}}+\frac{c x}{a \sqrt{a+\frac{b}{x}}} \]

Antiderivative was successfully verified.

[In]

Int[(c + d/x)/(a + b/x)^(3/2),x]

[Out]

(3*b*c - 2*a*d)/(a^2*Sqrt[a + b/x]) + (c*x)/(a*Sqrt[a + b/x]) - ((3*b*c - 2*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]
])/a^(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 78

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

Mathematica [C]  time = 0.0213857, size = 48, normalized size = 0.63 \[ \frac{(3 b c-2 a d) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{b}{a x}+1\right )+a c x}{a^2 \sqrt{a+\frac{b}{x}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.01, size = 387, normalized size = 5.1 \begin{align*}{\frac{x}{2\,b \left ( ax+b \right ) ^{2}}\sqrt{{\frac{ax+b}{x}}} \left ( 2\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+b \right ) x}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{2}{a}^{3}bd-3\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+b \right ) x}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{2}{a}^{2}{b}^{2}c-4\,{a}^{7/2}\sqrt{ \left ( ax+b \right ) x}{x}^{2}d+6\,{a}^{5/2}\sqrt{ \left ( ax+b \right ) x}{x}^{2}bc+4\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+b \right ) x}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) x{a}^{2}{b}^{2}d-6\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+b \right ) x}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) xa{b}^{3}c+4\,{a}^{5/2} \left ( \left ( ax+b \right ) x \right ) ^{3/2}d-4\,{a}^{3/2} \left ( \left ( ax+b \right ) x \right ) ^{3/2}bc-8\,{a}^{5/2}\sqrt{ \left ( ax+b \right ) x}xbd+12\,{a}^{3/2}\sqrt{ \left ( ax+b \right ) x}x{b}^{2}c+2\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+b \right ) x}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ) a{b}^{3}d-3\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+b \right ) x}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){b}^{4}c-4\,{a}^{3/2}\sqrt{ \left ( ax+b \right ) x}{b}^{2}d+6\,\sqrt{a}\sqrt{ \left ( ax+b \right ) x}{b}^{3}c \right ){a}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{ \left ( ax+b \right ) x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*((a*x+b)/x)^(1/2)*x/a^(5/2)*(2*ln(1/2*(2*((a*x+b)*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*x^2*a^3*b*d-3*ln(1/2*
(2*((a*x+b)*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*x^2*a^2*b^2*c-4*a^(7/2)*((a*x+b)*x)^(1/2)*x^2*d+6*a^(5/2)*((a*x
+b)*x)^(1/2)*x^2*b*c+4*ln(1/2*(2*((a*x+b)*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*x*a^2*b^2*d-6*ln(1/2*(2*((a*x+b)*
x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*x*a*b^3*c+4*a^(5/2)*((a*x+b)*x)^(3/2)*d-4*a^(3/2)*((a*x+b)*x)^(3/2)*b*c-8*a
^(5/2)*((a*x+b)*x)^(1/2)*x*b*d+12*a^(3/2)*((a*x+b)*x)^(1/2)*x*b^2*c+2*ln(1/2*(2*((a*x+b)*x)^(1/2)*a^(1/2)+2*a*
x+b)/a^(1/2))*a*b^3*d-3*ln(1/2*(2*((a*x+b)*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*b^4*c-4*a^(3/2)*((a*x+b)*x)^(1/2
)*b^2*d+6*a^(1/2)*((a*x+b)*x)^(1/2)*b^3*c)/((a*x+b)*x)^(1/2)/b/(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((c+d/x)/(a+b/x)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.35408, size = 473, normalized size = 6.22 \begin{align*} \left [-\frac{{\left (3 \, b^{2} c - 2 \, a b d +{\left (3 \, a b c - 2 \, a^{2} d\right )} x\right )} \sqrt{a} \log \left (2 \, a x + 2 \, \sqrt{a} x \sqrt{\frac{a x + b}{x}} + b\right ) - 2 \,{\left (a^{2} c x^{2} +{\left (3 \, a b c - 2 \, a^{2} d\right )} x\right )} \sqrt{\frac{a x + b}{x}}}{2 \,{\left (a^{4} x + a^{3} b\right )}}, \frac{{\left (3 \, b^{2} c - 2 \, a b d +{\left (3 \, a b c - 2 \, a^{2} d\right )} x\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}{a}\right ) +{\left (a^{2} c x^{2} +{\left (3 \, a b c - 2 \, a^{2} d\right )} x\right )} \sqrt{\frac{a x + b}{x}}}{a^{4} x + a^{3} b}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 21.5894, size = 224, normalized size = 2.95 \begin{align*} c \left (\frac{x^{\frac{3}{2}}}{a \sqrt{b} \sqrt{\frac{a x}{b} + 1}} + \frac{3 \sqrt{b} \sqrt{x}}{a^{2} \sqrt{\frac{a x}{b} + 1}} - \frac{3 b \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{a^{\frac{5}{2}}}\right ) + d \left (- \frac{2 a^{3} x \sqrt{1 + \frac{b}{a x}}}{a^{\frac{9}{2}} x + a^{\frac{7}{2}} b} - \frac{a^{3} x \log{\left (\frac{b}{a x} \right )}}{a^{\frac{9}{2}} x + a^{\frac{7}{2}} b} + \frac{2 a^{3} x \log{\left (\sqrt{1 + \frac{b}{a x}} + 1 \right )}}{a^{\frac{9}{2}} x + a^{\frac{7}{2}} b} - \frac{a^{2} b \log{\left (\frac{b}{a x} \right )}}{a^{\frac{9}{2}} x + a^{\frac{7}{2}} b} + \frac{2 a^{2} b \log{\left (\sqrt{1 + \frac{b}{a x}} + 1 \right )}}{a^{\frac{9}{2}} x + a^{\frac{7}{2}} b}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*(x**(3/2)/(a*sqrt(b)*sqrt(a*x/b + 1)) + 3*sqrt(b)*sqrt(x)/(a**2*sqrt(a*x/b + 1)) - 3*b*asinh(sqrt(a)*sqrt(x)
/sqrt(b))/a**(5/2)) + d*(-2*a**3*x*sqrt(1 + b/(a*x))/(a**(9/2)*x + a**(7/2)*b) - a**3*x*log(b/(a*x))/(a**(9/2)
*x + a**(7/2)*b) + 2*a**3*x*log(sqrt(1 + b/(a*x)) + 1)/(a**(9/2)*x + a**(7/2)*b) - a**2*b*log(b/(a*x))/(a**(9/
2)*x + a**(7/2)*b) + 2*a**2*b*log(sqrt(1 + b/(a*x)) + 1)/(a**(9/2)*x + a**(7/2)*b))

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Giac [A]  time = 1.19867, size = 165, normalized size = 2.17 \begin{align*} b{\left (\frac{{\left (3 \, b c - 2 \, a d\right )} \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2} b} + \frac{2 \, a b c - 2 \, a^{2} d - \frac{3 \,{\left (a x + b\right )} b c}{x} + \frac{2 \,{\left (a x + b\right )} a d}{x}}{{\left (a \sqrt{\frac{a x + b}{x}} - \frac{{\left (a x + b\right )} \sqrt{\frac{a x + b}{x}}}{x}\right )} a^{2} b}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*((3*b*c - 2*a*d)*arctan(sqrt((a*x + b)/x)/sqrt(-a))/(sqrt(-a)*a^2*b) + (2*a*b*c - 2*a^2*d - 3*(a*x + b)*b*c/
x + 2*(a*x + b)*a*d/x)/((a*sqrt((a*x + b)/x) - (a*x + b)*sqrt((a*x + b)/x)/x)*a^2*b))