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

Optimal. Leaf size=73 \[ -\frac{c (b c-4 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{3/2}}+\frac{c^2 x \sqrt{a+\frac{b}{x}}}{a}-\frac{2 d^2 \sqrt{a+\frac{b}{x}}}{b} \]

[Out]

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

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Rubi [A]  time = 0.0540239, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {375, 89, 80, 63, 208} \[ -\frac{c (b c-4 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{3/2}}+\frac{c^2 x \sqrt{a+\frac{b}{x}}}{a}-\frac{2 d^2 \sqrt{a+\frac{b}{x}}}{b} \]

Antiderivative was successfully verified.

[In]

Int[(c + d/x)^2/Sqrt[a + b/x],x]

[Out]

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

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 89

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c - a*
d)^2*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d^2*(d*e - c*f)*(n + 1)), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

Rule 80

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

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

Mathematica [A]  time = 0.0714159, size = 66, normalized size = 0.9 \[ \frac{c (4 a d-b c) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{3/2}}+\frac{\sqrt{a+\frac{b}{x}} \left (b c^2 x-2 a d^2\right )}{a b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.013, size = 348, normalized size = 4.8 \begin{align*} -{\frac{1}{2\,{b}^{2}x}\sqrt{{\frac{ax+b}{x}}} \left ( -2\,\sqrt{a{x}^{2}+bx}{a}^{5/2}{x}^{2}{d}^{2}-4\,\sqrt{a{x}^{2}+bx}{a}^{3/2}{x}^{2}bcd-\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b \right ){\frac{1}{\sqrt{a}}}} \right ){x}^{2}{a}^{2}b{d}^{2}-2\,\ln \left ( 1/2\,{\frac{2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{2}a{b}^{2}cd-2\,{a}^{5/2}\sqrt{ \left ( ax+b \right ) x}{x}^{2}{d}^{2}+4\,{a}^{3/2}\sqrt{ \left ( ax+b \right ) x}{x}^{2}bcd-2\,\sqrt{a}\sqrt{ \left ( ax+b \right ) x}{x}^{2}{b}^{2}{c}^{2}+\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{ \left ( ax+b \right ) x}\sqrt{a}+2\,ax+b \right ){\frac{1}{\sqrt{a}}}} \right ){x}^{2}{a}^{2}b{d}^{2}-2\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax+b \right ) x}\sqrt{a}+2\,ax+b}{\sqrt{a}}} \right ){x}^{2}a{b}^{2}cd+\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{ \left ( ax+b \right ) x}\sqrt{a}+2\,ax+b \right ){\frac{1}{\sqrt{a}}}} \right ){x}^{2}{b}^{3}{c}^{2}+4\, \left ( a{x}^{2}+bx \right ) ^{3/2}{a}^{3/2}{d}^{2} \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((c+d/x)^2/(a+b/x)^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.36408, size = 359, normalized size = 4.92 \begin{align*} \left [-\frac{{\left (b^{2} c^{2} - 4 \, a b c d\right )} \sqrt{a} \log \left (2 \, a x + 2 \, \sqrt{a} x \sqrt{\frac{a x + b}{x}} + b\right ) - 2 \,{\left (a b c^{2} x - 2 \, a^{2} d^{2}\right )} \sqrt{\frac{a x + b}{x}}}{2 \, a^{2} b}, \frac{{\left (b^{2} c^{2} - 4 \, a b c d\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}{a}\right ) +{\left (a b c^{2} x - 2 \, a^{2} d^{2}\right )} \sqrt{\frac{a x + b}{x}}}{a^{2} b}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 25.404, size = 114, normalized size = 1.56 \begin{align*} d^{2} \left (\begin{cases} - \frac{1}{\sqrt{a} x} & \text{for}\: b = 0 \\- \frac{2 \sqrt{a + \frac{b}{x}}}{b} & \text{otherwise} \end{cases}\right ) + \frac{\sqrt{b} c^{2} \sqrt{x} \sqrt{\frac{a x}{b} + 1}}{a} - \frac{4 c d \operatorname{atan}{\left (\frac{1}{\sqrt{- \frac{1}{a}} \sqrt{a + \frac{b}{x}}} \right )}}{a \sqrt{- \frac{1}{a}}} - \frac{b c^{2} \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{a^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d**2*Piecewise((-1/(sqrt(a)*x), Eq(b, 0)), (-2*sqrt(a + b/x)/b, True)) + sqrt(b)*c**2*sqrt(x)*sqrt(a*x/b + 1)/
a - 4*c*d*atan(1/(sqrt(-1/a)*sqrt(a + b/x)))/(a*sqrt(-1/a)) - b*c**2*asinh(sqrt(a)*sqrt(x)/sqrt(b))/a**(3/2)

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Giac [A]  time = 1.20205, size = 131, normalized size = 1.79 \begin{align*} -b{\left (\frac{c^{2} \sqrt{\frac{a x + b}{x}}}{{\left (a - \frac{a x + b}{x}\right )} a} + \frac{2 \, d^{2} \sqrt{\frac{a x + b}{x}}}{b^{2}} - \frac{{\left (b c^{2} - 4 \, a c d\right )} \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right )}{\sqrt{-a} a b}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b*(c^2*sqrt((a*x + b)/x)/((a - (a*x + b)/x)*a) + 2*d^2*sqrt((a*x + b)/x)/b^2 - (b*c^2 - 4*a*c*d)*arctan(sqrt(
(a*x + b)/x)/sqrt(-a))/(sqrt(-a)*a*b))