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

Optimal. Leaf size=51 \[ \frac {c x \sqrt {a+\frac {b}{x}}}{a}-\frac {(b c-2 a d) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {375, 78, 63, 208} \[ \frac {c x \sqrt {a+\frac {b}{x}}}{a}-\frac {(b c-2 a d) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(c + d/x)/Sqrt[a + b/x],x]

[Out]

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

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 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 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]

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]

Rubi steps

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

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Mathematica [A]  time = 0.04, size = 53, normalized size = 1.04 \[ \frac {2 \left (a d-\frac {b c}{2}\right ) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{3/2}}+\frac {c x \sqrt {a+\frac {b}{x}}}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d/x)/Sqrt[a + b/x],x]

[Out]

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

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fricas [A]  time = 0.98, size = 115, normalized size = 2.25 \[ \left [\frac {2 \, a c x \sqrt {\frac {a x + b}{x}} - {\left (b c - 2 \, a d\right )} \sqrt {a} \log \left (2 \, a x + 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right )}{2 \, a^{2}}, \frac {a c x \sqrt {\frac {a x + b}{x}} + {\left (b c - 2 \, a d\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a x + b}{x}}}{a}\right )}{a^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.20, size = 78, normalized size = 1.53 \[ -\frac {\frac {b^{2} c \sqrt {\frac {a x + b}{x}}}{{\left (a - \frac {a x + b}{x}\right )} a} - \frac {{\left (b^{2} c - 2 \, a b d\right )} \arctan \left (\frac {\sqrt {\frac {a x + b}{x}}}{\sqrt {-a}}\right )}{\sqrt {-a} a}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.06, size = 173, normalized size = 3.39 \[ \frac {\sqrt {\frac {a x +b}{x}}\, \left (a b d \ln \left (\frac {2 a x +b +2 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}}{2 \sqrt {a}}\right )+a b d \ln \left (\frac {2 a x +b +2 \sqrt {a \,x^{2}+b x}\, \sqrt {a}}{2 \sqrt {a}}\right )-b^{2} c \ln \left (\frac {2 a x +b +2 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}}{2 \sqrt {a}}\right )+2 \sqrt {a \,x^{2}+b x}\, a^{\frac {3}{2}} d -2 \sqrt {\left (a x +b \right ) x}\, a^{\frac {3}{2}} d +2 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}\, b c \right ) x}{2 \sqrt {\left (a x +b \right ) x}\, a^{\frac {3}{2}} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.30, size = 109, normalized size = 2.14 \[ \frac {1}{2} \, c {\left (\frac {2 \, \sqrt {a + \frac {b}{x}} b}{{\left (a + \frac {b}{x}\right )} a - a^{2}} + \frac {b \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right )}{a^{\frac {3}{2}}}\right )} - \frac {d \log \left (\frac {\sqrt {a + \frac {b}{x}} - \sqrt {a}}{\sqrt {a + \frac {b}{x}} + \sqrt {a}}\right )}{\sqrt {a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*c*(2*sqrt(a + b/x)*b/((a + b/x)*a - a^2) + b*log((sqrt(a + b/x) - sqrt(a))/(sqrt(a + b/x) + sqrt(a)))/a^(3
/2)) - d*log((sqrt(a + b/x) - sqrt(a))/(sqrt(a + b/x) + sqrt(a)))/sqrt(a)

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mupad [B]  time = 1.98, size = 88, normalized size = 1.73 \[ \frac {2\,d\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{\sqrt {a}}+\frac {2\,c\,x\,\left (\frac {3\,\sqrt {b}\,\sqrt {b+a\,x}}{2\,a\,x}+\frac {b^{3/2}\,\mathrm {asin}\left (\frac {\sqrt {a}\,\sqrt {x}\,1{}\mathrm {i}}{\sqrt {b}}\right )\,3{}\mathrm {i}}{2\,a^{3/2}\,x^{3/2}}\right )\,\sqrt {\frac {a\,x}{b}+1}}{3\,\sqrt {a+\frac {b}{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d/x)/(a + b/x)^(1/2),x)

[Out]

(2*d*atanh((a + b/x)^(1/2)/a^(1/2)))/a^(1/2) + (2*c*x*((3*b^(1/2)*(b + a*x)^(1/2))/(2*a*x) + (b^(3/2)*asin((a^
(1/2)*x^(1/2)*1i)/b^(1/2))*3i)/(2*a^(3/2)*x^(3/2)))*((a*x)/b + 1)^(1/2))/(3*(a + b/x)^(1/2))

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sympy [A]  time = 60.25, size = 82, normalized size = 1.61 \[ \frac {\sqrt {b} c \sqrt {x} \sqrt {\frac {a x}{b} + 1}}{a} - \frac {2 d \operatorname {atan}{\left (\frac {1}{\sqrt {- \frac {1}{a}} \sqrt {a + \frac {b}{x}}} \right )}}{a \sqrt {- \frac {1}{a}}} - \frac {b c \operatorname {asinh}{\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}} \right )}}{a^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(b)*c*sqrt(x)*sqrt(a*x/b + 1)/a - 2*d*atan(1/(sqrt(-1/a)*sqrt(a + b/x)))/(a*sqrt(-1/a)) - b*c*asinh(sqrt(a
)*sqrt(x)/sqrt(b))/a**(3/2)

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