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

Optimal. Leaf size=145 \[ -2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}+2 \sqrt {a} \tanh ^{-1}\left (\frac {2 a+b \sqrt {\frac {d}{x}}}{2 \sqrt {a} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )-\frac {b \sqrt {d} \tanh ^{-1}\left (\frac {b d+2 c \sqrt {\frac {d}{x}}}{2 \sqrt {c} \sqrt {d} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )}{\sqrt {c}} \]

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Rubi [A]  time = 0.21, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {1970, 1357, 734, 843, 621, 206, 724} \begin {gather*} -2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}+2 \sqrt {a} \tanh ^{-1}\left (\frac {2 a+b \sqrt {\frac {d}{x}}}{2 \sqrt {a} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )-\frac {b \sqrt {d} \tanh ^{-1}\left (\frac {b d+2 c \sqrt {\frac {d}{x}}}{2 \sqrt {c} \sqrt {d} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )}{\sqrt {c}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b*Sqrt[d/x] + c/x]/x,x]

[Out]

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

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 734

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
a + b*x + c*x^2)^p)/(e*(m + 2*p + 1)), x] - Dist[p/(e*(m + 2*p + 1)), Int[(d + e*x)^m*Simp[b*d - 2*a*e + (2*c*
d - b*e)*x, x]*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ
[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && GtQ[p, 0] && NeQ[m + 2*p + 1, 0] && ( !RationalQ[m] || Lt
Q[m, 1]) &&  !ILtQ[m + 2*p, 0] && IntQuadraticQ[a, b, c, d, e, m, p, x]

Rule 843

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + b*x + c*x^
2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&&  !IGtQ[m, 0]

Rule 1357

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && NeQ[
b^2 - 4*a*c, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 1970

Int[(x_)^(m_.)*((a_) + (b_.)*((d_.)/(x_))^(n_) + (c_.)*(x_)^(n2_.))^(p_.), x_Symbol] :> -Dist[d^(m + 1), Subst
[Int[(a + b*x^n + (c*x^(2*n))/d^(2*n))^p/x^(m + 2), x], x, d/x], x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[n2,
-2*n] && IntegerQ[2*n] && IntegerQ[m]

Rubi steps

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

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Mathematica [F]  time = 0.15, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[Sqrt[a + b*Sqrt[d/x] + c/x]/x,x]

[Out]

Integrate[Sqrt[a + b*Sqrt[d/x] + c/x]/x, x]

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IntegrateAlgebraic [A]  time = 0.58, size = 272, normalized size = 1.88 \begin {gather*} \frac {b d \sqrt {\frac {c}{d}} \log \left (-8 c d \sqrt {\frac {c}{d}} \sqrt {\frac {d}{x}} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}+4 a c d-b^2 d^2+4 b c d \sqrt {\frac {d}{x}}+\frac {8 c^2 d}{x}\right )}{2 c}-2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}+4 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{\sqrt {a}}-\frac {\sqrt {\frac {c}{d}} \sqrt {\frac {d}{x}}}{\sqrt {a}}\right )-\frac {b \sqrt {d} \tanh ^{-1}\left (\frac {2 \sqrt {c} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{b \sqrt {d}}-\frac {2 \sqrt {c} \sqrt {\frac {c}{d}} \sqrt {\frac {d}{x}}}{b \sqrt {d}}\right )}{\sqrt {c}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[a + b*Sqrt[d/x] + c/x]/x,x]

[Out]

-2*Sqrt[a + b*Sqrt[d/x] + c/x] + 4*Sqrt[a]*ArcTanh[Sqrt[a + b*Sqrt[d/x] + c/x]/Sqrt[a] - (Sqrt[c/d]*Sqrt[d/x])
/Sqrt[a]] - (b*Sqrt[d]*ArcTanh[(2*Sqrt[c]*Sqrt[a + b*Sqrt[d/x] + c/x])/(b*Sqrt[d]) - (2*Sqrt[c]*Sqrt[c/d]*Sqrt
[d/x])/(b*Sqrt[d])])/Sqrt[c] + (b*Sqrt[c/d]*d*Log[4*a*c*d - b^2*d^2 + 4*b*c*d*Sqrt[d/x] - 8*c*Sqrt[c/d]*d*Sqrt
[a + b*Sqrt[d/x] + c/x]*Sqrt[d/x] + (8*c^2*d)/x])/(2*c)

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [ab
s(x)]Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):
Check [abs(t_nostep)]Sign error (%%%{(-d*sqrt(a*d))*b+2*sqrt(c)*a*abs(d),0%%%}+%%%{(a*b*sqrt(c)*abs(d)-2*a*c*s
qrt(a*d))/c,1%%%}+%%%{(4*a^2*c*sqrt(c)*abs(d)-a*b^2*d*sqrt(c)*abs(d))/(4*c^2*d),2%%%}+%%%{undef,3%%%})Limit: M
ax order reached or unable to make series expansion Error: Bad Argument Value

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maple [B]  time = 0.16, size = 237, normalized size = 1.63 \begin {gather*} \frac {\sqrt {\frac {a x +\sqrt {\frac {d}{x}}\, b x +c}{x}}\, \left (-\sqrt {\frac {d}{x}}\, a^{\frac {3}{2}} b \sqrt {c}\, x \ln \left (\frac {\sqrt {\frac {d}{x}}\, b x +2 c +2 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, \sqrt {c}}{\sqrt {x}}\right )+2 a^{2} c \sqrt {x}\, \ln \left (\frac {2 a \sqrt {x}+\sqrt {\frac {d}{x}}\, b \sqrt {x}+2 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, \sqrt {a}}{2 \sqrt {a}}\right )+2 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, a^{\frac {5}{2}} x +2 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, \sqrt {\frac {d}{x}}\, a^{\frac {3}{2}} b x -2 \left (a x +\sqrt {\frac {d}{x}}\, b x +c \right )^{\frac {3}{2}} a^{\frac {3}{2}}\right )}{\sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, a^{\frac {3}{2}} c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((a*x+(d/x)^(1/2)*b*x+c)/x)^(1/2)*(-a^(3/2)*c^(1/2)*ln((2*c+(d/x)^(1/2)*b*x+2*c^(1/2)*(a*x+(d/x)^(1/2)*b*x+c)^
(1/2))/x^(1/2))*(d/x)^(1/2)*x*b+2*(a*x+(d/x)^(1/2)*b*x+c)^(1/2)*a^(3/2)*(d/x)^(1/2)*x*b+2*(a*x+(d/x)^(1/2)*b*x
+c)^(1/2)*a^(5/2)*x-2*(a*x+(d/x)^(1/2)*b*x+c)^(3/2)*a^(3/2)+2*ln(1/2*(2*a*x^(1/2)+(d/x)^(1/2)*b*x^(1/2)+2*(a*x
+(d/x)^(1/2)*b*x+c)^(1/2)*a^(1/2))/a^(1/2))*x^(1/2)*a^2*c)/(a*x+(d/x)^(1/2)*b*x+c)^(1/2)/c/a^(3/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {b \sqrt {\frac {d}{x}} + a + \frac {c}{x}}}{x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {a+\frac {c}{x}+b\,\sqrt {\frac {d}{x}}}}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a + b \sqrt {\frac {d}{x}} + \frac {c}{x}}}{x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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