∫ 1______∘ a+ b_ x2 √___

3.891 \(\int \frac {1}{\sqrt {a+\frac {b}{x^2}} \sqrt {c+d x^2}} \, dx\)

Optimal. Leaf size=70 \[ \frac {\sqrt {a x^2+b} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a x^2+b}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{\sqrt {a} \sqrt {d} x \sqrt {a+\frac {b}{x^2}}} \]

[Out]

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

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Rubi [A] time = 0.07, antiderivative size = 70, normalized size of antiderivative = 1.00, 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 = {435, 444, 63, 217, 206} \[ \frac {\sqrt {a x^2+b} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a x^2+b}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{\sqrt {a} \sqrt {d} x \sqrt {a+\frac {b}{x^2}}} \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

]*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 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 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,

b}, x] && !GtQ[a, 0]

Rule 435

Int[((c_) + (d_.)*(x_)^(mn_.))^(q_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Dist[(x^(n*FracPart[q])*(c +

d/x^n)^FracPart[q])/(d + c*x^n)^FracPart[q], Int[((a + b*x^n)^p*(d + c*x^n)^q)/x^(n*q), x], x] /; FreeQ[{a, b,

c, d, n, p, q}, x] && EqQ[mn, -n] && !IntegerQ[q] && !IntegerQ[p]

Rule 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int

[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m

- n + 1, 0]

Rubi steps

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

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Mathematica [A] time = 0.13, size = 105, normalized size = 1.50 \[ \frac {x \sqrt {a+\frac {b}{x^2}} \sqrt {c+d x^2} \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a x^2+b}}{\sqrt {a c-b d}}\right )}{\sqrt {d} \sqrt {a x^2+b} \sqrt {a c-b d} \sqrt {\frac {a \left (c+d x^2\right )}{a c-b d}}} \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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

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fricas [A] time = 0.44, size = 208, normalized size = 2.97 \[ \left [\frac {\sqrt {a d} \log \left (8 \, a^{2} d^{2} x^{4} + a^{2} c^{2} + 6 \, a b c d + b^{2} d^{2} + 8 \, {\left (a^{2} c d + a b d^{2}\right )} x^{2} + 4 \, {\left (2 \, a d x^{3} + {\left (a c + b d\right )} x\right )} \sqrt {d x^{2} + c} \sqrt {a d} \sqrt {\frac {a x^{2} + b}{x^{2}}}\right )}{4 \, a d}, -\frac {\sqrt {-a d} \arctan \left (\frac {{\left (2 \, a d x^{3} + {\left (a c + b d\right )} x\right )} \sqrt {d x^{2} + c} \sqrt {-a d} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{2 \, {\left (a^{2} d^{2} x^{4} + a b c d + {\left (a^{2} c d + a b d^{2}\right )} x^{2}\right )}}\right )}{2 \, a d}\right ] \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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

x^2))/(a*d)]

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^2)^(1/2)/(d*x^2+c)^(1/2),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(t_nostep)]Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is

real):Check [sign(t_nostep)]Warning, choosing root of [1,0,%%%{-2,[1,0,2]%%%}+%%%{-4,[0,1,0]%%%},0,%%%{1,[2,0

,4]%%%}] at parameters values [-22,93,91]Warning, choosing root of [1,0,%%%{-2,[1,0,2]%%%}+%%%{-4,[0,1,0]%%%},

0,%%%{1,[2,0,4]%%%}] at parameters values [31,-21,88]Warning, choosing root of [1,0,%%%{-2,[1,0,2]%%%}+%%%{-4,

[0,1,0]%%%},0,%%%{1,[2,0,4]%%%}] at parameters values [76,-66,66]Warning, choosing root of [1,0,%%%{-2,[1,0,2]

%%%}+%%%{-4,[0,1,0]%%%},0,%%%{1,[2,0,4]%%%}] at parameters values [5,-23,79]Warning, choosing root of [1,0,%%%

{-2,[1,0,2]%%%}+%%%{-4,[0,1,0]%%%},0,%%%{1,[2,0,4]%%%}] at parameters values [-88,9,6]Warning, choosing root o

f [1,0,%%%{-2,[1,0,2]%%%}+%%%{-4,[0,1,0]%%%},0,%%%{1,[2,0,4]%%%}] at parameters values [-69,-8,31]Warning, cho

osing root of [1,0,%%%{-2,[1,0,2]%%%}+%%%{-4,[0,1,0]%%%},0,%%%{1,[2,0,4]%%%}] at parameters values [2,90.21028

60468,-92]Warning, choosing root of [1,0,%%%{-2,[1,0,2]%%%}+%%%{-4,[0,1,0]%%%},0,%%%{1,[2,0,4]%%%}] at paramet

ers values [-17,26.2119182013,64]Warning, choosing root of [1,0,%%%{-2,[1,0,2]%%%}+%%%{-4,[0,1,0]%%%},0,%%%{1,

[2,0,4]%%%}] at parameters values [89,29.4664394325,-51]Warning, choosing root of [1,0,%%%{-2,[1,0,2]%%%}+%%%{

-4,[0,1,0]%%%},0,%%%{1,[2,0,4]%%%}] at parameters values [22,45.1969879479,76]Warning, choosing root of [1,0,%

%%{-2,[1,0,2]%%%}+%%%{-4,[0,1,0]%%%},0,%%%{1,[2,0,4]%%%}] at parameters values [-63,68.7323710029,13]Warning,

choosing root of [1,0,%%%{-2,[1,0,2]%%%}+%%%{-4,[0,1,0]%%%},0,%%%{1,[2,0,4]%%%}] at parameters values [43,85.9

758855961,12]Warning, choosing root of [1,0,%%%{-2,[1,0,2]%%%}+%%%{-4,[0,1,0]%%%},0,%%%{1,[2,0,4]%%%}] at para

meters values [72,95.2558838762,11]Warning, choosing root of [1,0,%%%{-2,[1,0,2]%%%}+%%%{-4,[0,1,0]%%%},0,%%%{

1,[2,0,4]%%%}] at parameters values [-9,91.3720739307,81]Warning, choosing root of [1,0,%%%{-2,[1,0,2]%%%}+%%%

{-4,[0,1,0]%%%},0,%%%{1,[2,0,4]%%%}] at parameters values [15,23.9552401127,-50]Warning, choosing root of [1,0

,%%%{-2,[1,0,2]%%%}+%%%{-4,[0,1,0]%%%},0,%%%{1,[2,0,4]%%%}] at parameters values [93,60.8246789905,-18]Warning

, choosing root of [1,0,%%%{-2,[1,0,2]%%%}+%%%{-4,[0,1,0]%%%},0,%%%{1,[2,0,4]%%%}] at parameters values [-88,6

3.3562821955,93]Warning, choosing root of [1,0,%%%{-2,[1,0,2]%%%}+%%%{-4,[0,1,0]%%%},0,%%%{1,[2,0,4]%%%}] at p

arameters values [-11,92.3620133325,-60]Warning, choosing root of [1,0,%%%{-2,[1,0,2]%%%}+%%%{-4,[0,1,0]%%%},0

,%%%{1,[2,0,4]%%%}] at parameters values [-88,51.1034068516,72]Warning, choosing root of [1,0,%%%{-2,[1,0,2]%%

%}+%%%{-4,[0,1,0]%%%},0,%%%{1,[2,0,4]%%%}] at parameters values [-90,71.1075269701,48]Warning, choosing root o

f [1,0,%%%{-2,[1,0,2]%%%}+%%%{-4,[0,1,0]%%%},0,%%%{1,[2,0,4]%%%}] at parameters values [-47,53.5483433446,-60]

Warning, choosing root of [1,0,%%%{-2,[1,0,2]%%%}+%%%{-4,[0,1,0]%%%},0,%%%{1,[2,0,4]%%%}] at parameters values

[70,89.9395644632,-32]Warning, choosing root of [1,0,%%%{-2,[1,0,2]%%%}+%%%{-4,[0,1,0]%%%},0,%%%{1,[2,0,4]%%%

}] at parameters values [-74,8.0407431256,-16]Warning, choosing root of [1,0,%%%{-2,[1,0,2]%%%}+%%%{-4,[0,1,0]

%%%},0,%%%{1,[2,0,4]%%%}] at parameters values [85,16.2654368887,-14]Warning, choosing root of [1,0,%%%{-2,[1,

0,2]%%%}+%%%{-4,[0,1,0]%%%},0,%%%{1,[2,0,4]%%%}] at parameters values [-77,83.0705981795,31]Warning, choosing

root of [1,0,%%%{-2,[1,0,2]%%%}+%%%{-4,[0,1,0]%%%},0,%%%{1,[2,0,4]%%%}] at parameters values [76,41.5291932677

,-48]Warning, choosing root of [1,0,%%%{-2,[1,0,2]%%%}+%%%{-4,[0,1,0]%%%},0,%%%{1,[2,0,4]%%%}] at parameters v

alues [-47,25.6140411007,23]Warning, choosing root of [1,0,%%%{-2,[1,0,2]%%%}+%%%{-4,[0,1,0]%%%},0,%%%{1,[2,0,

4]%%%}] at parameters values [32,76.6146142203,-62]Warning, choosing root of [1,0,%%%{-2,[1,0,2]%%%}+%%%{-4,[0

,1,0]%%%},0,%%%{1,[2,0,4]%%%}] at parameters values [66,99.6590219955,-30]Warning, choosing root of [1,0,%%%{-

2,[1,0,2]%%%}+%%%{-4,[0,1,0]%%%},0,%%%{1,[2,0,4]%%%}] at parameters values [-50,17.713730142,-44]Warning, choo

sing root of [1,0,%%%{-2,[1,0,2]%%%}+%%%{-4,[0,1,0]%%%},0,%%%{1,[2,0,4]%%%}] at parameters values [45,14.55155

09131,62]Warning, choosing root of [1,0,%%%{-2,[1,0,2]%%%}+%%%{-4,[0,1,0]%%%},0,%%%{1,[2,0,4]%%%}] at paramete

rs values [44,88.8429926978,11]Warning, choosing root of [1,0,%%%{-2,[1,0,2]%%%}+%%%{-4,[0,1,0]%%%},0,%%%{1,[2

,0,4]%%%}] at parameters values [-19,42.8964279308,-92]Warning, choosing root of [1,0,%%%{-2,[1,0,2]%%%}+%%%{-

4,[0,1,0]%%%},0,%%%{1,[2,0,4]%%%}] at parameters values [67,19.2648137459,-22]Warning, choosing root of [1,0,%

%%{-2,[1,0,2]%%%}+%%%{-4,[0,1,0]%%%},0,%%%{1,[2,0,4]%%%}] at parameters values [9,87.7979063494,94]Warning, ch

oosing root of [1,0,%%%{-2,[1,0,2]%%%}+%%%{-4,[0,1,0]%%%},0,%%%{1,[2,0,4]%%%}] at parameters values [95,98.958

2961812,-92]Warning, choosing root of [1,0,%%%{-2,[1,0,2]%%%}+%%%{-4,[0,1,0]%%%},0,%%%{1,[2,0,4]%%%}] at param

eters values [52,79.2538507222,22]Warning, choosing root of [1,0,%%%{-2,[1,0,2]%%%}+%%%{-4,[0,1,0]%%%},0,%%%{1

,[2,0,4]%%%}] at parameters values [-43,16.7638230952,21]Warning, choosing root of [1,0,%%%{-2,[1,0,2]%%%}+%%%

{-4,[0,1,0]%%%},0,%%%{1,[2,0,4]%%%}] at parameters values [71,61.8959259251,-83]Warning, choosing root of [1,0

,%%%{-2,[1,0,2]%%%}+%%%{-4,[0,1,0]%%%},0,%%%{1,[2,0,4]%%%}] at parameters values [-30,23.6526960679,40]sym2pol

y/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(const gen &

e,const index_m & i,const vecteur & l) Error: Bad Argument ValueWarning, choosing root of [1,0,%%%{-2,[1,0,0]

%%%}+%%%{-2,[0,1,2]%%%}+%%%{-2,[0,1,0]%%%},0,%%%{1,[2,0,0]%%%}+%%%{2,[1,1,2]%%%}+%%%{-2,[1,1,0]%%%}+%%%{1,[0,2

,4]%%%}+%%%{-2,[0,2,2]%%%}+%%%{1,[0,2,0]%%%}] at parameters values [0,56.2346625305,0]Warning, choosing root o

f [1,0,%%%{-2,[1,0,0]%%%}+%%%{-2,[0,1,2]%%%}+%%%{-2,[0,1,0]%%%},0,%%%{1,[2,0,0]%%%}+%%%{2,[1,1,2]%%%}+%%%{-2,[

1,1,0]%%%}+%%%{1,[0,2,4]%%%}+%%%{-2,[0,2,2]%%%}+%%%{1,[0,2,0]%%%}] at parameters values [-43,9.82222589385,18]

Warning, choosing root of [1,0,%%%{-2,[1,0,0]%%%}+%%%{-2,[0,1,2]%%%}+%%%{-2,[0,1,0]%%%},0,%%%{1,[2,0,0]%%%}+%%

%{2,[1,1,2]%%%}+%%%{-2,[1,1,0]%%%}+%%%{1,[0,2,4]%%%}+%%%{-2,[0,2,2]%%%}+%%%{1,[0,2,0]%%%}] at parameters value

s [95,6.65142845921,-60]Warning, choosing root of [1,0,%%%{-2,[1,0,0]%%%}+%%%{-2,[0,1,2]%%%}+%%%{-2,[0,1,0]%%%

},0,%%%{1,[2,0,0]%%%}+%%%{2,[1,1,2]%%%}+%%%{-2,[1,1,0]%%%}+%%%{1,[0,2,4]%%%}+%%%{-2,[0,2,2]%%%}+%%%{1,[0,2,0]%

%%}] at parameters values [65,77.8863785956,-16]Warning, choosing root of [1,0,%%%{-2,[1,0,0]%%%}+%%%{-2,[0,1,

2]%%%}+%%%{-2,[0,1,0]%%%},0,%%%{1,[2,0,0]%%%}+%%%{2,[1,1,2]%%%}+%%%{-2,[1,1,0]%%%}+%%%{1,[0,2,4]%%%}+%%%{-2,[0

,2,2]%%%}+%%%{1,[0,2,0]%%%}] at parameters values [-81,85.3390313801,-19]gen.cc:simplify/tmp.type!=_EXT Error:

Bad Argument ValueEvaluation time: 21.47Done

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maple [B] time = 0.05, size = 117, normalized size = 1.67 \[ \frac {\left (a \,x^{2}+b \right ) \sqrt {d \,x^{2}+c}\, \ln \left (\frac {2 a d \,x^{2}+a c +b d +2 \sqrt {a d \,x^{4}+a c \,x^{2}+b d \,x^{2}+b c}\, \sqrt {a d}}{2 \sqrt {a d}}\right )}{2 \sqrt {\frac {a \,x^{2}+b}{x^{2}}}\, \sqrt {a d}\, \sqrt {a d \,x^{4}+a c \,x^{2}+b d \,x^{2}+b c}\, x} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/((a*x^2+b)/x^2)^(1/2)/x*(a*x^2+b)*ln(1/2*(2*a*d*x^2+2*(a*d*x^4+a*c*x^2+b*d*x^2+b*c)^(1/2)*(a*d)^(1/2)+a*c+

b*d)/(a*d)^(1/2))*(d*x^2+c)^(1/2)/(a*d)^(1/2)/(a*d*x^4+a*c*x^2+b*d*x^2+b*c)^(1/2)

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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {d x^{2} + c} \sqrt {a + \frac {b}{x^{2}}}}\,{d x} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\sqrt {a+\frac {b}{x^2}}\,\sqrt {d\,x^2+c}} \,d x \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a + \frac {b}{x^{2}}} \sqrt {c + d x^{2}}}\, dx \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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