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

Optimal. Leaf size=47 \[ \frac {a x \sqrt {c+\frac {d}{x^2}}}{c}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {d}}{x \sqrt {c+\frac {d}{x^2}}}\right )}{\sqrt {d}} \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 47, 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, 451, 217, 206} \[ \frac {a x \sqrt {c+\frac {d}{x^2}}}{c}-\frac {b \tanh ^{-1}\left (\frac {\sqrt {d}}{x \sqrt {c+\frac {d}{x^2}}}\right )}{\sqrt {d}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^2)/Sqrt[c + d/x^2],x]

[Out]

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

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 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 451

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

Rubi steps

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

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Mathematica [A]  time = 0.04, size = 71, normalized size = 1.51 \[ \frac {a \sqrt {d} \left (c x^2+d\right )-b c \sqrt {c x^2+d} \tanh ^{-1}\left (\frac {\sqrt {c x^2+d}}{\sqrt {d}}\right )}{c \sqrt {d} x \sqrt {c+\frac {d}{x^2}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.74, size = 131, normalized size = 2.79 \[ \left [\frac {2 \, a d x \sqrt {\frac {c x^{2} + d}{x^{2}}} + b c \sqrt {d} \log \left (-\frac {c x^{2} - 2 \, \sqrt {d} x \sqrt {\frac {c x^{2} + d}{x^{2}}} + 2 \, d}{x^{2}}\right )}{2 \, c d}, \frac {a d x \sqrt {\frac {c x^{2} + d}{x^{2}}} + b c \sqrt {-d} \arctan \left (\frac {\sqrt {-d} x \sqrt {\frac {c x^{2} + d}{x^{2}}}}{c x^{2} + d}\right )}{c d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(2*a*d*x*sqrt((c*x^2 + d)/x^2) + b*c*sqrt(d)*log(-(c*x^2 - 2*sqrt(d)*x*sqrt((c*x^2 + d)/x^2) + 2*d)/x^2))
/(c*d), (a*d*x*sqrt((c*x^2 + d)/x^2) + b*c*sqrt(-d)*arctan(sqrt(-d)*x*sqrt((c*x^2 + d)/x^2)/(c*x^2 + d)))/(c*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((a+b/x^2)/(c+d/x^2)^(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),sign(t_nostep+sqrt(d)/c*sign(t_nostep))]sym2poly/r2sym(const gen & e,const index_
m & i,const vecteur & l) Error: Bad Argument Value

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maple [A]  time = 0.05, size = 73, normalized size = 1.55 \[ \frac {\sqrt {c \,x^{2}+d}\, \left (-b c \ln \left (\frac {2 d +2 \sqrt {c \,x^{2}+d}\, \sqrt {d}}{x}\right )+\sqrt {c \,x^{2}+d}\, a \sqrt {d}\right )}{\sqrt {\frac {c \,x^{2}+d}{x^{2}}}\, c \sqrt {d}\, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.28, size = 58, normalized size = 1.23 \[ \frac {a \sqrt {c + \frac {d}{x^{2}}} x}{c} + \frac {b \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} x - \sqrt {d}}{\sqrt {c + \frac {d}{x^{2}}} x + \sqrt {d}}\right )}{2 \, \sqrt {d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 5.00, size = 65, normalized size = 1.38 \[ \frac {a\,x\,\sqrt {\frac {c\,x^2}{d}+1}}{\sqrt {c+\frac {d}{x^2}}\,\left (\sqrt {\frac {c\,x^2}{d}+1}+1\right )}-\frac {b\,\ln \left (\sqrt {c+\frac {d}{x^2}}+\frac {\sqrt {d}}{x}\right )}{\sqrt {d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/x^2)/(c + d/x^2)^(1/2),x)

[Out]

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

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sympy [A]  time = 2.86, size = 39, normalized size = 0.83 \[ \frac {a \sqrt {d} \sqrt {\frac {c x^{2}}{d} + 1}}{c} - \frac {b \operatorname {asinh}{\left (\frac {\sqrt {d}}{\sqrt {c} x} \right )}}{\sqrt {d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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