3.920 \(\int \frac{\sqrt{a+\frac{c}{x^2}}}{d+e x} \, dx\)
Optimal. Leaf size=121 \[ -\frac{\sqrt{a d^2+c e^2} \tanh ^{-1}\left (\frac{a d-\frac{c e}{x}}{\sqrt{a+\frac{c}{x^2}} \sqrt{a d^2+c e^2}}\right )}{d e}-\frac{\sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c}}{x \sqrt{a+\frac{c}{x^2}}}\right )}{d}+\frac{\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+\frac{c}{x^2}}}{\sqrt{a}}\right )}{e} \]
[Out]
(Sqrt[a]*ArcTanh[Sqrt[a + c/x^2]/Sqrt[a]])/e - (Sqrt[a*d^2 + c*e^2]*ArcTanh[(a*d - (c*e)/x)/(Sqrt[a*d^2 + c*e^
2]*Sqrt[a + c/x^2])])/(d*e) - (Sqrt[c]*ArcTanh[Sqrt[c]/(Sqrt[a + c/x^2]*x)])/d
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Rubi [A] time = 0.164957, antiderivative size = 121, normalized size of antiderivative = 1.,
number of steps used = 11, number of rules used = 10, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.526, Rules used
= {1444, 1475, 896, 266, 63, 208, 844, 217, 206, 725} \[ -\frac{\sqrt{a d^2+c e^2} \tanh ^{-1}\left (\frac{a d-\frac{c e}{x}}{\sqrt{a+\frac{c}{x^2}} \sqrt{a d^2+c e^2}}\right )}{d e}-\frac{\sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c}}{x \sqrt{a+\frac{c}{x^2}}}\right )}{d}+\frac{\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+\frac{c}{x^2}}}{\sqrt{a}}\right )}{e} \]
Antiderivative was successfully verified.
[In]
Int[Sqrt[a + c/x^2]/(d + e*x),x]
[Out]
(Sqrt[a]*ArcTanh[Sqrt[a + c/x^2]/Sqrt[a]])/e - (Sqrt[a*d^2 + c*e^2]*ArcTanh[(a*d - (c*e)/x)/(Sqrt[a*d^2 + c*e^
2]*Sqrt[a + c/x^2])])/(d*e) - (Sqrt[c]*ArcTanh[Sqrt[c]/(Sqrt[a + c/x^2]*x)])/d
Rule 1444
Int[((d_) + (e_.)*(x_)^(mn_.))^(q_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.), x_Symbol] :> Int[x^(mn*q)*(e + d/x^mn)^
q*(a + c*x^n2)^p, x] /; FreeQ[{a, c, d, e, mn, p}, x] && EqQ[n2, -2*mn] && IntegerQ[q] && (PosQ[n2] || !Integ
erQ[p])
Rule 1475
Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(d + e*x)^q*(a + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, c, d, e, m, n, p, q}, x
] && EqQ[n2, 2*n] && IntegerQ[Simplify[(m + 1)/n]]
Rule 896
Int[((a_) + (c_.)*(x_)^2)^(p_)/(((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))), x_Symbol] :> Dist[(c*d^2 + a*e^2)/
(e*(e*f - d*g)), Int[(a + c*x^2)^(p - 1)/(d + e*x), x], x] - Dist[1/(e*(e*f - d*g)), Int[(Simp[c*d*f + a*e*g -
c*(e*f - d*g)*x, x]*(a + c*x^2)^(p - 1))/(f + g*x), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g,
0] && NeQ[c*d^2 + a*e^2, 0] && FractionQ[p] && GtQ[p, 0]
Rule 266
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
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]
Rule 844
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a,
c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && !IGtQ[m, 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 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 725
Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]
Rubi steps
\begin{align*} \int \frac{\sqrt{a+\frac{c}{x^2}}}{d+e x} \, dx &=\int \frac{\sqrt{a+\frac{c}{x^2}}}{\left (e+\frac{d}{x}\right ) x} \, dx\\ &=-\operatorname{Subst}\left (\int \frac{\sqrt{a+c x^2}}{x (e+d x)} \, dx,x,\frac{1}{x}\right )\\ &=\frac{\operatorname{Subst}\left (\int \frac{a d-c e x}{(e+d x) \sqrt{a+c x^2}} \, dx,x,\frac{1}{x}\right )}{e}-\frac{a \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+c x^2}} \, dx,x,\frac{1}{x}\right )}{e}\\ &=-\frac{c \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+c x^2}} \, dx,x,\frac{1}{x}\right )}{d}-\frac{a \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+c x}} \, dx,x,\frac{1}{x^2}\right )}{2 e}+\left (\frac{a d}{e}+\frac{c e}{d}\right ) \operatorname{Subst}\left (\int \frac{1}{(e+d x) \sqrt{a+c x^2}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{c \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{1}{\sqrt{a+\frac{c}{x^2}} x}\right )}{d}-\frac{a \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{c}+\frac{x^2}{c}} \, dx,x,\sqrt{a+\frac{c}{x^2}}\right )}{c e}+\left (-\frac{a d}{e}-\frac{c e}{d}\right ) \operatorname{Subst}\left (\int \frac{1}{a d^2+c e^2-x^2} \, dx,x,\frac{a d-\frac{c e}{x}}{\sqrt{a+\frac{c}{x^2}}}\right )\\ &=\frac{\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+\frac{c}{x^2}}}{\sqrt{a}}\right )}{e}-\frac{\sqrt{a d^2+c e^2} \tanh ^{-1}\left (\frac{a d-\frac{c e}{x}}{\sqrt{a d^2+c e^2} \sqrt{a+\frac{c}{x^2}}}\right )}{d e}-\frac{\sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c}}{\sqrt{a+\frac{c}{x^2}} x}\right )}{d}\\ \end{align*}
Mathematica [A] time = 0.0993253, size = 136, normalized size = 1.12 \[ \frac{x \sqrt{a+\frac{c}{x^2}} \left (\sqrt{a d^2+c e^2} \tanh ^{-1}\left (\frac{c e-a d x}{\sqrt{a x^2+c} \sqrt{a d^2+c e^2}}\right )+\sqrt{a} d \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+c}}\right )-\sqrt{c} e \tanh ^{-1}\left (\frac{\sqrt{a x^2+c}}{\sqrt{c}}\right )\right )}{d e \sqrt{a x^2+c}} \]
Antiderivative was successfully verified.
[In]
Integrate[Sqrt[a + c/x^2]/(d + e*x),x]
[Out]
(Sqrt[a + c/x^2]*x*(Sqrt[a]*d*ArcTanh[(Sqrt[a]*x)/Sqrt[c + a*x^2]] + Sqrt[a*d^2 + c*e^2]*ArcTanh[(c*e - a*d*x)
/(Sqrt[a*d^2 + c*e^2]*Sqrt[c + a*x^2])] - Sqrt[c]*e*ArcTanh[Sqrt[c + a*x^2]/Sqrt[c]]))/(d*e*Sqrt[c + a*x^2])
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Maple [B] time = 0.025, size = 244, normalized size = 2. \begin{align*}{\frac{x}{d{e}^{2}}\sqrt{{\frac{a{x}^{2}+c}{{x}^{2}}}} \left ( \sqrt{a}d\ln \left ({ \left ( \sqrt{a{x}^{2}+c}\sqrt{a}+ax \right ){\frac{1}{\sqrt{a}}}} \right ) e\sqrt{{\frac{a{d}^{2}+c{e}^{2}}{{e}^{2}}}}-\ln \left ( 2\,{\frac{\sqrt{c}\sqrt{a{x}^{2}+c}+c}{x}} \right ) \sqrt{c}\sqrt{{\frac{a{d}^{2}+c{e}^{2}}{{e}^{2}}}}{e}^{2}+\ln \left ( 2\,{\frac{1}{ex+d} \left ( \sqrt{a{x}^{2}+c}\sqrt{{\frac{a{d}^{2}+c{e}^{2}}{{e}^{2}}}}e-axd+ce \right ) } \right ) a{d}^{2}+\ln \left ( 2\,{\frac{1}{ex+d} \left ( \sqrt{a{x}^{2}+c}\sqrt{{\frac{a{d}^{2}+c{e}^{2}}{{e}^{2}}}}e-axd+ce \right ) } \right ) c{e}^{2} \right ){\frac{1}{\sqrt{a{x}^{2}+c}}}{\frac{1}{\sqrt{{\frac{a{d}^{2}+c{e}^{2}}{{e}^{2}}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+c/x^2)^(1/2)/(e*x+d),x)
[Out]
((a*x^2+c)/x^2)^(1/2)*x*(a^(1/2)*d*ln(((a*x^2+c)^(1/2)*a^(1/2)+a*x)/a^(1/2))*e*((a*d^2+c*e^2)/e^2)^(1/2)-ln(2*
(c^(1/2)*(a*x^2+c)^(1/2)+c)/x)*c^(1/2)*((a*d^2+c*e^2)/e^2)^(1/2)*e^2+ln(2*((a*x^2+c)^(1/2)*((a*d^2+c*e^2)/e^2)
^(1/2)*e-a*x*d+c*e)/(e*x+d))*a*d^2+ln(2*((a*x^2+c)^(1/2)*((a*d^2+c*e^2)/e^2)^(1/2)*e-a*x*d+c*e)/(e*x+d))*c*e^2
)/(a*x^2+c)^(1/2)/d/e^2/((a*d^2+c*e^2)/e^2)^(1/2)
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a + \frac{c}{x^{2}}}}{e x + d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+c/x^2)^(1/2)/(e*x+d),x, algorithm="maxima")
[Out]
integrate(sqrt(a + c/x^2)/(e*x + d), x)
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Fricas [A] time = 6.00252, size = 3367, normalized size = 27.83 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+c/x^2)^(1/2)/(e*x+d),x, algorithm="fricas")
[Out]
[1/2*(sqrt(a)*d*log(-2*a*x^2 - 2*sqrt(a)*x^2*sqrt((a*x^2 + c)/x^2) - c) + sqrt(c)*e*log(-(a*x^2 - 2*sqrt(c)*x*
sqrt((a*x^2 + c)/x^2) + 2*c)/x^2) + sqrt(a*d^2 + c*e^2)*log((2*a*c*d*e*x - a*c*d^2 - 2*c^2*e^2 - (2*a^2*d^2 +
a*c*e^2)*x^2 + 2*(a*d*x^2 - c*e*x)*sqrt(a*d^2 + c*e^2)*sqrt((a*x^2 + c)/x^2))/(e^2*x^2 + 2*d*e*x + d^2)))/(d*e
), -1/2*(2*sqrt(-a)*d*arctan(sqrt(-a)*x^2*sqrt((a*x^2 + c)/x^2)/(a*x^2 + c)) - sqrt(c)*e*log(-(a*x^2 - 2*sqrt(
c)*x*sqrt((a*x^2 + c)/x^2) + 2*c)/x^2) - sqrt(a*d^2 + c*e^2)*log((2*a*c*d*e*x - a*c*d^2 - 2*c^2*e^2 - (2*a^2*d
^2 + a*c*e^2)*x^2 + 2*(a*d*x^2 - c*e*x)*sqrt(a*d^2 + c*e^2)*sqrt((a*x^2 + c)/x^2))/(e^2*x^2 + 2*d*e*x + d^2)))
/(d*e), 1/2*(sqrt(a)*d*log(-2*a*x^2 - 2*sqrt(a)*x^2*sqrt((a*x^2 + c)/x^2) - c) + sqrt(c)*e*log(-(a*x^2 - 2*sqr
t(c)*x*sqrt((a*x^2 + c)/x^2) + 2*c)/x^2) + 2*sqrt(-a*d^2 - c*e^2)*arctan((a*d*x^2 - c*e*x)*sqrt(-a*d^2 - c*e^2
)*sqrt((a*x^2 + c)/x^2)/(a*c*d^2 + c^2*e^2 + (a^2*d^2 + a*c*e^2)*x^2)))/(d*e), -1/2*(2*sqrt(-a)*d*arctan(sqrt(
-a)*x^2*sqrt((a*x^2 + c)/x^2)/(a*x^2 + c)) - sqrt(c)*e*log(-(a*x^2 - 2*sqrt(c)*x*sqrt((a*x^2 + c)/x^2) + 2*c)/
x^2) - 2*sqrt(-a*d^2 - c*e^2)*arctan((a*d*x^2 - c*e*x)*sqrt(-a*d^2 - c*e^2)*sqrt((a*x^2 + c)/x^2)/(a*c*d^2 + c
^2*e^2 + (a^2*d^2 + a*c*e^2)*x^2)))/(d*e), 1/2*(2*sqrt(-c)*e*arctan(sqrt(-c)*x*sqrt((a*x^2 + c)/x^2)/(a*x^2 +
c)) + sqrt(a)*d*log(-2*a*x^2 - 2*sqrt(a)*x^2*sqrt((a*x^2 + c)/x^2) - c) + sqrt(a*d^2 + c*e^2)*log((2*a*c*d*e*x
- a*c*d^2 - 2*c^2*e^2 - (2*a^2*d^2 + a*c*e^2)*x^2 + 2*(a*d*x^2 - c*e*x)*sqrt(a*d^2 + c*e^2)*sqrt((a*x^2 + c)/
x^2))/(e^2*x^2 + 2*d*e*x + d^2)))/(d*e), -1/2*(2*sqrt(-a)*d*arctan(sqrt(-a)*x^2*sqrt((a*x^2 + c)/x^2)/(a*x^2 +
c)) - 2*sqrt(-c)*e*arctan(sqrt(-c)*x*sqrt((a*x^2 + c)/x^2)/(a*x^2 + c)) - sqrt(a*d^2 + c*e^2)*log((2*a*c*d*e*
x - a*c*d^2 - 2*c^2*e^2 - (2*a^2*d^2 + a*c*e^2)*x^2 + 2*(a*d*x^2 - c*e*x)*sqrt(a*d^2 + c*e^2)*sqrt((a*x^2 + c)
/x^2))/(e^2*x^2 + 2*d*e*x + d^2)))/(d*e), 1/2*(2*sqrt(-c)*e*arctan(sqrt(-c)*x*sqrt((a*x^2 + c)/x^2)/(a*x^2 + c
)) + sqrt(a)*d*log(-2*a*x^2 - 2*sqrt(a)*x^2*sqrt((a*x^2 + c)/x^2) - c) + 2*sqrt(-a*d^2 - c*e^2)*arctan((a*d*x^
2 - c*e*x)*sqrt(-a*d^2 - c*e^2)*sqrt((a*x^2 + c)/x^2)/(a*c*d^2 + c^2*e^2 + (a^2*d^2 + a*c*e^2)*x^2)))/(d*e), -
(sqrt(-a)*d*arctan(sqrt(-a)*x^2*sqrt((a*x^2 + c)/x^2)/(a*x^2 + c)) - sqrt(-c)*e*arctan(sqrt(-c)*x*sqrt((a*x^2
+ c)/x^2)/(a*x^2 + c)) - sqrt(-a*d^2 - c*e^2)*arctan((a*d*x^2 - c*e*x)*sqrt(-a*d^2 - c*e^2)*sqrt((a*x^2 + c)/x
^2)/(a*c*d^2 + c^2*e^2 + (a^2*d^2 + a*c*e^2)*x^2)))/(d*e)]
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a + \frac{c}{x^{2}}}}{d + e x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+c/x**2)**(1/2)/(e*x+d),x)
[Out]
Integral(sqrt(a + c/x**2)/(d + e*x), x)
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+c/x^2)^(1/2)/(e*x+d),x, algorithm="giac")
[Out]
Exception raised: TypeError