3.10.20 \(\int \frac {\sqrt {a+\frac {c}{x^2}}}{d+e x} \, dx\) [920]

Optimal. Leaf size=121 \[ \frac {\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+\frac {c}{x^2}}}{\sqrt {a}}\right )}{e}-\frac {\sqrt {a d^2+c e^2} \tanh ^{-1}\left (\frac {a d-\frac {c e}{x}}{\sqrt {a d^2+c e^2} \sqrt {a+\frac {c}{x^2}}}\right )}{d e}-\frac {\sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c}}{\sqrt {a+\frac {c}{x^2}} x}\right )}{d} \]

[Out]

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

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Rubi [A]
time = 0.11, antiderivative size = 121, normalized size of antiderivative = 1.00, 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 = {1458, 1489, 910, 272, 65, 214, 858, 223, 212, 739} \begin {gather*} -\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} \end {gather*}

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 65

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 212

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

Rule 214

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

Rule 223

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 272

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 739

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]

Rule 858

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 910

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 1458

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 1489

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]]

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\\ &=-\text {Subst}\left (\int \frac {\sqrt {a+c x^2}}{x (e+d x)} \, dx,x,\frac {1}{x}\right )\\ &=\frac {\text {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 \text {Subst}\left (\int \frac {1}{x \sqrt {a+c x^2}} \, dx,x,\frac {1}{x}\right )}{e}\\ &=-\frac {c \text {Subst}\left (\int \frac {1}{\sqrt {a+c x^2}} \, dx,x,\frac {1}{x}\right )}{d}-\frac {a \text {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 ) \text {Subst}\left (\int \frac {1}{(e+d x) \sqrt {a+c x^2}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {c \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {1}{\sqrt {a+\frac {c}{x^2}} x}\right )}{d}-\frac {a \text {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 ) \text {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*}

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Mathematica [A]
time = 0.26, size = 160, normalized size = 1.32 \begin {gather*} -\frac {\sqrt {a+\frac {c}{x^2}} x \left (2 \sqrt {-a d^2-c e^2} \tan ^{-1}\left (\frac {\sqrt {a} (d+e x)-e \sqrt {c+a x^2}}{\sqrt {-a d^2-c e^2}}\right )-2 \sqrt {c} e \tanh ^{-1}\left (\frac {\sqrt {a} x-\sqrt {c+a x^2}}{\sqrt {c}}\right )+\sqrt {a} d \log \left (-\sqrt {a} x+\sqrt {c+a x^2}\right )\right )}{d e \sqrt {c+a x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(246\) vs. \(2(103)=206\).
time = 0.26, size = 247, normalized size = 2.04

method	result	size
default	\(-\frac {\sqrt {\frac {a \,x^{2}+c}{x^{2}}}\, x \left (\sqrt {c}\, \sqrt {\frac {a \,d^{2}+c \,e^{2}}{e^{2}}}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {a \,x^{2}+c}}{x}\right ) e^{2}-\sqrt {a}\, d \ln \left (\frac {\sqrt {a}\, \sqrt {a \,x^{2}+c}+a x}{\sqrt {a}}\right ) e \sqrt {\frac {a \,d^{2}+c \,e^{2}}{e^{2}}}-\ln \left (\frac {2 \sqrt {\frac {a \,d^{2}+c \,e^{2}}{e^{2}}}\, \sqrt {a \,x^{2}+c}\, e -2 a d x +2 c e}{e x +d}\right ) a \,d^{2}-\ln \left (\frac {2 \sqrt {\frac {a \,d^{2}+c \,e^{2}}{e^{2}}}\, \sqrt {a \,x^{2}+c}\, e -2 a d x +2 c e}{e x +d}\right ) c \,e^{2}\right )}{\sqrt {a \,x^{2}+c}\, d \,e^{2} \sqrt {\frac {a \,d^{2}+c \,e^{2}}{e^{2}}}}\)	\(247\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+c/x^2)^(1/2)/(e*x+d),x,method=_RETURNVERBOSE)

[Out]

-((a*x^2+c)/x^2)^(1/2)*x*(c^(1/2)*((a*d^2+c*e^2)/e^2)^(1/2)*ln(2*(c^(1/2)*(a*x^2+c)^(1/2)+c)/x)*e^2-a^(1/2)*d*
ln((a^(1/2)*(a*x^2+c)^(1/2)+a*x)/a^(1/2))*e*((a*d^2+c*e^2)/e^2)^(1/2)-ln(2*(((a*d^2+c*e^2)/e^2)^(1/2)*(a*x^2+c
)^(1/2)*e-a*d*x+c*e)/(e*x+d))*a*d^2-ln(2*(((a*d^2+c*e^2)/e^2)^(1/2)*(a*x^2+c)^(1/2)*e-a*d*x+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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)/(x*e + d), x)

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Fricas [A]
time = 0.85, size = 1515, normalized size = 12.52 \begin {gather*} \left [\frac {{\left (\sqrt {a} d \log \left (-2 \, a x^{2} - 2 \, \sqrt {a} x^{2} \sqrt {\frac {a x^{2} + c}{x^{2}}} - c\right ) + \sqrt {c} e \log \left (-\frac {a x^{2} - 2 \, \sqrt {c} x \sqrt {\frac {a x^{2} + c}{x^{2}}} + 2 \, c}{x^{2}}\right ) + \sqrt {a d^{2} + c e^{2}} \log \left (-\frac {2 \, a^{2} d^{2} x^{2} - 2 \, a c d x e + a c d^{2} - 2 \, {\left (a d x^{2} - c x e\right )} \sqrt {a d^{2} + c e^{2}} \sqrt {\frac {a x^{2} + c}{x^{2}}} + {\left (a c x^{2} + 2 \, c^{2}\right )} e^{2}}{x^{2} e^{2} + 2 \, d x e + d^{2}}\right )\right )} e^{\left (-1\right )}}{2 \, d}, -\frac {{\left (2 \, \sqrt {-a} d \arctan \left (\frac {\sqrt {-a} x^{2} \sqrt {\frac {a x^{2} + c}{x^{2}}}}{a x^{2} + c}\right ) - \sqrt {c} e \log \left (-\frac {a x^{2} - 2 \, \sqrt {c} x \sqrt {\frac {a x^{2} + c}{x^{2}}} + 2 \, c}{x^{2}}\right ) - \sqrt {a d^{2} + c e^{2}} \log \left (-\frac {2 \, a^{2} d^{2} x^{2} - 2 \, a c d x e + a c d^{2} - 2 \, {\left (a d x^{2} - c x e\right )} \sqrt {a d^{2} + c e^{2}} \sqrt {\frac {a x^{2} + c}{x^{2}}} + {\left (a c x^{2} + 2 \, c^{2}\right )} e^{2}}{x^{2} e^{2} + 2 \, d x e + d^{2}}\right )\right )} e^{\left (-1\right )}}{2 \, d}, \frac {{\left (\sqrt {a} d \log \left (-2 \, a x^{2} - 2 \, \sqrt {a} x^{2} \sqrt {\frac {a x^{2} + c}{x^{2}}} - c\right ) + \sqrt {c} e \log \left (-\frac {a x^{2} - 2 \, \sqrt {c} x \sqrt {\frac {a x^{2} + c}{x^{2}}} + 2 \, c}{x^{2}}\right ) - 2 \, \sqrt {-a d^{2} - c e^{2}} \arctan \left (-\frac {{\left (a d x^{2} - c x e\right )} \sqrt {-a d^{2} - c e^{2}} \sqrt {\frac {a x^{2} + c}{x^{2}}}}{a^{2} d^{2} x^{2} + a c d^{2} + {\left (a c x^{2} + c^{2}\right )} e^{2}}\right )\right )} e^{\left (-1\right )}}{2 \, d}, -\frac {{\left (2 \, \sqrt {-a} d \arctan \left (\frac {\sqrt {-a} x^{2} \sqrt {\frac {a x^{2} + c}{x^{2}}}}{a x^{2} + c}\right ) - \sqrt {c} e \log \left (-\frac {a x^{2} - 2 \, \sqrt {c} x \sqrt {\frac {a x^{2} + c}{x^{2}}} + 2 \, c}{x^{2}}\right ) + 2 \, \sqrt {-a d^{2} - c e^{2}} \arctan \left (-\frac {{\left (a d x^{2} - c x e\right )} \sqrt {-a d^{2} - c e^{2}} \sqrt {\frac {a x^{2} + c}{x^{2}}}}{a^{2} d^{2} x^{2} + a c d^{2} + {\left (a c x^{2} + c^{2}\right )} e^{2}}\right )\right )} e^{\left (-1\right )}}{2 \, d}, \frac {{\left (2 \, \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x \sqrt {\frac {a x^{2} + c}{x^{2}}}}{a x^{2} + c}\right ) e + \sqrt {a} d \log \left (-2 \, a x^{2} - 2 \, \sqrt {a} x^{2} \sqrt {\frac {a x^{2} + c}{x^{2}}} - c\right ) + \sqrt {a d^{2} + c e^{2}} \log \left (-\frac {2 \, a^{2} d^{2} x^{2} - 2 \, a c d x e + a c d^{2} - 2 \, {\left (a d x^{2} - c x e\right )} \sqrt {a d^{2} + c e^{2}} \sqrt {\frac {a x^{2} + c}{x^{2}}} + {\left (a c x^{2} + 2 \, c^{2}\right )} e^{2}}{x^{2} e^{2} + 2 \, d x e + d^{2}}\right )\right )} e^{\left (-1\right )}}{2 \, d}, -\frac {{\left (2 \, \sqrt {-a} d \arctan \left (\frac {\sqrt {-a} x^{2} \sqrt {\frac {a x^{2} + c}{x^{2}}}}{a x^{2} + c}\right ) - 2 \, \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x \sqrt {\frac {a x^{2} + c}{x^{2}}}}{a x^{2} + c}\right ) e - \sqrt {a d^{2} + c e^{2}} \log \left (-\frac {2 \, a^{2} d^{2} x^{2} - 2 \, a c d x e + a c d^{2} - 2 \, {\left (a d x^{2} - c x e\right )} \sqrt {a d^{2} + c e^{2}} \sqrt {\frac {a x^{2} + c}{x^{2}}} + {\left (a c x^{2} + 2 \, c^{2}\right )} e^{2}}{x^{2} e^{2} + 2 \, d x e + d^{2}}\right )\right )} e^{\left (-1\right )}}{2 \, d}, \frac {{\left (2 \, \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x \sqrt {\frac {a x^{2} + c}{x^{2}}}}{a x^{2} + c}\right ) e + \sqrt {a} d \log \left (-2 \, a x^{2} - 2 \, \sqrt {a} x^{2} \sqrt {\frac {a x^{2} + c}{x^{2}}} - c\right ) - 2 \, \sqrt {-a d^{2} - c e^{2}} \arctan \left (-\frac {{\left (a d x^{2} - c x e\right )} \sqrt {-a d^{2} - c e^{2}} \sqrt {\frac {a x^{2} + c}{x^{2}}}}{a^{2} d^{2} x^{2} + a c d^{2} + {\left (a c x^{2} + c^{2}\right )} e^{2}}\right )\right )} e^{\left (-1\right )}}{2 \, d}, -\frac {{\left (\sqrt {-a} d \arctan \left (\frac {\sqrt {-a} x^{2} \sqrt {\frac {a x^{2} + c}{x^{2}}}}{a x^{2} + c}\right ) - \sqrt {-c} \arctan \left (\frac {\sqrt {-c} x \sqrt {\frac {a x^{2} + c}{x^{2}}}}{a x^{2} + c}\right ) e + \sqrt {-a d^{2} - c e^{2}} \arctan \left (-\frac {{\left (a d x^{2} - c x e\right )} \sqrt {-a d^{2} - c e^{2}} \sqrt {\frac {a x^{2} + c}{x^{2}}}}{a^{2} d^{2} x^{2} + a c d^{2} + {\left (a c x^{2} + c^{2}\right )} e^{2}}\right )\right )} e^{\left (-1\right )}}{d}\right ] \end {gather*}

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^2*d^2*x^2 - 2*a*c*d*x*e + a*c*d^2 - 2*(a*d*x
^2 - c*x*e)*sqrt(a*d^2 + c*e^2)*sqrt((a*x^2 + c)/x^2) + (a*c*x^2 + 2*c^2)*e^2)/(x^2*e^2 + 2*d*x*e + d^2)))*e^(
-1)/d, -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*s
qrt(c)*x*sqrt((a*x^2 + c)/x^2) + 2*c)/x^2) - sqrt(a*d^2 + c*e^2)*log(-(2*a^2*d^2*x^2 - 2*a*c*d*x*e + a*c*d^2 -
2*(a*d*x^2 - c*x*e)*sqrt(a*d^2 + c*e^2)*sqrt((a*x^2 + c)/x^2) + (a*c*x^2 + 2*c^2)*e^2)/(x^2*e^2 + 2*d*x*e + d
^2)))*e^(-1)/d, 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) - 2*sqrt(-a*d^2 - c*e^2)*arctan(-(a*d*x^2 - c*x*e)*sqrt(-a*d^
2 - c*e^2)*sqrt((a*x^2 + c)/x^2)/(a^2*d^2*x^2 + a*c*d^2 + (a*c*x^2 + c^2)*e^2)))*e^(-1)/d, -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*x*e)*sqrt(-a*d^2 - c*e^2)*sqrt((a*x^2 + c)/x^2)
/(a^2*d^2*x^2 + a*c*d^2 + (a*c*x^2 + c^2)*e^2)))*e^(-1)/d, 1/2*(2*sqrt(-c)*arctan(sqrt(-c)*x*sqrt((a*x^2 + c)/
x^2)/(a*x^2 + c))*e + 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^2*d^2*x^2 - 2*a*c*d*x*e + a*c*d^2 - 2*(a*d*x^2 - c*x*e)*sqrt(a*d^2 + c*e^2)*sqrt((a*x^2 + c)/x^2) +
(a*c*x^2 + 2*c^2)*e^2)/(x^2*e^2 + 2*d*x*e + d^2)))*e^(-1)/d, -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)*arctan(sqrt(-c)*x*sqrt((a*x^2 + c)/x^2)/(a*x^2 + c))*e - sqrt(a*d^2 + c*
e^2)*log(-(2*a^2*d^2*x^2 - 2*a*c*d*x*e + a*c*d^2 - 2*(a*d*x^2 - c*x*e)*sqrt(a*d^2 + c*e^2)*sqrt((a*x^2 + c)/x^
2) + (a*c*x^2 + 2*c^2)*e^2)/(x^2*e^2 + 2*d*x*e + d^2)))*e^(-1)/d, 1/2*(2*sqrt(-c)*arctan(sqrt(-c)*x*sqrt((a*x^
2 + c)/x^2)/(a*x^2 + c))*e + 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*x*e)*sqrt(-a*d^2 - c*e^2)*sqrt((a*x^2 + c)/x^2)/(a^2*d^2*x^2 + a*c*d^2 + (a*c*x
^2 + c^2)*e^2)))*e^(-1)/d, -(sqrt(-a)*d*arctan(sqrt(-a)*x^2*sqrt((a*x^2 + c)/x^2)/(a*x^2 + c)) - sqrt(-c)*arct
an(sqrt(-c)*x*sqrt((a*x^2 + c)/x^2)/(a*x^2 + c))*e + sqrt(-a*d^2 - c*e^2)*arctan(-(a*d*x^2 - c*x*e)*sqrt(-a*d^
2 - c*e^2)*sqrt((a*x^2 + c)/x^2)/(a^2*d^2*x^2 + a*c*d^2 + (a*c*x^2 + c^2)*e^2)))*e^(-1)/d]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a + \frac {c}{x^{2}}}}{d + e x}\, dx \end {gather*}

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 = 0.00, 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^2)^(1/2)/(e*x+d),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Ch
eck [abs(sa

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {a+\frac {c}{x^2}}}{d+e\,x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c/x^2)^(1/2)/(d + e*x),x)

[Out]

int((a + c/x^2)^(1/2)/(d + e*x), x)

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