\(\int \frac {\sqrt {d+e x^2}}{-c d^2+b d e+b e^2 x^2+c e^2 x^4} \, dx\) [222]

Optimal result

Integrand size = 41, antiderivative size = 76 \[ \int \frac {\sqrt {d+e x^2}}{-c d^2+b d e+b e^2 x^2+c e^2 x^4} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {e} \sqrt {2 c d-b e} x}{\sqrt {c d-b e} \sqrt {d+e x^2}}\right )}{\sqrt {e} \sqrt {c d-b e} \sqrt {2 c d-b e}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.073, Rules used = {1163, 385, 214} \[ \int \frac {\sqrt {d+e x^2}}{-c d^2+b d e+b e^2 x^2+c e^2 x^4} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {e} x \sqrt {2 c d-b e}}{\sqrt {d+e x^2} \sqrt {c d-b e}}\right )}{\sqrt {e} \sqrt {c d-b e} \sqrt {2 c d-b e}} \]

[In]

Int[Sqrt[d + e*x^2]/(-(c*d^2) + b*d*e + b*e^2*x^2 + c*e^2*x^4),x]

[Out]

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

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 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 1163

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[(d + e*x^2)^(p +
q)*(a/d + (c/e)*x^2)^p, x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2
, 0] && IntegerQ[p]

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

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.24 \[ \int \frac {\sqrt {d+e x^2}}{-c d^2+b d e+b e^2 x^2+c e^2 x^4} \, dx=-\frac {\text {arctanh}\left (\frac {-b e+c \left (d-e x^2+\sqrt {e} x \sqrt {d+e x^2}\right )}{\sqrt {2 c^2 d^2-3 b c d e+b^2 e^2}}\right )}{\sqrt {e} \sqrt {2 c^2 d^2-3 b c d e+b^2 e^2}} \]

[In]

Integrate[Sqrt[d + e*x^2]/(-(c*d^2) + b*d*e + b*e^2*x^2 + c*e^2*x^4),x]

[Out]

-(ArcTanh[(-(b*e) + c*(d - e*x^2 + Sqrt[e]*x*Sqrt[d + e*x^2]))/Sqrt[2*c^2*d^2 - 3*b*c*d*e + b^2*e^2]]/(Sqrt[e]
*Sqrt[2*c^2*d^2 - 3*b*c*d*e + b^2*e^2]))

Maple [A] (verified)

Time = 0.27 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.84

method	result	size
pseudoelliptic	\(\frac {\operatorname {arctanh}\left (\frac {\left (b e -c d \right ) \sqrt {e \,x^{2}+d}}{x \sqrt {e \left (b e -2 c d \right ) \left (b e -c d \right )}}\right )}{\sqrt {e \left (b e -2 c d \right ) \left (b e -c d \right )}}\)	\(64\)
default	\(\text {Expression too large to display}\)	\(1425\)

[In]

int((e*x^2+d)^(1/2)/(c*e^2*x^4+b*e^2*x^2+b*d*e-c*d^2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 188 vs. \(2 (62) = 124\).

Time = 0.30 (sec) , antiderivative size = 432, normalized size of antiderivative = 5.68 \[ \int \frac {\sqrt {d+e x^2}}{-c d^2+b d e+b e^2 x^2+c e^2 x^4} \, dx=\left [\frac {\log \left (\frac {c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2} + {\left (17 \, c^{2} d^{2} e^{2} - 24 \, b c d e^{3} + 8 \, b^{2} e^{4}\right )} x^{4} + 2 \, {\left (7 \, c^{2} d^{3} e - 11 \, b c d^{2} e^{2} + 4 \, b^{2} d e^{3}\right )} x^{2} - 4 \, \sqrt {2 \, c^{2} d^{2} e - 3 \, b c d e^{2} + b^{2} e^{3}} {\left ({\left (3 \, c d e - 2 \, b e^{2}\right )} x^{3} + {\left (c d^{2} - b d e\right )} x\right )} \sqrt {e x^{2} + d}}{c^{2} e^{2} x^{4} + c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2} - 2 \, {\left (c^{2} d e - b c e^{2}\right )} x^{2}}\right )}{4 \, \sqrt {2 \, c^{2} d^{2} e - 3 \, b c d e^{2} + b^{2} e^{3}}}, -\frac {\sqrt {-2 \, c^{2} d^{2} e + 3 \, b c d e^{2} - b^{2} e^{3}} \arctan \left (-\frac {\sqrt {-2 \, c^{2} d^{2} e + 3 \, b c d e^{2} - b^{2} e^{3}} {\left (c d^{2} - b d e + {\left (3 \, c d e - 2 \, b e^{2}\right )} x^{2}\right )} \sqrt {e x^{2} + d}}{2 \, {\left ({\left (2 \, c^{2} d^{2} e^{2} - 3 \, b c d e^{3} + b^{2} e^{4}\right )} x^{3} + {\left (2 \, c^{2} d^{3} e - 3 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x\right )}}\right )}{2 \, {\left (2 \, c^{2} d^{2} e - 3 \, b c d e^{2} + b^{2} e^{3}\right )}}\right ] \]

[In]

integrate((e*x^2+d)^(1/2)/(c*e^2*x^4+b*e^2*x^2+b*d*e-c*d^2),x, algorithm="fricas")

[Out]

[1/4*log((c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2 + (17*c^2*d^2*e^2 - 24*b*c*d*e^3 + 8*b^2*e^4)*x^4 + 2*(7*c^2*d^3
*e - 11*b*c*d^2*e^2 + 4*b^2*d*e^3)*x^2 - 4*sqrt(2*c^2*d^2*e - 3*b*c*d*e^2 + b^2*e^3)*((3*c*d*e - 2*b*e^2)*x^3
+ (c*d^2 - b*d*e)*x)*sqrt(e*x^2 + d))/(c^2*e^2*x^4 + c^2*d^2 - 2*b*c*d*e + b^2*e^2 - 2*(c^2*d*e - b*c*e^2)*x^2
))/sqrt(2*c^2*d^2*e - 3*b*c*d*e^2 + b^2*e^3), -1/2*sqrt(-2*c^2*d^2*e + 3*b*c*d*e^2 - b^2*e^3)*arctan(-1/2*sqrt
(-2*c^2*d^2*e + 3*b*c*d*e^2 - b^2*e^3)*(c*d^2 - b*d*e + (3*c*d*e - 2*b*e^2)*x^2)*sqrt(e*x^2 + d)/((2*c^2*d^2*e
^2 - 3*b*c*d*e^3 + b^2*e^4)*x^3 + (2*c^2*d^3*e - 3*b*c*d^2*e^2 + b^2*d*e^3)*x))/(2*c^2*d^2*e - 3*b*c*d*e^2 + b
^2*e^3)]

Sympy [F]

\[ \int \frac {\sqrt {d+e x^2}}{-c d^2+b d e+b e^2 x^2+c e^2 x^4} \, dx=\int \frac {1}{\sqrt {d + e x^{2}} \left (b e - c d + c e x^{2}\right )}\, dx \]

[In]

integrate((e*x**2+d)**(1/2)/(c*e**2*x**4+b*e**2*x**2+b*d*e-c*d**2),x)

[Out]

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

Maxima [F]

\[ \int \frac {\sqrt {d+e x^2}}{-c d^2+b d e+b e^2 x^2+c e^2 x^4} \, dx=\int { \frac {\sqrt {e x^{2} + d}}{c e^{2} x^{4} + b e^{2} x^{2} - c d^{2} + b d e} \,d x } \]

[In]

integrate((e*x^2+d)^(1/2)/(c*e^2*x^4+b*e^2*x^2+b*d*e-c*d^2),x, algorithm="maxima")

[Out]

integrate(sqrt(e*x^2 + d)/(c*e^2*x^4 + b*e^2*x^2 - c*d^2 + b*d*e), x)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.16 \[ \int \frac {\sqrt {d+e x^2}}{-c d^2+b d e+b e^2 x^2+c e^2 x^4} \, dx=-\frac {\arctan \left (\frac {{\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{2} c - 3 \, c d + 2 \, b e}{2 \, \sqrt {-2 \, c^{2} d^{2} + 3 \, b c d e - b^{2} e^{2}}}\right )}{\sqrt {-2 \, c^{2} d^{2} + 3 \, b c d e - b^{2} e^{2}} \sqrt {e}} \]

[In]

integrate((e*x^2+d)^(1/2)/(c*e^2*x^4+b*e^2*x^2+b*d*e-c*d^2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {d+e x^2}}{-c d^2+b d e+b e^2 x^2+c e^2 x^4} \, dx=\int \frac {\sqrt {e\,x^2+d}}{-c\,d^2+b\,d\,e+c\,e^2\,x^4+b\,e^2\,x^2} \,d x \]

[In]

int((d + e*x^2)^(1/2)/(b*e^2*x^2 - c*d^2 + c*e^2*x^4 + b*d*e),x)

[Out]

int((d + e*x^2)^(1/2)/(b*e^2*x^2 - c*d^2 + c*e^2*x^4 + b*d*e), x)