3.222 \(\int \frac{\sqrt{d+e x^2}}{-c d^2+b d e+b e^2 x^2+c e^2 x^4} \, dx\)
Optimal. Leaf size=76 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{e} x \sqrt{2 c d-b e}}{\sqrt{d+e x^2} \sqrt{c d-b e}}\right )}{\sqrt{e} \sqrt{c d-b e} \sqrt{2 c d-b e}} \]
[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]))
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Rubi [A] time = 0.0704533, antiderivative size = 76, normalized size of antiderivative = 1.,
number of steps used = 3, number of rules used = 3, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.073, Rules used =
{1149, 377, 208} \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{e} x \sqrt{2 c d-b e}}{\sqrt{d+e x^2} \sqrt{c d-b e}}\right )}{\sqrt{e} \sqrt{c d-b e} \sqrt{2 c d-b e}} \]
Antiderivative was successfully verified.
[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 1149
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*x^2)/e)^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]
Rule 377
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 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]
Rubi steps
\begin{align*} \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 (\frac{-c d^2+b d e}{d}+c e x^2\right )} \, dx\\ &=\operatorname{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] time = 0.067776, size = 76, normalized size = 1. \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{e} x \sqrt{2 c d-b e}}{\sqrt{d+e x^2} \sqrt{c d-b e}}\right )}{\sqrt{e} \sqrt{c d-b e} \sqrt{2 c d-b e}} \]
Antiderivative was successfully verified.
[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[(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]))
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Maple [B] time = 0.02, size = 2252, normalized size = 29.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
[Out]
1/2*c^2*e/((-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/(-(-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/(-(b*e-c*d)*c*e)^(1
/2)*((x-(-(b*e-c*d)*c*e)^(1/2)/c/e)^2*e+2*(-(b*e-c*d)*c*e)^(1/2)/c*(x-(-(b*e-c*d)*c*e)^(1/2)/c/e)-(b*e-2*c*d)/
c)^(1/2)+1/2*c*e^(1/2)/((-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/(-(-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))*ln(((-
(b*e-c*d)*c*e)^(1/2)/c+(x-(-(b*e-c*d)*c*e)^(1/2)/c/e)*e)/e^(1/2)+((x-(-(b*e-c*d)*c*e)^(1/2)/c/e)^2*e+2*(-(b*e-
c*d)*c*e)^(1/2)/c*(x-(-(b*e-c*d)*c*e)^(1/2)/c/e)-(b*e-2*c*d)/c)^(1/2))+1/2*c*e^2/((-d*e)^(1/2)*c+(-(b*e-c*d)*c
*e)^(1/2))/(-(-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/(-(b*e-c*d)*c*e)^(1/2)/(-(b*e-2*c*d)/c)^(1/2)*ln((-2*(b*e-
2*c*d)/c+2*(-(b*e-c*d)*c*e)^(1/2)/c*(x-(-(b*e-c*d)*c*e)^(1/2)/c/e)+2*(-(b*e-2*c*d)/c)^(1/2)*((x-(-(b*e-c*d)*c*
e)^(1/2)/c/e)^2*e+2*(-(b*e-c*d)*c*e)^(1/2)/c*(x-(-(b*e-c*d)*c*e)^(1/2)/c/e)-(b*e-2*c*d)/c)^(1/2))/(x-(-(b*e-c*
d)*c*e)^(1/2)/c/e))*b-c^2*e/((-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/(-(-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/(
-(b*e-c*d)*c*e)^(1/2)/(-(b*e-2*c*d)/c)^(1/2)*ln((-2*(b*e-2*c*d)/c+2*(-(b*e-c*d)*c*e)^(1/2)/c*(x-(-(b*e-c*d)*c*
e)^(1/2)/c/e)+2*(-(b*e-2*c*d)/c)^(1/2)*((x-(-(b*e-c*d)*c*e)^(1/2)/c/e)^2*e+2*(-(b*e-c*d)*c*e)^(1/2)/c*(x-(-(b*
e-c*d)*c*e)^(1/2)/c/e)-(b*e-2*c*d)/c)^(1/2))/(x-(-(b*e-c*d)*c*e)^(1/2)/c/e))*d+1/2*c*e/(-d*e)^(1/2)/((-d*e)^(1
/2)*c+(-(b*e-c*d)*c*e)^(1/2))/(-(-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))*((x+(-d*e)^(1/2)/e)^2*e-2*(-d*e)^(1/2)*
(x+(-d*e)^(1/2)/e))^(1/2)-1/2*c*e^(1/2)/((-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/(-(-d*e)^(1/2)*c+(-(b*e-c*d)*c
*e)^(1/2))*ln(((x+(-d*e)^(1/2)/e)*e-(-d*e)^(1/2))/e^(1/2)+((x+(-d*e)^(1/2)/e)^2*e-2*(-d*e)^(1/2)*(x+(-d*e)^(1/
2)/e))^(1/2))-1/2*c*e/(-d*e)^(1/2)/((-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/(-(-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(
1/2))*((x-(-d*e)^(1/2)/e)^2*e+2*(-d*e)^(1/2)*(x-(-d*e)^(1/2)/e))^(1/2)-1/2*c*e^(1/2)/((-d*e)^(1/2)*c+(-(b*e-c*
d)*c*e)^(1/2))/(-(-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))*ln(((x-(-d*e)^(1/2)/e)*e+(-d*e)^(1/2))/e^(1/2)+((x-(-d
*e)^(1/2)/e)^2*e+2*(-d*e)^(1/2)*(x-(-d*e)^(1/2)/e))^(1/2))-1/2*c^2*e/((-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/(
-(-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/(-(b*e-c*d)*c*e)^(1/2)*((x+(-(b*e-c*d)*c*e)^(1/2)/c/e)^2*e-2*(-(b*e-c*
d)*c*e)^(1/2)/c*(x+(-(b*e-c*d)*c*e)^(1/2)/c/e)-(b*e-2*c*d)/c)^(1/2)+1/2*c*e^(1/2)/((-d*e)^(1/2)*c+(-(b*e-c*d)*
c*e)^(1/2))/(-(-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))*ln((-(-(b*e-c*d)*c*e)^(1/2)/c+(x+(-(b*e-c*d)*c*e)^(1/2)/c
/e)*e)/e^(1/2)+((x+(-(b*e-c*d)*c*e)^(1/2)/c/e)^2*e-2*(-(b*e-c*d)*c*e)^(1/2)/c*(x+(-(b*e-c*d)*c*e)^(1/2)/c/e)-(
b*e-2*c*d)/c)^(1/2))-1/2*c*e^2/((-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/(-(-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2)
)/(-(b*e-c*d)*c*e)^(1/2)/(-(b*e-2*c*d)/c)^(1/2)*ln((-2*(b*e-2*c*d)/c-2*(-(b*e-c*d)*c*e)^(1/2)/c*(x+(-(b*e-c*d)
*c*e)^(1/2)/c/e)+2*(-(b*e-2*c*d)/c)^(1/2)*((x+(-(b*e-c*d)*c*e)^(1/2)/c/e)^2*e-2*(-(b*e-c*d)*c*e)^(1/2)/c*(x+(-
(b*e-c*d)*c*e)^(1/2)/c/e)-(b*e-2*c*d)/c)^(1/2))/(x+(-(b*e-c*d)*c*e)^(1/2)/c/e))*b+c^2*e/((-d*e)^(1/2)*c+(-(b*e
-c*d)*c*e)^(1/2))/(-(-d*e)^(1/2)*c+(-(b*e-c*d)*c*e)^(1/2))/(-(b*e-c*d)*c*e)^(1/2)/(-(b*e-2*c*d)/c)^(1/2)*ln((-
2*(b*e-2*c*d)/c-2*(-(b*e-c*d)*c*e)^(1/2)/c*(x+(-(b*e-c*d)*c*e)^(1/2)/c/e)+2*(-(b*e-2*c*d)/c)^(1/2)*((x+(-(b*e-
c*d)*c*e)^(1/2)/c/e)^2*e-2*(-(b*e-c*d)*c*e)^(1/2)/c*(x+(-(b*e-c*d)*c*e)^(1/2)/c/e)-(b*e-2*c*d)/c)^(1/2))/(x+(-
(b*e-c*d)*c*e)^(1/2)/c/e))*d
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
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Fricas [B] time = 2.31571, size = 907, normalized size = 11.93 \begin{align*} \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 ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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)]
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{d + e x^{2}} \left (b e - c d + c e x^{2}\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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]
Timed out