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

Optimal. Leaf size=76 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {e} x \sqrt {2 c d-b e}}{\sqrt {d+e x^2} \sqrt {c d-b e}}\right )}{\sqrt {e} \sqrt {c d-b e} \sqrt {2 c d-b e}} \]

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Rubi [A]  time = 0.07, antiderivative size = 76, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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}} \end {gather*}

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 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 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 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]

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*}

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Mathematica [A]  time = 0.07, size = 76, normalized size = 1.00 \begin {gather*} -\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}} \end {gather*}

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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IntegrateAlgebraic [B]  time = 0.19, size = 192, normalized size = 2.53 \begin {gather*} -\frac {\sqrt {b^2 e^2-3 b c d e+2 c^2 d^2} \tanh ^{-1}\left (-\frac {c e x^2}{\sqrt {b^2 e^2-3 b c d e+2 c^2 d^2}}+\frac {c \sqrt {e} x \sqrt {d+e x^2}}{\sqrt {b^2 e^2-3 b c d e+2 c^2 d^2}}+\frac {c d}{\sqrt {b^2 e^2-3 b c d e+2 c^2 d^2}}-\frac {b e}{\sqrt {b^2 e^2-3 b c d e+2 c^2 d^2}}\right )}{\sqrt {e} (b e-2 c d) (b e-c d)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 1.09, size = 432, normalized size = 5.68 \begin {gather*} \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 {gather*}

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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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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

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maple [B]  time = 0.02, size = 2252, normalized size = 29.63

result too large to display

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 {gather*}

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {d + e x^{2}} \left (b e - c d + c e x^{2}\right )}\, dx \end {gather*}

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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