\(\int \frac {1}{\sqrt {d+e x^2} (-c d^2+b d e+b e^2 x^2+c e^2 x^4)} \, dx\) [223]

Optimal result

Integrand size = 41, antiderivative size = 106 \[ \int \frac {1}{\sqrt {d+e x^2} \left (-c d^2+b d e+b e^2 x^2+c e^2 x^4\right )} \, dx=-\frac {x}{d (2 c d-b e) \sqrt {d+e x^2}}-\frac {c \text {arctanh}\left (\frac {\sqrt {e} \sqrt {2 c d-b e} x}{\sqrt {c d-b e} \sqrt {d+e x^2}}\right )}{\sqrt {e} \sqrt {c d-b e} (2 c d-b e)^{3/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.098, Rules used = {1163, 390, 385, 214} \[ \int \frac {1}{\sqrt {d+e x^2} \left (-c d^2+b d e+b e^2 x^2+c e^2 x^4\right )} \, dx=-\frac {c \text {arctanh}\left (\frac {\sqrt {e} x \sqrt {2 c d-b e}}{\sqrt {d+e x^2} \sqrt {c d-b e}}\right )}{\sqrt {e} \sqrt {c d-b e} (2 c d-b e)^{3/2}}-\frac {x}{d \sqrt {d+e x^2} (2 c d-b e)} \]

[In]

Int[1/(Sqrt[d + e*x^2]*(-(c*d^2) + b*d*e + b*e^2*x^2 + c*e^2*x^4)),x]

[Out]

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

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 390

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*x*(a + b*x^n)^(p + 1)*
((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Dist[(b*c + n*(p + 1)*(b*c - a*d))/(a*n*(p + 1)*(b*c - a
*d)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, q}, x] && NeQ[b*c - a*d, 0] && Eq
Q[n*(p + q + 2) + 1, 0] && (LtQ[p, -1] ||  !LtQ[q, -1]) && NeQ[p, -1]

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}{\left (d+e x^2\right )^{3/2} \left (\frac {-c d^2+b d e}{d}+c e x^2\right )} \, dx \\ & = -\frac {x}{d (2 c d-b e) \sqrt {d+e x^2}}+\frac {c \int \frac {1}{\sqrt {d+e x^2} \left (\frac {-c d^2+b d e}{d}+c e x^2\right )} \, dx}{2 c d-b e} \\ & = -\frac {x}{d (2 c d-b e) \sqrt {d+e x^2}}+\frac {c \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 )}{2 c d-b e} \\ & = -\frac {x}{d (2 c d-b e) \sqrt {d+e x^2}}-\frac {c \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} (2 c d-b e)^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.62 \[ \int \frac {1}{\sqrt {d+e x^2} \left (-c d^2+b d e+b e^2 x^2+c e^2 x^4\right )} \, dx=-\frac {\sqrt {e} \left (2 c^2 d^2-3 b c d e+b^2 e^2\right ) x+c d \sqrt {2 c^2 d^2-3 b c d e+b^2 e^2} \sqrt {d+e x^2} \text {arctanh}\left (\frac {-b e+c \left (d-e x^2+\sqrt {e} x \sqrt {d+e x^2}\right )}{\sqrt {2 c^2 d^2-3 b c d e+b^2 e^2}}\right )}{d \sqrt {e} (c d-b e) (-2 c d+b e)^2 \sqrt {d+e x^2}} \]

[In]

Integrate[1/(Sqrt[d + e*x^2]*(-(c*d^2) + b*d*e + b*e^2*x^2 + c*e^2*x^4)),x]

[Out]

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

Maple [A] (verified)

Time = 0.30 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.15

method	result	size
pseudoelliptic	\(\frac {-c d \,\operatorname {arctanh}\left (\frac {\left (b e -c d \right ) \sqrt {e \,x^{2}+d}}{x \sqrt {e \left (b e -2 c d \right ) \left (b e -c d \right )}}\right ) \sqrt {e \,x^{2}+d}+x \sqrt {e \left (b e -2 c d \right ) \left (b e -c d \right )}}{\left (b e -2 c d \right ) \sqrt {e \,x^{2}+d}\, \sqrt {e \left (b e -2 c d \right ) \left (b e -c d \right )}\, d}\)	\(122\)
default	\(-\frac {c \sqrt {\left (x +\frac {\sqrt {-e d}}{e}\right )^{2} e -2 \sqrt {-e d}\, \left (x +\frac {\sqrt {-e d}}{e}\right )}}{2 d \left (\sqrt {-e d}\, c +\sqrt {-\left (b e -c d \right ) e c}\right ) \left (-\sqrt {-e d}\, c +\sqrt {-\left (b e -c d \right ) e c}\right ) \left (x +\frac {\sqrt {-e d}}{e}\right )}-\frac {c \sqrt {\left (x -\frac {\sqrt {-e d}}{e}\right )^{2} e +2 \sqrt {-e d}\, \left (x -\frac {\sqrt {-e d}}{e}\right )}}{2 d \left (\sqrt {-e d}\, c +\sqrt {-\left (b e -c d \right ) e c}\right ) \left (-\sqrt {-e d}\, c +\sqrt {-\left (b e -c d \right ) e c}\right ) \left (x -\frac {\sqrt {-e d}}{e}\right )}+\frac {c^{2} e \ln \left (\frac {-\frac {2 \left (b e -2 c d \right )}{c}-\frac {2 \sqrt {-\left (b e -c d \right ) e c}\, \left (x +\frac {\sqrt {-\left (b e -c d \right ) e c}}{e c}\right )}{c}+2 \sqrt {-\frac {b e -2 c d}{c}}\, \sqrt {\left (x +\frac {\sqrt {-\left (b e -c d \right ) e c}}{e c}\right )^{2} e -\frac {2 \sqrt {-\left (b e -c d \right ) e c}\, \left (x +\frac {\sqrt {-\left (b e -c d \right ) e c}}{e c}\right )}{c}-\frac {b e -2 c d}{c}}}{x +\frac {\sqrt {-\left (b e -c d \right ) e c}}{e c}}\right )}{2 \left (\sqrt {-e d}\, c +\sqrt {-\left (b e -c d \right ) e c}\right ) \left (-\sqrt {-e d}\, c +\sqrt {-\left (b e -c d \right ) e c}\right ) \sqrt {-\left (b e -c d \right ) e c}\, \sqrt {-\frac {b e -2 c d}{c}}}-\frac {c^{2} e \ln \left (\frac {-\frac {2 \left (b e -2 c d \right )}{c}+\frac {2 \sqrt {-\left (b e -c d \right ) e c}\, \left (x -\frac {\sqrt {-\left (b e -c d \right ) e c}}{e c}\right )}{c}+2 \sqrt {-\frac {b e -2 c d}{c}}\, \sqrt {\left (x -\frac {\sqrt {-\left (b e -c d \right ) e c}}{e c}\right )^{2} e +\frac {2 \sqrt {-\left (b e -c d \right ) e c}\, \left (x -\frac {\sqrt {-\left (b e -c d \right ) e c}}{e c}\right )}{c}-\frac {b e -2 c d}{c}}}{x -\frac {\sqrt {-\left (b e -c d \right ) e c}}{e c}}\right )}{2 \left (\sqrt {-e d}\, c +\sqrt {-\left (b e -c d \right ) e c}\right ) \left (-\sqrt {-e d}\, c +\sqrt {-\left (b e -c d \right ) e c}\right ) \sqrt {-\left (b e -c d \right ) e c}\, \sqrt {-\frac {b e -2 c d}{c}}}\)	\(771\)

[In]

int(1/(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]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 309 vs. \(2 (90) = 180\).

Time = 0.34 (sec) , antiderivative size = 701, normalized size of antiderivative = 6.61 \[ \int \frac {1}{\sqrt {d+e x^2} \left (-c d^2+b d e+b e^2 x^2+c e^2 x^4\right )} \, dx=\left [-\frac {4 \, {\left (2 \, c^{2} d^{2} e - 3 \, b c d e^{2} + b^{2} e^{3}\right )} \sqrt {e x^{2} + d} x + \sqrt {2 \, c^{2} d^{2} e - 3 \, b c d e^{2} + b^{2} e^{3}} {\left (c d e x^{2} + c d^{2}\right )} \log \left (\frac {c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2} + {\left (17 \, c^{2} d^{2} e^{2} - 24 \, b c d e^{3} + 8 \, b^{2} e^{4}\right )} x^{4} + 2 \, {\left (7 \, c^{2} d^{3} e - 11 \, b c d^{2} e^{2} + 4 \, b^{2} d e^{3}\right )} x^{2} + 4 \, \sqrt {2 \, c^{2} d^{2} e - 3 \, b c d e^{2} + b^{2} e^{3}} {\left ({\left (3 \, c d e - 2 \, b e^{2}\right )} x^{3} + {\left (c d^{2} - b d e\right )} x\right )} \sqrt {e x^{2} + d}}{c^{2} e^{2} x^{4} + c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2} - 2 \, {\left (c^{2} d e - b c e^{2}\right )} x^{2}}\right )}{4 \, {\left (4 \, c^{3} d^{5} e - 8 \, b c^{2} d^{4} e^{2} + 5 \, b^{2} c d^{3} e^{3} - b^{3} d^{2} e^{4} + {\left (4 \, c^{3} d^{4} e^{2} - 8 \, b c^{2} d^{3} e^{3} + 5 \, b^{2} c d^{2} e^{4} - b^{3} d e^{5}\right )} x^{2}\right )}}, -\frac {2 \, {\left (2 \, c^{2} d^{2} e - 3 \, b c d e^{2} + b^{2} e^{3}\right )} \sqrt {e x^{2} + d} x + \sqrt {-2 \, c^{2} d^{2} e + 3 \, b c d e^{2} - b^{2} e^{3}} {\left (c d e x^{2} + c d^{2}\right )} \arctan \left (-\frac {\sqrt {-2 \, c^{2} d^{2} e + 3 \, b c d e^{2} - b^{2} e^{3}} {\left (c d^{2} - b d e + {\left (3 \, c d e - 2 \, b e^{2}\right )} x^{2}\right )} \sqrt {e x^{2} + d}}{2 \, {\left ({\left (2 \, c^{2} d^{2} e^{2} - 3 \, b c d e^{3} + b^{2} e^{4}\right )} x^{3} + {\left (2 \, c^{2} d^{3} e - 3 \, b c d^{2} e^{2} + b^{2} d e^{3}\right )} x\right )}}\right )}{2 \, {\left (4 \, c^{3} d^{5} e - 8 \, b c^{2} d^{4} e^{2} + 5 \, b^{2} c d^{3} e^{3} - b^{3} d^{2} e^{4} + {\left (4 \, c^{3} d^{4} e^{2} - 8 \, b c^{2} d^{3} e^{3} + 5 \, b^{2} c d^{2} e^{4} - b^{3} d e^{5}\right )} x^{2}\right )}}\right ] \]

[In]

integrate(1/(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*(4*(2*c^2*d^2*e - 3*b*c*d*e^2 + b^2*e^3)*sqrt(e*x^2 + d)*x + sqrt(2*c^2*d^2*e - 3*b*c*d*e^2 + b^2*e^3)*(
c*d*e*x^2 + c*d^2)*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)))/(4*c^3*d^5*e - 8*b*c^2*d^4*e^2 + 5*b^2*c*d^3*e^3 - b^3*d^2*e^4 + (4*c^3*d^4*e^2 - 8*b*c^2*d^
3*e^3 + 5*b^2*c*d^2*e^4 - b^3*d*e^5)*x^2), -1/2*(2*(2*c^2*d^2*e - 3*b*c*d*e^2 + b^2*e^3)*sqrt(e*x^2 + d)*x + s
qrt(-2*c^2*d^2*e + 3*b*c*d*e^2 - b^2*e^3)*(c*d*e*x^2 + c*d^2)*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)))/(4*c^3*d^5*e - 8*b*c^2*d^4*e^2 + 5*b^2*c*d^3*e^3 - b^3*d^2*e^
4 + (4*c^3*d^4*e^2 - 8*b*c^2*d^3*e^3 + 5*b^2*c*d^2*e^4 - b^3*d*e^5)*x^2)]

Sympy [F]

\[ \int \frac {1}{\sqrt {d+e x^2} \left (-c d^2+b d e+b e^2 x^2+c e^2 x^4\right )} \, dx=\int \frac {1}{\left (d + e x^{2}\right )^{\frac {3}{2}} \left (b e - c d + c e x^{2}\right )}\, dx \]

[In]

integrate(1/(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/((d + e*x**2)**(3/2)*(b*e - c*d + c*e*x**2)), x)

Maxima [F]

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

[In]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.22 \[ \int \frac {1}{\sqrt {d+e x^2} \left (-c d^2+b d e+b e^2 x^2+c e^2 x^4\right )} \, dx=\frac {c \sqrt {e} \arctan \left (-\frac {{\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{2} c - 3 \, c d + 2 \, b e}{2 \, \sqrt {-2 \, c^{2} d^{2} + 3 \, b c d e - b^{2} e^{2}}}\right )}{\sqrt {-2 \, c^{2} d^{2} + 3 \, b c d e - b^{2} e^{2}} {\left (2 \, c d e - b e^{2}\right )}} - \frac {x}{{\left (2 \, c d^{2} - b d e\right )} \sqrt {e x^{2} + d}} \]

[In]

integrate(1/(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]

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

Mupad [F(-1)]

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

[In]

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