3.1004 \(\int \frac {a-c x^4}{\sqrt {a+b x^2+c x^4} (a d+a e x^2+c d x^4)} \, dx\)

Optimal. Leaf size=54 \[ \frac {\tanh ^{-1}\left (\frac {x \sqrt {b d-a e}}{\sqrt {d} \sqrt {a+b x^2+c x^4}}\right )}{\sqrt {d} \sqrt {b d-a e}} \]

[Out]

arctanh(x*(-a*e+b*d)^(1/2)/d^(1/2)/(c*x^4+b*x^2+a)^(1/2))/d^(1/2)/(-a*e+b*d)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.25, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.047, Rules used = {2112, 208} \[ \frac {\tanh ^{-1}\left (\frac {x \sqrt {b d-a e}}{\sqrt {d} \sqrt {a+b x^2+c x^4}}\right )}{\sqrt {d} \sqrt {b d-a e}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTanh[(Sqrt[b*d - a*e]*x)/(Sqrt[d]*Sqrt[a + b*x^2 + c*x^4])]/(Sqrt[d]*Sqrt[b*d - a*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 2112

Int[((u_)*((A_) + (B_.)*(x_)^4))/Sqrt[v_], x_Symbol] :> With[{a = Coeff[v, x, 0], b = Coeff[v, x, 2], c = Coef
f[v, x, 4], d = Coeff[1/u, x, 0], e = Coeff[1/u, x, 2], f = Coeff[1/u, x, 4]}, Dist[A, Subst[Int[1/(d - (b*d -
a*e)*x^2), x], x, x/Sqrt[v]], x] /; EqQ[a*B + A*c, 0] && EqQ[c*d - a*f, 0]] /; FreeQ[{A, B}, x] && PolyQ[v, x
^2, 2] && PolyQ[1/u, x^2, 2]

Rubi steps

\begin {align*} \int \frac {a-c x^4}{\sqrt {a+b x^2+c x^4} \left (a d+a e x^2+c d x^4\right )} \, dx &=a \operatorname {Subst}\left (\int \frac {1}{a d-\left (a b d-a^2 e\right ) x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2+c x^4}}\right )\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {b d-a e} x}{\sqrt {d} \sqrt {a+b x^2+c x^4}}\right )}{\sqrt {d} \sqrt {b d-a e}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 1.96, size = 419, normalized size = 7.76 \[ \frac {i \sqrt {\frac {\sqrt {b^2-4 a c}+b+2 c x^2}{\sqrt {b^2-4 a c}+b}} \sqrt {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1} \left (-\Pi \left (\frac {\left (b+\sqrt {b^2-4 a c}\right ) d}{a e-\sqrt {a} \sqrt {a e^2-4 c d^2}};i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )-\Pi \left (\frac {\left (b+\sqrt {b^2-4 a c}\right ) d}{a e+\sqrt {a} \sqrt {a e^2-4 c d^2}};i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )+F\left (i \sinh ^{-1}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )\right )}{\sqrt {2} d \sqrt {\frac {c}{\sqrt {b^2-4 a c}+b}} \sqrt {a+b x^2+c x^4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(I*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]
*(EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*
c])] - EllipticPi[((b + Sqrt[b^2 - 4*a*c])*d)/(a*e - Sqrt[a]*Sqrt[-4*c*d^2 + a*e^2]), I*ArcSinh[Sqrt[2]*Sqrt[c
/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] - EllipticPi[((b + Sqrt[b^2 - 4
*a*c])*d)/(a*e + Sqrt[a]*Sqrt[-4*c*d^2 + a*e^2]), I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + S
qrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])]))/(Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*d*Sqrt[a + b*x^2 + c*x^4
])

________________________________________________________________________________________

fricas [A]  time = 23.15, size = 305, normalized size = 5.65 \[ \left [\frac {\log \left (-\frac {c^{2} d^{2} x^{8} + 2 \, {\left (4 \, b c d^{2} - 3 \, a c d e\right )} x^{6} - {\left (8 \, a b d e - a^{2} e^{2} - 2 \, {\left (4 \, b^{2} + a c\right )} d^{2}\right )} x^{4} + a^{2} d^{2} + 2 \, {\left (4 \, a b d^{2} - 3 \, a^{2} d e\right )} x^{2} + 4 \, {\left (c d x^{5} + {\left (2 \, b d - a e\right )} x^{3} + a d x\right )} \sqrt {c x^{4} + b x^{2} + a} \sqrt {b d^{2} - a d e}}{c^{2} d^{2} x^{8} + 2 \, a c d e x^{6} + 2 \, a^{2} d e x^{2} + {\left (2 \, a c d^{2} + a^{2} e^{2}\right )} x^{4} + a^{2} d^{2}}\right )}{4 \, \sqrt {b d^{2} - a d e}}, -\frac {\sqrt {-b d^{2} + a d e} \arctan \left (\frac {2 \, \sqrt {c x^{4} + b x^{2} + a} \sqrt {-b d^{2} + a d e} x}{c d x^{4} + {\left (2 \, b d - a e\right )} x^{2} + a d}\right )}{2 \, {\left (b d^{2} - a d e\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*log(-(c^2*d^2*x^8 + 2*(4*b*c*d^2 - 3*a*c*d*e)*x^6 - (8*a*b*d*e - a^2*e^2 - 2*(4*b^2 + a*c)*d^2)*x^4 + a^2
*d^2 + 2*(4*a*b*d^2 - 3*a^2*d*e)*x^2 + 4*(c*d*x^5 + (2*b*d - a*e)*x^3 + a*d*x)*sqrt(c*x^4 + b*x^2 + a)*sqrt(b*
d^2 - a*d*e))/(c^2*d^2*x^8 + 2*a*c*d*e*x^6 + 2*a^2*d*e*x^2 + (2*a*c*d^2 + a^2*e^2)*x^4 + a^2*d^2))/sqrt(b*d^2
- a*d*e), -1/2*sqrt(-b*d^2 + a*d*e)*arctan(2*sqrt(c*x^4 + b*x^2 + a)*sqrt(-b*d^2 + a*d*e)*x/(c*d*x^4 + (2*b*d
- a*e)*x^2 + a*d))/(b*d^2 - a*d*e)]

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {c x^{4} - a}{{\left (c d x^{4} + a e x^{2} + a d\right )} \sqrt {c x^{4} + b x^{2} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [C]  time = 0.09, size = 514, normalized size = 9.52 \[ -\frac {a \left (-\RootOf \left (c d \,\textit {\_Z}^{4}+a e \,\textit {\_Z}^{2}+a d \right )^{2} e -2 d \right ) \left (-\frac {\arctanh \left (\frac {2 \RootOf \left (c d \,\textit {\_Z}^{4}+a e \,\textit {\_Z}^{2}+a d \right )^{2} c \,x^{2}+b \RootOf \left (c d \,\textit {\_Z}^{4}+a e \,\textit {\_Z}^{2}+a d \right )^{2}+b \,x^{2}+2 a}{2 \sqrt {\frac {\left (-a e +b d \right ) \RootOf \left (c d \,\textit {\_Z}^{4}+a e \,\textit {\_Z}^{2}+a d \right )^{2}}{d}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}\right )}{\sqrt {\frac {\left (-a e +b d \right ) \RootOf \left (c d \,\textit {\_Z}^{4}+a e \,\textit {\_Z}^{2}+a d \right )^{2}}{d}}}+\frac {\sqrt {2}\, \left (\RootOf \left (c d \,\textit {\_Z}^{4}+a e \,\textit {\_Z}^{2}+a d \right )^{2} c d +a e \right ) \sqrt {\frac {b \,x^{2}}{a}-\frac {\sqrt {-4 a c +b^{2}}\, x^{2}}{a}+2}\, \sqrt {\frac {b \,x^{2}}{a}+\frac {\sqrt {-4 a c +b^{2}}\, x^{2}}{a}+2}\, \RootOf \left (c d \,\textit {\_Z}^{4}+a e \,\textit {\_Z}^{2}+a d \right ) \EllipticPi \left (\frac {\sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, x}{2}, \frac {\RootOf \left (c d \,\textit {\_Z}^{4}+a e \,\textit {\_Z}^{2}+a d \right )^{2} b c d +\sqrt {-4 a c +b^{2}}\, \RootOf \left (c d \,\textit {\_Z}^{4}+a e \,\textit {\_Z}^{2}+a d \right )^{2} c d +a b e +\sqrt {-4 a c +b^{2}}\, a e}{2 a c d}, \frac {\sqrt {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 a}}\, \sqrt {2}}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}\right )}{\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, a d}\right )}{4 d \left (2 \RootOf \left (c d \,\textit {\_Z}^{4}+a e \,\textit {\_Z}^{2}+a d \right )^{2} c d +a e \right ) \RootOf \left (c d \,\textit {\_Z}^{4}+a e \,\textit {\_Z}^{2}+a d \right )}-\frac {\sqrt {2}\, \sqrt {-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}+4}\, \sqrt {\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}+4}\, \EllipticF \left (\frac {\sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, x}{2}, \frac {\sqrt {\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) b}{a c}-4}}{2}\right )}{4 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4/d*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*(4-2*(-b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)*(4+2*(b+(-4*a*c+b^2
)^(1/2))/a*x^2)^(1/2)/(c*x^4+b*x^2+a)^(1/2)*EllipticF(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4+
2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))-1/4*a/d*sum((-_alpha^2*e-2*d)/_alpha/(2*_alpha^2*c*d+a*e)*(-1/(_alpha^2
/d*(-a*e+b*d))^(1/2)*arctanh(1/2*(2*_alpha^2*c*x^2+_alpha^2*b+b*x^2+2*a)/(_alpha^2/d*(-a*e+b*d))^(1/2)/(c*x^4+
b*x^2+a)^(1/2))+1/a/d*2^(1/2)*_alpha*(_alpha^2*c*d+a*e)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*(2+b*x^2/a-1/a*x^2*(
-4*a*c+b^2)^(1/2))^(1/2)*(2+b*x^2/a+1/a*x^2*(-4*a*c+b^2)^(1/2))^(1/2)/(c*x^4+b*x^2+a)^(1/2)*EllipticPi(1/2*x*2
^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(_alpha^2*(-4*a*c+b^2)^(1/2)*c*d+_alpha^2*b*c*d+(-4*a*c+b^2)^(1/2
)*a*e+a*b*e)/a/d/c,(-1/2*(b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2))),_alpha=Ro
otOf(_Z^4*c*d+_Z^2*a*e+a*d))

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {c x^{4} - a}{{\left (c d x^{4} + a e x^{2} + a d\right )} \sqrt {c x^{4} + b x^{2} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {a-c\,x^4}{\left (c\,d\,x^4+a\,e\,x^2+a\,d\right )\,\sqrt {c\,x^4+b\,x^2+a}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \int \left (- \frac {a}{a d \sqrt {a + b x^{2} + c x^{4}} + a e x^{2} \sqrt {a + b x^{2} + c x^{4}} + c d x^{4} \sqrt {a + b x^{2} + c x^{4}}}\right )\, dx - \int \frac {c x^{4}}{a d \sqrt {a + b x^{2} + c x^{4}} + a e x^{2} \sqrt {a + b x^{2} + c x^{4}} + c d x^{4} \sqrt {a + b x^{2} + c x^{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(-a/(a*d*sqrt(a + b*x**2 + c*x**4) + a*e*x**2*sqrt(a + b*x**2 + c*x**4) + c*d*x**4*sqrt(a + b*x**2 +
c*x**4)), x) - Integral(c*x**4/(a*d*sqrt(a + b*x**2 + c*x**4) + a*e*x**2*sqrt(a + b*x**2 + c*x**4) + c*d*x**4*
sqrt(a + b*x**2 + c*x**4)), x)

________________________________________________________________________________________