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

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Rubi [A]  time = 0.253317, antiderivative size = 54, normalized size of antiderivative = 1., 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 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]

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{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 = 1.82404, 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
])

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Maple [C]  time = 0.034, size = 514, normalized size = 9.5 \begin{align*} -{\frac{\sqrt{2}}{4\,d}\sqrt{4-2\,{\frac{ \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ){x}^{2}}{a}}}\sqrt{4+2\,{\frac{ \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ){x}^{2}}{a}}}{\it EllipticF} \left ({\frac{x\sqrt{2}}{2}\sqrt{{\frac{1}{a} \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) }}},{\frac{1}{2}\sqrt{-4+2\,{\frac{b \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) }{ac}}}} \right ){\frac{1}{\sqrt{{\frac{1}{a} \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) }}}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}-{\frac{a}{4\,d}\sum _{{\it \_alpha}={\it RootOf} \left ( cd{{\it \_Z}}^{4}+ae{{\it \_Z}}^{2}+ad \right ) }{\frac{-{{\it \_alpha}}^{2}e-2\,d}{{\it \_alpha}\, \left ( 2\,{{\it \_alpha}}^{2}cd+ae \right ) } \left ( -{{\it Artanh} \left ({\frac{2\,{{\it \_alpha}}^{2}c{x}^{2}+b{{\it \_alpha}}^{2}+b{x}^{2}+2\,a}{2}{\frac{1}{\sqrt{{\frac{{{\it \_alpha}}^{2} \left ( -ae+bd \right ) }{d}}}}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}} \right ){\frac{1}{\sqrt{{\frac{{{\it \_alpha}}^{2} \left ( -ae+bd \right ) }{d}}}}}}+{\frac{\sqrt{2}{\it \_alpha}\, \left ({{\it \_alpha}}^{2}cd+ae \right ) }{ad}\sqrt{2+{\frac{b{x}^{2}}{a}}-{\frac{{x}^{2}}{a}\sqrt{-4\,ac+{b}^{2}}}}\sqrt{2+{\frac{b{x}^{2}}{a}}+{\frac{{x}^{2}}{a}\sqrt{-4\,ac+{b}^{2}}}}{\it EllipticPi} \left ({\frac{x\sqrt{2}}{2}\sqrt{{\frac{1}{a} \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) }}},{\frac{1}{2\,acd} \left ({{\it \_alpha}}^{2}\sqrt{-4\,ac+{b}^{2}}cd+{{\it \_alpha}}^{2}bcd+\sqrt{-4\,ac+{b}^{2}}ae+abe \right ) },{\sqrt{2}\sqrt{-{\frac{1}{2\,a} \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) }}{\frac{1}{\sqrt{{\frac{1}{a} \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) }}}}} \right ){\frac{1}{\sqrt{{\frac{1}{a} \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) }}}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}} \right ) }} \end{align*}

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)/(1/a*(-b+(-4*a*c+b^2)^(1/2)))^(1/2)*(4-2/a*(-b+(-4*a*c+b^2)^(1/2))*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)*(1/a*(-b+(-4*a*c+b^2)^(1/2)))^(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/(_alp
ha^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)/(1/a*(-b+(-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)*(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)*(1/a*(-b+(-4*a*c+b^2)^(1/2)))^(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)/(1/a*(-b+(-4*a*c+b^2)^(1/2)))^(1/2)))
,_alpha=RootOf(_Z^4*c*d+_Z^2*a*e+a*d))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\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} \end{align*}

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)

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Fricas [A]  time = 111.351, size = 647, normalized size = 11.98 \begin{align*} \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 ] \end{align*}

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)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int - \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}}}\, 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 \end{align*}

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)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

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)