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

Optimal. Leaf size=80 \[ \frac{\tan ^{-1}\left (\frac{x \sqrt{a e^2-b d e+c d^2}}{\sqrt{d} \sqrt{e} \sqrt{a+b x^2+c x^4}}\right )}{\sqrt{d} \sqrt{e} \sqrt{a e^2-b d e+c d^2}} \]

[Out]

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

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Rubi [A]  time = 0.700983, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043 \[ \frac{\tan ^{-1}\left (\frac{x \sqrt{a e^2-b d e+c d^2}}{\sqrt{d} \sqrt{e} \sqrt{a+b x^2+c x^4}}\right )}{\sqrt{d} \sqrt{e} \sqrt{a e^2-b d e+c d^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-c*x**4+a)/(c*d*x**2+a*e)/(e*x**2+d)/(c*x**4+b*x**2+a)**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 0.770264, size = 383, normalized size = 4.79 \[ \frac{i \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{2 c x^2}{b-\sqrt{b^2-4 a c}}+1} \left (-\Pi \left (\frac{\left (b+\sqrt{b^2-4 a c}\right ) d}{2 a e};i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )-\Pi \left (\frac{\left (b+\sqrt{b^2-4 a c}\right ) e}{2 c d};i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )+F\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )\right )}{\sqrt{2} d e \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{a+b x^2+c x^4}} \]

Antiderivative was successfully verified.

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

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Maple [C]  time = 0.059, size = 555, normalized size = 6.9 \[ -{\frac{\sqrt{2}}{4\,de}\sqrt{4-2\,{\frac{ \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ){x}^{2}}{a}}}\sqrt{4+2\,{\frac{ \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ){x}^{2}}{a}}}{\it EllipticF} \left ({\frac{\sqrt{2}x}{2}\sqrt{{\frac{1}{a} \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) }}},{\frac{1}{2}\sqrt{-4+2\,{\frac{b \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) }{ac}}}} \right ){\frac{1}{\sqrt{{\frac{1}{a} \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) }}}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}+{\frac{\sqrt{2}}{de}\sqrt{1+{\frac{b{x}^{2}}{2\,a}}-{\frac{{x}^{2}}{2\,a}\sqrt{-4\,ac+{b}^{2}}}}\sqrt{1+{\frac{b{x}^{2}}{2\,a}}+{\frac{{x}^{2}}{2\,a}\sqrt{-4\,ac+{b}^{2}}}}{\it EllipticPi} \left ({\frac{\sqrt{2}x}{2}\sqrt{{\frac{1}{a} \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) }}},-2\,{\frac{ae}{ \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) d}},{\sqrt{2}\sqrt{-{\frac{1}{2\,a} \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) }}{\frac{1}{\sqrt{{\frac{1}{a} \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) }}}}} \right ){\frac{1}{\sqrt{-{\frac{b}{a}}+{\frac{1}{a}\sqrt{-4\,ac+{b}^{2}}}}}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}+{\frac{\sqrt{2}}{de}\sqrt{1+{\frac{b{x}^{2}}{2\,a}}-{\frac{{x}^{2}}{2\,a}\sqrt{-4\,ac+{b}^{2}}}}\sqrt{1+{\frac{b{x}^{2}}{2\,a}}+{\frac{{x}^{2}}{2\,a}\sqrt{-4\,ac+{b}^{2}}}}{\it EllipticPi} \left ({\frac{\sqrt{2}x}{2}\sqrt{{\frac{1}{a} \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) }}},-2\,{\frac{cd}{ \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) e}},{\sqrt{2}\sqrt{-{\frac{1}{2\,a} \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) }}{\frac{1}{\sqrt{{\frac{1}{a} \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) }}}}} \right ){\frac{1}{\sqrt{-{\frac{b}{a}}+{\frac{1}{a}\sqrt{-4\,ac+{b}^{2}}}}}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 44.3741, size = 1, normalized size = 0.01 \[ \left [\frac{\log \left (-\frac{{\left (c^{2} d^{2} e^{2} x^{8} - 2 \,{\left (3 \, c^{2} d^{3} e - 4 \, b c d^{2} e^{2} + 3 \, a c d e^{3}\right )} x^{6} + a^{2} d^{2} e^{2} +{\left (c^{2} d^{4} - 8 \, b c d^{3} e - 8 \, a b d e^{3} + a^{2} e^{4} + 4 \,{\left (2 \, b^{2} + a c\right )} d^{2} e^{2}\right )} x^{4} - 2 \,{\left (3 \, a c d^{3} e - 4 \, a b d^{2} e^{2} + 3 \, a^{2} d e^{3}\right )} x^{2}\right )} \sqrt{-c d^{3} e + b d^{2} e^{2} - a d e^{3}} - 4 \,{\left ({\left (c^{2} d^{4} e^{2} - b c d^{3} e^{3} + a c d^{2} e^{4}\right )} x^{5} -{\left (c^{2} d^{5} e - 3 \, b c d^{4} e^{2} - 3 \, a b d^{2} e^{4} + a^{2} d e^{5} + 2 \,{\left (b^{2} + a c\right )} d^{3} e^{3}\right )} x^{3} +{\left (a c d^{4} e^{2} - a b d^{3} e^{3} + a^{2} d^{2} e^{4}\right )} x\right )} \sqrt{c x^{4} + b x^{2} + a}}{c^{2} d^{2} e^{2} x^{8} + 2 \,{\left (c^{2} d^{3} e + a c d e^{3}\right )} x^{6} + a^{2} d^{2} e^{2} +{\left (c^{2} d^{4} + 4 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} x^{4} + 2 \,{\left (a c d^{3} e + a^{2} d e^{3}\right )} x^{2}}\right )}{4 \, \sqrt{-c d^{3} e + b d^{2} e^{2} - a d e^{3}}}, \frac{\arctan \left (\frac{2 \, \sqrt{c d^{3} e - b d^{2} e^{2} + a d e^{3}} \sqrt{c x^{4} + b x^{2} + a} x}{c d e x^{4} + a d e -{\left (c d^{2} - 2 \, b d e + a e^{2}\right )} x^{2}}\right )}{2 \, \sqrt{c d^{3} e - b d^{2} e^{2} + a d e^{3}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError