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

Optimal. Leaf size=53 \[ \frac{\tan ^{-1}\left (\frac{x \sqrt{a e+b d}}{\sqrt{d} \sqrt{a-b x^2+c x^4}}\right )}{\sqrt{d} \sqrt{a e+b d}} \]

[Out]

ArcTan[(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 = 0.418844, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ \frac{\tan ^{-1}\left (\frac{x \sqrt{a e+b d}}{\sqrt{d} \sqrt{a-b x^2+c x^4}}\right )}{\sqrt{d} \sqrt{a e+b d}} \]

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]

ArcTan[(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 in Sympy [A]  time = 25.7817, size = 48, normalized size = 0.91 \[ \frac{\operatorname{atan}{\left (\frac{x \sqrt{a e + b d}}{\sqrt{d} \sqrt{a - b x^{2} + c x^{4}}} \right )}}{\sqrt{d} \sqrt{a e + b d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_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]

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

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Mathematica [C]  time = 1.17526, size = 416, normalized size = 7.85 \[ \frac{i \sqrt{\frac{4 c x^2}{\sqrt{b^2-4 a c}-b}+2} \sqrt{1-\frac{2 c x^2}{\sqrt{b^2-4 a c}+b}} \left (-\Pi \left (\frac{\left (b-\sqrt{b^2-4 a c}\right ) d}{\sqrt{a} \sqrt{a e^2-4 c d^2}-a e};i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{\sqrt{b^2-4 a c}-b}} x\right )|\frac{b-\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}\right )-\Pi \left (\frac{\left (\sqrt{b^2-4 a c}-b\right ) d}{a e+\sqrt{a} \sqrt{a e^2-4 c d^2}};i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{\sqrt{b^2-4 a c}-b}} x\right )|\frac{b-\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}\right )+F\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{\sqrt{b^2-4 a c}-b}} x\right )|\frac{b-\sqrt{b^2-4 a c}}{b+\sqrt{b^2-4 a c}}\right )\right )}{2 d \sqrt{\frac{c}{\sqrt{b^2-4 a c}-b}} \sqrt{a-b x^2+c x^4}} \]

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/2)*Sqrt[2 + (4*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 - Sqrt[b^2 - 4*a*c])/
(b + Sqrt[b^2 - 4*a*c])]))/(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 = 0.067, size = 517, normalized size = 9.8 \[ -{\frac{\sqrt{2}}{4\,d}\sqrt{4-2\,{\frac{ \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ){x}^{2}}{a}}}\sqrt{4+2\,{\frac{ \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ){x}^{2}}{a}}}{\it EllipticF} \left ({\frac{\sqrt{2}x}{2}\sqrt{{\frac{1}{a} \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) }}},{\frac{1}{2}\sqrt{-4-2\,{\frac{b \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) }{ac}}}} \right ){\frac{1}{\sqrt{{\frac{1}{a} \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) }}}}{\frac{1}{\sqrt{c{x}^{4}-b{x}^{2}+a}}}}-{\frac{a}{4\,d}\sum _{{\it \_alpha}={\it RootOf} \left ({{\it \_Z}}^{4}cd+{{\it \_Z}}^{2}ae+ad \right ) }{\frac{-{{\it \_alpha}}^{2}e-2\,d}{{\it \_alpha}\, \left ( 2\,{{\it \_alpha}}^{2}cd+ae \right ) } \left ( -{1{\it Artanh} \left ({\frac{2\,{{\it \_alpha}}^{2}c{x}^{2}-b{{\it \_alpha}}^{2}-b{x}^{2}+2\,a}{2}{\frac{1}{\sqrt{-{\frac{{{\it \_alpha}}^{2} \left ( ae+bd \right ) }{d}}}}}{\frac{1}{\sqrt{c{x}^{4}-b{x}^{2}+a}}}} \right ){\frac{1}{\sqrt{-{\frac{{{\it \_alpha}}^{2} \left ( ae+bd \right ) }{d}}}}}}+{\frac{\sqrt{2}{\it \_alpha}\, \left ({{\it \_alpha}}^{2}cd+ae \right ) }{ad}\sqrt{2-{\frac{b{x}^{2}}{a}}-{\frac{{x}^{2}}{a}\sqrt{-4\,ac+{b}^{2}}}}\sqrt{2-{\frac{b{x}^{2}}{a}}+{\frac{{x}^{2}}{a}\sqrt{-4\,ac+{b}^{2}}}}{\it EllipticPi} \left ({\frac{\sqrt{2}x}{2}\sqrt{{\frac{1}{a} \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) }}},-{\frac{1}{2\,acd} \left ( -{{\it \_alpha}}^{2}\sqrt{-4\,ac+{b}^{2}}cd+{{\it \_alpha}}^{2}bcd-\sqrt{-4\,ac+{b}^{2}}ae+abe \right ) },{\sqrt{2}\sqrt{-{\frac{1}{2\,a} \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) }}{\frac{1}{\sqrt{{\frac{1}{a} \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) }}}}} \right ){\frac{1}{\sqrt{{\frac{1}{a} \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) }}}}{\frac{1}{\sqrt{c{x}^{4}-b{x}^{2}+a}}}} \right ) }} \]

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)*Ellipti
cF(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*(a*e+b*d)/d)^(1/2)*arctanh(1/2*(2*_alpha^2*c*x^2-_alpha^2*b-b*x^2+2*a
)/(-_alpha^2*(a*e+b*d)/d)^(1/2)/(c*x^4-b*x^2+a)^(1/2))+1/a/d*2^(1/2)*_alpha*(_al
pha^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/c/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))),_alpha=Roo
tOf(_Z^4*c*d+_Z^2*a*e+a*d))

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

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)*sqrt(c*x^4 - b*x^2 + a)),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 = 12.8586, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (\frac{4 \,{\left ({\left (b c d^{3} + a c d^{2} e\right )} x^{5} -{\left (2 \, b^{2} d^{3} + 3 \, a b d^{2} e + a^{2} d e^{2}\right )} x^{3} +{\left (a b d^{3} + a^{2} d^{2} e\right )} x\right )} \sqrt{c x^{4} - b x^{2} + a} -{\left (c^{2} d^{2} x^{8} - 2 \,{\left (4 \, b c d^{2} + 3 \, a c d e\right )} x^{6} +{\left (8 \, a b d e + a^{2} e^{2} + 2 \,{\left (4 \, b^{2} + a c\right )} d^{2}\right )} x^{4} + a^{2} d^{2} - 2 \,{\left (4 \, a b d^{2} + 3 \, a^{2} d e\right )} x^{2}\right )} \sqrt{-b d^{2} - a d e}}{c^{2} d^{2} x^{8} + 2 \, a c d e x^{6} + 2 \, a^{2} d e x^{2} +{\left (2 \, a c d^{2} + a^{2} e^{2}\right )} x^{4} + a^{2} d^{2}}\right )}{4 \, \sqrt{-b d^{2} - a d e}}, \frac{\arctan \left (\frac{2 \, \sqrt{c x^{4} - b x^{2} + a} \sqrt{b d^{2} + a d e} x}{c d x^{4} -{\left (2 \, b d + a e\right )} x^{2} + a d}\right )}{2 \, \sqrt{b d^{2} + a d e}}\right ] \]

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)*sqrt(c*x^4 - b*x^2 + a)),x, algorithm="fricas")

[Out]

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

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

[Out]

Exception raised: TypeError