3.930 \(\int x^4 \sqrt{a+b x^2+c x^4} \, dx\)

Optimal. Leaf size=395 \[ \frac{b x \left (8 b^2-29 a c\right ) \sqrt{a+b x^2+c x^4}}{105 c^{5/2} \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{\sqrt [4]{a} b \left (8 b^2-29 a c\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{105 c^{11/4} \sqrt{a+b x^2+c x^4}}-\frac{2 x \left (2 b^2-5 a c\right ) \sqrt{a+b x^2+c x^4}}{105 c^2}+\frac{\sqrt [4]{a} \left (2 \sqrt{a} \sqrt{c} \left (2 b^2-5 a c\right )-29 a b c+8 b^3\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{210 c^{11/4} \sqrt{a+b x^2+c x^4}}+\frac{x^3 \left (b+5 c x^2\right ) \sqrt{a+b x^2+c x^4}}{35 c} \]

[Out]

(-2*(2*b^2 - 5*a*c)*x*Sqrt[a + b*x^2 + c*x^4])/(105*c^2) + (b*(8*b^2 - 29*a*c)*x
*Sqrt[a + b*x^2 + c*x^4])/(105*c^(5/2)*(Sqrt[a] + Sqrt[c]*x^2)) + (x^3*(b + 5*c*
x^2)*Sqrt[a + b*x^2 + c*x^4])/(35*c) - (a^(1/4)*b*(8*b^2 - 29*a*c)*(Sqrt[a] + Sq
rt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTa
n[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(105*c^(11/4)*Sqrt[a + b*x
^2 + c*x^4]) + (a^(1/4)*(8*b^3 - 29*a*b*c + 2*Sqrt[a]*Sqrt[c]*(2*b^2 - 5*a*c))*(
Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*Ellip
ticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(210*c^(11/4)*
Sqrt[a + b*x^2 + c*x^4])

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Rubi [A]  time = 0.661045, antiderivative size = 395, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{b x \left (8 b^2-29 a c\right ) \sqrt{a+b x^2+c x^4}}{105 c^{5/2} \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{\sqrt [4]{a} b \left (8 b^2-29 a c\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{105 c^{11/4} \sqrt{a+b x^2+c x^4}}-\frac{2 x \left (2 b^2-5 a c\right ) \sqrt{a+b x^2+c x^4}}{105 c^2}+\frac{\sqrt [4]{a} \left (2 \sqrt{a} \sqrt{c} \left (2 b^2-5 a c\right )-29 a b c+8 b^3\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{210 c^{11/4} \sqrt{a+b x^2+c x^4}}+\frac{x^3 \left (b+5 c x^2\right ) \sqrt{a+b x^2+c x^4}}{35 c} \]

Warning: Unable to verify antiderivative.

[In]  Int[x^4*Sqrt[a + b*x^2 + c*x^4],x]

[Out]

(-2*(2*b^2 - 5*a*c)*x*Sqrt[a + b*x^2 + c*x^4])/(105*c^2) + (b*(8*b^2 - 29*a*c)*x
*Sqrt[a + b*x^2 + c*x^4])/(105*c^(5/2)*(Sqrt[a] + Sqrt[c]*x^2)) + (x^3*(b + 5*c*
x^2)*Sqrt[a + b*x^2 + c*x^4])/(35*c) - (a^(1/4)*b*(8*b^2 - 29*a*c)*(Sqrt[a] + Sq
rt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTa
n[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(105*c^(11/4)*Sqrt[a + b*x
^2 + c*x^4]) + (a^(1/4)*(8*b^3 - 29*a*b*c + 2*Sqrt[a]*Sqrt[c]*(2*b^2 - 5*a*c))*(
Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*Ellip
ticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(210*c^(11/4)*
Sqrt[a + b*x^2 + c*x^4])

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Rubi in Sympy [A]  time = 67.2463, size = 364, normalized size = 0.92 \[ - \frac{\sqrt [4]{a} b \sqrt{\frac{a + b x^{2} + c x^{4}}{\left (\sqrt{a} + \sqrt{c} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x^{2}\right ) \left (- 29 a c + 8 b^{2}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2} - \frac{b}{4 \sqrt{a} \sqrt{c}}\right )}{105 c^{\frac{11}{4}} \sqrt{a + b x^{2} + c x^{4}}} + \frac{\sqrt [4]{a} \sqrt{\frac{a + b x^{2} + c x^{4}}{\left (\sqrt{a} + \sqrt{c} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x^{2}\right ) \left (\sqrt{a} \sqrt{c} \left (- 10 a c + 4 b^{2}\right ) + b \left (- 29 a c + 8 b^{2}\right )\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2} - \frac{b}{4 \sqrt{a} \sqrt{c}}\right )}{210 c^{\frac{11}{4}} \sqrt{a + b x^{2} + c x^{4}}} + \frac{b x \left (- 29 a c + 8 b^{2}\right ) \sqrt{a + b x^{2} + c x^{4}}}{105 c^{\frac{5}{2}} \left (\sqrt{a} + \sqrt{c} x^{2}\right )} + \frac{x^{3} \left (b + 5 c x^{2}\right ) \sqrt{a + b x^{2} + c x^{4}}}{35 c} - \frac{2 x \left (- 5 a c + 2 b^{2}\right ) \sqrt{a + b x^{2} + c x^{4}}}{105 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**(1/4)*b*sqrt((a + b*x**2 + c*x**4)/(sqrt(a) + sqrt(c)*x**2)**2)*(sqrt(a) + s
qrt(c)*x**2)*(-29*a*c + 8*b**2)*elliptic_e(2*atan(c**(1/4)*x/a**(1/4)), 1/2 - b/
(4*sqrt(a)*sqrt(c)))/(105*c**(11/4)*sqrt(a + b*x**2 + c*x**4)) + a**(1/4)*sqrt((
a + b*x**2 + c*x**4)/(sqrt(a) + sqrt(c)*x**2)**2)*(sqrt(a) + sqrt(c)*x**2)*(sqrt
(a)*sqrt(c)*(-10*a*c + 4*b**2) + b*(-29*a*c + 8*b**2))*elliptic_f(2*atan(c**(1/4
)*x/a**(1/4)), 1/2 - b/(4*sqrt(a)*sqrt(c)))/(210*c**(11/4)*sqrt(a + b*x**2 + c*x
**4)) + b*x*(-29*a*c + 8*b**2)*sqrt(a + b*x**2 + c*x**4)/(105*c**(5/2)*(sqrt(a)
+ sqrt(c)*x**2)) + x**3*(b + 5*c*x**2)*sqrt(a + b*x**2 + c*x**4)/(35*c) - 2*x*(-
5*a*c + 2*b**2)*sqrt(a + b*x**2 + c*x**4)/(105*c**2)

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Mathematica [C]  time = 2.87946, size = 538, normalized size = 1.36 \[ \frac{4 c x \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \left (10 a^2 c+a \left (-4 b^2+13 b c x^2+25 c^2 x^4\right )-4 b^3 x^2-b^2 c x^4+18 b c^2 x^6+15 c^3 x^8\right )-i \left (-20 a^2 c^2+37 a b^2 c-29 a b c \sqrt{b^2-4 a c}+8 b^3 \sqrt{b^2-4 a c}-8 b^4\right ) \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{-2 \sqrt{b^2-4 a c}+2 b+4 c x^2}{b-\sqrt{b^2-4 a c}}} F\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )+i b \left (8 b^2-29 a c\right ) \left (\sqrt{b^2-4 a c}-b\right ) \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{-2 \sqrt{b^2-4 a c}+2 b+4 c x^2}{b-\sqrt{b^2-4 a c}}} E\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )}{420 c^3 \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{a+b x^2+c x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4*Sqrt[a + b*x^2 + c*x^4],x]

[Out]

(4*c*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x*(10*a^2*c - 4*b^3*x^2 - b^2*c*x^4 + 18*b*
c^2*x^6 + 15*c^3*x^8 + a*(-4*b^2 + 13*b*c*x^2 + 25*c^2*x^4)) + I*b*(8*b^2 - 29*a
*c)*(-b + Sqrt[b^2 - 4*a*c])*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^
2 - 4*a*c])]*Sqrt[(2*b - 2*Sqrt[b^2 - 4*a*c] + 4*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]
*EllipticE[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])] - I*(-8*b^4 + 37*a*b^2*c - 20*a^2*c^2 + 8*b^3*
Sqrt[b^2 - 4*a*c] - 29*a*b*c*Sqrt[b^2 - 4*a*c])*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*
c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[(2*b - 2*Sqrt[b^2 - 4*a*c] + 4*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])])/(420*c^3*Sqrt[c/(b + Sqrt[
b^2 - 4*a*c])]*Sqrt[a + b*x^2 + c*x^4])

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Maple [A]  time = 0.053, size = 476, normalized size = 1.2 \[{\frac{{x}^{5}}{7}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{b{x}^{3}}{35\,c}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{x}{3\,c} \left ({\frac{2\,a}{7}}-{\frac{4\,{b}^{2}}{35\,c}} \right ) \sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{a\sqrt{2}}{12\,c} \left ({\frac{2\,a}{7}}-{\frac{4\,{b}^{2}}{35\,c}} \right ) \sqrt{4-2\,{\frac{ \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ){x}^{2}}{a}}}\sqrt{4+2\,{\frac{ \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ){x}^{2}}{a}}}{\it EllipticF} \left ({\frac{\sqrt{2}x}{2}\sqrt{{\frac{1}{a} \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) }}},{\frac{1}{2}\sqrt{-4+2\,{\frac{b \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) }{ac}}}} \right ){\frac{1}{\sqrt{{\frac{1}{a} \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) }}}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}-{\frac{a\sqrt{2}}{2} \left ( -{\frac{3\,ab}{35\,c}}-{\frac{2\,b}{3\,c} \left ({\frac{2\,a}{7}}-{\frac{4\,{b}^{2}}{35\,c}} \right ) } \right ) \sqrt{4-2\,{\frac{ \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ){x}^{2}}{a}}}\sqrt{4+2\,{\frac{ \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ){x}^{2}}{a}}} \left ({\it EllipticF} \left ({\frac{\sqrt{2}x}{2}\sqrt{{\frac{1}{a} \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) }}},{\frac{1}{2}\sqrt{-4+2\,{\frac{b \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) }{ac}}}} \right ) -{\it EllipticE} \left ({\frac{\sqrt{2}x}{2}\sqrt{{\frac{1}{a} \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) }}},{\frac{1}{2}\sqrt{-4+2\,{\frac{b \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) }{ac}}}} \right ) \right ){\frac{1}{\sqrt{{\frac{1}{a} \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) }}}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}} \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) ^{-1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7*x^5*(c*x^4+b*x^2+a)^(1/2)+1/35*b/c*x^3*(c*x^4+b*x^2+a)^(1/2)+1/3*(2/7*a-4/35
*b^2/c)/c*x*(c*x^4+b*x^2+a)^(1/2)-1/12*(2/7*a-4/35*b^2/c)/c*a*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))-1/2*(-
3/35*b/c*a-2/3*(2/7*a-4/35*b^2/c)/c*b)*a*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)/(b+(-4*a*c+b^2)^(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))-Ell
ipticE(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)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{c x^{4} + b x^{2} + a} x^{4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(c*x^4 + b*x^2 + a)*x^4, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\sqrt{c x^{4} + b x^{2} + a} x^{4}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(sqrt(c*x^4 + b*x^2 + a)*x^4, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{4} \sqrt{a + b x^{2} + c x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**4*sqrt(a + b*x**2 + c*x**4), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{c x^{4} + b x^{2} + a} x^{4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(c*x^4 + b*x^2 + a)*x^4, x)