3.952 \(\int \frac{\left (a+b x^2+c x^4\right )^{3/2}}{x^6} \, dx\)

Optimal. Leaf size=400 \[ \frac{\sqrt [4]{c} \left (8 \sqrt{a} b \sqrt{c}+12 a c+b^2\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{10 a^{3/4} \sqrt{a+b x^2+c x^4}}-\frac{\sqrt [4]{c} \left (12 a c+b^2\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{5 a^{3/4} \sqrt{a+b x^2+c x^4}}-\frac{\left (12 a c+b^2\right ) \sqrt{a+b x^2+c x^4}}{5 a x}+\frac{\sqrt{c} x \left (12 a c+b^2\right ) \sqrt{a+b x^2+c x^4}}{5 a \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{\left (a+b x^2+c x^4\right )^{3/2}}{5 x^5}-\frac{\left (b-6 c x^2\right ) \sqrt{a+b x^2+c x^4}}{5 x^3} \]

[Out]

-((b^2 + 12*a*c)*Sqrt[a + b*x^2 + c*x^4])/(5*a*x) + (Sqrt[c]*(b^2 + 12*a*c)*x*Sq
rt[a + b*x^2 + c*x^4])/(5*a*(Sqrt[a] + Sqrt[c]*x^2)) - ((b - 6*c*x^2)*Sqrt[a + b
*x^2 + c*x^4])/(5*x^3) - (a + b*x^2 + c*x^4)^(3/2)/(5*x^5) - (c^(1/4)*(b^2 + 12*
a*c)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]
*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(5*a^(3/
4)*Sqrt[a + b*x^2 + c*x^4]) + (c^(1/4)*(b^2 + 8*Sqrt[a]*b*Sqrt[c] + 12*a*c)*(Sqr
t[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*Elliptic
F[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(10*a^(3/4)*Sqrt[
a + b*x^2 + c*x^4])

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Rubi [A]  time = 0.613534, antiderivative size = 400, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ \frac{\sqrt [4]{c} \left (8 \sqrt{a} b \sqrt{c}+12 a c+b^2\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{10 a^{3/4} \sqrt{a+b x^2+c x^4}}-\frac{\sqrt [4]{c} \left (12 a c+b^2\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{5 a^{3/4} \sqrt{a+b x^2+c x^4}}-\frac{\left (12 a c+b^2\right ) \sqrt{a+b x^2+c x^4}}{5 a x}+\frac{\sqrt{c} x \left (12 a c+b^2\right ) \sqrt{a+b x^2+c x^4}}{5 a \left (\sqrt{a}+\sqrt{c} x^2\right )}-\frac{\left (a+b x^2+c x^4\right )^{3/2}}{5 x^5}-\frac{\left (b-6 c x^2\right ) \sqrt{a+b x^2+c x^4}}{5 x^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2 + c*x^4)^(3/2)/x^6,x]

[Out]

-((b^2 + 12*a*c)*Sqrt[a + b*x^2 + c*x^4])/(5*a*x) + (Sqrt[c]*(b^2 + 12*a*c)*x*Sq
rt[a + b*x^2 + c*x^4])/(5*a*(Sqrt[a] + Sqrt[c]*x^2)) - ((b - 6*c*x^2)*Sqrt[a + b
*x^2 + c*x^4])/(5*x^3) - (a + b*x^2 + c*x^4)^(3/2)/(5*x^5) - (c^(1/4)*(b^2 + 12*
a*c)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]
*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(5*a^(3/
4)*Sqrt[a + b*x^2 + c*x^4]) + (c^(1/4)*(b^2 + 8*Sqrt[a]*b*Sqrt[c] + 12*a*c)*(Sqr
t[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*Elliptic
F[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(10*a^(3/4)*Sqrt[
a + b*x^2 + c*x^4])

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Rubi in Sympy [A]  time = 77.0867, size = 362, normalized size = 0.9 \[ - \frac{\left (b - 6 c x^{2}\right ) \sqrt{a + b x^{2} + c x^{4}}}{5 x^{3}} - \frac{\left (a + b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{5 x^{5}} + \frac{\sqrt{c} x \left (12 a c + b^{2}\right ) \sqrt{a + b x^{2} + c x^{4}}}{5 a \left (\sqrt{a} + \sqrt{c} x^{2}\right )} - \frac{\left (12 a c + b^{2}\right ) \sqrt{a + b x^{2} + c x^{4}}}{5 a x} - \frac{\sqrt [4]{c} \sqrt{\frac{a + b x^{2} + c x^{4}}{\left (\sqrt{a} + \sqrt{c} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x^{2}\right ) \left (12 a c + b^{2}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2} - \frac{b}{4 \sqrt{a} \sqrt{c}}\right )}{5 a^{\frac{3}{4}} \sqrt{a + b x^{2} + c x^{4}}} + \frac{\sqrt [4]{c} \sqrt{\frac{a + b x^{2} + c x^{4}}{\left (\sqrt{a} + \sqrt{c} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x^{2}\right ) \left (8 \sqrt{a} b \sqrt{c} + 12 a c + b^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2} - \frac{b}{4 \sqrt{a} \sqrt{c}}\right )}{10 a^{\frac{3}{4}} \sqrt{a + b x^{2} + c x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+b*x**2+a)**(3/2)/x**6,x)

[Out]

-(b - 6*c*x**2)*sqrt(a + b*x**2 + c*x**4)/(5*x**3) - (a + b*x**2 + c*x**4)**(3/2
)/(5*x**5) + sqrt(c)*x*(12*a*c + b**2)*sqrt(a + b*x**2 + c*x**4)/(5*a*(sqrt(a) +
sqrt(c)*x**2)) - (12*a*c + b**2)*sqrt(a + b*x**2 + c*x**4)/(5*a*x) - c**(1/4)*s
qrt((a + b*x**2 + c*x**4)/(sqrt(a) + sqrt(c)*x**2)**2)*(sqrt(a) + sqrt(c)*x**2)*
(12*a*c + b**2)*elliptic_e(2*atan(c**(1/4)*x/a**(1/4)), 1/2 - b/(4*sqrt(a)*sqrt(
c)))/(5*a**(3/4)*sqrt(a + b*x**2 + c*x**4)) + c**(1/4)*sqrt((a + b*x**2 + c*x**4
)/(sqrt(a) + sqrt(c)*x**2)**2)*(sqrt(a) + sqrt(c)*x**2)*(8*sqrt(a)*b*sqrt(c) + 1
2*a*c + b**2)*elliptic_f(2*atan(c**(1/4)*x/a**(1/4)), 1/2 - b/(4*sqrt(a)*sqrt(c)
))/(10*a**(3/4)*sqrt(a + b*x**2 + c*x**4))

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Mathematica [C]  time = 2.55586, size = 527, normalized size = 1.32 \[ \frac{-4 \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \left (a^3+a^2 \left (3 b x^2+8 c x^4\right )+a \left (3 b^2 x^4+9 b c x^6+7 c^2 x^8\right )+b^2 x^6 \left (b+c x^2\right )\right )+i x^5 \left (12 a c+b^2\right ) \left (\sqrt{b^2-4 a c}-b\right ) \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{-2 \sqrt{b^2-4 a c}+2 b+4 c x^2}{b-\sqrt{b^2-4 a c}}} E\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )-i x^5 \left (b^2 \sqrt{b^2-4 a c}+12 a c \sqrt{b^2-4 a c}+4 a b c-b^3\right ) \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{-2 \sqrt{b^2-4 a c}+2 b+4 c x^2}{b-\sqrt{b^2-4 a c}}} F\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )}{20 a x^5 \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{a+b x^2+c x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2 + c*x^4)^(3/2)/x^6,x]

[Out]

(-4*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*(a^3 + b^2*x^6*(b + c*x^2) + a^2*(3*b*x^2 +
8*c*x^4) + a*(3*b^2*x^4 + 9*b*c*x^6 + 7*c^2*x^8)) + I*(b^2 + 12*a*c)*(-b + Sqrt[
b^2 - 4*a*c])*x^5*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*(-b^3 + 4*a*b*c + b^2*Sqrt[b^2 - 4*a*c] + 12*a*c*Sqrt
[b^2 - 4*a*c])*x^5*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])])/(20*a*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x^5*Sqrt[a + b*x^
2 + c*x^4])

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Maple [A]  time = 0.023, size = 450, normalized size = 1.1 \[ -{\frac{a}{5\,{x}^{5}}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{2\,b}{5\,{x}^{3}}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{7\,ac+{b}^{2}}{5\,ax}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{2\,bc\sqrt{2}}{5}\sqrt{4-2\,{\frac{ \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ){x}^{2}}{a}}}\sqrt{4+2\,{\frac{ \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ){x}^{2}}{a}}}{\it EllipticF} \left ({\frac{\sqrt{2}x}{2}\sqrt{{\frac{1}{a} \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) }}},{\frac{1}{2}\sqrt{-4+2\,{\frac{b \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) }{ac}}}} \right ){\frac{1}{\sqrt{{\frac{1}{a} \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) }}}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}-{\frac{a\sqrt{2}}{2} \left ({c}^{2}+{\frac{c \left ( 7\,ac+{b}^{2} \right ) }{5\,a}} \right ) \sqrt{4-2\,{\frac{ \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ){x}^{2}}{a}}}\sqrt{4+2\,{\frac{ \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ){x}^{2}}{a}}} \left ({\it EllipticF} \left ({\frac{\sqrt{2}x}{2}\sqrt{{\frac{1}{a} \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) }}},{\frac{1}{2}\sqrt{-4+2\,{\frac{b \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) }{ac}}}} \right ) -{\it EllipticE} \left ({\frac{\sqrt{2}x}{2}\sqrt{{\frac{1}{a} \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) }}},{\frac{1}{2}\sqrt{-4+2\,{\frac{b \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) }{ac}}}} \right ) \right ){\frac{1}{\sqrt{{\frac{1}{a} \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) }}}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}} \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) ^{-1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^2+a)^(3/2)/x^6,x)

[Out]

-1/5*a*(c*x^4+b*x^2+a)^(1/2)/x^5-2/5*b*(c*x^4+b*x^2+a)^(1/2)/x^3-1/5*(7*a*c+b^2)
/a*(c*x^4+b*x^2+a)^(1/2)/x+2/5*b*c*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*(c^2+1/5*c*(7*a*c+b^2)/a)*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))-EllipticE(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. \[ \int \frac{{\left (c x^{4} + b x^{2} + a\right )}^{\frac{3}{2}}}{x^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^(3/2)/x^6,x, algorithm="maxima")

[Out]

integrate((c*x^4 + b*x^2 + a)^(3/2)/x^6, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (c x^{4} + b x^{2} + a\right )}^{\frac{3}{2}}}{x^{6}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^(3/2)/x^6,x, algorithm="fricas")

[Out]

integral((c*x^4 + b*x^2 + a)^(3/2)/x^6, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{x^{6}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**4+b*x**2+a)**(3/2)/x**6,x)

[Out]

Integral((a + b*x**2 + c*x**4)**(3/2)/x**6, x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{4} + b x^{2} + a\right )}^{\frac{3}{2}}}{x^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^(3/2)/x^6,x, algorithm="giac")

[Out]

integrate((c*x^4 + b*x^2 + a)^(3/2)/x^6, x)