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

Optimal. Leaf size=443 \[ \frac{x \left (84 a^2 c^2-57 a b^2 c+8 b^4\right ) \sqrt{a+b x^2+c x^4}}{315 c^{5/2} \left (\sqrt{a}+\sqrt{c} x^2\right )}+\frac{\sqrt [4]{a} \left (84 a^2 c^2-57 a b^2 c+4 \sqrt{a} b \sqrt{c} \left (b^2-6 a c\right )+8 b^4\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{630 c^{11/4} \sqrt{a+b x^2+c x^4}}-\frac{\sqrt [4]{a} \left (84 a^2 c^2-57 a b^2 c+8 b^4\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{315 c^{11/4} \sqrt{a+b x^2+c x^4}}-\frac{x \left (6 c x^2 \left (2 b^2-7 a c\right )+b \left (4 b^2-9 a c\right )\right ) \sqrt{a+b x^2+c x^4}}{315 c^2}+\frac{x \left (3 b+7 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{63 c} \]

[Out]

((8*b^4 - 57*a*b^2*c + 84*a^2*c^2)*x*Sqrt[a + b*x^2 + c*x^4])/(315*c^(5/2)*(Sqrt
[a] + Sqrt[c]*x^2)) - (x*(b*(4*b^2 - 9*a*c) + 6*c*(2*b^2 - 7*a*c)*x^2)*Sqrt[a +
b*x^2 + c*x^4])/(315*c^2) + (x*(3*b + 7*c*x^2)*(a + b*x^2 + c*x^4)^(3/2))/(63*c)
- (a^(1/4)*(8*b^4 - 57*a*b^2*c + 84*a^2*c^2)*(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])/(315*c^(11/4)*Sqrt[a + b*x^2 + c*x^4]) + (a^(1/
4)*(8*b^4 - 57*a*b^2*c + 84*a^2*c^2 + 4*Sqrt[a]*b*Sqrt[c]*(b^2 - 6*a*c))*(Sqrt[a
] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2
*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(630*c^(11/4)*Sqrt[a
+ b*x^2 + c*x^4])

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Rubi [A]  time = 0.581442, antiderivative size = 443, 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 \[ \frac{x \left (84 a^2 c^2-57 a b^2 c+8 b^4\right ) \sqrt{a+b x^2+c x^4}}{315 c^{5/2} \left (\sqrt{a}+\sqrt{c} x^2\right )}+\frac{\sqrt [4]{a} \left (84 a^2 c^2-57 a b^2 c+4 \sqrt{a} b \sqrt{c} \left (b^2-6 a c\right )+8 b^4\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{630 c^{11/4} \sqrt{a+b x^2+c x^4}}-\frac{\sqrt [4]{a} \left (84 a^2 c^2-57 a b^2 c+8 b^4\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{315 c^{11/4} \sqrt{a+b x^2+c x^4}}-\frac{x \left (6 c x^2 \left (2 b^2-7 a c\right )+b \left (4 b^2-9 a c\right )\right ) \sqrt{a+b x^2+c x^4}}{315 c^2}+\frac{x \left (3 b+7 c x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{63 c} \]

Warning: Unable to verify antiderivative.

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

[Out]

((8*b^4 - 57*a*b^2*c + 84*a^2*c^2)*x*Sqrt[a + b*x^2 + c*x^4])/(315*c^(5/2)*(Sqrt
[a] + Sqrt[c]*x^2)) - (x*(b*(4*b^2 - 9*a*c) + 6*c*(2*b^2 - 7*a*c)*x^2)*Sqrt[a +
b*x^2 + c*x^4])/(315*c^2) + (x*(3*b + 7*c*x^2)*(a + b*x^2 + c*x^4)^(3/2))/(63*c)
- (a^(1/4)*(8*b^4 - 57*a*b^2*c + 84*a^2*c^2)*(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])/(315*c^(11/4)*Sqrt[a + b*x^2 + c*x^4]) + (a^(1/
4)*(8*b^4 - 57*a*b^2*c + 84*a^2*c^2 + 4*Sqrt[a]*b*Sqrt[c]*(b^2 - 6*a*c))*(Sqrt[a
] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2
*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(630*c^(11/4)*Sqrt[a
+ b*x^2 + c*x^4])

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Rubi in Sympy [A]  time = 66.3376, size = 413, normalized size = 0.93 \[ - \frac{\sqrt [4]{a} \sqrt{\frac{a + b x^{2} + c x^{4}}{\left (\sqrt{a} + \sqrt{c} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x^{2}\right ) \left (84 a^{2} c^{2} - 57 a b^{2} c + 8 b^{4}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2} - \frac{b}{4 \sqrt{a} \sqrt{c}}\right )}{315 c^{\frac{11}{4}} \sqrt{a + b x^{2} + c x^{4}}} + \frac{\sqrt [4]{a} \sqrt{\frac{a + b x^{2} + c x^{4}}{\left (\sqrt{a} + \sqrt{c} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x^{2}\right ) \left (4 \sqrt{a} b \sqrt{c} \left (- 6 a c + b^{2}\right ) + 84 a^{2} c^{2} - 57 a b^{2} c + 8 b^{4}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2} - \frac{b}{4 \sqrt{a} \sqrt{c}}\right )}{630 c^{\frac{11}{4}} \sqrt{a + b x^{2} + c x^{4}}} + \frac{x \left (3 b + 7 c x^{2}\right ) \left (a + b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{63 c} - \frac{x \left (b \left (- 9 a c + 4 b^{2}\right ) + 6 c x^{2} \left (- 7 a c + 2 b^{2}\right )\right ) \sqrt{a + b x^{2} + c x^{4}}}{315 c^{2}} + \frac{x \sqrt{a + b x^{2} + c x^{4}} \left (84 a^{2} c^{2} - 57 a b^{2} c + 8 b^{4}\right )}{315 c^{\frac{5}{2}} \left (\sqrt{a} + \sqrt{c} x^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**(1/4)*sqrt((a + b*x**2 + c*x**4)/(sqrt(a) + sqrt(c)*x**2)**2)*(sqrt(a) + sqr
t(c)*x**2)*(84*a**2*c**2 - 57*a*b**2*c + 8*b**4)*elliptic_e(2*atan(c**(1/4)*x/a*
*(1/4)), 1/2 - b/(4*sqrt(a)*sqrt(c)))/(315*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) + sq
rt(c)*x**2)*(4*sqrt(a)*b*sqrt(c)*(-6*a*c + b**2) + 84*a**2*c**2 - 57*a*b**2*c +
8*b**4)*elliptic_f(2*atan(c**(1/4)*x/a**(1/4)), 1/2 - b/(4*sqrt(a)*sqrt(c)))/(63
0*c**(11/4)*sqrt(a + b*x**2 + c*x**4)) + x*(3*b + 7*c*x**2)*(a + b*x**2 + c*x**4
)**(3/2)/(63*c) - x*(b*(-9*a*c + 4*b**2) + 6*c*x**2*(-7*a*c + 2*b**2))*sqrt(a +
b*x**2 + c*x**4)/(315*c**2) + x*sqrt(a + b*x**2 + c*x**4)*(84*a**2*c**2 - 57*a*b
**2*c + 8*b**4)/(315*c**(5/2)*(sqrt(a) + sqrt(c)*x**2))

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Mathematica [C]  time = 3.64503, size = 602, normalized size = 1.36 \[ \frac{i \left (84 a^2 c^2-57 a b^2 c+8 b^4\right ) \left (\sqrt{b^2-4 a c}-b\right ) \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{-2 \sqrt{b^2-4 a c}+2 b+4 c x^2}{b-\sqrt{b^2-4 a c}}} E\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )+4 c x \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \left (a^2 c \left (24 b+77 c x^2\right )+a \left (-4 b^3+27 b^2 c x^2+151 b c^2 x^4+112 c^3 x^6\right )-4 b^4 x^2-b^3 c x^4+53 b^2 c^2 x^6+85 b c^3 x^8+35 c^4 x^{10}\right )-i \left (84 a^2 c^2 \sqrt{b^2-4 a c}-132 a^2 b c^2+65 a b^3 c-57 a b^2 c \sqrt{b^2-4 a c}+8 b^4 \sqrt{b^2-4 a c}-8 b^5\right ) \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{-2 \sqrt{b^2-4 a c}+2 b+4 c x^2}{b-\sqrt{b^2-4 a c}}} F\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )}{1260 c^3 \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{a+b x^2+c x^4}} \]

Antiderivative was successfully verified.

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

[Out]

(4*c*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x*(-4*b^4*x^2 - b^3*c*x^4 + 53*b^2*c^2*x^6
+ 85*b*c^3*x^8 + 35*c^4*x^10 + a^2*c*(24*b + 77*c*x^2) + a*(-4*b^3 + 27*b^2*c*x^
2 + 151*b*c^2*x^4 + 112*c^3*x^6)) + I*(8*b^4 - 57*a*b^2*c + 84*a^2*c^2)*(-b + Sq
rt[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^5 + 65*a*b^3*c - 132*a^2*b*c^2 + 8*b^4*Sqrt[b^2
- 4*a*c] - 57*a*b^2*c*Sqrt[b^2 - 4*a*c] + 84*a^2*c^2*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])])/(1
260*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 = 0.016, size = 545, normalized size = 1.2 \[{\frac{c{x}^{7}}{9}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{10\,b{x}^{5}}{63}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{{x}^{3}}{5\,c} \left ({\frac{11\,ac}{9}}+{\frac{{b}^{2}}{21}} \right ) \sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{x}{3\,c} \left ({\frac{76\,ab}{63}}-{\frac{4\,b}{5\,c} \left ({\frac{11\,ac}{9}}+{\frac{{b}^{2}}{21}} \right ) } \right ) \sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{a\sqrt{2}}{12\,c} \left ({\frac{76\,ab}{63}}-{\frac{4\,b}{5\,c} \left ({\frac{11\,ac}{9}}+{\frac{{b}^{2}}{21}} \right ) } \right ) \sqrt{4-2\,{\frac{ \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ){x}^{2}}{a}}}\sqrt{4+2\,{\frac{ \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ){x}^{2}}{a}}}{\it EllipticF} \left ({\frac{\sqrt{2}x}{2}\sqrt{{\frac{1}{a} \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) }}},{\frac{1}{2}\sqrt{-4+2\,{\frac{b \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) }{ac}}}} \right ){\frac{1}{\sqrt{{\frac{1}{a} \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) }}}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}}-{\frac{a\sqrt{2}}{2} \left ({a}^{2}-{\frac{3\,a}{5\,c} \left ({\frac{11\,ac}{9}}+{\frac{{b}^{2}}{21}} \right ) }-{\frac{2\,b}{3\,c} \left ({\frac{76\,ab}{63}}-{\frac{4\,b}{5\,c} \left ({\frac{11\,ac}{9}}+{\frac{{b}^{2}}{21}} \right ) } \right ) } \right ) \sqrt{4-2\,{\frac{ \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ){x}^{2}}{a}}}\sqrt{4+2\,{\frac{ \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ){x}^{2}}{a}}} \left ({\it EllipticF} \left ({\frac{\sqrt{2}x}{2}\sqrt{{\frac{1}{a} \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) }}},{\frac{1}{2}\sqrt{-4+2\,{\frac{b \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) }{ac}}}} \right ) -{\it EllipticE} \left ({\frac{\sqrt{2}x}{2}\sqrt{{\frac{1}{a} \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) }}},{\frac{1}{2}\sqrt{-4+2\,{\frac{b \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) }{ac}}}} \right ) \right ){\frac{1}{\sqrt{{\frac{1}{a} \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) }}}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}} \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) ^{-1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/9*c*x^7*(c*x^4+b*x^2+a)^(1/2)+10/63*b*x^5*(c*x^4+b*x^2+a)^(1/2)+1/5*(11/9*a*c+
1/21*b^2)/c*x^3*(c*x^4+b*x^2+a)^(1/2)+1/3*(76/63*a*b-4/5*(11/9*a*c+1/21*b^2)/c*b
)/c*x*(c*x^4+b*x^2+a)^(1/2)-1/12*(76/63*a*b-4/5*(11/9*a*c+1/21*b^2)/c*b)/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*(a^2-3/5*(11/9*a*c+1/21*b^2)/c*a-2/3*(76/63*a*b-4/5*(11/9*a*c+1/21*
b^2)/c*b)/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))-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. \[ \int{\left (c x^{4} + b x^{2} + a\right )}^{\frac{3}{2}} x^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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