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

Optimal. Leaf size=234 \[ \frac{a^{5/4} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 c^{3/4} \sqrt{a+c x^4}}-\frac{2 a^{5/4} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 c^{3/4} \sqrt{a+c x^4}}+\frac{1}{5} x^3 \sqrt{a+c x^4}+\frac{2 a x \sqrt{a+c x^4}}{5 \sqrt{c} \left (\sqrt{a}+\sqrt{c} x^2\right )} \]

[Out]

(x^3*Sqrt[a + c*x^4])/5 + (2*a*x*Sqrt[a + c*x^4])/(5*Sqrt[c]*(Sqrt[a] + Sqrt[c]*
x^2)) - (2*a^(5/4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x
^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(5*c^(3/4)*Sqrt[a + c*x^4]
) + (a^(5/4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]
*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(5*c^(3/4)*Sqrt[a + c*x^4])

_______________________________________________________________________________________

Rubi [A]  time = 0.189373, antiderivative size = 234, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{a^{5/4} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 c^{3/4} \sqrt{a+c x^4}}-\frac{2 a^{5/4} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{5 c^{3/4} \sqrt{a+c x^4}}+\frac{1}{5} x^3 \sqrt{a+c x^4}+\frac{2 a x \sqrt{a+c x^4}}{5 \sqrt{c} \left (\sqrt{a}+\sqrt{c} x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[x^2*Sqrt[a + c*x^4],x]

[Out]

(x^3*Sqrt[a + c*x^4])/5 + (2*a*x*Sqrt[a + c*x^4])/(5*Sqrt[c]*(Sqrt[a] + Sqrt[c]*
x^2)) - (2*a^(5/4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x
^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(5*c^(3/4)*Sqrt[a + c*x^4]
) + (a^(5/4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]
*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(5*c^(3/4)*Sqrt[a + c*x^4])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 22.3875, size = 211, normalized size = 0.9 \[ - \frac{2 a^{\frac{5}{4}} \sqrt{\frac{a + c x^{4}}{\left (\sqrt{a} + \sqrt{c} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x^{2}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{5 c^{\frac{3}{4}} \sqrt{a + c x^{4}}} + \frac{a^{\frac{5}{4}} \sqrt{\frac{a + c x^{4}}{\left (\sqrt{a} + \sqrt{c} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{5 c^{\frac{3}{4}} \sqrt{a + c x^{4}}} + \frac{2 a x \sqrt{a + c x^{4}}}{5 \sqrt{c} \left (\sqrt{a} + \sqrt{c} x^{2}\right )} + \frac{x^{3} \sqrt{a + c x^{4}}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a**(5/4)*sqrt((a + c*x**4)/(sqrt(a) + sqrt(c)*x**2)**2)*(sqrt(a) + sqrt(c)*x*
*2)*elliptic_e(2*atan(c**(1/4)*x/a**(1/4)), 1/2)/(5*c**(3/4)*sqrt(a + c*x**4)) +
a**(5/4)*sqrt((a + c*x**4)/(sqrt(a) + sqrt(c)*x**2)**2)*(sqrt(a) + sqrt(c)*x**2
)*elliptic_f(2*atan(c**(1/4)*x/a**(1/4)), 1/2)/(5*c**(3/4)*sqrt(a + c*x**4)) + 2
*a*x*sqrt(a + c*x**4)/(5*sqrt(c)*(sqrt(a) + sqrt(c)*x**2)) + x**3*sqrt(a + c*x**
4)/5

_______________________________________________________________________________________

Mathematica [C]  time = 0.546719, size = 121, normalized size = 0.52 \[ \frac{x^3 \left (a+c x^4\right )+\frac{2 i a \sqrt{\frac{c x^4}{a}+1} \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )\right )}{\left (\frac{i \sqrt{c}}{\sqrt{a}}\right )^{3/2}}}{5 \sqrt{a+c x^4}} \]

Antiderivative was successfully verified.

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

[Out]

(x^3*(a + c*x^4) + ((2*I)*a*Sqrt[1 + (c*x^4)/a]*(EllipticE[I*ArcSinh[Sqrt[(I*Sqr
t[c])/Sqrt[a]]*x], -1] - EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1]))
/((I*Sqrt[c])/Sqrt[a])^(3/2))/(5*Sqrt[a + c*x^4])

_______________________________________________________________________________________

Maple [C]  time = 0.012, size = 112, normalized size = 0.5 \[{\frac{{x}^{3}}{5}\sqrt{c{x}^{4}+a}}+{{\frac{2\,i}{5}}{a}^{{\frac{3}{2}}}\sqrt{1-{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}} \left ({\it EllipticF} \left ( x\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ) -{\it EllipticE} \left ( x\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ) \right ){\frac{1}{\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{c{x}^{4}+a}}}{\frac{1}{\sqrt{c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(c*x^4+a)^(1/2),x)

[Out]

1/5*x^3*(c*x^4+a)^(1/2)+2/5*I*a^(3/2)/(I/a^(1/2)*c^(1/2))^(1/2)*(1-I/a^(1/2)*c^(
1/2)*x^2)^(1/2)*(1+I/a^(1/2)*c^(1/2)*x^2)^(1/2)/(c*x^4+a)^(1/2)/c^(1/2)*(Ellipti
cF(x*(I/a^(1/2)*c^(1/2))^(1/2),I)-EllipticE(x*(I/a^(1/2)*c^(1/2))^(1/2),I))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(c*x^4 + a)*x^2, x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(sqrt(c*x^4 + a)*x^2, x)

_______________________________________________________________________________________

Sympy [A]  time = 2.24364, size = 39, normalized size = 0.17 \[ \frac{\sqrt{a} x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{c x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{7}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(a)*x**3*gamma(3/4)*hyper((-1/2, 3/4), (7/4,), c*x**4*exp_polar(I*pi)/a)/(4*
gamma(7/4))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(c*x^4 + a)*x^2, x)