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

Optimal. Leaf size=255 \[ \frac{2 a^{9/4} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 c^{3/4} \sqrt{a+c x^4}}-\frac{4 a^{9/4} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 c^{3/4} \sqrt{a+c x^4}}+\frac{4 a^2 x \sqrt{a+c x^4}}{15 \sqrt{c} \left (\sqrt{a}+\sqrt{c} x^2\right )}+\frac{1}{9} x^3 \left (a+c x^4\right )^{3/2}+\frac{2}{15} a x^3 \sqrt{a+c x^4} \]

[Out]

(2*a*x^3*Sqrt[a + c*x^4])/15 + (4*a^2*x*Sqrt[a + c*x^4])/(15*Sqrt[c]*(Sqrt[a] +
Sqrt[c]*x^2)) + (x^3*(a + c*x^4)^(3/2))/9 - (4*a^(9/4)*(Sqrt[a] + Sqrt[c]*x^2)*S
qrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4
)], 1/2])/(15*c^(3/4)*Sqrt[a + c*x^4]) + (2*a^(9/4)*(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])/(15*c^(3/4)*Sqrt[a + c*x^4])

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Rubi [A]  time = 0.236141, antiderivative size = 255, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{2 a^{9/4} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 c^{3/4} \sqrt{a+c x^4}}-\frac{4 a^{9/4} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 c^{3/4} \sqrt{a+c x^4}}+\frac{4 a^2 x \sqrt{a+c x^4}}{15 \sqrt{c} \left (\sqrt{a}+\sqrt{c} x^2\right )}+\frac{1}{9} x^3 \left (a+c x^4\right )^{3/2}+\frac{2}{15} a x^3 \sqrt{a+c x^4} \]

Antiderivative was successfully verified.

[In]  Int[x^2*(a + c*x^4)^(3/2),x]

[Out]

(2*a*x^3*Sqrt[a + c*x^4])/15 + (4*a^2*x*Sqrt[a + c*x^4])/(15*Sqrt[c]*(Sqrt[a] +
Sqrt[c]*x^2)) + (x^3*(a + c*x^4)^(3/2))/9 - (4*a^(9/4)*(Sqrt[a] + Sqrt[c]*x^2)*S
qrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4
)], 1/2])/(15*c^(3/4)*Sqrt[a + c*x^4]) + (2*a^(9/4)*(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])/(15*c^(3/4)*Sqrt[a + c*x^4])

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Rubi in Sympy [A]  time = 28.0885, size = 233, normalized size = 0.91 \[ - \frac{4 a^{\frac{9}{4}} \sqrt{\frac{a + c x^{4}}{\left (\sqrt{a} + \sqrt{c} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x^{2}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{15 c^{\frac{3}{4}} \sqrt{a + c x^{4}}} + \frac{2 a^{\frac{9}{4}} \sqrt{\frac{a + c x^{4}}{\left (\sqrt{a} + \sqrt{c} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{15 c^{\frac{3}{4}} \sqrt{a + c x^{4}}} + \frac{4 a^{2} x \sqrt{a + c x^{4}}}{15 \sqrt{c} \left (\sqrt{a} + \sqrt{c} x^{2}\right )} + \frac{2 a x^{3} \sqrt{a + c x^{4}}}{15} + \frac{x^{3} \left (a + c x^{4}\right )^{\frac{3}{2}}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*a**(9/4)*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)/(15*c**(3/4)*sqrt(a + c*x**4))
+ 2*a**(9/4)*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)/(15*c**(3/4)*sqrt(a + c*x**4))
+ 4*a**2*x*sqrt(a + c*x**4)/(15*sqrt(c)*(sqrt(a) + sqrt(c)*x**2)) + 2*a*x**3*sq
rt(a + c*x**4)/15 + x**3*(a + c*x**4)**(3/2)/9

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Mathematica [C]  time = 0.538826, size = 133, normalized size = 0.52 \[ \frac{\left (a+c x^4\right ) \left (11 a x^3+5 c x^7\right )+\frac{12 i a^2 \sqrt{\frac{c x^4}{a}+1} \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )\right )}{\left (\frac{i \sqrt{c}}{\sqrt{a}}\right )^{3/2}}}{45 \sqrt{a+c x^4}} \]

Antiderivative was successfully verified.

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

[Out]

((a + c*x^4)*(11*a*x^3 + 5*c*x^7) + ((12*I)*a^2*Sqrt[1 + (c*x^4)/a]*(EllipticE[I
*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] - EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c]
)/Sqrt[a]]*x], -1]))/((I*Sqrt[c])/Sqrt[a])^(3/2))/(45*Sqrt[a + c*x^4])

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Maple [C]  time = 0.013, size = 128, normalized size = 0.5 \[{\frac{c{x}^{7}}{9}\sqrt{c{x}^{4}+a}}+{\frac{11\,a{x}^{3}}{45}\sqrt{c{x}^{4}+a}}+{{\frac{4\,i}{15}}{a}^{{\frac{5}{2}}}\sqrt{1-{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}} \left ({\it EllipticF} \left ( x\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ) -{\it EllipticE} \left ( x\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ) \right ){\frac{1}{\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{c{x}^{4}+a}}}{\frac{1}{\sqrt{c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(c*x^4+a)^(3/2),x)

[Out]

1/9*c*x^7*(c*x^4+a)^(1/2)+11/45*a*x^3*(c*x^4+a)^(1/2)+4/15*I*a^(5/2)/(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)*(EllipticF(x*(I/a^(1/2)*c^(1/2))^(1/2),I)-EllipticE(x*(I/
a^(1/2)*c^(1/2))^(1/2),I))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x^4 + a)^(3/2)*x^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.17425, size = 39, normalized size = 0.15 \[ \frac{a^{\frac{3}{2}} x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{c x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{7}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x^4 + a)^(3/2)*x^2, x)