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

Optimal. Leaf size=127 \[ -\frac{a^{7/4} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{21 c^{5/4} \sqrt{a+c x^4}}+\frac{2 a x \sqrt{a+c x^4}}{21 c}+\frac{1}{7} x^5 \sqrt{a+c x^4} \]

[Out]

(2*a*x*Sqrt[a + c*x^4])/(21*c) + (x^5*Sqrt[a + c*x^4])/7 - (a^(7/4)*(Sqrt[a] + S
qrt[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])/(21*c^(5/4)*Sqrt[a + c*x^4])

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Rubi [A]  time = 0.10449, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{a^{7/4} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{21 c^{5/4} \sqrt{a+c x^4}}+\frac{2 a x \sqrt{a+c x^4}}{21 c}+\frac{1}{7} x^5 \sqrt{a+c x^4} \]

Antiderivative was successfully verified.

[In]  Int[x^4*Sqrt[a + c*x^4],x]

[Out]

(2*a*x*Sqrt[a + c*x^4])/(21*c) + (x^5*Sqrt[a + c*x^4])/7 - (a^(7/4)*(Sqrt[a] + S
qrt[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])/(21*c^(5/4)*Sqrt[a + c*x^4])

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Rubi in Sympy [A]  time = 11.1112, size = 112, normalized size = 0.88 \[ - \frac{a^{\frac{7}{4}} \sqrt{\frac{a + c x^{4}}{\left (\sqrt{a} + \sqrt{c} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{21 c^{\frac{5}{4}} \sqrt{a + c x^{4}}} + \frac{2 a x \sqrt{a + c x^{4}}}{21 c} + \frac{x^{5} \sqrt{a + c x^{4}}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**(7/4)*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)/(21*c**(5/4)*sqrt(a + c*x**4)) +
2*a*x*sqrt(a + c*x**4)/(21*c) + x**5*sqrt(a + c*x**4)/7

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Mathematica [C]  time = 0.281049, size = 106, normalized size = 0.83 \[ \frac{\frac{2 i a^2 \sqrt{\frac{c x^4}{a}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )}{\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}}}+2 a^2 x+5 a c x^5+3 c^2 x^9}{21 c \sqrt{a+c x^4}} \]

Antiderivative was successfully verified.

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

[Out]

(2*a^2*x + 5*a*c*x^5 + 3*c^2*x^9 + ((2*I)*a^2*Sqrt[1 + (c*x^4)/a]*EllipticF[I*Ar
cSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1])/Sqrt[(I*Sqrt[c])/Sqrt[a]])/(21*c*Sqrt[a
+ c*x^4])

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Maple [C]  time = 0.05, size = 108, normalized size = 0.9 \[{\frac{{x}^{5}}{7}\sqrt{c{x}^{4}+a}}+{\frac{2\,ax}{21\,c}\sqrt{c{x}^{4}+a}}-{\frac{2\,{a}^{2}}{21\,c}\sqrt{1-{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{c{x}^{4}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(c*x^4+a)^(1/2),x)

[Out]

1/7*x^5*(c*x^4+a)^(1/2)+2/21*a*x*(c*x^4+a)^(1/2)/c-2/21*a^2/c/(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)*EllipticF(x*(I/a^(1/2)*c^(1/2))^(1/2),I)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(c*x^4 + a)*x^4, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(sqrt(c*x^4 + a)*x^4, x)

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Sympy [A]  time = 2.46479, size = 39, normalized size = 0.31 \[ \frac{\sqrt{a} x^{5} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{c x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{9}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(a)*x**5*gamma(5/4)*hyper((-1/2, 5/4), (9/4,), c*x**4*exp_polar(I*pi)/a)/(4*
gamma(9/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(c*x^4 + a)*x^4, x)