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

Optimal. Leaf size=148 \[ -\frac{2 a^{11/4} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{77 c^{5/4} \sqrt{a+c x^4}}+\frac{4 a^2 x \sqrt{a+c x^4}}{77 c}+\frac{1}{11} x^5 \left (a+c x^4\right )^{3/2}+\frac{6}{77} a x^5 \sqrt{a+c x^4} \]

[Out]

(4*a^2*x*Sqrt[a + c*x^4])/(77*c) + (6*a*x^5*Sqrt[a + c*x^4])/77 + (x^5*(a + c*x^
4)^(3/2))/11 - (2*a^(11/4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + S
qrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(77*c^(5/4)*Sqrt[a
+ c*x^4])

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Rubi [A]  time = 0.135681, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{2 a^{11/4} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{77 c^{5/4} \sqrt{a+c x^4}}+\frac{4 a^2 x \sqrt{a+c x^4}}{77 c}+\frac{1}{11} x^5 \left (a+c x^4\right )^{3/2}+\frac{6}{77} a x^5 \sqrt{a+c x^4} \]

Antiderivative was successfully verified.

[In]  Int[x^4*(a + c*x^4)^(3/2),x]

[Out]

(4*a^2*x*Sqrt[a + c*x^4])/(77*c) + (6*a*x^5*Sqrt[a + c*x^4])/77 + (x^5*(a + c*x^
4)^(3/2))/11 - (2*a^(11/4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + S
qrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(77*c^(5/4)*Sqrt[a
+ c*x^4])

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Rubi in Sympy [A]  time = 15.0443, size = 134, normalized size = 0.91 \[ - \frac{2 a^{\frac{11}{4}} \sqrt{\frac{a + c x^{4}}{\left (\sqrt{a} + \sqrt{c} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{c} x^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{77 c^{\frac{5}{4}} \sqrt{a + c x^{4}}} + \frac{4 a^{2} x \sqrt{a + c x^{4}}}{77 c} + \frac{6 a x^{5} \sqrt{a + c x^{4}}}{77} + \frac{x^{5} \left (a + c x^{4}\right )^{\frac{3}{2}}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a**(11/4)*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)/(77*c**(5/4)*sqrt(a + c*x**4))
+ 4*a**2*x*sqrt(a + c*x**4)/(77*c) + 6*a*x**5*sqrt(a + c*x**4)/77 + x**5*(a + c
*x**4)**(3/2)/11

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Mathematica [C]  time = 0.332878, size = 117, normalized size = 0.79 \[ \frac{\frac{4 i a^3 \sqrt{\frac{c x^4}{a}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}} x\right )\right |-1\right )}{\sqrt{\frac{i \sqrt{c}}{\sqrt{a}}}}+4 a^3 x+17 a^2 c x^5+20 a c^2 x^9+7 c^3 x^{13}}{77 c \sqrt{a+c x^4}} \]

Antiderivative was successfully verified.

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

[Out]

(4*a^3*x + 17*a^2*c*x^5 + 20*a*c^2*x^9 + 7*c^3*x^13 + ((4*I)*a^3*Sqrt[1 + (c*x^4
)/a]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1])/Sqrt[(I*Sqrt[c])/Sqr
t[a]])/(77*c*Sqrt[a + c*x^4])

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Maple [C]  time = 0.012, size = 126, normalized size = 0.9 \[{\frac{c{x}^{9}}{11}\sqrt{c{x}^{4}+a}}+{\frac{13\,a{x}^{5}}{77}\sqrt{c{x}^{4}+a}}+{\frac{4\,x{a}^{2}}{77\,c}\sqrt{c{x}^{4}+a}}-{\frac{4\,{a}^{3}}{77\,c}\sqrt{1-{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{c{x}^{4}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(c*x^4+a)^(3/2),x)

[Out]

1/11*c*x^9*(c*x^4+a)^(1/2)+13/77*a*x^5*(c*x^4+a)^(1/2)+4/77*a^2*x*(c*x^4+a)^(1/2
)/c-4/77*a^3/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. \[ \int{\left (c x^{4} + a\right )}^{\frac{3}{2}} x^{4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x^4 + a)^(3/2)*x^4, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (c x^{8} + a x^{4}\right )} \sqrt{c x^{4} + a}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((c*x^8 + a*x^4)*sqrt(c*x^4 + a), x)

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Sympy [A]  time = 3.98772, size = 39, normalized size = 0.26 \[ \frac{a^{\frac{3}{2}} x^{5} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{c x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{9}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**(3/2)*x**5*gamma(5/4)*hyper((-3/2, 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. \[ \int{\left (c x^{4} + a\right )}^{\frac{3}{2}} x^{4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((c*x^4 + a)^(3/2)*x^4, x)