3.801 \(\int \left (1+x^4\right )^{3/2} \, dx\)

Optimal. Leaf size=72 \[ \frac{1}{7} x \left (x^4+1\right )^{3/2}+\frac{2}{7} x \sqrt{x^4+1}+\frac{2 \left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{7 \sqrt{x^4+1}} \]

[Out]

(2*x*Sqrt[1 + x^4])/7 + (x*(1 + x^4)^(3/2))/7 + (2*(1 + x^2)*Sqrt[(1 + x^4)/(1 +
x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(7*Sqrt[1 + x^4])

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Rubi [A]  time = 0.0327541, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{1}{7} x \left (x^4+1\right )^{3/2}+\frac{2}{7} x \sqrt{x^4+1}+\frac{2 \left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{7 \sqrt{x^4+1}} \]

Antiderivative was successfully verified.

[In]  Int[(1 + x^4)^(3/2),x]

[Out]

(2*x*Sqrt[1 + x^4])/7 + (x*(1 + x^4)^(3/2))/7 + (2*(1 + x^2)*Sqrt[(1 + x^4)/(1 +
x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(7*Sqrt[1 + x^4])

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Rubi in Sympy [A]  time = 1.68956, size = 65, normalized size = 0.9 \[ \frac{x \left (x^{4} + 1\right )^{\frac{3}{2}}}{7} + \frac{2 x \sqrt{x^{4} + 1}}{7} + \frac{2 \sqrt{\frac{x^{4} + 1}{\left (x^{2} + 1\right )^{2}}} \left (x^{2} + 1\right ) F\left (2 \operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{7 \sqrt{x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**4+1)**(3/2),x)

[Out]

x*(x**4 + 1)**(3/2)/7 + 2*x*sqrt(x**4 + 1)/7 + 2*sqrt((x**4 + 1)/(x**2 + 1)**2)*
(x**2 + 1)*elliptic_f(2*atan(x), 1/2)/(7*sqrt(x**4 + 1))

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Mathematica [C]  time = 0.0403646, size = 55, normalized size = 0.76 \[ \frac{x^9+4 x^5-4 \sqrt [4]{-1} \sqrt{x^4+1} F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )+3 x}{7 \sqrt{x^4+1}} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + x^4)^(3/2),x]

[Out]

(3*x + 4*x^5 + x^9 - 4*(-1)^(1/4)*Sqrt[1 + x^4]*EllipticF[I*ArcSinh[(-1)^(1/4)*x
], -1])/(7*Sqrt[1 + x^4])

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Maple [C]  time = 0.128, size = 84, normalized size = 1.2 \[{\frac{{x}^{5}}{7}\sqrt{{x}^{4}+1}}+{\frac{3\,x}{7}\sqrt{{x}^{4}+1}}+{\frac{4\,{\it EllipticF} \left ( x \left ( 1/2\,\sqrt{2}+i/2\sqrt{2} \right ) ,i \right ) }{{\frac{7\,\sqrt{2}}{2}}+{\frac{7\,i}{2}}\sqrt{2}}\sqrt{1-i{x}^{2}}\sqrt{1+i{x}^{2}}{\frac{1}{\sqrt{{x}^{4}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^4+1)^(3/2),x)

[Out]

1/7*x^5*(x^4+1)^(1/2)+3/7*x*(x^4+1)^(1/2)+4/7/(1/2*2^(1/2)+1/2*I*2^(1/2))*(1-I*x
^2)^(1/2)*(1+I*x^2)^(1/2)/(x^4+1)^(1/2)*EllipticF(x*(1/2*2^(1/2)+1/2*I*2^(1/2)),
I)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (x^{4} + 1\right )}^{\frac{3}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^4 + 1)^(3/2),x, algorithm="maxima")

[Out]

integrate((x^4 + 1)^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (x^{4} + 1\right )}^{\frac{3}{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^4 + 1)^(3/2),x, algorithm="fricas")

[Out]

integral((x^4 + 1)^(3/2), x)

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Sympy [A]  time = 2.298, size = 29, normalized size = 0.4 \[ \frac{x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac{5}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**4+1)**(3/2),x)

[Out]

x*gamma(1/4)*hyper((-3/2, 1/4), (5/4,), x**4*exp_polar(I*pi))/(4*gamma(5/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (x^{4} + 1\right )}^{\frac{3}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^4 + 1)^(3/2),x, algorithm="giac")

[Out]

integrate((x^4 + 1)^(3/2), x)