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

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

[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 = 0.0107769, 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, Rules used = {195, 220} \[ \frac{1}{7} x \left (x^4+1\right )^{3/2}+\frac{2}{7} x \sqrt{x^4+1}+\frac{2 \left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{7 \sqrt{x^4+1}} \]

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])

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 220

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[((1 + q^2*x^2)*Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]*EllipticF[2*ArcTan[q*x], 1/2])/(2*q*Sqrt[a + b*x^4]), x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rubi steps

\begin{align*} \int \left (1+x^4\right )^{3/2} \, dx &=\frac{1}{7} x \left (1+x^4\right )^{3/2}+\frac{6}{7} \int \sqrt{1+x^4} \, dx\\ &=\frac{2}{7} x \sqrt{1+x^4}+\frac{1}{7} x \left (1+x^4\right )^{3/2}+\frac{4}{7} \int \frac{1}{\sqrt{1+x^4}} \, dx\\ &=\frac{2}{7} x \sqrt{1+x^4}+\frac{1}{7} x \left (1+x^4\right )^{3/2}+\frac{2 \left (1+x^2\right ) \sqrt{\frac{1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{7 \sqrt{1+x^4}}\\ \end{align*}

Mathematica [C]  time = 0.001885, size = 17, normalized size = 0.24 \[ x \, _2F_1\left (-\frac{3}{2},\frac{1}{4};\frac{5}{4};-x^4\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + x^4)^(3/2),x]

[Out]

x*Hypergeometric2F1[-3/2, 1/4, 5/4, -x^4]

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Maple [C]  time = 0.005, size = 84, normalized size = 1.2 \begin{align*}{\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}}}} \end{align*}

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. \begin{align*} \int{\left (x^{4} + 1\right )}^{\frac{3}{2}}\,{d x} \end{align*}

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. \begin{align*}{\rm integral}\left ({\left (x^{4} + 1\right )}^{\frac{3}{2}}, x\right ) \end{align*}

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 [C]  time = 0.832735, size = 29, normalized size = 0.4 \begin{align*} \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 )} \end{align*}

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

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)