3.305 \(\int x^6 \sqrt [4]{3+4 x^4} \, dx\)

Optimal. Leaf size=93 \[ \frac{1}{8} \sqrt [4]{4 x^4+3} x^7+\frac{3}{128} \sqrt [4]{4 x^4+3} x^3+\frac{27 \tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt [4]{4 x^4+3}}\right )}{512 \sqrt{2}}-\frac{27 \tanh ^{-1}\left (\frac{\sqrt{2} x}{\sqrt [4]{4 x^4+3}}\right )}{512 \sqrt{2}} \]

[Out]

(3*x^3*(3 + 4*x^4)^(1/4))/128 + (x^7*(3 + 4*x^4)^(1/4))/8 + (27*ArcTan[(Sqrt[2]*x)/(3 + 4*x^4)^(1/4)])/(512*Sq
rt[2]) - (27*ArcTanh[(Sqrt[2]*x)/(3 + 4*x^4)^(1/4)])/(512*Sqrt[2])

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Rubi [A]  time = 0.0285984, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {279, 321, 331, 298, 203, 206} \[ \frac{1}{8} \sqrt [4]{4 x^4+3} x^7+\frac{3}{128} \sqrt [4]{4 x^4+3} x^3+\frac{27 \tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt [4]{4 x^4+3}}\right )}{512 \sqrt{2}}-\frac{27 \tanh ^{-1}\left (\frac{\sqrt{2} x}{\sqrt [4]{4 x^4+3}}\right )}{512 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Int[x^6*(3 + 4*x^4)^(1/4),x]

[Out]

(3*x^3*(3 + 4*x^4)^(1/4))/128 + (x^7*(3 + 4*x^4)^(1/4))/8 + (27*ArcTan[(Sqrt[2]*x)/(3 + 4*x^4)^(1/4)])/(512*Sq
rt[2]) - (27*ArcTanh[(Sqrt[2]*x)/(3 + 4*x^4)^(1/4)])/(512*Sqrt[2])

Rule 279

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
n*p + 1)), x] + Dist[(a*n*p)/(m + n*p + 1), Int[(c*x)^m*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x]
&& IGtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 331

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + (m + 1)/n), Subst[Int[x^m/(1 - b*x^n)^(
p + (m + 1)/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[
p, -2^(-1)] && IntegersQ[m, p + (m + 1)/n]

Rule 298

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b),
2]]}, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &
&  !GtQ[a/b, 0]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int x^6 \sqrt [4]{3+4 x^4} \, dx &=\frac{1}{8} x^7 \sqrt [4]{3+4 x^4}+\frac{3}{8} \int \frac{x^6}{\left (3+4 x^4\right )^{3/4}} \, dx\\ &=\frac{3}{128} x^3 \sqrt [4]{3+4 x^4}+\frac{1}{8} x^7 \sqrt [4]{3+4 x^4}-\frac{27}{128} \int \frac{x^2}{\left (3+4 x^4\right )^{3/4}} \, dx\\ &=\frac{3}{128} x^3 \sqrt [4]{3+4 x^4}+\frac{1}{8} x^7 \sqrt [4]{3+4 x^4}-\frac{27}{128} \operatorname{Subst}\left (\int \frac{x^2}{1-4 x^4} \, dx,x,\frac{x}{\sqrt [4]{3+4 x^4}}\right )\\ &=\frac{3}{128} x^3 \sqrt [4]{3+4 x^4}+\frac{1}{8} x^7 \sqrt [4]{3+4 x^4}-\frac{27}{512} \operatorname{Subst}\left (\int \frac{1}{1-2 x^2} \, dx,x,\frac{x}{\sqrt [4]{3+4 x^4}}\right )+\frac{27}{512} \operatorname{Subst}\left (\int \frac{1}{1+2 x^2} \, dx,x,\frac{x}{\sqrt [4]{3+4 x^4}}\right )\\ &=\frac{3}{128} x^3 \sqrt [4]{3+4 x^4}+\frac{1}{8} x^7 \sqrt [4]{3+4 x^4}+\frac{27 \tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt [4]{3+4 x^4}}\right )}{512 \sqrt{2}}-\frac{27 \tanh ^{-1}\left (\frac{\sqrt{2} x}{\sqrt [4]{3+4 x^4}}\right )}{512 \sqrt{2}}\\ \end{align*}

Mathematica [C]  time = 0.0136698, size = 43, normalized size = 0.46 \[ \frac{1}{32} x^3 \left (\left (4 x^4+3\right )^{5/4}-3 \sqrt [4]{3} \, _2F_1\left (-\frac{1}{4},\frac{3}{4};\frac{7}{4};-\frac{4 x^4}{3}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^6*(3 + 4*x^4)^(1/4),x]

[Out]

(x^3*((3 + 4*x^4)^(5/4) - 3*3^(1/4)*Hypergeometric2F1[-1/4, 3/4, 7/4, (-4*x^4)/3]))/32

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Maple [C]  time = 0.02, size = 42, normalized size = 0.5 \begin{align*}{\frac{{x}^{3} \left ( 16\,{x}^{4}+3 \right ) }{128}\sqrt [4]{4\,{x}^{4}+3}}-{\frac{3\,\sqrt [4]{3}{x}^{3}}{128}{\mbox{$_2$F$_1$}({\frac{3}{4}},{\frac{3}{4}};\,{\frac{7}{4}};\,-{\frac{4\,{x}^{4}}{3}})}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(4*x^4+3)^(1/4),x)

[Out]

1/128*x^3*(16*x^4+3)*(4*x^4+3)^(1/4)-3/128*3^(1/4)*x^3*hypergeom([3/4,3/4],[7/4],-4/3*x^4)

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Maxima [A]  time = 1.43927, size = 174, normalized size = 1.87 \begin{align*} -\frac{27}{1024} \, \sqrt{2} \arctan \left (\frac{\sqrt{2}{\left (4 \, x^{4} + 3\right )}^{\frac{1}{4}}}{2 \, x}\right ) + \frac{27}{2048} \, \sqrt{2} \log \left (-\frac{\sqrt{2} - \frac{{\left (4 \, x^{4} + 3\right )}^{\frac{1}{4}}}{x}}{\sqrt{2} + \frac{{\left (4 \, x^{4} + 3\right )}^{\frac{1}{4}}}{x}}\right ) - \frac{9 \,{\left (\frac{12 \,{\left (4 \, x^{4} + 3\right )}^{\frac{1}{4}}}{x} + \frac{{\left (4 \, x^{4} + 3\right )}^{\frac{5}{4}}}{x^{5}}\right )}}{128 \,{\left (\frac{8 \,{\left (4 \, x^{4} + 3\right )}}{x^{4}} - \frac{{\left (4 \, x^{4} + 3\right )}^{2}}{x^{8}} - 16\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(4*x^4+3)^(1/4),x, algorithm="maxima")

[Out]

-27/1024*sqrt(2)*arctan(1/2*sqrt(2)*(4*x^4 + 3)^(1/4)/x) + 27/2048*sqrt(2)*log(-(sqrt(2) - (4*x^4 + 3)^(1/4)/x
)/(sqrt(2) + (4*x^4 + 3)^(1/4)/x)) - 9/128*(12*(4*x^4 + 3)^(1/4)/x + (4*x^4 + 3)^(5/4)/x^5)/(8*(4*x^4 + 3)/x^4
- (4*x^4 + 3)^2/x^8 - 16)

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Fricas [A]  time = 1.89777, size = 302, normalized size = 3.25 \begin{align*} -\frac{27}{1024} \, \sqrt{2} \arctan \left (\frac{\sqrt{2}{\left (4 \, x^{4} + 3\right )}^{\frac{1}{4}}}{2 \, x}\right ) + \frac{27}{2048} \, \sqrt{2} \log \left (8 \, x^{4} - 4 \, \sqrt{2}{\left (4 \, x^{4} + 3\right )}^{\frac{1}{4}} x^{3} + 4 \, \sqrt{4 \, x^{4} + 3} x^{2} - 2 \, \sqrt{2}{\left (4 \, x^{4} + 3\right )}^{\frac{3}{4}} x + 3\right ) + \frac{1}{128} \,{\left (16 \, x^{7} + 3 \, x^{3}\right )}{\left (4 \, x^{4} + 3\right )}^{\frac{1}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(4*x^4+3)^(1/4),x, algorithm="fricas")

[Out]

-27/1024*sqrt(2)*arctan(1/2*sqrt(2)*(4*x^4 + 3)^(1/4)/x) + 27/2048*sqrt(2)*log(8*x^4 - 4*sqrt(2)*(4*x^4 + 3)^(
1/4)*x^3 + 4*sqrt(4*x^4 + 3)*x^2 - 2*sqrt(2)*(4*x^4 + 3)^(3/4)*x + 3) + 1/128*(16*x^7 + 3*x^3)*(4*x^4 + 3)^(1/
4)

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Sympy [C]  time = 1.59965, size = 39, normalized size = 0.42 \begin{align*} \frac{\sqrt [4]{3} x^{7} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{4 x^{4} e^{i \pi }}{3}} \right )}}{4 \Gamma \left (\frac{11}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6*(4*x**4+3)**(1/4),x)

[Out]

3**(1/4)*x**7*gamma(7/4)*hyper((-1/4, 7/4), (11/4,), 4*x**4*exp_polar(I*pi)/3)/(4*gamma(11/4))

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Giac [A]  time = 1.10721, size = 147, normalized size = 1.58 \begin{align*} \frac{1}{128} \, x^{8}{\left (\frac{{\left (4 \, x^{4} + 3\right )}^{\frac{1}{4}}{\left (\frac{3}{x^{4}} + 4\right )}}{x} + \frac{12 \,{\left (4 \, x^{4} + 3\right )}^{\frac{1}{4}}}{x}\right )} - \frac{27}{1024} \, \sqrt{2} \arctan \left (\frac{\sqrt{2}{\left (4 \, x^{4} + 3\right )}^{\frac{1}{4}}}{2 \, x}\right ) + \frac{27}{2048} \, \sqrt{2} \log \left (-\frac{\sqrt{2} - \frac{{\left (4 \, x^{4} + 3\right )}^{\frac{1}{4}}}{x}}{\sqrt{2} + \frac{{\left (4 \, x^{4} + 3\right )}^{\frac{1}{4}}}{x}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(4*x^4+3)^(1/4),x, algorithm="giac")

[Out]

1/128*x^8*((4*x^4 + 3)^(1/4)*(3/x^4 + 4)/x + 12*(4*x^4 + 3)^(1/4)/x) - 27/1024*sqrt(2)*arctan(1/2*sqrt(2)*(4*x
^4 + 3)^(1/4)/x) + 27/2048*sqrt(2)*log(-(sqrt(2) - (4*x^4 + 3)^(1/4)/x)/(sqrt(2) + (4*x^4 + 3)^(1/4)/x))