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

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

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Rubi [A]  time = 0.03, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, 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 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])

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 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 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]

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*}

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Mathematica [C]  time = 0.01, 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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IntegrateAlgebraic [A]  time = 0.18, size = 83, normalized size = 0.89 \[ \frac {27 \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt [4]{4 x^4+3}}\right )}{512 \sqrt {2}}-\frac {27 \tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt [4]{4 x^4+3}}\right )}{512 \sqrt {2}}+\frac {1}{128} \sqrt [4]{4 x^4+3} \left (16 x^7+3 x^3\right ) \]

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^6*(3 + 4*x^4)^(1/4),x]

[Out]

((3 + 4*x^4)^(1/4)*(3*x^3 + 16*x^7))/128 + (27*ArcTan[(Sqrt[2]*x)/(3 + 4*x^4)^(1/4)])/(512*Sqrt[2]) - (27*ArcT
anh[(Sqrt[2]*x)/(3 + 4*x^4)^(1/4)])/(512*Sqrt[2])

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fricas [A]  time = 0.51, size = 105, normalized size = 1.13 \[ -\frac {27}{1024} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{2 \, x}\right ) + \frac {27}{2048} \, \sqrt {2} \log \left (8 \, x^{4} - 4 \, \sqrt {2} {\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {4 \, x^{4} + 3} x^{2} - 2 \, \sqrt {2} {\left (4 \, x^{4} + 3\right )}^{\frac {3}{4}} x + 3\right ) + \frac {1}{128} \, {\left (16 \, x^{7} + 3 \, x^{3}\right )} {\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}} \]

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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giac [A]  time = 0.67, size = 109, normalized size = 1.17 \[ \frac {1}{128} \, x^{8} {\left (\frac {{\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}} {\left (\frac {3}{x^{4}} + 4\right )}}{x} + \frac {12 \, {\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{x}\right )} - \frac {27}{1024} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{2 \, x}\right ) + \frac {27}{2048} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - \frac {{\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{x}}{\sqrt {2} + \frac {{\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{x}}\right ) \]

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))

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maple [C]  time = 3.32, size = 20, normalized size = 0.22

method	result	size
meijerg	\(\frac {3^{\frac {1}{4}} x^{7} \hypergeom \left (\left [-\frac {1}{4}, \frac {7}{4}\right ], \left [\frac {11}{4}\right ], -\frac {4 x^{4}}{3}\right )}{7}\)	\(20\)
risch	\(\frac {x^{3} \left (16 x^{4}+3\right ) \left (4 x^{4}+3\right )^{\frac {1}{4}}}{128}-\frac {3 \,3^{\frac {1}{4}} x^{3} \hypergeom \left (\left [\frac {3}{4}, \frac {3}{4}\right ], \left [\frac {7}{4}\right ], -\frac {4 x^{4}}{3}\right )}{128}\)	\(42\)
trager	\(\frac {x^{3} \left (16 x^{4}+3\right ) \left (4 x^{4}+3\right )^{\frac {1}{4}}}{128}+\frac {27 \RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (-4 \RootOf \left (\textit {\_Z}^{2}-2\right ) \sqrt {4 x^{4}+3}\, x^{2}-8 \RootOf \left (\textit {\_Z}^{2}-2\right ) x^{4}+4 \left (4 x^{4}+3\right )^{\frac {3}{4}} x +8 x^{3} \left (4 x^{4}+3\right )^{\frac {1}{4}}-3 \RootOf \left (\textit {\_Z}^{2}-2\right )\right )}{2048}+\frac {27 \RootOf \left (\textit {\_Z}^{2}+2\right ) \ln \left (-4 \RootOf \left (\textit {\_Z}^{2}+2\right ) \sqrt {4 x^{4}+3}\, x^{2}+8 \RootOf \left (\textit {\_Z}^{2}+2\right ) x^{4}+4 \left (4 x^{4}+3\right )^{\frac {3}{4}} x -8 x^{3} \left (4 x^{4}+3\right )^{\frac {1}{4}}+3 \RootOf \left (\textit {\_Z}^{2}+2\right )\right )}{2048}\)	\(173\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*3^(1/4)*x^7*hypergeom([-1/4,7/4],[11/4],-4/3*x^4)

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maxima [A]  time = 1.20, size = 129, normalized size = 1.39 \[ -\frac {27}{1024} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{2 \, x}\right ) + \frac {27}{2048} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - \frac {{\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{x}}{\sqrt {2} + \frac {{\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{x}}\right ) - \frac {9 \, {\left (\frac {12 \, {\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{x} + \frac {{\left (4 \, x^{4} + 3\right )}^{\frac {5}{4}}}{x^{5}}\right )}}{128 \, {\left (\frac {8 \, {\left (4 \, x^{4} + 3\right )}}{x^{4}} - \frac {{\left (4 \, x^{4} + 3\right )}^{2}}{x^{8}} - 16\right )}} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x^6\,{\left (4\,x^4+3\right )}^{1/4} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(4*x^4 + 3)^(1/4),x)

[Out]

int(x^6*(4*x^4 + 3)^(1/4), x)

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sympy [C]  time = 1.32, size = 39, normalized size = 0.42 \[ \frac {\sqrt [4]{3} x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {4 x^{4} e^{i \pi }}{3}} \right )}}{4 \Gamma \left (\frac {11}{4}\right )} \]

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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