Integrand size = 15, antiderivative size = 93 \[ \int x^6 \sqrt [4]{3+4 x^4} \, dx=\frac {3}{128} x^3 \sqrt [4]{3+4 x^4}+\frac {1}{8} x^7 \sqrt [4]{3+4 x^4}+\frac {27 \arctan \left (\frac {\sqrt {2} x}{\sqrt [4]{3+4 x^4}}\right )}{512 \sqrt {2}}-\frac {27 \text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt [4]{3+4 x^4}}\right )}{512 \sqrt {2}} \] Output:
3/128*x^3*(4*x^4+3)^(1/4)+1/8*x^7*(4*x^4+3)^(1/4)+27/1024*arctan(x*2^(1/2) /(4*x^4+3)^(1/4))*2^(1/2)-27/1024*arctanh(x*2^(1/2)/(4*x^4+3)^(1/4))*2^(1/ 2)
Time = 0.22 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.88 \[ \int x^6 \sqrt [4]{3+4 x^4} \, dx=\frac {1}{128} x^3 \sqrt [4]{3+4 x^4} \left (3+16 x^4\right )+\frac {27 \arctan \left (\frac {\sqrt {2} x}{\sqrt [4]{3+4 x^4}}\right )}{512 \sqrt {2}}-\frac {27 \text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt [4]{3+4 x^4}}\right )}{512 \sqrt {2}} \] Input:
Integrate[x^6*(3 + 4*x^4)^(1/4),x]
Output:
(x^3*(3 + 4*x^4)^(1/4)*(3 + 16*x^4))/128 + (27*ArcTan[(Sqrt[2]*x)/(3 + 4*x ^4)^(1/4)])/(512*Sqrt[2]) - (27*ArcTanh[(Sqrt[2]*x)/(3 + 4*x^4)^(1/4)])/(5 12*Sqrt[2])
Time = 0.20 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.11, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {811, 843, 854, 827, 216, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int x^6 \sqrt [4]{4 x^4+3} \, dx\) |
\(\Big \downarrow \) 811 |
\(\displaystyle \frac {3}{8} \int \frac {x^6}{\left (4 x^4+3\right )^{3/4}}dx+\frac {1}{8} \sqrt [4]{4 x^4+3} x^7\) |
\(\Big \downarrow \) 843 |
\(\displaystyle \frac {3}{8} \left (\frac {1}{16} x^3 \sqrt [4]{4 x^4+3}-\frac {9}{16} \int \frac {x^2}{\left (4 x^4+3\right )^{3/4}}dx\right )+\frac {1}{8} \sqrt [4]{4 x^4+3} x^7\) |
\(\Big \downarrow \) 854 |
\(\displaystyle \frac {3}{8} \left (\frac {1}{16} x^3 \sqrt [4]{4 x^4+3}-\frac {9}{16} \int \frac {x^2}{\sqrt {4 x^4+3} \left (1-\frac {4 x^4}{4 x^4+3}\right )}d\frac {x}{\sqrt [4]{4 x^4+3}}\right )+\frac {1}{8} \sqrt [4]{4 x^4+3} x^7\) |
\(\Big \downarrow \) 827 |
\(\displaystyle \frac {3}{8} \left (\frac {1}{16} x^3 \sqrt [4]{4 x^4+3}-\frac {9}{16} \left (\frac {1}{4} \int \frac {1}{1-\frac {2 x^2}{\sqrt {4 x^4+3}}}d\frac {x}{\sqrt [4]{4 x^4+3}}-\frac {1}{4} \int \frac {1}{\frac {2 x^2}{\sqrt {4 x^4+3}}+1}d\frac {x}{\sqrt [4]{4 x^4+3}}\right )\right )+\frac {1}{8} \sqrt [4]{4 x^4+3} x^7\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {3}{8} \left (\frac {1}{16} x^3 \sqrt [4]{4 x^4+3}-\frac {9}{16} \left (\frac {1}{4} \int \frac {1}{1-\frac {2 x^2}{\sqrt {4 x^4+3}}}d\frac {x}{\sqrt [4]{4 x^4+3}}-\frac {\arctan \left (\frac {\sqrt {2} x}{\sqrt [4]{4 x^4+3}}\right )}{4 \sqrt {2}}\right )\right )+\frac {1}{8} \sqrt [4]{4 x^4+3} x^7\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {3}{8} \left (\frac {1}{16} x^3 \sqrt [4]{4 x^4+3}-\frac {9}{16} \left (\frac {\text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt [4]{4 x^4+3}}\right )}{4 \sqrt {2}}-\frac {\arctan \left (\frac {\sqrt {2} x}{\sqrt [4]{4 x^4+3}}\right )}{4 \sqrt {2}}\right )\right )+\frac {1}{8} \sqrt [4]{4 x^4+3} x^7\) |
Input:
Int[x^6*(3 + 4*x^4)^(1/4),x]
Output:
(x^7*(3 + 4*x^4)^(1/4))/8 + (3*((x^3*(3 + 4*x^4)^(1/4))/16 - (9*(-1/4*ArcT an[(Sqrt[2]*x)/(3 + 4*x^4)^(1/4)]/Sqrt[2] + ArcTanh[(Sqrt[2]*x)/(3 + 4*x^4 )^(1/4)]/(4*Sqrt[2])))/16))/8
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
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] + Simp[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] && I GtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b, c, n, m , p, x]
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ [a/b, 0]
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] - Simp[ 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]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[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]
Result contains higher order function than in optimal. Order 5 vs. order 3.
Time = 2.33 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.22
method | result | size |
meijerg | \(\frac {3^{\frac {1}{4}} x^{7} \operatorname {hypergeom}\left (\left [-\frac {1}{4}, \frac {7}{4}\right ], \left [\frac {11}{4}\right ], -\frac {4 x^{4}}{3}\right )}{7}\) | \(20\) |
risch | \(\frac {x^{3} \left (16 x^{4}+3\right ) \left (4 x^{4}+3\right )^{\frac {1}{4}}}{128}-\frac {3 \,3^{\frac {1}{4}} x^{3} \operatorname {hypergeom}\left (\left [\frac {3}{4}, \frac {3}{4}\right ], \left [\frac {7}{4}\right ], -\frac {4 x^{4}}{3}\right )}{128}\) | \(42\) |
pseudoelliptic | \(-\frac {9 \left (-128 x^{7} \left (4 x^{4}+3\right )^{\frac {1}{4}}-24 x^{3} \left (4 x^{4}+3\right )^{\frac {1}{4}}+27 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (4 x^{4}+3\right )^{\frac {1}{4}} \sqrt {2}}{2 x}\right )+27 \sqrt {2}\, \arctan \left (\frac {\left (4 x^{4}+3\right )^{\frac {1}{4}} \sqrt {2}}{2 x}\right )\right )}{1024 \left (-2 x^{2}+\sqrt {4 x^{4}+3}\right )^{2} \left (2 x^{2}+\sqrt {4 x^{4}+3}\right )^{2}}\) | \(112\) |
trager | \(\frac {x^{3} \left (16 x^{4}+3\right ) \left (4 x^{4}+3\right )^{\frac {1}{4}}}{128}-\frac {27 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \left (4 x^{4}+3\right )^{\frac {3}{4}} x -4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \left (4 x^{4}+3\right )^{\frac {1}{4}} x^{3}-4 \sqrt {4 x^{4}+3}\, x^{2}-8 x^{4}-3\right )}{2048}+\frac {27 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \ln \left (-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \sqrt {4 x^{4}+3}\, x^{2}+8 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x^{4}+4 \left (4 x^{4}+3\right )^{\frac {3}{4}} x -8 x^{3} \left (4 x^{4}+3\right )^{\frac {1}{4}}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right )\right )}{2048}\) | \(166\) |
Input:
int(x^6*(4*x^4+3)^(1/4),x,method=_RETURNVERBOSE)
Output:
1/7*3^(1/4)*x^7*hypergeom([-1/4,7/4],[11/4],-4/3*x^4)
Time = 0.07 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.10 \[ \int x^6 \sqrt [4]{3+4 x^4} \, dx=\frac {27}{1024} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} x}{{\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}\right ) + \frac {27}{2048} \, \sqrt {2} \log \left (8 \, x^{4} - 4 \, \sqrt {2} {\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {4 \, x^{4} + 3} x^{2} - 2 \, \sqrt {2} {\left (4 \, x^{4} + 3\right )}^{\frac {3}{4}} x + 3\right ) + \frac {1}{128} \, {\left (16 \, x^{7} + 3 \, x^{3}\right )} {\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}} \] Input:
integrate(x^6*(4*x^4+3)^(1/4),x, algorithm="fricas")
Output:
27/1024*sqrt(2)*arctan(sqrt(2)*x/(4*x^4 + 3)^(1/4)) + 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)
Result contains complex when optimal does not.
Time = 0.98 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.42 \[ \int x^6 \sqrt [4]{3+4 x^4} \, dx=\frac {\sqrt [4]{3} x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {4 x^{4} e^{i \pi }}{3}} \right )}}{4 \Gamma \left (\frac {11}{4}\right )} \] Input:
integrate(x**6*(4*x**4+3)**(1/4),x)
Output:
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))
Time = 0.11 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.39 \[ \int x^6 \sqrt [4]{3+4 x^4} \, dx=-\frac {27}{1024} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{2 \, x}\right ) + \frac {27}{2048} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - \frac {{\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{x}}{\sqrt {2} + \frac {{\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{x}}\right ) - \frac {9 \, {\left (\frac {12 \, {\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{x} + \frac {{\left (4 \, x^{4} + 3\right )}^{\frac {5}{4}}}{x^{5}}\right )}}{128 \, {\left (\frac {8 \, {\left (4 \, x^{4} + 3\right )}}{x^{4}} - \frac {{\left (4 \, x^{4} + 3\right )}^{2}}{x^{8}} - 16\right )}} \] Input:
integrate(x^6*(4*x^4+3)^(1/4),x, algorithm="maxima")
Output:
-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)
Time = 0.13 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.17 \[ \int x^6 \sqrt [4]{3+4 x^4} \, dx=\frac {1}{128} \, x^{8} {\left (\frac {{\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}} {\left (\frac {3}{x^{4}} + 4\right )}}{x} + \frac {12 \, {\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{x}\right )} - \frac {27}{1024} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{2 \, x}\right ) + \frac {27}{2048} \, \sqrt {2} \log \left (-\frac {\sqrt {2} - \frac {{\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{x}}{\sqrt {2} + \frac {{\left (4 \, x^{4} + 3\right )}^{\frac {1}{4}}}{x}}\right ) \] Input:
integrate(x^6*(4*x^4+3)^(1/4),x, algorithm="giac")
Output:
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))
Timed out. \[ \int x^6 \sqrt [4]{3+4 x^4} \, dx=\int x^6\,{\left (4\,x^4+3\right )}^{1/4} \,d x \] Input:
int(x^6*(4*x^4 + 3)^(1/4),x)
Output:
int(x^6*(4*x^4 + 3)^(1/4), x)
\[ \int x^6 \sqrt [4]{3+4 x^4} \, dx=\frac {\left (4 x^{4}+3\right )^{\frac {1}{4}} x^{7}}{8}+\frac {3 \left (4 x^{4}+3\right )^{\frac {1}{4}} x^{3}}{128}-\frac {27 \left (\int \frac {x^{2}}{\left (4 x^{4}+3\right )^{\frac {3}{4}}}d x \right )}{128} \] Input:
int(x^6*(4*x^4+3)^(1/4),x)
Output:
(16*(4*x**4 + 3)**(1/4)*x**7 + 3*(4*x**4 + 3)**(1/4)*x**3 - 27*int(((4*x** 4 + 3)**(1/4)*x**2)/(4*x**4 + 3),x))/128