Integrand size = 20, antiderivative size = 72 \[ \int \frac {x^4 \sqrt [4]{x^3+x^4}}{1+x} \, dx=\frac {\sqrt [4]{x^3+x^4} \left (7315-4180 x+3040 x^2-2432 x^3+2048 x^4\right )}{10240}+\frac {4389 \arctan \left (\frac {x}{\sqrt [4]{x^3+x^4}}\right )}{4096}-\frac {4389 \text {arctanh}\left (\frac {x}{\sqrt [4]{x^3+x^4}}\right )}{4096} \]
1/10240*(x^4+x^3)^(1/4)*(2048*x^4-2432*x^3+3040*x^2-4180*x+7315)+4389/4096 *arctan(x/(x^4+x^3)^(1/4))-4389/4096*arctanh(x/(x^4+x^3)^(1/4))
Time = 0.15 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.32 \[ \int \frac {x^4 \sqrt [4]{x^3+x^4}}{1+x} \, dx=\frac {x^{9/4} \left (2 x^{3/4} \left (7315+3135 x-1140 x^2+608 x^3-384 x^4+2048 x^5\right )-21945 (1+x)^{3/4} \arctan \left (\frac {1}{\sqrt [4]{\frac {x}{1+x}}}\right )-21945 (1+x)^{3/4} \text {arctanh}\left (\frac {1}{\sqrt [4]{\frac {x}{1+x}}}\right )\right )}{20480 \left (x^3 (1+x)\right )^{3/4}} \]
(x^(9/4)*(2*x^(3/4)*(7315 + 3135*x - 1140*x^2 + 608*x^3 - 384*x^4 + 2048*x ^5) - 21945*(1 + x)^(3/4)*ArcTan[(x/(1 + x))^(-1/4)] - 21945*(1 + x)^(3/4) *ArcTanh[(x/(1 + x))^(-1/4)]))/(20480*(x^3*(1 + x))^(3/4))
Leaf count is larger than twice the leaf count of optimal. \(161\) vs. \(2(72)=144\).
Time = 0.32 (sec) , antiderivative size = 161, normalized size of antiderivative = 2.24, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {2467, 60, 60, 60, 60, 60, 73, 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.
| \(\displaystyle \int \frac {x^4 \sqrt [4]{x^4+x^3}}{x+1} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [4]{x^4+x^3} \int \frac {x^{19/4}}{(x+1)^{3/4}}dx}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {\sqrt [4]{x^4+x^3} \left (\frac {1}{5} x^{19/4} \sqrt [4]{x+1}-\frac {19}{20} \int \frac {x^{15/4}}{(x+1)^{3/4}}dx\right )}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {\sqrt [4]{x^4+x^3} \left (\frac {1}{5} x^{19/4} \sqrt [4]{x+1}-\frac {19}{20} \left (\frac {1}{4} x^{15/4} \sqrt [4]{x+1}-\frac {15}{16} \int \frac {x^{11/4}}{(x+1)^{3/4}}dx\right )\right )}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {\sqrt [4]{x^4+x^3} \left (\frac {1}{5} x^{19/4} \sqrt [4]{x+1}-\frac {19}{20} \left (\frac {1}{4} x^{15/4} \sqrt [4]{x+1}-\frac {15}{16} \left (\frac {1}{3} x^{11/4} \sqrt [4]{x+1}-\frac {11}{12} \int \frac {x^{7/4}}{(x+1)^{3/4}}dx\right )\right )\right )}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {\sqrt [4]{x^4+x^3} \left (\frac {1}{5} x^{19/4} \sqrt [4]{x+1}-\frac {19}{20} \left (\frac {1}{4} x^{15/4} \sqrt [4]{x+1}-\frac {15}{16} \left (\frac {1}{3} x^{11/4} \sqrt [4]{x+1}-\frac {11}{12} \left (\frac {1}{2} x^{7/4} \sqrt [4]{x+1}-\frac {7}{8} \int \frac {x^{3/4}}{(x+1)^{3/4}}dx\right )\right )\right )\right )}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {\sqrt [4]{x^4+x^3} \left (\frac {1}{5} x^{19/4} \sqrt [4]{x+1}-\frac {19}{20} \left (\frac {1}{4} x^{15/4} \sqrt [4]{x+1}-\frac {15}{16} \left (\frac {1}{3} x^{11/4} \sqrt [4]{x+1}-\frac {11}{12} \left (\frac {1}{2} x^{7/4} \sqrt [4]{x+1}-\frac {7}{8} \left (x^{3/4} \sqrt [4]{x+1}-\frac {3}{4} \int \frac {1}{\sqrt [4]{x} (x+1)^{3/4}}dx\right )\right )\right )\right )\right )}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\sqrt [4]{x^4+x^3} \left (\frac {1}{5} x^{19/4} \sqrt [4]{x+1}-\frac {19}{20} \left (\frac {1}{4} x^{15/4} \sqrt [4]{x+1}-\frac {15}{16} \left (\frac {1}{3} x^{11/4} \sqrt [4]{x+1}-\frac {11}{12} \left (\frac {1}{2} x^{7/4} \sqrt [4]{x+1}-\frac {7}{8} \left (x^{3/4} \sqrt [4]{x+1}-3 \int \frac {\sqrt {x}}{(x+1)^{3/4}}d\sqrt [4]{x}\right )\right )\right )\right )\right )}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 854 |
| \(\displaystyle \frac {\sqrt [4]{x^4+x^3} \left (\frac {1}{5} x^{19/4} \sqrt [4]{x+1}-\frac {19}{20} \left (\frac {1}{4} x^{15/4} \sqrt [4]{x+1}-\frac {15}{16} \left (\frac {1}{3} x^{11/4} \sqrt [4]{x+1}-\frac {11}{12} \left (\frac {1}{2} x^{7/4} \sqrt [4]{x+1}-\frac {7}{8} \left (x^{3/4} \sqrt [4]{x+1}-3 \int \frac {\sqrt {x}}{1-x}d\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )\right )\right )\right )\right )}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle \frac {\sqrt [4]{x^4+x^3} \left (\frac {1}{5} x^{19/4} \sqrt [4]{x+1}-\frac {19}{20} \left (\frac {1}{4} x^{15/4} \sqrt [4]{x+1}-\frac {15}{16} \left (\frac {1}{3} x^{11/4} \sqrt [4]{x+1}-\frac {11}{12} \left (\frac {1}{2} x^{7/4} \sqrt [4]{x+1}-\frac {7}{8} \left (x^{3/4} \sqrt [4]{x+1}-3 \left (\frac {1}{2} \int \frac {1}{1-\sqrt {x}}d\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}-\frac {1}{2} \int \frac {1}{\sqrt {x}+1}d\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )\right )\right )\right )\right )\right )}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\sqrt [4]{x^4+x^3} \left (\frac {1}{5} x^{19/4} \sqrt [4]{x+1}-\frac {19}{20} \left (\frac {1}{4} x^{15/4} \sqrt [4]{x+1}-\frac {15}{16} \left (\frac {1}{3} x^{11/4} \sqrt [4]{x+1}-\frac {11}{12} \left (\frac {1}{2} x^{7/4} \sqrt [4]{x+1}-\frac {7}{8} \left (x^{3/4} \sqrt [4]{x+1}-3 \left (\frac {1}{2} \int \frac {1}{1-\sqrt {x}}d\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )\right )\right )\right )\right )\right )\right )}{x^{3/4} \sqrt [4]{x+1}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\sqrt [4]{x^4+x^3} \left (\frac {1}{5} x^{19/4} \sqrt [4]{x+1}-\frac {19}{20} \left (\frac {1}{4} x^{15/4} \sqrt [4]{x+1}-\frac {15}{16} \left (\frac {1}{3} x^{11/4} \sqrt [4]{x+1}-\frac {11}{12} \left (\frac {1}{2} x^{7/4} \sqrt [4]{x+1}-\frac {7}{8} \left (x^{3/4} \sqrt [4]{x+1}-3 \left (\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )\right )\right )\right )\right )\right )\right )}{x^{3/4} \sqrt [4]{x+1}}\) |
((x^3 + x^4)^(1/4)*((x^(19/4)*(1 + x)^(1/4))/5 - (19*((x^(15/4)*(1 + x)^(1 /4))/4 - (15*((x^(11/4)*(1 + x)^(1/4))/3 - (11*((x^(7/4)*(1 + x)^(1/4))/2 - (7*(x^(3/4)*(1 + x)^(1/4) - 3*(-1/2*ArcTan[x^(1/4)/(1 + x)^(1/4)] + ArcT anh[x^(1/4)/(1 + x)^(1/4)]/2)))/8))/12))/16))/20))/(x^(3/4)*(1 + x)^(1/4))
3.10.49.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
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[(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[(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]
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
Result contains higher order function than in optimal. Order 5 vs. order 3.
Time = 1.57 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.21
| method | result | size |
| meijerg | \(\frac {4 x^{\frac {23}{4}} \operatorname {hypergeom}\left (\left [\frac {3}{4}, \frac {23}{4}\right ], \left [\frac {27}{4}\right ], -x \right )}{23}\) | \(15\) |
| trager | \(\left (\frac {1}{5} x^{4}-\frac {19}{80} x^{3}+\frac {19}{64} x^{2}-\frac {209}{512} x +\frac {1463}{2048}\right ) \left (x^{4}+x^{3}\right )^{\frac {1}{4}}-\frac {4389 \ln \left (\frac {2 \left (x^{4}+x^{3}\right )^{\frac {3}{4}}+2 \sqrt {x^{4}+x^{3}}\, x +2 \left (x^{4}+x^{3}\right )^{\frac {1}{4}} x^{2}+2 x^{3}+x^{2}}{x^{2}}\right )}{8192}-\frac {4389 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {2 \sqrt {x^{4}+x^{3}}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x -2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+2 \left (x^{4}+x^{3}\right )^{\frac {3}{4}}-2 \left (x^{4}+x^{3}\right )^{\frac {1}{4}} x^{2}}{x^{2}}\right )}{8192}\) | \(164\) |
| pseudoelliptic | \(\frac {x^{15} \left (8192 x^{4} \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}-9728 x^{3} \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}+12160 \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}} x^{2}-16720 x \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}+21945 \ln \left (\frac {\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}-x}{x}\right )-43890 \arctan \left (\frac {\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{x}\right )-21945 \ln \left (\frac {\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}+x}{x}\right )+29260 \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}\right )}{40960 {\left (\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}-x \right )}^{5} \left (x^{2}+\sqrt {x^{3} \left (1+x \right )}\right )^{5} {\left (\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}+x \right )}^{5}}\) | \(169\) |
| risch | \(\frac {\left (2048 x^{4}-2432 x^{3}+3040 x^{2}-4180 x +7315\right ) \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{10240}+\frac {\left (\frac {4389 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {2 \sqrt {x^{4}+3 x^{3}+3 x^{2}+x}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x -2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \sqrt {x^{4}+3 x^{3}+3 x^{2}+x}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+2 \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {3}{4}}-2 \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {1}{4}} x^{2}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x -4 \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {1}{4}} x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )-2 \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {1}{4}}}{\left (1+x \right )^{2}}\right )}{8192}+\frac {4389 \ln \left (\frac {2 \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {3}{4}}-2 \sqrt {x^{4}+3 x^{3}+3 x^{2}+x}\, x +2 \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {1}{4}} x^{2}-2 x^{3}-2 \sqrt {x^{4}+3 x^{3}+3 x^{2}+x}+4 \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {1}{4}} x -5 x^{2}+2 \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {1}{4}}-4 x -1}{\left (1+x \right )^{2}}\right )}{8192}\right ) \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}} \left (\left (1+x \right )^{3} x \right )^{\frac {1}{4}}}{x \left (1+x \right )}\) | \(390\) |
Time = 0.24 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.21 \[ \int \frac {x^4 \sqrt [4]{x^3+x^4}}{1+x} \, dx=\frac {1}{10240} \, {\left (2048 \, x^{4} - 2432 \, x^{3} + 3040 \, x^{2} - 4180 \, x + 7315\right )} {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}} - \frac {4389}{4096} \, \arctan \left (\frac {{\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {4389}{8192} \, \log \left (\frac {x + {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {4389}{8192} \, \log \left (-\frac {x - {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) \]
1/10240*(2048*x^4 - 2432*x^3 + 3040*x^2 - 4180*x + 7315)*(x^4 + x^3)^(1/4) - 4389/4096*arctan((x^4 + x^3)^(1/4)/x) - 4389/8192*log((x + (x^4 + x^3)^ (1/4))/x) + 4389/8192*log(-(x - (x^4 + x^3)^(1/4))/x)
\[ \int \frac {x^4 \sqrt [4]{x^3+x^4}}{1+x} \, dx=\int \frac {x^{4} \sqrt [4]{x^{3} \left (x + 1\right )}}{x + 1}\, dx \]
\[ \int \frac {x^4 \sqrt [4]{x^3+x^4}}{1+x} \, dx=\int { \frac {{\left (x^{4} + x^{3}\right )}^{\frac {1}{4}} x^{4}}{x + 1} \,d x } \]
Time = 0.29 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.21 \[ \int \frac {x^4 \sqrt [4]{x^3+x^4}}{1+x} \, dx=\frac {1}{10240} \, {\left (7315 \, {\left (\frac {1}{x} + 1\right )}^{\frac {17}{4}} - 33440 \, {\left (\frac {1}{x} + 1\right )}^{\frac {13}{4}} + 59470 \, {\left (\frac {1}{x} + 1\right )}^{\frac {9}{4}} - 50312 \, {\left (\frac {1}{x} + 1\right )}^{\frac {5}{4}} + 19015 \, {\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right )} x^{5} - \frac {4389}{4096} \, \arctan \left ({\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) - \frac {4389}{8192} \, \log \left ({\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {4389}{8192} \, \log \left ({\left | {\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}} - 1 \right |}\right ) \]
1/10240*(7315*(1/x + 1)^(17/4) - 33440*(1/x + 1)^(13/4) + 59470*(1/x + 1)^ (9/4) - 50312*(1/x + 1)^(5/4) + 19015*(1/x + 1)^(1/4))*x^5 - 4389/4096*arc tan((1/x + 1)^(1/4)) - 4389/8192*log((1/x + 1)^(1/4) + 1) + 4389/8192*log( abs((1/x + 1)^(1/4) - 1))
Timed out. \[ \int \frac {x^4 \sqrt [4]{x^3+x^4}}{1+x} \, dx=\int \frac {x^4\,{\left (x^4+x^3\right )}^{1/4}}{x+1} \,d x \]