Integrand size = 22, antiderivative size = 138 \[ \int \frac {x^2 \sqrt [4]{-x^3+x^4}}{2+x} \, dx=\frac {1}{96} \left (401-100 x+32 x^2\right ) \sqrt [4]{-x^3+x^4}+\frac {1135}{64} \arctan \left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )-8\ 2^{3/4} \sqrt [4]{3} \arctan \left (\frac {\sqrt [4]{\frac {3}{2}} x}{\sqrt [4]{-x^3+x^4}}\right )-\frac {1135}{64} \text {arctanh}\left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )+8\ 2^{3/4} \sqrt [4]{3} \text {arctanh}\left (\frac {\sqrt [4]{\frac {3}{2}} x}{\sqrt [4]{-x^3+x^4}}\right ) \]
1/96*(32*x^2-100*x+401)*(x^4-x^3)^(1/4)+1135/64*arctan(x/(x^4-x^3)^(1/4))- 8*2^(3/4)*3^(1/4)*arctan(1/2*3^(1/4)*2^(3/4)*x/(x^4-x^3)^(1/4))-1135/64*ar ctanh(x/(x^4-x^3)^(1/4))+8*2^(3/4)*3^(1/4)*arctanh(1/2*3^(1/4)*2^(3/4)*x/( x^4-x^3)^(1/4))
Time = 0.48 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.17 \[ \int \frac {x^2 \sqrt [4]{-x^3+x^4}}{2+x} \, dx=\frac {(-1+x)^{3/4} x^{9/4} \left (802 \sqrt [4]{-1+x} x^{3/4}-200 \sqrt [4]{-1+x} x^{7/4}+64 \sqrt [4]{-1+x} x^{11/4}+3405 \arctan \left (\frac {1}{\sqrt [4]{\frac {-1+x}{x}}}\right )-1536\ 2^{3/4} \sqrt [4]{3} \arctan \left (\frac {\sqrt [4]{\frac {3}{2}}}{\sqrt [4]{\frac {-1+x}{x}}}\right )-3405 \text {arctanh}\left (\frac {1}{\sqrt [4]{\frac {-1+x}{x}}}\right )+1536\ 2^{3/4} \sqrt [4]{3} \text {arctanh}\left (\frac {\sqrt [4]{\frac {3}{2}}}{\sqrt [4]{\frac {-1+x}{x}}}\right )\right )}{192 \left ((-1+x) x^3\right )^{3/4}} \]
((-1 + x)^(3/4)*x^(9/4)*(802*(-1 + x)^(1/4)*x^(3/4) - 200*(-1 + x)^(1/4)*x ^(7/4) + 64*(-1 + x)^(1/4)*x^(11/4) + 3405*ArcTan[((-1 + x)/x)^(-1/4)] - 1 536*2^(3/4)*3^(1/4)*ArcTan[(3/2)^(1/4)/((-1 + x)/x)^(1/4)] - 3405*ArcTanh[ ((-1 + x)/x)^(-1/4)] + 1536*2^(3/4)*3^(1/4)*ArcTanh[(3/2)^(1/4)/((-1 + x)/ x)^(1/4)]))/(192*((-1 + x)*x^3)^(3/4))
Time = 0.37 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.41, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.773, Rules used = {1948, 112, 27, 173, 25, 104, 827, 216, 219, 1194, 27, 90, 73, 770, 756, 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.
| \(\displaystyle \int \frac {x^2 \sqrt [4]{x^4-x^3}}{x+2} \, dx\) |
| \(\Big \downarrow \) 1948 |
| \(\displaystyle \frac {\sqrt [4]{x^4-x^3} \int \frac {\sqrt [4]{x-1} x^{11/4}}{x+2}dx}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 112 |
| \(\displaystyle \frac {\sqrt [4]{x^4-x^3} \left (\frac {1}{3} \sqrt [4]{x-1} x^{11/4}-\frac {1}{3} \int -\frac {(22-25 x) x^{7/4}}{4 (x-1)^{3/4} (x+2)}dx\right )}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt [4]{x^4-x^3} \left (\frac {1}{12} \int \frac {(22-25 x) x^{7/4}}{(x-1)^{3/4} (x+2)}dx+\frac {1}{3} \sqrt [4]{x-1} x^{11/4}\right )}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 173 |
| \(\displaystyle \frac {\sqrt [4]{x^4-x^3} \left (\frac {1}{12} \left (\int -\frac {25 x^2-72 x+144}{(x-1)^{3/4} \sqrt [4]{x}}dx+288 \int \frac {1}{(x-1)^{3/4} \sqrt [4]{x} (x+2)}dx\right )+\frac {1}{3} \sqrt [4]{x-1} x^{11/4}\right )}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt [4]{x^4-x^3} \left (\frac {1}{12} \left (288 \int \frac {1}{(x-1)^{3/4} \sqrt [4]{x} (x+2)}dx-\int \frac {25 x^2-72 x+144}{(x-1)^{3/4} \sqrt [4]{x}}dx\right )+\frac {1}{3} \sqrt [4]{x-1} x^{11/4}\right )}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {\sqrt [4]{x^4-x^3} \left (\frac {1}{12} \left (1152 \int \frac {\sqrt {x}}{\sqrt {x-1} \left (2-\frac {3 x}{x-1}\right )}d\frac {\sqrt [4]{x}}{\sqrt [4]{x-1}}-\int \frac {25 x^2-72 x+144}{(x-1)^{3/4} \sqrt [4]{x}}dx\right )+\frac {1}{3} \sqrt [4]{x-1} x^{11/4}\right )}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle \frac {\sqrt [4]{x^4-x^3} \left (\frac {1}{12} \left (1152 \left (\frac {\int \frac {1}{\sqrt {2}-\frac {\sqrt {3} \sqrt {x}}{\sqrt {x-1}}}d\frac {\sqrt [4]{x}}{\sqrt [4]{x-1}}}{2 \sqrt {3}}-\frac {\int \frac {1}{\frac {\sqrt {3} \sqrt {x}}{\sqrt {x-1}}+\sqrt {2}}d\frac {\sqrt [4]{x}}{\sqrt [4]{x-1}}}{2 \sqrt {3}}\right )-\int \frac {25 x^2-72 x+144}{(x-1)^{3/4} \sqrt [4]{x}}dx\right )+\frac {1}{3} \sqrt [4]{x-1} x^{11/4}\right )}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\sqrt [4]{x^4-x^3} \left (\frac {1}{12} \left (1152 \left (\frac {\int \frac {1}{\sqrt {2}-\frac {\sqrt {3} \sqrt {x}}{\sqrt {x-1}}}d\frac {\sqrt [4]{x}}{\sqrt [4]{x-1}}}{2 \sqrt {3}}-\frac {\arctan \left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{2 \sqrt [4]{2} 3^{3/4}}\right )-\int \frac {25 x^2-72 x+144}{(x-1)^{3/4} \sqrt [4]{x}}dx\right )+\frac {1}{3} \sqrt [4]{x-1} x^{11/4}\right )}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\sqrt [4]{x^4-x^3} \left (\frac {1}{12} \left (1152 \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{2 \sqrt [4]{2} 3^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{2 \sqrt [4]{2} 3^{3/4}}\right )-\int \frac {25 x^2-72 x+144}{(x-1)^{3/4} \sqrt [4]{x}}dx\right )+\frac {1}{3} \sqrt [4]{x-1} x^{11/4}\right )}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 1194 |
| \(\displaystyle \frac {\sqrt [4]{x^4-x^3} \left (\frac {1}{12} \left (-\frac {1}{2} \int \frac {1077-301 x}{4 (x-1)^{3/4} \sqrt [4]{x}}dx+1152 \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{2 \sqrt [4]{2} 3^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{2 \sqrt [4]{2} 3^{3/4}}\right )-\frac {25}{2} x^{3/4} (x-1)^{5/4}\right )+\frac {1}{3} \sqrt [4]{x-1} x^{11/4}\right )}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt [4]{x^4-x^3} \left (\frac {1}{12} \left (-\frac {1}{8} \int \frac {1077-301 x}{(x-1)^{3/4} \sqrt [4]{x}}dx+1152 \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{2 \sqrt [4]{2} 3^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{2 \sqrt [4]{2} 3^{3/4}}\right )-\frac {25}{2} x^{3/4} (x-1)^{5/4}\right )+\frac {1}{3} \sqrt [4]{x-1} x^{11/4}\right )}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 90 |
| \(\displaystyle \frac {\sqrt [4]{x^4-x^3} \left (\frac {1}{12} \left (\frac {1}{8} \left (301 \sqrt [4]{x-1} x^{3/4}-\frac {3405}{4} \int \frac {1}{(x-1)^{3/4} \sqrt [4]{x}}dx\right )+1152 \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{2 \sqrt [4]{2} 3^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{2 \sqrt [4]{2} 3^{3/4}}\right )-\frac {25}{2} x^{3/4} (x-1)^{5/4}\right )+\frac {1}{3} \sqrt [4]{x-1} x^{11/4}\right )}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\sqrt [4]{x^4-x^3} \left (\frac {1}{12} \left (\frac {1}{8} \left (301 \sqrt [4]{x-1} x^{3/4}-3405 \int \frac {1}{\sqrt [4]{x}}d\sqrt [4]{x-1}\right )+1152 \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{2 \sqrt [4]{2} 3^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{2 \sqrt [4]{2} 3^{3/4}}\right )-\frac {25}{2} x^{3/4} (x-1)^{5/4}\right )+\frac {1}{3} \sqrt [4]{x-1} x^{11/4}\right )}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 770 |
| \(\displaystyle \frac {\sqrt [4]{x^4-x^3} \left (\frac {1}{12} \left (\frac {1}{8} \left (301 \sqrt [4]{x-1} x^{3/4}-3405 \int \frac {1}{2-x}d\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )+1152 \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{2 \sqrt [4]{2} 3^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{2 \sqrt [4]{2} 3^{3/4}}\right )-\frac {25}{2} x^{3/4} (x-1)^{5/4}\right )+\frac {1}{3} \sqrt [4]{x-1} x^{11/4}\right )}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle \frac {\sqrt [4]{x^4-x^3} \left (\frac {1}{12} \left (\frac {1}{8} \left (301 \sqrt [4]{x-1} x^{3/4}-3405 \left (\frac {1}{2} \int \frac {1}{1-\sqrt {x-1}}d\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}+\frac {1}{2} \int \frac {1}{\sqrt {x-1}+1}d\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )\right )+1152 \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{2 \sqrt [4]{2} 3^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{2 \sqrt [4]{2} 3^{3/4}}\right )-\frac {25}{2} x^{3/4} (x-1)^{5/4}\right )+\frac {1}{3} \sqrt [4]{x-1} x^{11/4}\right )}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\sqrt [4]{x^4-x^3} \left (\frac {1}{12} \left (\frac {1}{8} \left (301 \sqrt [4]{x-1} x^{3/4}-3405 \left (\frac {1}{2} \int \frac {1}{1-\sqrt {x-1}}d\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}+\frac {1}{2} \arctan \left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )\right )\right )+1152 \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{2 \sqrt [4]{2} 3^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{2 \sqrt [4]{2} 3^{3/4}}\right )-\frac {25}{2} x^{3/4} (x-1)^{5/4}\right )+\frac {1}{3} \sqrt [4]{x-1} x^{11/4}\right )}{\sqrt [4]{x-1} x^{3/4}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\sqrt [4]{x^4-x^3} \left (\frac {1}{12} \left (\frac {1}{8} \left (301 \sqrt [4]{x-1} x^{3/4}-3405 \left (\frac {1}{2} \arctan \left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )\right )\right )+1152 \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{2 \sqrt [4]{2} 3^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{\frac {3}{2}} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{2 \sqrt [4]{2} 3^{3/4}}\right )-\frac {25}{2} x^{3/4} (x-1)^{5/4}\right )+\frac {1}{3} \sqrt [4]{x-1} x^{11/4}\right )}{\sqrt [4]{x-1} x^{3/4}}\) |
((-x^3 + x^4)^(1/4)*(((-1 + x)^(1/4)*x^(11/4))/3 + ((-25*(-1 + x)^(5/4)*x^ (3/4))/2 + (301*(-1 + x)^(1/4)*x^(3/4) - 3405*(ArcTan[(-1 + x)^(1/4)/x^(1/ 4)]/2 + ArcTanh[(-1 + x)^(1/4)/x^(1/4)]/2))/8 + 1152*(-1/2*ArcTan[((3/2)^( 1/4)*x^(1/4))/(-1 + x)^(1/4)]/(2^(1/4)*3^(3/4)) + ArcTanh[((3/2)^(1/4)*x^( 1/4))/(-1 + x)^(1/4)]/(2*2^(1/4)*3^(3/4))))/12))/((-1 + x)^(1/4)*x^(3/4))
3.20.57.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^m*(c + d*x)^n*((e + f*x)^(p + 1)/(f*(m + n + p + 1))), x] - Simp[1/(f*(m + n + p + 1)) Int[(a + b*x)^(m - 1)*(c + d*x) ^(n - 1)*(e + f*x)^p*Simp[c*m*(b*e - a*f) + a*n*(d*e - c*f) + (d*m*(b*e - a *f) + b*n*(d*e - c*f))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[m, 0] && GtQ[n, 0] && NeQ[m + n + p + 1, 0] && (IntegersQ[2*m, 2*n, 2*p ] || (IntegersQ[m, n + p] || IntegersQ[p, m + n]))
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((g_.) + (h_.)*(x_ )))/((e_.) + (f_.)*(x_)), x_] :> Simp[(f*g - e*h)*((c*f - d*e)^(m + n + 1)/ f^(m + n + 2)) Int[(a + b*x)^m/((c + d*x)^(m + 1)*(e + f*x)), x], x] + Si mp[1/f^(m + n + 2) Int[((a + b*x)^m/(c + d*x)^(m + 1))*ExpandToSum[(f^(m + n + 2)*(c + d*x)^(m + n + 1)*(g + h*x) - (f*g - e*h)*(c*f - d*e)^(m + n + 1))/(e + f*x), x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && IGtQ[m + n + 1, 0] && (LtQ[m, 0] || SumSimplerQ[m, 1] || !SumSimplerQ[n, 1])
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[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 ]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ[a /b, 0]
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + 1/n) Subst[In t[1/(1 - b*x^n)^(p + 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)] && IntegerQ[p + 1 /n]
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[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[c^p*(d + e*x)^(m + 2*p)*((f + g*x )^(n + 1)/(g*e^(2*p)*(m + n + 2*p + 1))), x] + Simp[1/(g*e^(2*p)*(m + n + 2 *p + 1)) Int[(d + e*x)^m*(f + g*x)^n*ExpandToSum[g*(m + n + 2*p + 1)*(e^( 2*p)*(a + b*x + c*x^2)^p - c^p*(d + e*x)^(2*p)) - c^p*(e*f - d*g)*(m + 2*p) *(d + e*x)^(2*p - 1), x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ [p, 0] && !IntegerQ[m] && !IntegerQ[n] && NeQ[m + n + 2*p + 1, 0]
Int[((e_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(jn_.))^(p_)*((c_) + (d_.)*(x_)^(n_.))^(q_.), x_Symbol] :> Simp[e^IntPart[m]*(e*x)^FracPart[m]*( (a*x^j + b*x^(j + n))^FracPart[p]/(x^(FracPart[m] + j*FracPart[p])*(a + b*x ^n)^FracPart[p])) Int[x^(m + j*p)*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, j, m, n, p, q}, x] && EqQ[jn, j + n] && !IntegerQ[p] && NeQ[b*c - a*d, 0] && !(EqQ[n, 1] && EqQ[j, 1])
Time = 6.69 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.62
| method | result | size |
| pseudoelliptic | \(-\frac {x^{9} \left (1536 \,2^{\frac {3}{4}} 3^{\frac {1}{4}} \ln \left (\frac {-2^{\frac {3}{4}} 3^{\frac {1}{4}} x -2 \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{2^{\frac {3}{4}} 3^{\frac {1}{4}} x -2 \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}\right )+3072 \,2^{\frac {3}{4}} 3^{\frac {1}{4}} \arctan \left (\frac {\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}} 2^{\frac {1}{4}} 3^{\frac {3}{4}}}{3 x}\right )+128 \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}} x^{2}-400 \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}} x +3405 \ln \left (\frac {\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}-x}{x}\right )-6810 \arctan \left (\frac {\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{x}\right )-3405 \ln \left (\frac {x +\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{x}\right )+1604 \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}\right )}{384 {\left (x +\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}\right )}^{3} {\left (\left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}-x \right )}^{3} \left (x^{2}+\sqrt {x^{3} \left (-1+x \right )}\right )^{3}}\) | \(223\) |
| trager | \(\left (\frac {1}{3} x^{2}-\frac {25}{24} x +\frac {401}{96}\right ) \left (x^{4}-x^{3}\right )^{\frac {1}{4}}-\frac {1135 \ln \left (\frac {2 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}+2 \sqrt {x^{4}-x^{3}}\, x +2 x^{2} \left (x^{4}-x^{3}\right )^{\frac {1}{4}}+2 x^{3}-x^{2}}{x^{2}}\right )}{128}+4 \operatorname {RootOf}\left (\textit {\_Z}^{4}-24\right ) \ln \left (\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{4}-24\right )^{3} x^{3}+12 \left (x^{4}-x^{3}\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-24\right )^{2} x^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}-24\right )^{3} x^{2}+24 \sqrt {x^{4}-x^{3}}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-24\right ) x +48 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}}{x^{2} \left (2+x \right )}\right )-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-24\right )^{2}\right ) \ln \left (-\frac {-5 \operatorname {RootOf}\left (\textit {\_Z}^{4}-24\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-24\right )^{2}\right ) x^{3}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}-24\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-24\right )^{2}\right ) x^{2}+12 \left (x^{4}-x^{3}\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-24\right )^{2} x^{2}+24 \sqrt {x^{4}-x^{3}}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-24\right )^{2}\right ) x -48 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}}{x^{2} \left (2+x \right )}\right )+\frac {1135 \operatorname {RootOf}\left (\textit {\_Z}^{4}-24\right )^{3} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-24\right )^{2}\right ) \ln \left (-\frac {2 \sqrt {x^{4}-x^{3}}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-24\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-24\right )^{3} x -2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-24\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-24\right )^{3} x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-24\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-24\right )^{3} x^{2}-48 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}+48 x^{2} \left (x^{4}-x^{3}\right )^{\frac {1}{4}}}{x^{2}}\right )}{3072}\) | \(475\) |
| risch | \(\text {Expression too large to display}\) | \(925\) |
-1/384*x^9*(1536*2^(3/4)*3^(1/4)*ln((-2^(3/4)*3^(1/4)*x-2*(x^3*(-1+x))^(1/ 4))/(2^(3/4)*3^(1/4)*x-2*(x^3*(-1+x))^(1/4)))+3072*2^(3/4)*3^(1/4)*arctan( 1/3*(x^3*(-1+x))^(1/4)/x*2^(1/4)*3^(3/4))+128*(x^3*(-1+x))^(1/4)*x^2-400*( x^3*(-1+x))^(1/4)*x+3405*ln(((x^3*(-1+x))^(1/4)-x)/x)-6810*arctan((x^3*(-1 +x))^(1/4)/x)-3405*ln((x+(x^3*(-1+x))^(1/4))/x)+1604*(x^3*(-1+x))^(1/4))/( x+(x^3*(-1+x))^(1/4))^3/((x^3*(-1+x))^(1/4)-x)^3/(x^2+(x^3*(-1+x))^(1/2))^ 3
Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.48 \[ \int \frac {x^2 \sqrt [4]{-x^3+x^4}}{2+x} \, dx=\frac {1}{96} \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (32 \, x^{2} - 100 \, x + 401\right )} + 4 \cdot 24^{\frac {1}{4}} \log \left (\frac {24^{\frac {1}{4}} x + 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - 4 \cdot 24^{\frac {1}{4}} \log \left (-\frac {24^{\frac {1}{4}} x - 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + 4 i \cdot 24^{\frac {1}{4}} \log \left (\frac {i \cdot 24^{\frac {1}{4}} x + 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - 4 i \cdot 24^{\frac {1}{4}} \log \left (\frac {-i \cdot 24^{\frac {1}{4}} x + 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1135}{64} \, \arctan \left (\frac {{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1135}{128} \, \log \left (\frac {x + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1135}{128} \, \log \left (-\frac {x - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) \]
1/96*(x^4 - x^3)^(1/4)*(32*x^2 - 100*x + 401) + 4*24^(1/4)*log((24^(1/4)*x + 2*(x^4 - x^3)^(1/4))/x) - 4*24^(1/4)*log(-(24^(1/4)*x - 2*(x^4 - x^3)^( 1/4))/x) + 4*I*24^(1/4)*log((I*24^(1/4)*x + 2*(x^4 - x^3)^(1/4))/x) - 4*I* 24^(1/4)*log((-I*24^(1/4)*x + 2*(x^4 - x^3)^(1/4))/x) - 1135/64*arctan((x^ 4 - x^3)^(1/4)/x) - 1135/128*log((x + (x^4 - x^3)^(1/4))/x) + 1135/128*log (-(x - (x^4 - x^3)^(1/4))/x)
\[ \int \frac {x^2 \sqrt [4]{-x^3+x^4}}{2+x} \, dx=\int \frac {x^{2} \sqrt [4]{x^{3} \left (x - 1\right )}}{x + 2}\, dx \]
\[ \int \frac {x^2 \sqrt [4]{-x^3+x^4}}{2+x} \, dx=\int { \frac {{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} x^{2}}{x + 2} \,d x } \]
Time = 0.32 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.08 \[ \int \frac {x^2 \sqrt [4]{-x^3+x^4}}{2+x} \, dx=\frac {1}{96} \, {\left (401 \, {\left (\frac {1}{x} - 1\right )}^{2} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} - 702 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {5}{4}} + 333 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right )} x^{3} + 8 \cdot 24^{\frac {1}{4}} \arctan \left (\frac {2}{3} \, \left (\frac {3}{2}\right )^{\frac {3}{4}} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + 4 \cdot 24^{\frac {1}{4}} \log \left (\left (\frac {3}{2}\right )^{\frac {1}{4}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) - 4 \cdot 24^{\frac {1}{4}} \log \left ({\left | -\left (\frac {3}{2}\right )^{\frac {1}{4}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} \right |}\right ) - \frac {1135}{64} \, \arctan \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1135}{128} \, \log \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {1135}{128} \, \log \left ({\left | {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} - 1 \right |}\right ) \]
1/96*(401*(1/x - 1)^2*(-1/x + 1)^(1/4) - 702*(-1/x + 1)^(5/4) + 333*(-1/x + 1)^(1/4))*x^3 + 8*24^(1/4)*arctan(2/3*(3/2)^(3/4)*(-1/x + 1)^(1/4)) + 4* 24^(1/4)*log((3/2)^(1/4) + (-1/x + 1)^(1/4)) - 4*24^(1/4)*log(abs(-(3/2)^( 1/4) + (-1/x + 1)^(1/4))) - 1135/64*arctan((-1/x + 1)^(1/4)) - 1135/128*lo g((-1/x + 1)^(1/4) + 1) + 1135/128*log(abs((-1/x + 1)^(1/4) - 1))
Timed out. \[ \int \frac {x^2 \sqrt [4]{-x^3+x^4}}{2+x} \, dx=\int \frac {x^2\,{\left (x^4-x^3\right )}^{1/4}}{x+2} \,d x \]