Integrand size = 24, antiderivative size = 257 \[ \int x^2 \sqrt {a x^2+b x^3+c x^4} \, dx=\frac {b \left (35 b^2-116 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{960 c^3}-\frac {\left (105 b^4-460 a b^2 c+256 a^2 c^2\right ) \sqrt {a x^2+b x^3+c x^4}}{1920 c^4 x}-\frac {\left (7 b^2-16 a c\right ) x \sqrt {a x^2+b x^3+c x^4}}{240 c^2}+\frac {x^2 (b+8 c x) \sqrt {a x^2+b x^3+c x^4}}{40 c}+\frac {b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right ) x \sqrt {a+b x+c x^2} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{256 c^{9/2} \sqrt {a x^2+b x^3+c x^4}} \]
1/256*b*(-12*a*c+7*b^2)*(-4*a*c+b^2)*x*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^ 2+b*x+a)^(1/2))*(c*x^2+b*x+a)^(1/2)/c^(9/2)/(c*x^4+b*x^3+a*x^2)^(1/2)+1/96 0*b*(-116*a*c+35*b^2)*(c*x^4+b*x^3+a*x^2)^(1/2)/c^3-1/1920*(256*a^2*c^2-46 0*a*b^2*c+105*b^4)*(c*x^4+b*x^3+a*x^2)^(1/2)/c^4/x-1/240*(-16*a*c+7*b^2)*x *(c*x^4+b*x^3+a*x^2)^(1/2)/c^2+1/40*x^2*(8*c*x+b)*(c*x^4+b*x^3+a*x^2)^(1/2 )/c
Time = 0.42 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.72 \[ \int x^2 \sqrt {a x^2+b x^3+c x^4} \, dx=\frac {2 \sqrt {c} x (a+x (b+c x)) \left (-105 b^4+70 b^3 c x+4 b^2 c \left (115 a-14 c x^2\right )+8 b c^2 x \left (-29 a+6 c x^2\right )+128 c^2 \left (-2 a^2+a c x^2+3 c^2 x^4\right )\right )-15 \left (7 b^5-40 a b^3 c+48 a^2 b c^2\right ) x \sqrt {a+x (b+c x)} \log \left (c^4 \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )\right )}{3840 c^{9/2} \sqrt {x^2 (a+x (b+c x))}} \]
(2*Sqrt[c]*x*(a + x*(b + c*x))*(-105*b^4 + 70*b^3*c*x + 4*b^2*c*(115*a - 1 4*c*x^2) + 8*b*c^2*x*(-29*a + 6*c*x^2) + 128*c^2*(-2*a^2 + a*c*x^2 + 3*c^2 *x^4)) - 15*(7*b^5 - 40*a*b^3*c + 48*a^2*b*c^2)*x*Sqrt[a + x*(b + c*x)]*Lo g[c^4*(b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/(3840*c^(9/2)*Sqrt[x ^2*(a + x*(b + c*x))])
Time = 0.65 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.08, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {1966, 27, 1996, 27, 1996, 27, 1996, 27, 1961, 1092, 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.
| \(\displaystyle \int x^2 \sqrt {a x^2+b x^3+c x^4} \, dx\) |
| \(\Big \downarrow \) 1966 |
| \(\displaystyle \frac {\int -\frac {x^3 \left (6 a b+\left (7 b^2-16 a c\right ) x\right )}{2 \sqrt {c x^4+b x^3+a x^2}}dx}{40 c}+\frac {x^2 (b+8 c x) \sqrt {a x^2+b x^3+c x^4}}{40 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^2 (b+8 c x) \sqrt {a x^2+b x^3+c x^4}}{40 c}-\frac {\int \frac {x^3 \left (6 a b+\left (7 b^2-16 a c\right ) x\right )}{\sqrt {c x^4+b x^3+a x^2}}dx}{80 c}\) |
| \(\Big \downarrow \) 1996 |
| \(\displaystyle \frac {x^2 (b+8 c x) \sqrt {a x^2+b x^3+c x^4}}{40 c}-\frac {\frac {x \left (7 b^2-16 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{3 c}-\frac {\int \frac {x^2 \left (4 a \left (7 b^2-16 a c\right )+b \left (35 b^2-116 a c\right ) x\right )}{2 \sqrt {c x^4+b x^3+a x^2}}dx}{3 c}}{80 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^2 (b+8 c x) \sqrt {a x^2+b x^3+c x^4}}{40 c}-\frac {\frac {x \left (7 b^2-16 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{3 c}-\frac {\int \frac {x^2 \left (4 a \left (7 b^2-16 a c\right )+b \left (35 b^2-116 a c\right ) x\right )}{\sqrt {c x^4+b x^3+a x^2}}dx}{6 c}}{80 c}\) |
| \(\Big \downarrow \) 1996 |
| \(\displaystyle \frac {x^2 (b+8 c x) \sqrt {a x^2+b x^3+c x^4}}{40 c}-\frac {\frac {x \left (7 b^2-16 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{3 c}-\frac {\frac {b \left (35 b^2-116 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{2 c}-\frac {\int \frac {x \left (2 a b \left (35 b^2-116 a c\right )+\left (105 b^4-460 a c b^2+256 a^2 c^2\right ) x\right )}{2 \sqrt {c x^4+b x^3+a x^2}}dx}{2 c}}{6 c}}{80 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^2 (b+8 c x) \sqrt {a x^2+b x^3+c x^4}}{40 c}-\frac {\frac {x \left (7 b^2-16 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{3 c}-\frac {\frac {b \left (35 b^2-116 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{2 c}-\frac {\int \frac {x \left (2 a b \left (35 b^2-116 a c\right )+\left (105 b^4-460 a c b^2+256 a^2 c^2\right ) x\right )}{\sqrt {c x^4+b x^3+a x^2}}dx}{4 c}}{6 c}}{80 c}\) |
| \(\Big \downarrow \) 1996 |
| \(\displaystyle \frac {x^2 (b+8 c x) \sqrt {a x^2+b x^3+c x^4}}{40 c}-\frac {\frac {x \left (7 b^2-16 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{3 c}-\frac {\frac {b \left (35 b^2-116 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{2 c}-\frac {\frac {\left (256 a^2 c^2-460 a b^2 c+105 b^4\right ) \sqrt {a x^2+b x^3+c x^4}}{c x}-\frac {\int \frac {15 b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right ) x}{2 \sqrt {c x^4+b x^3+a x^2}}dx}{c}}{4 c}}{6 c}}{80 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^2 (b+8 c x) \sqrt {a x^2+b x^3+c x^4}}{40 c}-\frac {\frac {x \left (7 b^2-16 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{3 c}-\frac {\frac {b \left (35 b^2-116 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{2 c}-\frac {\frac {\left (256 a^2 c^2-460 a b^2 c+105 b^4\right ) \sqrt {a x^2+b x^3+c x^4}}{c x}-\frac {15 b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right ) \int \frac {x}{\sqrt {c x^4+b x^3+a x^2}}dx}{2 c}}{4 c}}{6 c}}{80 c}\) |
| \(\Big \downarrow \) 1961 |
| \(\displaystyle \frac {x^2 (b+8 c x) \sqrt {a x^2+b x^3+c x^4}}{40 c}-\frac {\frac {x \left (7 b^2-16 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{3 c}-\frac {\frac {b \left (35 b^2-116 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{2 c}-\frac {\frac {\left (256 a^2 c^2-460 a b^2 c+105 b^4\right ) \sqrt {a x^2+b x^3+c x^4}}{c x}-\frac {15 b x \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{2 c \sqrt {a x^2+b x^3+c x^4}}}{4 c}}{6 c}}{80 c}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {x^2 (b+8 c x) \sqrt {a x^2+b x^3+c x^4}}{40 c}-\frac {\frac {x \left (7 b^2-16 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{3 c}-\frac {\frac {b \left (35 b^2-116 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{2 c}-\frac {\frac {\left (256 a^2 c^2-460 a b^2 c+105 b^4\right ) \sqrt {a x^2+b x^3+c x^4}}{c x}-\frac {15 b x \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{c \sqrt {a x^2+b x^3+c x^4}}}{4 c}}{6 c}}{80 c}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {x^2 (b+8 c x) \sqrt {a x^2+b x^3+c x^4}}{40 c}-\frac {\frac {x \left (7 b^2-16 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{3 c}-\frac {\frac {b \left (35 b^2-116 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{2 c}-\frac {\frac {\left (256 a^2 c^2-460 a b^2 c+105 b^4\right ) \sqrt {a x^2+b x^3+c x^4}}{c x}-\frac {15 b x \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 c^{3/2} \sqrt {a x^2+b x^3+c x^4}}}{4 c}}{6 c}}{80 c}\) |
(x^2*(b + 8*c*x)*Sqrt[a*x^2 + b*x^3 + c*x^4])/(40*c) - (((7*b^2 - 16*a*c)* x*Sqrt[a*x^2 + b*x^3 + c*x^4])/(3*c) - ((b*(35*b^2 - 116*a*c)*Sqrt[a*x^2 + b*x^3 + c*x^4])/(2*c) - (((105*b^4 - 460*a*b^2*c + 256*a^2*c^2)*Sqrt[a*x^ 2 + b*x^3 + c*x^4])/(c*x) - (15*b*(7*b^2 - 12*a*c)*(b^2 - 4*a*c)*x*Sqrt[a + b*x + c*x^2]*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(2* c^(3/2)*Sqrt[a*x^2 + b*x^3 + c*x^4]))/(4*c))/(6*c))/(80*c)
3.1.29.3.1 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_)^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[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
Int[(x_)^(m_.)/Sqrt[(b_.)*(x_)^(n_.) + (a_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.)] , x_Symbol] :> Simp[x^(q/2)*(Sqrt[a + b*x^(n - q) + c*x^(2*(n - q))]/Sqrt[a *x^q + b*x^n + c*x^(2*n - q)]) Int[x^(m - q/2)/Sqrt[a + b*x^(n - q) + c*x ^(2*(n - q))], x], x] /; FreeQ[{a, b, c, m, n, q}, x] && EqQ[r, 2*n - q] && PosQ[n - q] && ((EqQ[m, 1] && EqQ[n, 3] && EqQ[q, 2]) || ((EqQ[m + 1/2] || EqQ[m, 3/2] || EqQ[m, 1/2] || EqQ[m, 5/2]) && EqQ[n, 3] && EqQ[q, 1]))
Int[(x_)^(m_.)*((b_.)*(x_)^(n_.) + (a_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(p_ ), x_Symbol] :> Simp[x^(m - n + q + 1)*(b*(n - q)*p + c*(m + p*q + (n - q)* (2*p - 1) + 1)*x^(n - q))*((a*x^q + b*x^n + c*x^(2*n - q))^p/(c*(m + p*(2*n - q) + 1)*(m + p*q + (n - q)*(2*p - 1) + 1))), x] + Simp[(n - q)*(p/(c*(m + p*(2*n - q) + 1)*(m + p*q + (n - q)*(2*p - 1) + 1))) Int[x^(m - (n - 2* q))*Simp[(-a)*b*(m + p*q - n + q + 1) + (2*a*c*(m + p*q + (n - q)*(2*p - 1) + 1) - b^2*(m + p*q + (n - q)*(p - 1) + 1))*x^(n - q), x]*(a*x^q + b*x^n + c*x^(2*n - q))^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && EqQ[r, 2*n - q] & & PosQ[n - q] && !IntegerQ[p] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GtQ[ p, 0] && RationalQ[m, q] && GtQ[m + p*q + 1, n - q] && NeQ[m + p*(2*n - q) + 1, 0] && NeQ[m + p*q + (n - q)*(2*p - 1) + 1, 0]
Int[(x_)^(m_.)*((c_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.) + (a_.)*(x_)^(q_.))^(p_ .)*((A_) + (B_.)*(x_)^(r_.)), x_Symbol] :> Simp[B*x^(m - n + 1)*((a*x^q + b *x^n + c*x^(2*n - q))^(p + 1)/(c*(m + p*q + (n - q)*(2*p + 1) + 1))), x] - Simp[1/(c*(m + p*q + (n - q)*(2*p + 1) + 1)) Int[x^(m - n + q)*Simp[a*B*( m + p*q - n + q + 1) + (b*B*(m + p*q + (n - q)*p + 1) - A*c*(m + p*q + (n - q)*(2*p + 1) + 1))*x^(n - q), x]*(a*x^q + b*x^n + c*x^(2*n - q))^p, x], x] /; FreeQ[{a, b, c, A, B}, x] && EqQ[r, n - q] && EqQ[j, 2*n - q] && !Inte gerQ[p] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && GeQ[p, -1] && LtQ[p, 0] && RationalQ[m, q] && GeQ[m + p*q, n - q - 1] && NeQ[m + p*q + (n - q)*(2*p + 1) + 1, 0]
Time = 0.23 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.52
| method | result | size |
| pseudoelliptic | \(-\frac {2 \left (\left (-\frac {45}{32} a^{2} b \,c^{2}+\frac {75}{64} a \,b^{3} c -\frac {105}{512} b^{5}\right ) \ln \left (2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b \right )+\sqrt {c \,x^{2}+b x +a}\, \left (\left (\frac {7}{32} b^{2} x^{2}+\frac {29}{32} a b x +a^{2}\right ) c^{\frac {5}{2}}-\frac {115 \left (\frac {7 b x}{46}+a \right ) b^{2} c^{\frac {3}{2}}}{64}-\frac {\left (\frac {3 b x}{8}+a \right ) x^{2} c^{\frac {7}{2}}}{2}-\frac {3 c^{\frac {9}{2}} x^{4}}{2}+\frac {105 \sqrt {c}\, b^{4}}{256}\right )\right )}{15 c^{\frac {9}{2}}}\) | \(133\) |
| risch | \(-\frac {\left (-384 c^{4} x^{4}-48 b \,c^{3} x^{3}-128 a \,c^{3} x^{2}+56 b^{2} c^{2} x^{2}+232 a b \,c^{2} x -70 b^{3} c x +256 a^{2} c^{2}-460 a \,b^{2} c +105 b^{4}\right ) \sqrt {x^{2} \left (c \,x^{2}+b x +a \right )}}{1920 c^{4} x}+\frac {b \left (48 a^{2} c^{2}-40 a \,b^{2} c +7 b^{4}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) \sqrt {x^{2} \left (c \,x^{2}+b x +a \right )}}{256 c^{\frac {9}{2}} x \sqrt {c \,x^{2}+b x +a}}\) | \(182\) |
| default | \(\frac {\sqrt {c \,x^{4}+b \,x^{3}+a \,x^{2}}\, \left (768 x^{2} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} c^{\frac {9}{2}}-672 c^{\frac {7}{2}} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b x -512 c^{\frac {7}{2}} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} a +720 c^{\frac {7}{2}} \sqrt {c \,x^{2}+b x +a}\, a b x +560 c^{\frac {5}{2}} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} b^{2}-420 c^{\frac {5}{2}} \sqrt {c \,x^{2}+b x +a}\, b^{3} x +360 c^{\frac {5}{2}} \sqrt {c \,x^{2}+b x +a}\, a \,b^{2}-210 c^{\frac {3}{2}} \sqrt {c \,x^{2}+b x +a}\, b^{4}+720 \ln \left (\frac {2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b}{2 \sqrt {c}}\right ) a^{2} b \,c^{3}-600 \ln \left (\frac {2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b}{2 \sqrt {c}}\right ) a \,b^{3} c^{2}+105 \ln \left (\frac {2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {c}+2 c x +b}{2 \sqrt {c}}\right ) b^{5} c \right )}{3840 x \sqrt {c \,x^{2}+b x +a}\, c^{\frac {11}{2}}}\) | \(310\) |
-2/15/c^(9/2)*((-45/32*a^2*b*c^2+75/64*a*b^3*c-105/512*b^5)*ln(2*(c*x^2+b* x+a)^(1/2)*c^(1/2)+2*c*x+b)+(c*x^2+b*x+a)^(1/2)*((7/32*b^2*x^2+29/32*a*b*x +a^2)*c^(5/2)-115/64*(7/46*b*x+a)*b^2*c^(3/2)-1/2*(3/8*b*x+a)*x^2*c^(7/2)- 3/2*c^(9/2)*x^4+105/256*c^(1/2)*b^4))
Time = 0.29 (sec) , antiderivative size = 390, normalized size of antiderivative = 1.52 \[ \int x^2 \sqrt {a x^2+b x^3+c x^4} \, dx=\left [\frac {15 \, {\left (7 \, b^{5} - 40 \, a b^{3} c + 48 \, a^{2} b c^{2}\right )} \sqrt {c} x \log \left (-\frac {8 \, c^{2} x^{3} + 8 \, b c x^{2} + 4 \, \sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {c} + {\left (b^{2} + 4 \, a c\right )} x}{x}\right ) + 4 \, {\left (384 \, c^{5} x^{4} + 48 \, b c^{4} x^{3} - 105 \, b^{4} c + 460 \, a b^{2} c^{2} - 256 \, a^{2} c^{3} - 8 \, {\left (7 \, b^{2} c^{3} - 16 \, a c^{4}\right )} x^{2} + 2 \, {\left (35 \, b^{3} c^{2} - 116 \, a b c^{3}\right )} x\right )} \sqrt {c x^{4} + b x^{3} + a x^{2}}}{7680 \, c^{5} x}, -\frac {15 \, {\left (7 \, b^{5} - 40 \, a b^{3} c + 48 \, a^{2} b c^{2}\right )} \sqrt {-c} x \arctan \left (\frac {\sqrt {c x^{4} + b x^{3} + a x^{2}} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{3} + b c x^{2} + a c x\right )}}\right ) - 2 \, {\left (384 \, c^{5} x^{4} + 48 \, b c^{4} x^{3} - 105 \, b^{4} c + 460 \, a b^{2} c^{2} - 256 \, a^{2} c^{3} - 8 \, {\left (7 \, b^{2} c^{3} - 16 \, a c^{4}\right )} x^{2} + 2 \, {\left (35 \, b^{3} c^{2} - 116 \, a b c^{3}\right )} x\right )} \sqrt {c x^{4} + b x^{3} + a x^{2}}}{3840 \, c^{5} x}\right ] \]
[1/7680*(15*(7*b^5 - 40*a*b^3*c + 48*a^2*b*c^2)*sqrt(c)*x*log(-(8*c^2*x^3 + 8*b*c*x^2 + 4*sqrt(c*x^4 + b*x^3 + a*x^2)*(2*c*x + b)*sqrt(c) + (b^2 + 4 *a*c)*x)/x) + 4*(384*c^5*x^4 + 48*b*c^4*x^3 - 105*b^4*c + 460*a*b^2*c^2 - 256*a^2*c^3 - 8*(7*b^2*c^3 - 16*a*c^4)*x^2 + 2*(35*b^3*c^2 - 116*a*b*c^3)* x)*sqrt(c*x^4 + b*x^3 + a*x^2))/(c^5*x), -1/3840*(15*(7*b^5 - 40*a*b^3*c + 48*a^2*b*c^2)*sqrt(-c)*x*arctan(1/2*sqrt(c*x^4 + b*x^3 + a*x^2)*(2*c*x + b)*sqrt(-c)/(c^2*x^3 + b*c*x^2 + a*c*x)) - 2*(384*c^5*x^4 + 48*b*c^4*x^3 - 105*b^4*c + 460*a*b^2*c^2 - 256*a^2*c^3 - 8*(7*b^2*c^3 - 16*a*c^4)*x^2 + 2*(35*b^3*c^2 - 116*a*b*c^3)*x)*sqrt(c*x^4 + b*x^3 + a*x^2))/(c^5*x)]
\[ \int x^2 \sqrt {a x^2+b x^3+c x^4} \, dx=\int x^{2} \sqrt {x^{2} \left (a + b x + c x^{2}\right )}\, dx \]
\[ \int x^2 \sqrt {a x^2+b x^3+c x^4} \, dx=\int { \sqrt {c x^{4} + b x^{3} + a x^{2}} x^{2} \,d x } \]
Time = 0.34 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.07 \[ \int x^2 \sqrt {a x^2+b x^3+c x^4} \, dx=\frac {1}{1920} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (6 \, {\left (8 \, x \mathrm {sgn}\left (x\right ) + \frac {b \mathrm {sgn}\left (x\right )}{c}\right )} x - \frac {7 \, b^{2} c^{2} \mathrm {sgn}\left (x\right ) - 16 \, a c^{3} \mathrm {sgn}\left (x\right )}{c^{4}}\right )} x + \frac {35 \, b^{3} c \mathrm {sgn}\left (x\right ) - 116 \, a b c^{2} \mathrm {sgn}\left (x\right )}{c^{4}}\right )} x - \frac {105 \, b^{4} \mathrm {sgn}\left (x\right ) - 460 \, a b^{2} c \mathrm {sgn}\left (x\right ) + 256 \, a^{2} c^{2} \mathrm {sgn}\left (x\right )}{c^{4}}\right )} - \frac {{\left (7 \, b^{5} \mathrm {sgn}\left (x\right ) - 40 \, a b^{3} c \mathrm {sgn}\left (x\right ) + 48 \, a^{2} b c^{2} \mathrm {sgn}\left (x\right )\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{256 \, c^{\frac {9}{2}}} + \frac {{\left (105 \, b^{5} \log \left ({\left | b - 2 \, \sqrt {a} \sqrt {c} \right |}\right ) - 600 \, a b^{3} c \log \left ({\left | b - 2 \, \sqrt {a} \sqrt {c} \right |}\right ) + 720 \, a^{2} b c^{2} \log \left ({\left | b - 2 \, \sqrt {a} \sqrt {c} \right |}\right ) + 210 \, \sqrt {a} b^{4} \sqrt {c} - 920 \, a^{\frac {3}{2}} b^{2} c^{\frac {3}{2}} + 512 \, a^{\frac {5}{2}} c^{\frac {5}{2}}\right )} \mathrm {sgn}\left (x\right )}{3840 \, c^{\frac {9}{2}}} \]
1/1920*sqrt(c*x^2 + b*x + a)*(2*(4*(6*(8*x*sgn(x) + b*sgn(x)/c)*x - (7*b^2 *c^2*sgn(x) - 16*a*c^3*sgn(x))/c^4)*x + (35*b^3*c*sgn(x) - 116*a*b*c^2*sgn (x))/c^4)*x - (105*b^4*sgn(x) - 460*a*b^2*c*sgn(x) + 256*a^2*c^2*sgn(x))/c ^4) - 1/256*(7*b^5*sgn(x) - 40*a*b^3*c*sgn(x) + 48*a^2*b*c^2*sgn(x))*log(a bs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) + b))/c^(9/2) + 1/3840*(1 05*b^5*log(abs(b - 2*sqrt(a)*sqrt(c))) - 600*a*b^3*c*log(abs(b - 2*sqrt(a) *sqrt(c))) + 720*a^2*b*c^2*log(abs(b - 2*sqrt(a)*sqrt(c))) + 210*sqrt(a)*b ^4*sqrt(c) - 920*a^(3/2)*b^2*c^(3/2) + 512*a^(5/2)*c^(5/2))*sgn(x)/c^(9/2)
Timed out. \[ \int x^2 \sqrt {a x^2+b x^3+c x^4} \, dx=\int x^2\,\sqrt {c\,x^4+b\,x^3+a\,x^2} \,d x \]