Integrand size = 24, antiderivative size = 271 \[ \int \frac {1}{x \left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx=\frac {2 \left (b^2-2 a c+b c x\right )}{a \left (b^2-4 a c\right ) x^2 \sqrt {a x^2+b x^3+c x^4}}-\frac {\left (7 b^2-16 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{3 a^2 \left (b^2-4 a c\right ) x^4}+\frac {b \left (35 b^2-116 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{12 a^3 \left (b^2-4 a c\right ) x^3}-\frac {\left (105 b^4-460 a b^2 c+256 a^2 c^2\right ) \sqrt {a x^2+b x^3+c x^4}}{24 a^4 \left (b^2-4 a c\right ) x^2}+\frac {5 b \left (7 b^2-12 a c\right ) \text {arctanh}\left (\frac {x (2 a+b x)}{2 \sqrt {a} \sqrt {a x^2+b x^3+c x^4}}\right )}{16 a^{9/2}} \] Output:
2*(b*c*x-2*a*c+b^2)/a/(-4*a*c+b^2)/x^2/(c*x^4+b*x^3+a*x^2)^(1/2)-1/3*(-16* a*c+7*b^2)*(c*x^4+b*x^3+a*x^2)^(1/2)/a^2/(-4*a*c+b^2)/x^4+1/12*b*(-116*a*c +35*b^2)*(c*x^4+b*x^3+a*x^2)^(1/2)/a^3/(-4*a*c+b^2)/x^3-1/24*(256*a^2*c^2- 460*a*b^2*c+105*b^4)*(c*x^4+b*x^3+a*x^2)^(1/2)/a^4/(-4*a*c+b^2)/x^2+5/16*b *(-12*a*c+7*b^2)*arctanh(1/2*x*(b*x+2*a)/a^(1/2)/(c*x^4+b*x^3+a*x^2)^(1/2) )/a^(9/2)
Time = 1.61 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.83 \[ \int \frac {1}{x \left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx=\frac {\sqrt {a} \left (-32 a^4 c+105 b^4 x^3 (b+c x)+5 a b^2 x^2 \left (7 b^2-106 b c x-92 c^2 x^2\right )+8 a^3 \left (b^2+7 b c x+16 c^2 x^2\right )+2 a^2 x \left (-7 b^3-86 b^2 c x+244 b c^2 x^2+128 c^3 x^3\right )\right )+15 b \left (7 b^4-40 a b^2 c+48 a^2 c^2\right ) x^3 \sqrt {a+x (b+c x)} \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{24 a^{9/2} \left (-b^2+4 a c\right ) x^2 \sqrt {x^2 (a+x (b+c x))}} \] Input:
Integrate[1/(x*(a*x^2 + b*x^3 + c*x^4)^(3/2)),x]
Output:
(Sqrt[a]*(-32*a^4*c + 105*b^4*x^3*(b + c*x) + 5*a*b^2*x^2*(7*b^2 - 106*b*c *x - 92*c^2*x^2) + 8*a^3*(b^2 + 7*b*c*x + 16*c^2*x^2) + 2*a^2*x*(-7*b^3 - 86*b^2*c*x + 244*b*c^2*x^2 + 128*c^3*x^3)) + 15*b*(7*b^4 - 40*a*b^2*c + 48 *a^2*c^2)*x^3*Sqrt[a + x*(b + c*x)]*ArcTanh[(Sqrt[c]*x - Sqrt[a + x*(b + c *x)])/Sqrt[a]])/(24*a^(9/2)*(-b^2 + 4*a*c)*x^2*Sqrt[x^2*(a + x*(b + c*x))] )
Time = 0.59 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.03, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {1971, 27, 1998, 27, 1998, 27, 1998, 27, 1951, 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.
| \(\displaystyle \int \frac {1}{x \left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1971 |
| \(\displaystyle \frac {2 \left (-2 a c+b^2+b c x\right )}{a x^2 \left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}-\frac {2 \int -\frac {7 b^2+6 c x b-16 a c}{2 x^3 \sqrt {c x^4+b x^3+a x^2}}dx}{a \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {7 b^2+6 c x b-16 a c}{x^3 \sqrt {c x^4+b x^3+a x^2}}dx}{a \left (b^2-4 a c\right )}+\frac {2 \left (-2 a c+b^2+b c x\right )}{a x^2 \left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}\) |
| \(\Big \downarrow \) 1998 |
| \(\displaystyle \frac {-\frac {\int \frac {b \left (35 b^2-116 a c\right )+4 c \left (7 b^2-16 a c\right ) x}{2 x^2 \sqrt {c x^4+b x^3+a x^2}}dx}{3 a}-\frac {\left (7 b^2-16 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{3 a x^4}}{a \left (b^2-4 a c\right )}+\frac {2 \left (-2 a c+b^2+b c x\right )}{a x^2 \left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\int \frac {b \left (35 b^2-116 a c\right )+4 c \left (7 b^2-16 a c\right ) x}{x^2 \sqrt {c x^4+b x^3+a x^2}}dx}{6 a}-\frac {\left (7 b^2-16 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{3 a x^4}}{a \left (b^2-4 a c\right )}+\frac {2 \left (-2 a c+b^2+b c x\right )}{a x^2 \left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}\) |
| \(\Big \downarrow \) 1998 |
| \(\displaystyle \frac {-\frac {-\frac {\int \frac {105 b^4-460 a c b^2+2 c \left (35 b^2-116 a c\right ) x b+256 a^2 c^2}{2 x \sqrt {c x^4+b x^3+a x^2}}dx}{2 a}-\frac {b \left (35 b^2-116 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{2 a x^3}}{6 a}-\frac {\left (7 b^2-16 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{3 a x^4}}{a \left (b^2-4 a c\right )}+\frac {2 \left (-2 a c+b^2+b c x\right )}{a x^2 \left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {-\frac {\int \frac {105 b^4-460 a c b^2+2 c \left (35 b^2-116 a c\right ) x b+256 a^2 c^2}{x \sqrt {c x^4+b x^3+a x^2}}dx}{4 a}-\frac {b \left (35 b^2-116 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{2 a x^3}}{6 a}-\frac {\left (7 b^2-16 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{3 a x^4}}{a \left (b^2-4 a c\right )}+\frac {2 \left (-2 a c+b^2+b c x\right )}{a x^2 \left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}\) |
| \(\Big \downarrow \) 1998 |
| \(\displaystyle \frac {-\frac {-\frac {-\frac {\int \frac {15 b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right )}{2 \sqrt {c x^4+b x^3+a x^2}}dx}{a}-\frac {\left (256 a^2 c^2-460 a b^2 c+105 b^4\right ) \sqrt {a x^2+b x^3+c x^4}}{a x^2}}{4 a}-\frac {b \left (35 b^2-116 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{2 a x^3}}{6 a}-\frac {\left (7 b^2-16 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{3 a x^4}}{a \left (b^2-4 a c\right )}+\frac {2 \left (-2 a c+b^2+b c x\right )}{a x^2 \left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {-\frac {-\frac {15 b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right ) \int \frac {1}{\sqrt {c x^4+b x^3+a x^2}}dx}{2 a}-\frac {\left (256 a^2 c^2-460 a b^2 c+105 b^4\right ) \sqrt {a x^2+b x^3+c x^4}}{a x^2}}{4 a}-\frac {b \left (35 b^2-116 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{2 a x^3}}{6 a}-\frac {\left (7 b^2-16 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{3 a x^4}}{a \left (b^2-4 a c\right )}+\frac {2 \left (-2 a c+b^2+b c x\right )}{a x^2 \left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}\) |
| \(\Big \downarrow \) 1951 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {15 b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right ) \int \frac {1}{4 a-\frac {x^2 (2 a+b x)^2}{c x^4+b x^3+a x^2}}d\frac {x (2 a+b x)}{\sqrt {c x^4+b x^3+a x^2}}}{a}-\frac {\left (256 a^2 c^2-460 a b^2 c+105 b^4\right ) \sqrt {a x^2+b x^3+c x^4}}{a x^2}}{4 a}-\frac {b \left (35 b^2-116 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{2 a x^3}}{6 a}-\frac {\left (7 b^2-16 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{3 a x^4}}{a \left (b^2-4 a c\right )}+\frac {2 \left (-2 a c+b^2+b c x\right )}{a x^2 \left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {15 b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right ) \text {arctanh}\left (\frac {x (2 a+b x)}{2 \sqrt {a} \sqrt {a x^2+b x^3+c x^4}}\right )}{2 a^{3/2}}-\frac {\left (256 a^2 c^2-460 a b^2 c+105 b^4\right ) \sqrt {a x^2+b x^3+c x^4}}{a x^2}}{4 a}-\frac {b \left (35 b^2-116 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{2 a x^3}}{6 a}-\frac {\left (7 b^2-16 a c\right ) \sqrt {a x^2+b x^3+c x^4}}{3 a x^4}}{a \left (b^2-4 a c\right )}+\frac {2 \left (-2 a c+b^2+b c x\right )}{a x^2 \left (b^2-4 a c\right ) \sqrt {a x^2+b x^3+c x^4}}\) |
Input:
Int[1/(x*(a*x^2 + b*x^3 + c*x^4)^(3/2)),x]
Output:
(2*(b^2 - 2*a*c + b*c*x))/(a*(b^2 - 4*a*c)*x^2*Sqrt[a*x^2 + b*x^3 + c*x^4] ) + (-1/3*((7*b^2 - 16*a*c)*Sqrt[a*x^2 + b*x^3 + c*x^4])/(a*x^4) - (-1/2*( b*(35*b^2 - 116*a*c)*Sqrt[a*x^2 + b*x^3 + c*x^4])/(a*x^3) - (-(((105*b^4 - 460*a*b^2*c + 256*a^2*c^2)*Sqrt[a*x^2 + b*x^3 + c*x^4])/(a*x^2)) + (15*b* (7*b^2 - 12*a*c)*(b^2 - 4*a*c)*ArcTanh[(x*(2*a + b*x))/(2*Sqrt[a]*Sqrt[a*x ^2 + b*x^3 + c*x^4])])/(2*a^(3/2)))/(4*a))/(6*a))/(a*(b^2 - 4*a*c))
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_.)*(x_)^2 + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(r_.)], x_Symbol] : > Simp[-2/(n - 2) Subst[Int[1/(4*a - x^2), x], x, x*((2*a + b*x^(n - 2))/ Sqrt[a*x^2 + b*x^n + c*x^r])], x] /; FreeQ[{a, b, c, n, r}, x] && EqQ[r, 2* n - 2] && PosQ[n - 2] && NeQ[b^2 - 4*a*c, 0]
Int[(x_)^(m_.)*((b_.)*(x_)^(n_.) + (a_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(p_ ), x_Symbol] :> Simp[(-x^(m - q + 1))*(b^2 - 2*a*c + b*c*x^(n - q))*((a*x^q + b*x^n + c*x^(2*n - q))^(p + 1)/(a*(n - q)*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(a*(n - q)*(p + 1)*(b^2 - 4*a*c)) Int[x^(m - q)*(b^2*(m + p*q + (n - q)*(p + 1) + 1) - 2*a*c*(m + p*q + 2*(n - q)*(p + 1) + 1) + b*c*(m + p*q + (n - q)*(2*p + 3) + 1)*x^(n - q))*(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] && !Int egerQ[p] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && LtQ[p, -1] && RationalQ[m, q] && LtQ[m + p*q + 1, n - q]
Int[(x_)^(m_.)*((c_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.) + (a_.)*(x_)^(q_.))^(p_ .)*((A_) + (B_.)*(x_)^(r_.)), x_Symbol] :> Simp[A*x^(m - q + 1)*((a*x^q + b *x^n + c*x^(2*n - q))^(p + 1)/(a*(m + p*q + 1))), x] + Simp[1/(a*(m + p*q + 1)) Int[x^(m + n - q)*Simp[a*B*(m + p*q + 1) - A*b*(m + p*q + (n - q)*(p + 1) + 1) - A*c*(m + p*q + 2*(n - q)*(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] && !IntegerQ[p] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && RationalQ[m, p, q] && ((GeQ[p, -1] && LtQ[p, 0]) || EqQ[m + p*q + (n - q)*(2*p + 1) + 1, 0]) && LeQ[m + p*q, -(n - q)] && NeQ[m + p*q + 1, 0]
Time = 0.30 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.79
| method | result | size |
| pseudoelliptic | \(-\frac {15 \left (-\frac {7 \left (-\frac {92}{7} c^{2} x^{2}-\frac {106}{7} b c x +b^{2}\right ) b^{2} x^{2} a^{\frac {3}{2}}}{72}+\frac {7 \left (-\frac {128}{7} c^{3} x^{3}-\frac {244}{7} b \,c^{2} x^{2}+\frac {86}{7} b^{2} c x +b^{3}\right ) x \,a^{\frac {5}{2}}}{180}+\frac {\left (-16 c^{2} x^{2}-7 b c x -b^{2}\right ) a^{\frac {7}{2}}}{45}+\frac {4 a^{\frac {9}{2}} c}{45}+b \left (-\frac {7 b^{3} \left (c x +b \right ) \sqrt {a}}{24}+\sqrt {c \,x^{2}+b x +a}\, \left (-\ln \left (2\right )+\ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x \sqrt {a}}\right )\right ) \left (a c -\frac {7 b^{2}}{12}\right ) \left (a c -\frac {b^{2}}{4}\right )\right ) x^{3}\right )}{\sqrt {c \,x^{2}+b x +a}\, a^{\frac {9}{2}} \left (4 a c -b^{2}\right ) x^{3}}\) | \(213\) |
| default | \(-\frac {\left (c \,x^{2}+b x +a \right ) \left (-512 a^{\frac {7}{2}} c^{3} x^{4}+920 a^{\frac {5}{2}} b^{2} c^{2} x^{4}-210 a^{\frac {3}{2}} b^{4} c \,x^{4}-976 a^{\frac {7}{2}} b \,c^{2} x^{3}+1060 a^{\frac {5}{2}} b^{3} c \,x^{3}-210 a^{\frac {3}{2}} b^{5} x^{3}+720 \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) \sqrt {c \,x^{2}+b x +a}\, a^{3} b \,c^{2} x^{3}-600 \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) \sqrt {c \,x^{2}+b x +a}\, a^{2} b^{3} c \,x^{3}+105 \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) \sqrt {c \,x^{2}+b x +a}\, a \,b^{5} x^{3}-256 a^{\frac {9}{2}} c^{2} x^{2}+344 a^{\frac {7}{2}} b^{2} c \,x^{2}-70 a^{\frac {5}{2}} b^{4} x^{2}-112 a^{\frac {9}{2}} b c x +28 a^{\frac {7}{2}} b^{3} x +64 a^{\frac {11}{2}} c -16 a^{\frac {9}{2}} b^{2}\right )}{48 \left (c \,x^{4}+b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}} a^{\frac {11}{2}} \left (4 a c -b^{2}\right )}\) | \(340\) |
| risch | \(-\frac {\left (c \,x^{2}+b x +a \right ) \left (-40 a c \,x^{2}+57 b^{2} x^{2}-22 a b x +8 a^{2}\right )}{24 a^{4} x^{2} \sqrt {x^{2} \left (c \,x^{2}+b x +a \right )}}+\frac {\left (\frac {2 b^{4} c x}{a^{4} \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\frac {b^{5}}{a^{4} \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\frac {4 c^{3} x}{a^{2} \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\frac {2 c^{2} b}{a^{2} \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {b^{3}}{a^{4} \sqrt {c \,x^{2}+b x +a}}-\frac {8 b^{2} c^{2} x}{a^{3} \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {4 b^{3} c}{a^{3} \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}+\frac {2 b c}{a^{3} \sqrt {c \,x^{2}+b x +a}}-\frac {15 b \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right ) c}{4 a^{\frac {7}{2}}}+\frac {35 b^{3} \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{16 a^{\frac {9}{2}}}\right ) x \sqrt {c \,x^{2}+b x +a}}{\sqrt {x^{2} \left (c \,x^{2}+b x +a \right )}}\) | \(405\) |
Input:
int(1/x/(c*x^4+b*x^3+a*x^2)^(3/2),x,method=_RETURNVERBOSE)
Output:
-15/(c*x^2+b*x+a)^(1/2)/a^(9/2)*(-7/72*(-92/7*c^2*x^2-106/7*b*c*x+b^2)*b^2 *x^2*a^(3/2)+7/180*(-128/7*c^3*x^3-244/7*b*c^2*x^2+86/7*b^2*c*x+b^3)*x*a^( 5/2)+1/45*(-16*c^2*x^2-7*b*c*x-b^2)*a^(7/2)+4/45*a^(9/2)*c+b*(-7/24*b^3*(c *x+b)*a^(1/2)+(c*x^2+b*x+a)^(1/2)*(-ln(2)+ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x +a)^(1/2))/x/a^(1/2)))*(a*c-7/12*b^2)*(a*c-1/4*b^2))*x^3)/(4*a*c-b^2)/x^3
Time = 0.28 (sec) , antiderivative size = 716, normalized size of antiderivative = 2.64 \[ \int \frac {1}{x \left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
integrate(1/x/(c*x^4+b*x^3+a*x^2)^(3/2),x, algorithm="fricas")
Output:
[-1/96*(15*((7*b^5*c - 40*a*b^3*c^2 + 48*a^2*b*c^3)*x^6 + (7*b^6 - 40*a*b^ 4*c + 48*a^2*b^2*c^2)*x^5 + (7*a*b^5 - 40*a^2*b^3*c + 48*a^3*b*c^2)*x^4)*s qrt(a)*log(-(8*a*b*x^2 + (b^2 + 4*a*c)*x^3 + 8*a^2*x - 4*sqrt(c*x^4 + b*x^ 3 + a*x^2)*(b*x + 2*a)*sqrt(a))/x^3) + 4*(8*a^4*b^2 - 32*a^5*c + (105*a*b^ 4*c - 460*a^2*b^2*c^2 + 256*a^3*c^3)*x^4 + (105*a*b^5 - 530*a^2*b^3*c + 48 8*a^3*b*c^2)*x^3 + (35*a^2*b^4 - 172*a^3*b^2*c + 128*a^4*c^2)*x^2 - 14*(a^ 3*b^3 - 4*a^4*b*c)*x)*sqrt(c*x^4 + b*x^3 + a*x^2))/((a^5*b^2*c - 4*a^6*c^2 )*x^6 + (a^5*b^3 - 4*a^6*b*c)*x^5 + (a^6*b^2 - 4*a^7*c)*x^4), -1/48*(15*(( 7*b^5*c - 40*a*b^3*c^2 + 48*a^2*b*c^3)*x^6 + (7*b^6 - 40*a*b^4*c + 48*a^2* b^2*c^2)*x^5 + (7*a*b^5 - 40*a^2*b^3*c + 48*a^3*b*c^2)*x^4)*sqrt(-a)*arcta n(1/2*sqrt(c*x^4 + b*x^3 + a*x^2)*(b*x + 2*a)*sqrt(-a)/(a*c*x^3 + a*b*x^2 + a^2*x)) + 2*(8*a^4*b^2 - 32*a^5*c + (105*a*b^4*c - 460*a^2*b^2*c^2 + 256 *a^3*c^3)*x^4 + (105*a*b^5 - 530*a^2*b^3*c + 488*a^3*b*c^2)*x^3 + (35*a^2* b^4 - 172*a^3*b^2*c + 128*a^4*c^2)*x^2 - 14*(a^3*b^3 - 4*a^4*b*c)*x)*sqrt( c*x^4 + b*x^3 + a*x^2))/((a^5*b^2*c - 4*a^6*c^2)*x^6 + (a^5*b^3 - 4*a^6*b* c)*x^5 + (a^6*b^2 - 4*a^7*c)*x^4)]
\[ \int \frac {1}{x \left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx=\int \frac {1}{x \left (x^{2} \left (a + b x + c x^{2}\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(1/x/(c*x**4+b*x**3+a*x**2)**(3/2),x)
Output:
Integral(1/(x*(x**2*(a + b*x + c*x**2))**(3/2)), x)
\[ \int \frac {1}{x \left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (c x^{4} + b x^{3} + a x^{2}\right )}^{\frac {3}{2}} x} \,d x } \] Input:
integrate(1/x/(c*x^4+b*x^3+a*x^2)^(3/2),x, algorithm="maxima")
Output:
integrate(1/((c*x^4 + b*x^3 + a*x^2)^(3/2)*x), x)
Timed out. \[ \int \frac {1}{x \left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate(1/x/(c*x^4+b*x^3+a*x^2)^(3/2),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {1}{x \left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx=\int \frac {1}{x\,{\left (c\,x^4+b\,x^3+a\,x^2\right )}^{3/2}} \,d x \] Input:
int(1/(x*(a*x^2 + b*x^3 + c*x^4)^(3/2)),x)
Output:
int(1/(x*(a*x^2 + b*x^3 + c*x^4)^(3/2)), x)
Time = 0.24 (sec) , antiderivative size = 796, normalized size of antiderivative = 2.94 \[ \int \frac {1}{x \left (a x^2+b x^3+c x^4\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
int(1/x/(c*x^4+b*x^3+a*x^2)^(3/2),x)
Output:
( - 64*sqrt(a + b*x + c*x**2)*a**5*c + 16*sqrt(a + b*x + c*x**2)*a**4*b**2 + 112*sqrt(a + b*x + c*x**2)*a**4*b*c*x + 256*sqrt(a + b*x + c*x**2)*a**4 *c**2*x**2 - 28*sqrt(a + b*x + c*x**2)*a**3*b**3*x - 344*sqrt(a + b*x + c* x**2)*a**3*b**2*c*x**2 + 976*sqrt(a + b*x + c*x**2)*a**3*b*c**2*x**3 + 512 *sqrt(a + b*x + c*x**2)*a**3*c**3*x**4 + 70*sqrt(a + b*x + c*x**2)*a**2*b* *4*x**2 - 1060*sqrt(a + b*x + c*x**2)*a**2*b**3*c*x**3 - 920*sqrt(a + b*x + c*x**2)*a**2*b**2*c**2*x**4 + 210*sqrt(a + b*x + c*x**2)*a*b**5*x**3 + 2 10*sqrt(a + b*x + c*x**2)*a*b**4*c*x**4 + 720*sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b*x)*a**3*b*c**2*x**3 - 600*sqrt(a)*log(2*sqrt(a) *sqrt(a + b*x + c*x**2) - 2*a - b*x)*a**2*b**3*c*x**3 + 720*sqrt(a)*log(2* sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b*x)*a**2*b**2*c**2*x**4 + 720*sqrt (a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b*x)*a**2*b*c**3*x**5 + 1 05*sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b*x)*a*b**5*x**3 - 600*sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b*x)*a*b**4*c*x* *4 - 600*sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b*x)*a*b**3* c**2*x**5 + 105*sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b*x)* b**6*x**4 + 105*sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b*x)* b**5*c*x**5 - 720*sqrt(a)*log(x)*a**3*b*c**2*x**3 + 600*sqrt(a)*log(x)*a** 2*b**3*c*x**3 - 720*sqrt(a)*log(x)*a**2*b**2*c**2*x**4 - 720*sqrt(a)*log(x )*a**2*b*c**3*x**5 - 105*sqrt(a)*log(x)*a*b**5*x**3 + 600*sqrt(a)*log(x...