Integrand size = 16, antiderivative size = 334 \[ \int \frac {1}{x^2 \left (a+b x+c x^2\right )^4} \, dx=-\frac {1}{a^4 x}-\frac {b \left (b^2-3 a c\right )+c \left (b^2-2 a c\right ) x}{3 a^2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}-\frac {b \left (3 b^2-7 a c\right ) \left (b^2-5 a c\right )+c \left (3 b^4-20 a b^2 c+22 a^2 c^2\right ) x}{3 a^3 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}-\frac {b \left (3 b^6-34 a b^4 c+124 a^2 b^2 c^2-134 a^3 c^3\right )+c \left (3 b^6-32 a b^4 c+104 a^2 b^2 c^2-76 a^3 c^3\right ) x}{a^4 \left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}-\frac {4 \left (b^8-14 a b^6 c+70 a^2 b^4 c^2-140 a^3 b^2 c^3+70 a^4 c^4\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a^5 \left (b^2-4 a c\right )^{7/2}}-\frac {4 b \log (x)}{a^5}+\frac {2 b \log \left (a+b x+c x^2\right )}{a^5} \] Output:
-1/a^4/x-1/3*(b*(-3*a*c+b^2)+c*(-2*a*c+b^2)*x)/a^2/(-4*a*c+b^2)/(c*x^2+b*x +a)^3-1/3*(b*(-7*a*c+3*b^2)*(-5*a*c+b^2)+c*(22*a^2*c^2-20*a*b^2*c+3*b^4)*x )/a^3/(-4*a*c+b^2)^2/(c*x^2+b*x+a)^2-(b*(-134*a^3*c^3+124*a^2*b^2*c^2-34*a *b^4*c+3*b^6)+c*(-76*a^3*c^3+104*a^2*b^2*c^2-32*a*b^4*c+3*b^6)*x)/a^4/(-4* a*c+b^2)^3/(c*x^2+b*x+a)-4*(70*a^4*c^4-140*a^3*b^2*c^3+70*a^2*b^4*c^2-14*a *b^6*c+b^8)*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2))/a^5/(-4*a*c+b^2)^(7/2)-4 *b*ln(x)/a^5+2*b*ln(c*x^2+b*x+a)/a^5
Time = 0.56 (sec) , antiderivative size = 329, normalized size of antiderivative = 0.99 \[ \int \frac {1}{x^2 \left (a+b x+c x^2\right )^4} \, dx=\frac {-\frac {3 a}{x}+\frac {a^3 \left (b^3-3 a b c+b^2 c x-2 a c^2 x\right )}{\left (-b^2+4 a c\right ) (a+x (b+c x))^3}-\frac {a^2 \left (3 b^5-22 a b^3 c+35 a^2 b c^2+3 b^4 c x-20 a b^2 c^2 x+22 a^2 c^3 x\right )}{\left (b^2-4 a c\right )^2 (a+x (b+c x))^2}+\frac {3 a \left (-3 b^7+34 a b^5 c-124 a^2 b^3 c^2+134 a^3 b c^3-3 b^6 c x+32 a b^4 c^2 x-104 a^2 b^2 c^3 x+76 a^3 c^4 x\right )}{\left (b^2-4 a c\right )^3 (a+x (b+c x))}-\frac {12 \left (b^8-14 a b^6 c+70 a^2 b^4 c^2-140 a^3 b^2 c^3+70 a^4 c^4\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{7/2}}-12 b \log (x)+6 b \log (a+x (b+c x))}{3 a^5} \] Input:
Integrate[1/(x^2*(a + b*x + c*x^2)^4),x]
Output:
((-3*a)/x + (a^3*(b^3 - 3*a*b*c + b^2*c*x - 2*a*c^2*x))/((-b^2 + 4*a*c)*(a + x*(b + c*x))^3) - (a^2*(3*b^5 - 22*a*b^3*c + 35*a^2*b*c^2 + 3*b^4*c*x - 20*a*b^2*c^2*x + 22*a^2*c^3*x))/((b^2 - 4*a*c)^2*(a + x*(b + c*x))^2) + ( 3*a*(-3*b^7 + 34*a*b^5*c - 124*a^2*b^3*c^2 + 134*a^3*b*c^3 - 3*b^6*c*x + 3 2*a*b^4*c^2*x - 104*a^2*b^2*c^3*x + 76*a^3*c^4*x))/((b^2 - 4*a*c)^3*(a + x *(b + c*x))) - (12*(b^8 - 14*a*b^6*c + 70*a^2*b^4*c^2 - 140*a^3*b^2*c^3 + 70*a^4*c^4)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(7/2) - 12*b*Log[x] + 6*b*Log[a + x*(b + c*x)])/(3*a^5)
Time = 1.33 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.23, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1165, 27, 1235, 27, 1235, 27, 1200, 2009}
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.
| \(\displaystyle \int \frac {1}{x^2 \left (a+b x+c x^2\right )^4} \, dx\) |
| \(\Big \downarrow \) 1165 |
| \(\displaystyle \frac {-2 a c+b^2+b c x}{3 a x \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}-\frac {\int -\frac {2 \left (2 b^2+3 c x b-7 a c\right )}{x^2 \left (c x^2+b x+a\right )^3}dx}{3 a \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \int \frac {2 b^2+3 c x b-7 a c}{x^2 \left (c x^2+b x+a\right )^3}dx}{3 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x}{3 a x \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle \frac {2 \left (\frac {2 \left (7 a^2 c^2-7 a b^2 c+b^4\right )+b c x \left (2 b^2-13 a c\right )}{2 a x \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {\int -\frac {2 \left (\left (3 b^2-7 a c\right ) \left (b^2-5 a c\right )+2 b c \left (2 b^2-13 a c\right ) x\right )}{x^2 \left (c x^2+b x+a\right )^2}dx}{2 a \left (b^2-4 a c\right )}\right )}{3 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x}{3 a x \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (\frac {\int \frac {\left (3 b^2-7 a c\right ) \left (b^2-5 a c\right )+2 b c \left (2 b^2-13 a c\right ) x}{x^2 \left (c x^2+b x+a\right )^2}dx}{a \left (b^2-4 a c\right )}+\frac {2 \left (7 a^2 c^2-7 a b^2 c+b^4\right )+b c x \left (2 b^2-13 a c\right )}{2 a x \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\right )}{3 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x}{3 a x \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle \frac {2 \left (\frac {\frac {-70 a^3 c^3+105 a^2 b^2 c^2+3 b c x \left (29 a^2 c^2-10 a b^2 c+b^4\right )-32 a b^4 c+3 b^6}{a x \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {\int -\frac {6 \left (b^6-11 a c b^4+38 a^2 c^2 b^2+c \left (b^4-10 a c b^2+29 a^2 c^2\right ) x b-35 a^3 c^3\right )}{x^2 \left (c x^2+b x+a\right )}dx}{a \left (b^2-4 a c\right )}}{a \left (b^2-4 a c\right )}+\frac {2 \left (7 a^2 c^2-7 a b^2 c+b^4\right )+b c x \left (2 b^2-13 a c\right )}{2 a x \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\right )}{3 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x}{3 a x \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (\frac {\frac {6 \int \frac {b^6-11 a c b^4+38 a^2 c^2 b^2+c \left (b^4-10 a c b^2+29 a^2 c^2\right ) x b-35 a^3 c^3}{x^2 \left (c x^2+b x+a\right )}dx}{a \left (b^2-4 a c\right )}+\frac {-70 a^3 c^3+105 a^2 b^2 c^2+3 b c x \left (29 a^2 c^2-10 a b^2 c+b^4\right )-32 a b^4 c+3 b^6}{a x \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}}{a \left (b^2-4 a c\right )}+\frac {2 \left (7 a^2 c^2-7 a b^2 c+b^4\right )+b c x \left (2 b^2-13 a c\right )}{2 a x \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\right )}{3 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x}{3 a x \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}\) |
| \(\Big \downarrow \) 1200 |
| \(\displaystyle \frac {2 \left (\frac {\frac {6 \int \left (\frac {b \left (4 a c-b^2\right )^3}{a^2 x}+\frac {b c x \left (b^2-4 a c\right )^3+\left (b^2-5 a c\right ) \left (b^6-8 a c b^4+19 a^2 c^2 b^2-7 a^3 c^3\right )}{a^2 \left (c x^2+b x+a\right )}+\frac {b^6-11 a c b^4+38 a^2 c^2 b^2-35 a^3 c^3}{a x^2}\right )dx}{a \left (b^2-4 a c\right )}+\frac {-70 a^3 c^3+105 a^2 b^2 c^2+3 b c x \left (29 a^2 c^2-10 a b^2 c+b^4\right )-32 a b^4 c+3 b^6}{a x \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}}{a \left (b^2-4 a c\right )}+\frac {2 \left (7 a^2 c^2-7 a b^2 c+b^4\right )+b c x \left (2 b^2-13 a c\right )}{2 a x \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\right )}{3 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x}{3 a x \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \left (\frac {2 \left (7 a^2 c^2-7 a b^2 c+b^4\right )+b c x \left (2 b^2-13 a c\right )}{2 a x \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}+\frac {\frac {-70 a^3 c^3+105 a^2 b^2 c^2+3 b c x \left (29 a^2 c^2-10 a b^2 c+b^4\right )-32 a b^4 c+3 b^6}{a x \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {6 \left (\frac {b \left (b^2-4 a c\right )^3 \log \left (a+b x+c x^2\right )}{2 a^2}-\frac {b \log (x) \left (b^2-4 a c\right )^3}{a^2}-\frac {-35 a^3 c^3+38 a^2 b^2 c^2-11 a b^4 c+b^6}{a x}-\frac {\left (70 a^4 c^4-140 a^3 b^2 c^3+70 a^2 b^4 c^2-14 a b^6 c+b^8\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a^2 \sqrt {b^2-4 a c}}\right )}{a \left (b^2-4 a c\right )}}{a \left (b^2-4 a c\right )}\right )}{3 a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x}{3 a x \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3}\) |
Input:
Int[1/(x^2*(a + b*x + c*x^2)^4),x]
Output:
(b^2 - 2*a*c + b*c*x)/(3*a*(b^2 - 4*a*c)*x*(a + b*x + c*x^2)^3) + (2*((2*( b^4 - 7*a*b^2*c + 7*a^2*c^2) + b*c*(2*b^2 - 13*a*c)*x)/(2*a*(b^2 - 4*a*c)* x*(a + b*x + c*x^2)^2) + ((3*b^6 - 32*a*b^4*c + 105*a^2*b^2*c^2 - 70*a^3*c ^3 + 3*b*c*(b^4 - 10*a*b^2*c + 29*a^2*c^2)*x)/(a*(b^2 - 4*a*c)*x*(a + b*x + c*x^2)) + (6*(-((b^6 - 11*a*b^4*c + 38*a^2*b^2*c^2 - 35*a^3*c^3)/(a*x)) - ((b^8 - 14*a*b^6*c + 70*a^2*b^4*c^2 - 140*a^3*b^2*c^3 + 70*a^4*c^4)*ArcT anh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(a^2*Sqrt[b^2 - 4*a*c]) - (b*(b^2 - 4* a*c)^3*Log[x])/a^2 + (b*(b^2 - 4*a*c)^3*Log[a + b*x + c*x^2])/(2*a^2)))/(a *(b^2 - 4*a*c)))/(a*(b^2 - 4*a*c))))/(3*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[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(d + e*x)^(m + 1)*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e) *x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^ 2))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^m*Simp[b*c*d*e*(2*p - m + 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]
Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_.) + (b_.)* (x_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*((f + g* x)^n/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && In tegersQ[n]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2 *a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^m *(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d* m + b*e*m) - b*d*(3*c*d - b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p] )
Leaf count of result is larger than twice the leaf count of optimal. \(711\) vs. \(2(326)=652\).
Time = 0.84 (sec) , antiderivative size = 712, normalized size of antiderivative = 2.13
| method | result | size |
| default | \(-\frac {1}{a^{4} x}-\frac {4 b \ln \left (x \right )}{a^{5}}-\frac {\frac {\frac {c^{3} a \left (76 a^{3} c^{3}-104 a^{2} b^{2} c^{2}+32 a \,b^{4} c -3 b^{6}\right ) x^{5}}{64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}+\frac {b \,c^{2} a \left (286 a^{3} c^{3}-332 a^{2} b^{2} c^{2}+98 a \,b^{4} c -9 b^{6}\right ) x^{4}}{64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}+\frac {a c \left (544 a^{4} c^{4}+306 a^{3} b^{2} c^{3}-832 a^{2} b^{4} c^{2}+279 a \,b^{6} c -27 b^{8}\right ) x^{3}}{192 a^{3} c^{3}-144 a^{2} b^{2} c^{2}+36 a \,b^{4} c -3 b^{6}}+\frac {b a \left (496 a^{4} c^{4}-397 a^{3} b^{2} c^{3}+30 a^{2} b^{4} c^{2}+20 a \,b^{6} c -3 b^{8}\right ) x^{2}}{64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}+\frac {a^{2} \left (116 a^{4} c^{4}+166 a^{3} b^{2} c^{3}-243 a^{2} b^{4} c^{2}+75 a \,b^{6} c -7 b^{8}\right ) x}{64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}+\frac {\left (590 a^{3} c^{3}-535 a^{2} b^{2} c^{2}+147 a \,b^{4} c -13 b^{6}\right ) a^{3} b}{192 a^{3} c^{3}-144 a^{2} b^{2} c^{2}+36 a \,b^{4} c -3 b^{6}}}{\left (c \,x^{2}+b x +a \right )^{3}}+\frac {\frac {2 \left (-64 a^{3} b \,c^{4}+48 a^{2} b^{3} c^{3}-12 a \,b^{5} c^{2}+b^{7} c \right ) \ln \left (c \,x^{2}+b x +a \right )}{c}+\frac {8 \left (35 a^{4} c^{4}-102 a^{3} b^{2} c^{3}+59 a^{2} b^{4} c^{2}-13 a \,b^{6} c +b^{8}-\frac {\left (-64 a^{3} b \,c^{4}+48 a^{2} b^{3} c^{3}-12 a \,b^{5} c^{2}+b^{7} c \right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}}{a^{5}}\) | \(712\) |
| risch | \(\text {Expression too large to display}\) | \(1183\) |
Input:
int(1/x^2/(c*x^2+b*x+a)^4,x,method=_RETURNVERBOSE)
Output:
-1/a^4/x-4*b*ln(x)/a^5-1/a^5*((c^3*a*(76*a^3*c^3-104*a^2*b^2*c^2+32*a*b^4* c-3*b^6)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x^5+b*c^2*a*(286*a^3*c ^3-332*a^2*b^2*c^2+98*a*b^4*c-9*b^6)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c -b^6)*x^4+1/3*a*c*(544*a^4*c^4+306*a^3*b^2*c^3-832*a^2*b^4*c^2+279*a*b^6*c -27*b^8)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x^3+b*a*(496*a^4*c^4-3 97*a^3*b^2*c^3+30*a^2*b^4*c^2+20*a*b^6*c-3*b^8)/(64*a^3*c^3-48*a^2*b^2*c^2 +12*a*b^4*c-b^6)*x^2+a^2*(116*a^4*c^4+166*a^3*b^2*c^3-243*a^2*b^4*c^2+75*a *b^6*c-7*b^8)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)*x+1/3*(590*a^3*c^ 3-535*a^2*b^2*c^2+147*a*b^4*c-13*b^6)*a^3*b/(64*a^3*c^3-48*a^2*b^2*c^2+12* a*b^4*c-b^6))/(c*x^2+b*x+a)^3+4/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6) *(1/2*(-64*a^3*b*c^4+48*a^2*b^3*c^3-12*a*b^5*c^2+b^7*c)/c*ln(c*x^2+b*x+a)+ 2*(35*a^4*c^4-102*a^3*b^2*c^3+59*a^2*b^4*c^2-13*a*b^6*c+b^8-1/2*(-64*a^3*b *c^4+48*a^2*b^3*c^3-12*a*b^5*c^2+b^7*c)*b/c)/(4*a*c-b^2)^(1/2)*arctan((2*c *x+b)/(4*a*c-b^2)^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 1957 vs. \(2 (326) = 652\).
Time = 1.52 (sec) , antiderivative size = 3934, normalized size of antiderivative = 11.78 \[ \int \frac {1}{x^2 \left (a+b x+c x^2\right )^4} \, dx=\text {Too large to display} \] Input:
integrate(1/x^2/(c*x^2+b*x+a)^4,x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {1}{x^2 \left (a+b x+c x^2\right )^4} \, dx=\text {Timed out} \] Input:
integrate(1/x**2/(c*x**2+b*x+a)**4,x)
Output:
Timed out
Exception generated. \[ \int \frac {1}{x^2 \left (a+b x+c x^2\right )^4} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(1/x^2/(c*x^2+b*x+a)^4,x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for more deta
Time = 0.21 (sec) , antiderivative size = 496, normalized size of antiderivative = 1.49 \[ \int \frac {1}{x^2 \left (a+b x+c x^2\right )^4} \, dx=\frac {4 \, {\left (b^{8} - 14 \, a b^{6} c + 70 \, a^{2} b^{4} c^{2} - 140 \, a^{3} b^{2} c^{3} + 70 \, a^{4} c^{4}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (a^{5} b^{6} - 12 \, a^{6} b^{4} c + 48 \, a^{7} b^{2} c^{2} - 64 \, a^{8} c^{3}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {2 \, b \log \left (c x^{2} + b x + a\right )}{a^{5}} - \frac {4 \, b \log \left ({\left | x \right |}\right )}{a^{5}} - \frac {3 \, a^{4} b^{6} - 36 \, a^{5} b^{4} c + 144 \, a^{6} b^{2} c^{2} - 192 \, a^{7} c^{3} + 12 \, {\left (a b^{6} c^{3} - 11 \, a^{2} b^{4} c^{4} + 38 \, a^{3} b^{2} c^{5} - 35 \, a^{4} c^{6}\right )} x^{6} + 6 \, {\left (6 \, a b^{7} c^{2} - 67 \, a^{2} b^{5} c^{3} + 238 \, a^{3} b^{3} c^{4} - 239 \, a^{4} b c^{5}\right )} x^{5} + 2 \, {\left (18 \, a b^{8} c - 189 \, a^{2} b^{6} c^{2} + 578 \, a^{3} b^{4} c^{3} - 225 \, a^{4} b^{2} c^{4} - 560 \, a^{5} c^{5}\right )} x^{4} + 3 \, {\left (4 \, a b^{9} - 26 \, a^{2} b^{7} c - 54 \, a^{3} b^{5} c^{2} + 621 \, a^{4} b^{3} c^{3} - 880 \, a^{5} b c^{4}\right )} x^{3} + 3 \, {\left (10 \, a^{2} b^{8} - 108 \, a^{3} b^{6} c + 351 \, a^{4} b^{4} c^{2} - 214 \, a^{5} b^{2} c^{3} - 308 \, a^{6} c^{4}\right )} x^{2} + {\left (22 \, a^{3} b^{7} - 255 \, a^{4} b^{5} c + 967 \, a^{5} b^{3} c^{2} - 1166 \, a^{6} b c^{3}\right )} x}{3 \, {\left (c x^{2} + b x + a\right )}^{3} {\left (b^{2} - 4 \, a c\right )}^{3} a^{5} x} \] Input:
integrate(1/x^2/(c*x^2+b*x+a)^4,x, algorithm="giac")
Output:
4*(b^8 - 14*a*b^6*c + 70*a^2*b^4*c^2 - 140*a^3*b^2*c^3 + 70*a^4*c^4)*arcta n((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((a^5*b^6 - 12*a^6*b^4*c + 48*a^7*b^2*c^ 2 - 64*a^8*c^3)*sqrt(-b^2 + 4*a*c)) + 2*b*log(c*x^2 + b*x + a)/a^5 - 4*b*l og(abs(x))/a^5 - 1/3*(3*a^4*b^6 - 36*a^5*b^4*c + 144*a^6*b^2*c^2 - 192*a^7 *c^3 + 12*(a*b^6*c^3 - 11*a^2*b^4*c^4 + 38*a^3*b^2*c^5 - 35*a^4*c^6)*x^6 + 6*(6*a*b^7*c^2 - 67*a^2*b^5*c^3 + 238*a^3*b^3*c^4 - 239*a^4*b*c^5)*x^5 + 2*(18*a*b^8*c - 189*a^2*b^6*c^2 + 578*a^3*b^4*c^3 - 225*a^4*b^2*c^4 - 560* a^5*c^5)*x^4 + 3*(4*a*b^9 - 26*a^2*b^7*c - 54*a^3*b^5*c^2 + 621*a^4*b^3*c^ 3 - 880*a^5*b*c^4)*x^3 + 3*(10*a^2*b^8 - 108*a^3*b^6*c + 351*a^4*b^4*c^2 - 214*a^5*b^2*c^3 - 308*a^6*c^4)*x^2 + (22*a^3*b^7 - 255*a^4*b^5*c + 967*a^ 5*b^3*c^2 - 1166*a^6*b*c^3)*x)/((c*x^2 + b*x + a)^3*(b^2 - 4*a*c)^3*a^5*x)
Time = 11.34 (sec) , antiderivative size = 1856, normalized size of antiderivative = 5.56 \[ \int \frac {1}{x^2 \left (a+b x+c x^2\right )^4} \, dx=\text {Too large to display} \] Input:
int(1/(x^2*(a + b*x + c*x^2)^4),x)
Output:
(2*log(2*a*b^8*(-(4*a*c - b^2)^7)^(1/2) - 2*b^16*x - 2*a*b^15 + 55*a^2*b^1 3*c + 26816*a^8*b*c^7 - 4480*a^8*c^8*x + 2*b^9*x*(-(4*a*c - b^2)^7)^(1/2) - 647*a^3*b^11*c^2 + 4218*a^4*b^9*c^3 - 16443*a^5*b^7*c^4 + 38276*a^6*b^5* c^5 - 49168*a^7*b^3*c^6 + 35*a^5*c^4*(-(4*a*c - b^2)^7)^(1/2) - 25*a^2*b^6 *c*(-(4*a*c - b^2)^7)^(1/2) - 673*a^2*b^12*c^2*x + 4504*a^3*b^10*c^3*x - 1 8159*a^4*b^8*c^4*x + 44282*a^5*b^6*c^5*x - 61208*a^6*b^4*c^6*x + 39136*a^7 *b^2*c^7*x + 56*a*b^14*c*x + 107*a^3*b^4*c^2*(-(4*a*c - b^2)^7)^(1/2) - 16 6*a^4*b^2*c^3*(-(4*a*c - b^2)^7)^(1/2) - 28*a*b^7*c*x*(-(4*a*c - b^2)^7)^( 1/2) + 227*a^4*b*c^4*x*(-(4*a*c - b^2)^7)^(1/2) + 143*a^2*b^5*c^2*x*(-(4*a *c - b^2)^7)^(1/2) - 310*a^3*b^3*c^3*x*(-(4*a*c - b^2)^7)^(1/2))*(b^8*(-(4 *a*c - b^2)^7)^(1/2) - b^15 + 16384*a^7*b*c^7 - 336*a^2*b^11*c^2 + 2240*a^ 3*b^9*c^3 - 8960*a^4*b^7*c^4 + 21504*a^5*b^5*c^5 - 28672*a^6*b^3*c^6 + 70* a^4*c^4*(-(4*a*c - b^2)^7)^(1/2) + 28*a*b^13*c + 70*a^2*b^4*c^2*(-(4*a*c - b^2)^7)^(1/2) - 140*a^3*b^2*c^3*(-(4*a*c - b^2)^7)^(1/2) - 14*a*b^6*c*(-( 4*a*c - b^2)^7)^(1/2)))/(a^5*(4*a*c - b^2)^7) - (4*b*log(x))/a^5 - (2*log( 2*a*b^15 + 2*b^16*x + 2*a*b^8*(-(4*a*c - b^2)^7)^(1/2) - 55*a^2*b^13*c - 2 6816*a^8*b*c^7 + 4480*a^8*c^8*x + 2*b^9*x*(-(4*a*c - b^2)^7)^(1/2) + 647*a ^3*b^11*c^2 - 4218*a^4*b^9*c^3 + 16443*a^5*b^7*c^4 - 38276*a^6*b^5*c^5 + 4 9168*a^7*b^3*c^6 + 35*a^5*c^4*(-(4*a*c - b^2)^7)^(1/2) - 25*a^2*b^6*c*(-(4 *a*c - b^2)^7)^(1/2) + 673*a^2*b^12*c^2*x - 4504*a^3*b^10*c^3*x + 18159...
Time = 0.22 (sec) , antiderivative size = 3927, normalized size of antiderivative = 11.76 \[ \int \frac {1}{x^2 \left (a+b x+c x^2\right )^4} \, dx =\text {Too large to display} \] Input:
int(1/x^2/(c*x^2+b*x+a)^4,x)
Output:
( - 840*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**7*b*c** 4*x + 1680*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**6*b* *3*c**3*x - 2520*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a **6*b**2*c**4*x**2 - 2520*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**6*b*c**5*x**3 - 840*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4* a*c - b**2))*a**5*b**5*c**2*x + 5040*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/s qrt(4*a*c - b**2))*a**5*b**4*c**3*x**2 + 2520*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**5*b**3*c**4*x**3 - 5040*sqrt(4*a*c - b**2)* atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**5*b**2*c**5*x**4 - 2520*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**5*b*c**6*x**5 + 168*sqrt( 4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**4*b**7*c*x - 2520*sq rt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**4*b**6*c**2*x**2 + 2520*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**4*b**5*c **3*x**3 + 9240*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a* *4*b**4*c**4*x**4 + 2520*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**4*b**3*c**5*x**5 - 2520*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt (4*a*c - b**2))*a**4*b**2*c**6*x**6 - 840*sqrt(4*a*c - b**2)*atan((b + 2*c *x)/sqrt(4*a*c - b**2))*a**4*b*c**7*x**7 - 12*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b**9*x + 504*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a**3*b**8*c*x**2 - 2016*sqrt(4*a*c - b**2)*...