Integrand size = 19, antiderivative size = 173 \[ \int \frac {\text {b1}+\text {c1} x}{\left (a+2 b x+c x^2\right )^4} \, dx=-\frac {b \text {b1}-a \text {c1}+(\text {b1} c-b \text {c1}) x}{6 \left (b^2-a c\right ) \left (a+2 b x+c x^2\right )^3}+\frac {5 (\text {b1} c-b \text {c1}) (b+c x)}{24 \left (b^2-a c\right )^2 \left (a+2 b x+c x^2\right )^2}-\frac {5 c (\text {b1} c-b \text {c1}) (b+c x)}{16 \left (b^2-a c\right )^3 \left (a+2 b x+c x^2\right )}+\frac {5 c^2 (\text {b1} c-b \text {c1}) \text {arctanh}\left (\frac {b+c x}{\sqrt {b^2-a c}}\right )}{16 \left (b^2-a c\right )^{7/2}} \]
1/6*(-b*b1+a*c1-(-b*c1+b1*c)*x)/(-a*c+b^2)/(c*x^2+2*b*x+a)^3+5/24*(-b*c1+b 1*c)*(c*x+b)/(-a*c+b^2)^2/(c*x^2+2*b*x+a)^2-5/16*c*(-b*c1+b1*c)*(c*x+b)/(- a*c+b^2)^3/(c*x^2+2*b*x+a)+5/16*c^2*(-b*c1+b1*c)*arctanh((c*x+b)/(-a*c+b^2 )^(1/2))/(-a*c+b^2)^(7/2)
Time = 0.19 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.97 \[ \int \frac {\text {b1}+\text {c1} x}{\left (a+2 b x+c x^2\right )^4} \, dx=\frac {\frac {8 \left (b^2-a c\right )^2 (-b \text {b1}+a \text {c1}-\text {b1} c x+b \text {c1} x)}{(a+x (2 b+c x))^3}-\frac {10 \left (b^2-a c\right ) (-\text {b1} c+b \text {c1}) (b+c x)}{(a+x (2 b+c x))^2}+\frac {15 c (-\text {b1} c+b \text {c1}) (b+c x)}{a+x (2 b+c x)}+\frac {15 c^2 (-\text {b1} c+b \text {c1}) \arctan \left (\frac {b+c x}{\sqrt {-b^2+a c}}\right )}{\sqrt {-b^2+a c}}}{48 \left (b^2-a c\right )^3} \]
((8*(b^2 - a*c)^2*(-(b*b1) + a*c1 - b1*c*x + b*c1*x))/(a + x*(2*b + c*x))^ 3 - (10*(b^2 - a*c)*(-(b1*c) + b*c1)*(b + c*x))/(a + x*(2*b + c*x))^2 + (1 5*c*(-(b1*c) + b*c1)*(b + c*x))/(a + x*(2*b + c*x)) + (15*c^2*(-(b1*c) + b *c1)*ArcTan[(b + c*x)/Sqrt[-b^2 + a*c]])/Sqrt[-b^2 + a*c])/(48*(b^2 - a*c) ^3)
Time = 0.28 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.10, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {1159, 1086, 1086, 1083, 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.
| \(\displaystyle \int \frac {\text {b1}+\text {c1} x}{\left (a+2 b x+c x^2\right )^4} \, dx\) |
| \(\Big \downarrow \) 1159 |
| \(\displaystyle -\frac {5 (\text {b1} c-b \text {c1}) \int \frac {1}{\left (c x^2+2 b x+a\right )^3}dx}{6 \left (b^2-a c\right )}-\frac {-a \text {c1}+x (\text {b1} c-b \text {c1})+b \text {b1}}{6 \left (b^2-a c\right ) \left (a+2 b x+c x^2\right )^3}\) |
| \(\Big \downarrow \) 1086 |
| \(\displaystyle -\frac {5 (\text {b1} c-b \text {c1}) \left (-\frac {3 c \int \frac {1}{\left (c x^2+2 b x+a\right )^2}dx}{4 \left (b^2-a c\right )}-\frac {b+c x}{4 \left (b^2-a c\right ) \left (a+2 b x+c x^2\right )^2}\right )}{6 \left (b^2-a c\right )}-\frac {-a \text {c1}+x (\text {b1} c-b \text {c1})+b \text {b1}}{6 \left (b^2-a c\right ) \left (a+2 b x+c x^2\right )^3}\) |
| \(\Big \downarrow \) 1086 |
| \(\displaystyle -\frac {5 (\text {b1} c-b \text {c1}) \left (-\frac {3 c \left (-\frac {c \int \frac {1}{c x^2+2 b x+a}dx}{2 \left (b^2-a c\right )}-\frac {b+c x}{2 \left (b^2-a c\right ) \left (a+2 b x+c x^2\right )}\right )}{4 \left (b^2-a c\right )}-\frac {b+c x}{4 \left (b^2-a c\right ) \left (a+2 b x+c x^2\right )^2}\right )}{6 \left (b^2-a c\right )}-\frac {-a \text {c1}+x (\text {b1} c-b \text {c1})+b \text {b1}}{6 \left (b^2-a c\right ) \left (a+2 b x+c x^2\right )^3}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle -\frac {5 (\text {b1} c-b \text {c1}) \left (-\frac {3 c \left (\frac {c \int \frac {1}{4 \left (b^2-a c\right )-(2 b+2 c x)^2}d(2 b+2 c x)}{b^2-a c}-\frac {b+c x}{2 \left (b^2-a c\right ) \left (a+2 b x+c x^2\right )}\right )}{4 \left (b^2-a c\right )}-\frac {b+c x}{4 \left (b^2-a c\right ) \left (a+2 b x+c x^2\right )^2}\right )}{6 \left (b^2-a c\right )}-\frac {-a \text {c1}+x (\text {b1} c-b \text {c1})+b \text {b1}}{6 \left (b^2-a c\right ) \left (a+2 b x+c x^2\right )^3}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {5 (\text {b1} c-b \text {c1}) \left (-\frac {3 c \left (\frac {c \text {arctanh}\left (\frac {2 b+2 c x}{2 \sqrt {b^2-a c}}\right )}{2 \left (b^2-a c\right )^{3/2}}-\frac {b+c x}{2 \left (b^2-a c\right ) \left (a+2 b x+c x^2\right )}\right )}{4 \left (b^2-a c\right )}-\frac {b+c x}{4 \left (b^2-a c\right ) \left (a+2 b x+c x^2\right )^2}\right )}{6 \left (b^2-a c\right )}-\frac {-a \text {c1}+x (\text {b1} c-b \text {c1})+b \text {b1}}{6 \left (b^2-a c\right ) \left (a+2 b x+c x^2\right )^3}\) |
-1/6*(b*b1 - a*c1 + (b1*c - b*c1)*x)/((b^2 - a*c)*(a + 2*b*x + c*x^2)^3) - (5*(b1*c - b*c1)*(-1/4*(b + c*x)/((b^2 - a*c)*(a + 2*b*x + c*x^2)^2) - (3 *c*(-1/2*(b + c*x)/((b^2 - a*c)*(a + 2*b*x + c*x^2)) + (c*ArcTanh[(2*b + 2 *c*x)/(2*Sqrt[b^2 - a*c])])/(2*(b^2 - a*c)^(3/2))))/(4*(b^2 - a*c))))/(6*( b^2 - a*c))
3.2.97.3.1 Defintions of rubi rules used
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_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) *((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] - Simp[2*c*((2*p + 3)/((p + 1)*(b^2 - 4*a*c))) Int[(a + b*x + c*x^2)^(p + 1), x], x] /; Fre eQ[{a, b, c}, x] && ILtQ[p, -1]
Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol ] :> Simp[((b*d - 2*a*e + (2*c*d - b*e)*x)/((p + 1)*(b^2 - 4*a*c)))*(a + b* x + c*x^2)^(p + 1), x] - Simp[(2*p + 3)*((2*c*d - b*e)/((p + 1)*(b^2 - 4*a* c))) Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] & & LtQ[p, -1] && NeQ[p, -3/2]
Time = 0.48 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.19
| method | result | size |
| default | \(\frac {\left (-2 b \operatorname {c1} +2 \operatorname {b1} c \right ) x +2 b \operatorname {b1} -2 a \operatorname {c1}}{3 \left (4 a c -4 b^{2}\right ) \left (c \,x^{2}+2 b x +a \right )^{3}}+\frac {5 \left (-2 b \operatorname {c1} +2 \operatorname {b1} c \right ) \left (\frac {2 c x +2 b}{2 \left (4 a c -4 b^{2}\right ) \left (c \,x^{2}+2 b x +a \right )^{2}}+\frac {3 c \left (\frac {2 c x +2 b}{\left (4 a c -4 b^{2}\right ) \left (c \,x^{2}+2 b x +a \right )}+\frac {2 c \arctan \left (\frac {2 c x +2 b}{2 \sqrt {a c -b^{2}}}\right )}{\left (4 a c -4 b^{2}\right ) \sqrt {a c -b^{2}}}\right )}{4 a c -4 b^{2}}\right )}{3 \left (4 a c -4 b^{2}\right )}\) | \(206\) |
| risch | \(\frac {-\frac {5 c^{4} \left (b \operatorname {c1} -\operatorname {b1} c \right ) x^{5}}{16 \left (a^{3} c^{3}-3 b^{2} c^{2} a^{2}+3 a \,b^{4} c -b^{6}\right )}-\frac {25 c^{3} \left (b \operatorname {c1} -\operatorname {b1} c \right ) b \,x^{4}}{16 \left (a^{3} c^{3}-3 b^{2} c^{2} a^{2}+3 a \,b^{4} c -b^{6}\right )}-\frac {5 \left (4 a c +11 b^{2}\right ) c^{2} \left (b \operatorname {c1} -\operatorname {b1} c \right ) x^{3}}{24 \left (a^{3} c^{3}-3 b^{2} c^{2} a^{2}+3 a \,b^{4} c -b^{6}\right )}-\frac {5 b \left (4 a c +b^{2}\right ) c \left (b \operatorname {c1} -\operatorname {b1} c \right ) x^{2}}{8 \left (a^{3} c^{3}-3 b^{2} c^{2} a^{2}+3 a \,b^{4} c -b^{6}\right )}-\frac {\left (11 a^{2} b \,c^{2} \operatorname {c1} -11 a^{2} \operatorname {b1} \,c^{3}+18 a \,b^{3} c \operatorname {c1} -18 a \,b^{2} \operatorname {b1} \,c^{2}-4 b^{5} \operatorname {c1} +4 b^{4} \operatorname {b1} c \right ) x}{16 \left (a^{3} c^{3}-3 b^{2} c^{2} a^{2}+3 a \,b^{4} c -b^{6}\right )}-\frac {8 a^{3} c^{2} \operatorname {c1} +9 a^{2} b^{2} c \operatorname {c1} -33 a^{2} b \operatorname {b1} \,c^{2}-2 a \,b^{4} \operatorname {c1} +26 a \,b^{3} \operatorname {b1} c -8 b^{5} \operatorname {b1}}{48 \left (a^{3} c^{3}-3 b^{2} c^{2} a^{2}+3 a \,b^{4} c -b^{6}\right )}}{\left (c \,x^{2}+2 b x +a \right )^{3}}+\frac {5 c^{2} \ln \left (\left (-a^{3} c^{4}+3 a^{2} b^{2} c^{3}-3 a \,b^{4} c^{2}+b^{6} c \right ) x -\left (-a c +b^{2}\right )^{\frac {7}{2}}-a^{3} b \,c^{3}+3 a^{2} b^{3} c^{2}-3 a \,b^{5} c +b^{7}\right ) b \operatorname {c1}}{32 \left (-a c +b^{2}\right )^{\frac {7}{2}}}-\frac {5 c^{3} \ln \left (\left (-a^{3} c^{4}+3 a^{2} b^{2} c^{3}-3 a \,b^{4} c^{2}+b^{6} c \right ) x -\left (-a c +b^{2}\right )^{\frac {7}{2}}-a^{3} b \,c^{3}+3 a^{2} b^{3} c^{2}-3 a \,b^{5} c +b^{7}\right ) \operatorname {b1}}{32 \left (-a c +b^{2}\right )^{\frac {7}{2}}}-\frac {5 c^{2} \ln \left (\left (a^{3} c^{4}-3 a^{2} b^{2} c^{3}+3 a \,b^{4} c^{2}-b^{6} c \right ) x -\left (-a c +b^{2}\right )^{\frac {7}{2}}+a^{3} b \,c^{3}-3 a^{2} b^{3} c^{2}+3 a \,b^{5} c -b^{7}\right ) b \operatorname {c1}}{32 \left (-a c +b^{2}\right )^{\frac {7}{2}}}+\frac {5 c^{3} \ln \left (\left (a^{3} c^{4}-3 a^{2} b^{2} c^{3}+3 a \,b^{4} c^{2}-b^{6} c \right ) x -\left (-a c +b^{2}\right )^{\frac {7}{2}}+a^{3} b \,c^{3}-3 a^{2} b^{3} c^{2}+3 a \,b^{5} c -b^{7}\right ) \operatorname {b1}}{32 \left (-a c +b^{2}\right )^{\frac {7}{2}}}\) | \(792\) |
1/3*((-2*b*c1+2*b1*c)*x+2*b*b1-2*a*c1)/(4*a*c-4*b^2)/(c*x^2+2*b*x+a)^3+5/3 *(-2*b*c1+2*b1*c)/(4*a*c-4*b^2)*(1/2*(2*c*x+2*b)/(4*a*c-4*b^2)/(c*x^2+2*b* x+a)^2+3*c/(4*a*c-4*b^2)*((2*c*x+2*b)/(4*a*c-4*b^2)/(c*x^2+2*b*x+a)+2*c/(4 *a*c-4*b^2)/(a*c-b^2)^(1/2)*arctan(1/2*(2*c*x+2*b)/(a*c-b^2)^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 962 vs. \(2 (161) = 322\).
Time = 0.30 (sec) , antiderivative size = 1950, normalized size of antiderivative = 11.27 \[ \int \frac {\text {b1}+\text {c1} x}{\left (a+2 b x+c x^2\right )^4} \, dx=\text {Too large to display} \]
[-1/96*(16*b^7*b1 - 68*a*b^5*b1*c + 118*a^2*b^3*b1*c^2 - 66*a^3*b*b1*c^3 + 30*(b^2*b1*c^5 - a*b1*c^6 - (b^3*c^4 - a*b*c^5)*c1)*x^5 + 150*(b^3*b1*c^4 - a*b*b1*c^5 - (b^4*c^3 - a*b^2*c^4)*c1)*x^4 + 20*(11*b^4*b1*c^3 - 7*a*b^ 2*b1*c^4 - 4*a^2*b1*c^5 - (11*b^5*c^2 - 7*a*b^3*c^3 - 4*a^2*b*c^4)*c1)*x^3 + 60*(b^5*b1*c^2 + 3*a*b^3*b1*c^3 - 4*a^2*b*b1*c^4 - (b^6*c + 3*a*b^4*c^2 - 4*a^2*b^2*c^3)*c1)*x^2 - 15*(a^3*b1*c^3 - a^3*b*c^2*c1 + (b1*c^6 - b*c^ 5*c1)*x^6 + 6*(b*b1*c^5 - b^2*c^4*c1)*x^5 + 3*(4*b^2*b1*c^4 + a*b1*c^5 - ( 4*b^3*c^3 + a*b*c^4)*c1)*x^4 + 4*(2*b^3*b1*c^3 + 3*a*b*b1*c^4 - (2*b^4*c^2 + 3*a*b^2*c^3)*c1)*x^3 + 3*(4*a*b^2*b1*c^3 + a^2*b1*c^4 - (4*a*b^3*c^2 + a^2*b*c^3)*c1)*x^2 + 6*(a^2*b*b1*c^3 - a^2*b^2*c^2*c1)*x)*sqrt(b^2 - a*c)* log((c^2*x^2 + 2*b*c*x + 2*b^2 - a*c + 2*sqrt(b^2 - a*c)*(c*x + b))/(c*x^2 + 2*b*x + a)) + 2*(2*a*b^6 - 11*a^2*b^4*c + a^3*b^2*c^2 + 8*a^4*c^3)*c1 - 6*(4*b^6*b1*c - 22*a*b^4*b1*c^2 + 7*a^2*b^2*b1*c^3 + 11*a^3*b1*c^4 - (4*b ^7 - 22*a*b^5*c + 7*a^2*b^3*c^2 + 11*a^3*b*c^3)*c1)*x)/(a^3*b^8 - 4*a^4*b^ 6*c + 6*a^5*b^4*c^2 - 4*a^6*b^2*c^3 + a^7*c^4 + (b^8*c^3 - 4*a*b^6*c^4 + 6 *a^2*b^4*c^5 - 4*a^3*b^2*c^6 + a^4*c^7)*x^6 + 6*(b^9*c^2 - 4*a*b^7*c^3 + 6 *a^2*b^5*c^4 - 4*a^3*b^3*c^5 + a^4*b*c^6)*x^5 + 3*(4*b^10*c - 15*a*b^8*c^2 + 20*a^2*b^6*c^3 - 10*a^3*b^4*c^4 + a^5*c^6)*x^4 + 4*(2*b^11 - 5*a*b^9*c + 10*a^3*b^5*c^3 - 10*a^4*b^3*c^4 + 3*a^5*b*c^5)*x^3 + 3*(4*a*b^10 - 15*a^ 2*b^8*c + 20*a^3*b^6*c^2 - 10*a^4*b^4*c^3 + a^6*c^5)*x^2 + 6*(a^2*b^9 -...
Leaf count of result is larger than twice the leaf count of optimal. 1027 vs. \(2 (158) = 316\).
Time = 1.53 (sec) , antiderivative size = 1027, normalized size of antiderivative = 5.94 \[ \int \frac {\text {b1}+\text {c1} x}{\left (a+2 b x+c x^2\right )^4} \, dx =\text {Too large to display} \]
5*c**2*sqrt(-1/(a*c - b**2)**7)*(b*c1 - b1*c)*log(x + (-5*a**4*c**6*sqrt(- 1/(a*c - b**2)**7)*(b*c1 - b1*c) + 20*a**3*b**2*c**5*sqrt(-1/(a*c - b**2)* *7)*(b*c1 - b1*c) - 30*a**2*b**4*c**4*sqrt(-1/(a*c - b**2)**7)*(b*c1 - b1* c) + 20*a*b**6*c**3*sqrt(-1/(a*c - b**2)**7)*(b*c1 - b1*c) - 5*b**8*c**2*s qrt(-1/(a*c - b**2)**7)*(b*c1 - b1*c) + 5*b**2*c**2*c1 - 5*b*b1*c**3)/(5*b *c**3*c1 - 5*b1*c**4))/32 - 5*c**2*sqrt(-1/(a*c - b**2)**7)*(b*c1 - b1*c)* log(x + (5*a**4*c**6*sqrt(-1/(a*c - b**2)**7)*(b*c1 - b1*c) - 20*a**3*b**2 *c**5*sqrt(-1/(a*c - b**2)**7)*(b*c1 - b1*c) + 30*a**2*b**4*c**4*sqrt(-1/( a*c - b**2)**7)*(b*c1 - b1*c) - 20*a*b**6*c**3*sqrt(-1/(a*c - b**2)**7)*(b *c1 - b1*c) + 5*b**8*c**2*sqrt(-1/(a*c - b**2)**7)*(b*c1 - b1*c) + 5*b**2* c**2*c1 - 5*b*b1*c**3)/(5*b*c**3*c1 - 5*b1*c**4))/32 + (-8*a**3*c**2*c1 - 9*a**2*b**2*c*c1 + 33*a**2*b*b1*c**2 + 2*a*b**4*c1 - 26*a*b**3*b1*c + 8*b* *5*b1 + x**5*(-15*b*c**4*c1 + 15*b1*c**5) + x**4*(-75*b**2*c**3*c1 + 75*b* b1*c**4) + x**3*(-40*a*b*c**3*c1 + 40*a*b1*c**4 - 110*b**3*c**2*c1 + 110*b **2*b1*c**3) + x**2*(-120*a*b**2*c**2*c1 + 120*a*b*b1*c**3 - 30*b**4*c*c1 + 30*b**3*b1*c**2) + x*(-33*a**2*b*c**2*c1 + 33*a**2*b1*c**3 - 54*a*b**3*c *c1 + 54*a*b**2*b1*c**2 + 12*b**5*c1 - 12*b**4*b1*c))/(48*a**6*c**3 - 144* a**5*b**2*c**2 + 144*a**4*b**4*c - 48*a**3*b**6 + x**6*(48*a**3*c**6 - 144 *a**2*b**2*c**5 + 144*a*b**4*c**4 - 48*b**6*c**3) + x**5*(288*a**3*b*c**5 - 864*a**2*b**3*c**4 + 864*a*b**5*c**3 - 288*b**7*c**2) + x**4*(144*a**...
Exception generated. \[ \int \frac {\text {b1}+\text {c1} x}{\left (a+2 b x+c x^2\right )^4} \, dx=\text {Exception raised: ValueError} \]
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*b^2-4*a*c>0)', see `assume?` f or more de
Leaf count of result is larger than twice the leaf count of optimal. 363 vs. \(2 (161) = 322\).
Time = 0.31 (sec) , antiderivative size = 363, normalized size of antiderivative = 2.10 \[ \int \frac {\text {b1}+\text {c1} x}{\left (a+2 b x+c x^2\right )^4} \, dx=-\frac {5 \, {\left (b_{1} c^{3} - b c^{2} c_{1}\right )} \arctan \left (\frac {c x + b}{\sqrt {-b^{2} + a c}}\right )}{16 \, {\left (b^{6} - 3 \, a b^{4} c + 3 \, a^{2} b^{2} c^{2} - a^{3} c^{3}\right )} \sqrt {-b^{2} + a c}} - \frac {15 \, b_{1} c^{5} x^{5} - 15 \, b c^{4} c_{1} x^{5} + 75 \, b b_{1} c^{4} x^{4} - 75 \, b^{2} c^{3} c_{1} x^{4} + 110 \, b^{2} b_{1} c^{3} x^{3} + 40 \, a b_{1} c^{4} x^{3} - 110 \, b^{3} c^{2} c_{1} x^{3} - 40 \, a b c^{3} c_{1} x^{3} + 30 \, b^{3} b_{1} c^{2} x^{2} + 120 \, a b b_{1} c^{3} x^{2} - 30 \, b^{4} c c_{1} x^{2} - 120 \, a b^{2} c^{2} c_{1} x^{2} - 12 \, b^{4} b_{1} c x + 54 \, a b^{2} b_{1} c^{2} x + 33 \, a^{2} b_{1} c^{3} x + 12 \, b^{5} c_{1} x - 54 \, a b^{3} c c_{1} x - 33 \, a^{2} b c^{2} c_{1} x + 8 \, b^{5} b_{1} - 26 \, a b^{3} b_{1} c + 33 \, a^{2} b b_{1} c^{2} + 2 \, a b^{4} c_{1} - 9 \, a^{2} b^{2} c c_{1} - 8 \, a^{3} c^{2} c_{1}}{48 \, {\left (b^{6} - 3 \, a b^{4} c + 3 \, a^{2} b^{2} c^{2} - a^{3} c^{3}\right )} {\left (c x^{2} + 2 \, b x + a\right )}^{3}} \]
-5/16*(b1*c^3 - b*c^2*c1)*arctan((c*x + b)/sqrt(-b^2 + a*c))/((b^6 - 3*a*b ^4*c + 3*a^2*b^2*c^2 - a^3*c^3)*sqrt(-b^2 + a*c)) - 1/48*(15*b1*c^5*x^5 - 15*b*c^4*c1*x^5 + 75*b*b1*c^4*x^4 - 75*b^2*c^3*c1*x^4 + 110*b^2*b1*c^3*x^3 + 40*a*b1*c^4*x^3 - 110*b^3*c^2*c1*x^3 - 40*a*b*c^3*c1*x^3 + 30*b^3*b1*c^ 2*x^2 + 120*a*b*b1*c^3*x^2 - 30*b^4*c*c1*x^2 - 120*a*b^2*c^2*c1*x^2 - 12*b ^4*b1*c*x + 54*a*b^2*b1*c^2*x + 33*a^2*b1*c^3*x + 12*b^5*c1*x - 54*a*b^3*c *c1*x - 33*a^2*b*c^2*c1*x + 8*b^5*b1 - 26*a*b^3*b1*c + 33*a^2*b*b1*c^2 + 2 *a*b^4*c1 - 9*a^2*b^2*c*c1 - 8*a^3*c^2*c1)/((b^6 - 3*a*b^4*c + 3*a^2*b^2*c ^2 - a^3*c^3)*(c*x^2 + 2*b*x + a)^3)
Time = 0.73 (sec) , antiderivative size = 640, normalized size of antiderivative = 3.70 \[ \int \frac {\text {b1}+\text {c1} x}{\left (a+2 b x+c x^2\right )^4} \, dx=\frac {\frac {5\,c^4\,x^5\,\left (b\,c_{1}-b_{1}\,c\right )}{16\,\left (-a^3\,c^3+3\,a^2\,b^2\,c^2-3\,a\,b^4\,c+b^6\right )}-\frac {-8\,c_{1}\,a^3\,c^2-9\,c_{1}\,a^2\,b^2\,c+33\,b_{1}\,a^2\,b\,c^2+2\,c_{1}\,a\,b^4-26\,b_{1}\,a\,b^3\,c+8\,b_{1}\,b^5}{48\,\left (-a^3\,c^3+3\,a^2\,b^2\,c^2-3\,a\,b^4\,c+b^6\right )}+\frac {x\,\left (b\,c_{1}-b_{1}\,c\right )\,\left (11\,a^2\,c^2+18\,a\,b^2\,c-4\,b^4\right )}{16\,\left (-a^3\,c^3+3\,a^2\,b^2\,c^2-3\,a\,b^4\,c+b^6\right )}+\frac {5\,c\,x^3\,\left (11\,b^2\,c+4\,a\,c^2\right )\,\left (b\,c_{1}-b_{1}\,c\right )}{24\,\left (-a^3\,c^3+3\,a^2\,b^2\,c^2-3\,a\,b^4\,c+b^6\right )}+\frac {5\,c\,x^2\,\left (b^3+4\,a\,c\,b\right )\,\left (b\,c_{1}-b_{1}\,c\right )}{8\,\left (-a^3\,c^3+3\,a^2\,b^2\,c^2-3\,a\,b^4\,c+b^6\right )}+\frac {25\,b\,c^3\,x^4\,\left (b\,c_{1}-b_{1}\,c\right )}{16\,\left (-a^3\,c^3+3\,a^2\,b^2\,c^2-3\,a\,b^4\,c+b^6\right )}}{x^3\,\left (8\,b^3+12\,a\,c\,b\right )+x^2\,\left (3\,c\,a^2+12\,a\,b^2\right )+x^4\,\left (12\,b^2\,c+3\,a\,c^2\right )+a^3+c^3\,x^6+6\,b\,c^2\,x^5+6\,a^2\,b\,x}-\frac {5\,c^2\,\mathrm {atan}\left (\frac {16\,\left (\frac {5\,c^3\,x\,\left (b\,c_{1}-b_{1}\,c\right )}{16\,{\left (a\,c-b^2\right )}^{7/2}}+\frac {5\,c^2\,\left (b\,c_{1}-b_{1}\,c\right )\,\left (-32\,a^3\,b\,c^3+96\,a^2\,b^3\,c^2-96\,a\,b^5\,c+32\,b^7\right )}{512\,{\left (a\,c-b^2\right )}^{7/2}\,\left (-a^3\,c^3+3\,a^2\,b^2\,c^2-3\,a\,b^4\,c+b^6\right )}\right )\,\left (-a^3\,c^3+3\,a^2\,b^2\,c^2-3\,a\,b^4\,c+b^6\right )}{5\,b_{1}\,c^3-5\,b\,c^2\,c_{1}}\right )\,\left (b\,c_{1}-b_{1}\,c\right )}{16\,{\left (a\,c-b^2\right )}^{7/2}} \]
((5*c^4*x^5*(b*c1 - b1*c))/(16*(b^6 - a^3*c^3 + 3*a^2*b^2*c^2 - 3*a*b^4*c) ) - (8*b^5*b1 - 8*a^3*c^2*c1 + 2*a*b^4*c1 - 26*a*b^3*b1*c + 33*a^2*b*b1*c^ 2 - 9*a^2*b^2*c*c1)/(48*(b^6 - a^3*c^3 + 3*a^2*b^2*c^2 - 3*a*b^4*c)) + (x* (b*c1 - b1*c)*(11*a^2*c^2 - 4*b^4 + 18*a*b^2*c))/(16*(b^6 - a^3*c^3 + 3*a^ 2*b^2*c^2 - 3*a*b^4*c)) + (5*c*x^3*(4*a*c^2 + 11*b^2*c)*(b*c1 - b1*c))/(24 *(b^6 - a^3*c^3 + 3*a^2*b^2*c^2 - 3*a*b^4*c)) + (5*c*x^2*(b^3 + 4*a*b*c)*( b*c1 - b1*c))/(8*(b^6 - a^3*c^3 + 3*a^2*b^2*c^2 - 3*a*b^4*c)) + (25*b*c^3* x^4*(b*c1 - b1*c))/(16*(b^6 - a^3*c^3 + 3*a^2*b^2*c^2 - 3*a*b^4*c)))/(x^3* (8*b^3 + 12*a*b*c) + x^2*(12*a*b^2 + 3*a^2*c) + x^4*(3*a*c^2 + 12*b^2*c) + a^3 + c^3*x^6 + 6*b*c^2*x^5 + 6*a^2*b*x) - (5*c^2*atan((16*((5*c^3*x*(b*c 1 - b1*c))/(16*(a*c - b^2)^(7/2)) + (5*c^2*(b*c1 - b1*c)*(32*b^7 - 32*a^3* b*c^3 + 96*a^2*b^3*c^2 - 96*a*b^5*c))/(512*(a*c - b^2)^(7/2)*(b^6 - a^3*c^ 3 + 3*a^2*b^2*c^2 - 3*a*b^4*c)))*(b^6 - a^3*c^3 + 3*a^2*b^2*c^2 - 3*a*b^4* c))/(5*b1*c^3 - 5*b*c^2*c1))*(b*c1 - b1*c))/(16*(a*c - b^2)^(7/2))
Time = 0.00 (sec) , antiderivative size = 1706, normalized size of antiderivative = 9.86 \[ \int \frac {\text {b1}+\text {c1} x}{\left (a+2 b x+c x^2\right )^4} \, dx =\text {Too large to display} \]
int((b1 + c1*x)/(a**4 + 8*a**3*b*x + 4*a**3*c*x**2 + 24*a**2*b**2*x**2 + 2 4*a**2*b*c*x**3 + 6*a**2*c**2*x**4 + 32*a*b**3*x**3 + 48*a*b**2*c*x**4 + 2 4*a*b*c**2*x**5 + 4*a*c**3*x**6 + 16*b**4*x**4 + 32*b**3*c*x**5 + 24*b**2* c**2*x**6 + 8*b*c**3*x**7 + c**4*x**8),x)
( - 30*sqrt(a*c - b**2)*atan((b + c*x)/sqrt(a*c - b**2))*a**3*b**2*c**2*c1 + 30*sqrt(a*c - b**2)*atan((b + c*x)/sqrt(a*c - b**2))*a**3*b*b1*c**3 - 1 80*sqrt(a*c - b**2)*atan((b + c*x)/sqrt(a*c - b**2))*a**2*b**3*c**2*c1*x + 180*sqrt(a*c - b**2)*atan((b + c*x)/sqrt(a*c - b**2))*a**2*b**2*b1*c**3*x - 90*sqrt(a*c - b**2)*atan((b + c*x)/sqrt(a*c - b**2))*a**2*b**2*c**3*c1* x**2 + 90*sqrt(a*c - b**2)*atan((b + c*x)/sqrt(a*c - b**2))*a**2*b*b1*c**4 *x**2 - 360*sqrt(a*c - b**2)*atan((b + c*x)/sqrt(a*c - b**2))*a*b**4*c**2* c1*x**2 + 360*sqrt(a*c - b**2)*atan((b + c*x)/sqrt(a*c - b**2))*a*b**3*b1* c**3*x**2 - 360*sqrt(a*c - b**2)*atan((b + c*x)/sqrt(a*c - b**2))*a*b**3*c **3*c1*x**3 + 360*sqrt(a*c - b**2)*atan((b + c*x)/sqrt(a*c - b**2))*a*b**2 *b1*c**4*x**3 - 90*sqrt(a*c - b**2)*atan((b + c*x)/sqrt(a*c - b**2))*a*b** 2*c**4*c1*x**4 + 90*sqrt(a*c - b**2)*atan((b + c*x)/sqrt(a*c - b**2))*a*b* b1*c**5*x**4 - 240*sqrt(a*c - b**2)*atan((b + c*x)/sqrt(a*c - b**2))*b**5* c**2*c1*x**3 + 240*sqrt(a*c - b**2)*atan((b + c*x)/sqrt(a*c - b**2))*b**4* b1*c**3*x**3 - 360*sqrt(a*c - b**2)*atan((b + c*x)/sqrt(a*c - b**2))*b**4* c**3*c1*x**4 + 360*sqrt(a*c - b**2)*atan((b + c*x)/sqrt(a*c - b**2))*b**3* b1*c**4*x**4 - 180*sqrt(a*c - b**2)*atan((b + c*x)/sqrt(a*c - b**2))*b**3* c**4*c1*x**5 + 180*sqrt(a*c - b**2)*atan((b + c*x)/sqrt(a*c - b**2))*b**2* b1*c**5*x**5 - 30*sqrt(a*c - b**2)*atan((b + c*x)/sqrt(a*c - b**2))*b**2*c **5*c1*x**6 + 30*sqrt(a*c - b**2)*atan((b + c*x)/sqrt(a*c - b**2))*b*b1...