Integrand size = 20, antiderivative size = 388 \[ \int \frac {(d+e x)^5}{\left (a+b x+c x^2\right )^3} \, dx=-\frac {e^2 (2 c d-b e) \left (3 c^2 d^2-b^2 e^2-c e (3 b d-7 a e)\right ) x}{c^2 \left (b^2-4 a c\right )^2}-\frac {(d+e x)^4 (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {(d+e x)^2 \left (8 a c e \left (c d^2+2 a e^2\right )-6 b c d \left (c d^2+3 a e^2\right )+b^2 \left (7 c d^2 e-a e^3\right )-(2 c d-b e) \left (6 c^2 d^2-b^2 e^2-2 c e (3 b d-5 a e)\right ) x\right )}{2 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )}-\frac {\left (12 c^5 d^5-b^5 e^5+10 a b^3 c e^5-30 a^2 b c^2 e^5-10 c^4 d^3 e (3 b d-4 a e)+20 c^3 d e^2 \left (b^2 d^2-3 a b d e+3 a^2 e^2\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^3 \left (b^2-4 a c\right )^{5/2}}+\frac {e^5 \log \left (a+b x+c x^2\right )}{2 c^3} \]
-e^2*(-b*e+2*c*d)*(3*c^2*d^2-b^2*e^2-c*e*(-7*a*e+3*b*d))*x/c^2/(-4*a*c+b^2 )^2-1/2*(e*x+d)^4*(b*d-2*a*e+(-b*e+2*c*d)*x)/(-4*a*c+b^2)/(c*x^2+b*x+a)^2- 1/2*(e*x+d)^2*(8*a*c*e*(2*a*e^2+c*d^2)-6*b*c*d*(3*a*e^2+c*d^2)+b^2*(-a*e^3 +7*c*d^2*e)-(-b*e+2*c*d)*(6*c^2*d^2-b^2*e^2-2*c*e*(-5*a*e+3*b*d))*x)/c/(-4 *a*c+b^2)^2/(c*x^2+b*x+a)-(12*c^5*d^5-b^5*e^5+10*a*b^3*c*e^5-30*a^2*b*c^2* e^5-10*c^4*d^3*e*(-4*a*e+3*b*d)+20*c^3*d*e^2*(3*a^2*e^2-3*a*b*d*e+b^2*d^2) )*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2))/c^3/(-4*a*c+b^2)^(5/2)+1/2*e^5*ln( c*x^2+b*x+a)/c^3
Time = 0.56 (sec) , antiderivative size = 628, normalized size of antiderivative = 1.62 \[ \int \frac {(d+e x)^5}{\left (a+b x+c x^2\right )^3} \, dx=\frac {\frac {b^5 e^5 x+b^4 e^4 (a e-5 c d x)-5 b^3 c e^3 \left (-2 c d^2 x+a e (d+e x)\right )-2 b^2 c e^2 \left (2 a^2 e^3+5 c^2 d^3 x-5 a c d e (d+2 e x)\right )+2 c^2 \left (a^3 e^5-c^3 d^5 x-5 a^2 c d e^3 (2 d+e x)+5 a c^2 d^3 e (d+2 e x)\right )+b c^2 \left (-c^2 d^4 (d-5 e x)+5 a^2 e^4 (3 d+e x)-10 a c d^2 e^2 (d+3 e x)\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))^2}+\frac {-b^6 e^5+b^5 c e^4 (5 d+4 e x)+b^4 c e^3 \left (11 a e^2-10 c d (d+e x)\right )+10 b^3 c^2 e^2 \left (c d^3-a e^2 (4 d+3 e x)\right )+4 c^3 \left (8 a^3 e^5+3 c^3 d^5 x+10 a c^2 d^3 e^2 x-5 a^2 c d e^3 (8 d+5 e x)\right )+2 b c^3 \left (3 c^2 d^4 (d-5 e x)+10 a c d^2 e^2 (d-3 e x)+5 a^2 e^4 (11 d+5 e x)\right )+b^2 c^2 e \left (-39 a^2 e^4-5 c^2 d^3 (3 d-4 e x)+10 a c d e^2 (5 d+8 e x)\right )}{\left (b^2-4 a c\right )^2 (a+x (b+c x))}+\frac {2 c (2 c d-b e) \left (6 c^4 d^4+b^4 e^4+2 b^2 c e^3 (b d-5 a e)-4 c^3 d^2 e (3 b d-5 a e)+2 c^2 e^2 \left (2 b^2 d^2-10 a b d e+15 a^2 e^2\right )\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{5/2}}+c e^5 \log (a+x (b+c x))}{2 c^4} \]
((b^5*e^5*x + b^4*e^4*(a*e - 5*c*d*x) - 5*b^3*c*e^3*(-2*c*d^2*x + a*e*(d + e*x)) - 2*b^2*c*e^2*(2*a^2*e^3 + 5*c^2*d^3*x - 5*a*c*d*e*(d + 2*e*x)) + 2 *c^2*(a^3*e^5 - c^3*d^5*x - 5*a^2*c*d*e^3*(2*d + e*x) + 5*a*c^2*d^3*e*(d + 2*e*x)) + b*c^2*(-(c^2*d^4*(d - 5*e*x)) + 5*a^2*e^4*(3*d + e*x) - 10*a*c* d^2*e^2*(d + 3*e*x)))/((b^2 - 4*a*c)*(a + x*(b + c*x))^2) + (-(b^6*e^5) + b^5*c*e^4*(5*d + 4*e*x) + b^4*c*e^3*(11*a*e^2 - 10*c*d*(d + e*x)) + 10*b^3 *c^2*e^2*(c*d^3 - a*e^2*(4*d + 3*e*x)) + 4*c^3*(8*a^3*e^5 + 3*c^3*d^5*x + 10*a*c^2*d^3*e^2*x - 5*a^2*c*d*e^3*(8*d + 5*e*x)) + 2*b*c^3*(3*c^2*d^4*(d - 5*e*x) + 10*a*c*d^2*e^2*(d - 3*e*x) + 5*a^2*e^4*(11*d + 5*e*x)) + b^2*c^ 2*e*(-39*a^2*e^4 - 5*c^2*d^3*(3*d - 4*e*x) + 10*a*c*d*e^2*(5*d + 8*e*x)))/ ((b^2 - 4*a*c)^2*(a + x*(b + c*x))) + (2*c*(2*c*d - b*e)*(6*c^4*d^4 + b^4* e^4 + 2*b^2*c*e^3*(b*d - 5*a*e) - 4*c^3*d^2*e*(3*b*d - 5*a*e) + 2*c^2*e^2* (2*b^2*d^2 - 10*a*b*d*e + 15*a^2*e^2))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a* c]])/(-b^2 + 4*a*c)^(5/2) + c*e^5*Log[a + x*(b + c*x)])/(2*c^4)
Time = 0.99 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.07, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1164, 1233, 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.
\(\displaystyle \int \frac {(d+e x)^5}{\left (a+b x+c x^2\right )^3} \, dx\) |
\(\Big \downarrow \) 1164 |
\(\displaystyle -\frac {\int \frac {(d+e x)^3 \left (6 c d^2-e (7 b d-8 a e)-e (2 c d-b e) x\right )}{\left (c x^2+b x+a\right )^2}dx}{2 \left (b^2-4 a c\right )}-\frac {(d+e x)^4 (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\) |
\(\Big \downarrow \) 1233 |
\(\displaystyle -\frac {\frac {\int -\frac {2 (d+e x) \left (6 c^3 d^4-c^2 e (15 b d-14 a e) d^2-a b^2 e^4+c e^2 \left (10 b^2 d^2-21 a b e d+16 a^2 e^2\right )-e (2 c d-b e) \left (3 c^2 d^2-b^2 e^2-c e (3 b d-7 a e)\right ) x\right )}{c x^2+b x+a}dx}{c \left (b^2-4 a c\right )}+\frac {(d+e x)^2 \left (-x (2 c d-b e) \left (-2 c e (3 b d-5 a e)-b^2 e^2+6 c^2 d^2\right )+b^2 \left (7 c d^2 e-a e^3\right )-6 b c d \left (3 a e^2+c d^2\right )+8 a c e \left (2 a e^2+c d^2\right )\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}}{2 \left (b^2-4 a c\right )}-\frac {(d+e x)^4 (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\frac {(d+e x)^2 \left (-x (2 c d-b e) \left (-2 c e (3 b d-5 a e)-b^2 e^2+6 c^2 d^2\right )+b^2 \left (7 c d^2 e-a e^3\right )-6 b c d \left (3 a e^2+c d^2\right )+8 a c e \left (2 a e^2+c d^2\right )\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {2 \int \frac {(d+e x) \left (6 c^3 d^4-c^2 e (15 b d-14 a e) d^2-a b^2 e^4+c e^2 \left (10 b^2 d^2-21 a b e d+16 a^2 e^2\right )-e (2 c d-b e) \left (3 c^2 d^2-b^2 e^2-c e (3 b d-7 a e)\right ) x\right )}{c x^2+b x+a}dx}{c \left (b^2-4 a c\right )}}{2 \left (b^2-4 a c\right )}-\frac {(d+e x)^4 (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\) |
\(\Big \downarrow \) 1200 |
\(\displaystyle -\frac {\frac {(d+e x)^2 \left (-x (2 c d-b e) \left (-2 c e (3 b d-5 a e)-b^2 e^2+6 c^2 d^2\right )+b^2 \left (7 c d^2 e-a e^3\right )-6 b c d \left (3 a e^2+c d^2\right )+8 a c e \left (2 a e^2+c d^2\right )\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {2 \int \left (\frac {6 c^4 d^5-5 c^3 e (3 b d-4 a e) d^3+10 c^2 e^2 \left (b^2 d^2-3 a b e d+3 a^2 e^2\right ) d+a b^3 e^5-7 a^2 b c e^5+\left (b^2-4 a c\right )^2 e^5 x}{c \left (c x^2+b x+a\right )}-\frac {e^2 (2 c d-b e) \left (3 c^2 d^2-b^2 e^2-c e (3 b d-7 a e)\right )}{c}\right )dx}{c \left (b^2-4 a c\right )}}{2 \left (b^2-4 a c\right )}-\frac {(d+e x)^4 (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\frac {(d+e x)^2 \left (-x (2 c d-b e) \left (-2 c e (3 b d-5 a e)-b^2 e^2+6 c^2 d^2\right )+b^2 \left (7 c d^2 e-a e^3\right )-6 b c d \left (3 a e^2+c d^2\right )+8 a c e \left (2 a e^2+c d^2\right )\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {2 \left (-\frac {\text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (20 c^3 d e^2 \left (3 a^2 e^2-3 a b d e+b^2 d^2\right )-30 a^2 b c^2 e^5+10 a b^3 c e^5-10 c^4 d^3 e (3 b d-4 a e)-b^5 e^5+12 c^5 d^5\right )}{c^2 \sqrt {b^2-4 a c}}-\frac {e^2 x (2 c d-b e) \left (-c e (3 b d-7 a e)-b^2 e^2+3 c^2 d^2\right )}{c}+\frac {e^5 \left (b^2-4 a c\right )^2 \log \left (a+b x+c x^2\right )}{2 c^2}\right )}{c \left (b^2-4 a c\right )}}{2 \left (b^2-4 a c\right )}-\frac {(d+e x)^4 (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\) |
-1/2*((d + e*x)^4*(b*d - 2*a*e + (2*c*d - b*e)*x))/((b^2 - 4*a*c)*(a + b*x + c*x^2)^2) - (((d + e*x)^2*(8*a*c*e*(c*d^2 + 2*a*e^2) - 6*b*c*d*(c*d^2 + 3*a*e^2) + b^2*(7*c*d^2*e - a*e^3) - (2*c*d - b*e)*(6*c^2*d^2 - b^2*e^2 - 2*c*e*(3*b*d - 5*a*e))*x))/(c*(b^2 - 4*a*c)*(a + b*x + c*x^2)) - (2*(-((e ^2*(2*c*d - b*e)*(3*c^2*d^2 - b^2*e^2 - c*e*(3*b*d - 7*a*e))*x)/c) - ((12* c^5*d^5 - b^5*e^5 + 10*a*b^3*c*e^5 - 30*a^2*b*c^2*e^5 - 10*c^4*d^3*e*(3*b* d - 4*a*e) + 20*c^3*d*e^2*(b^2*d^2 - 3*a*b*d*e + 3*a^2*e^2))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(c^2*Sqrt[b^2 - 4*a*c]) + ((b^2 - 4*a*c)^2*e^5* Log[a + b*x + c*x^2])/(2*c^2)))/(c*(b^2 - 4*a*c)))/(2*(b^2 - 4*a*c))
3.23.1.3.1 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[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(d + e*x)^(m - 1)*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a* c)) Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2* c*d^2*(2*p + 3) + e*(b*e - 2*d*c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[p, -1] && GtQ[m, 1] && Int QuadraticQ[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))*(a + b*x + c*x^2) ^(p + 1)*((2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g - c *(b*e*f + b*d*g + 2*a*e*g))*x)/(c*(p + 1)*(b^2 - 4*a*c))), x] - Simp[1/(c*( p + 1)*(b^2 - 4*a*c)) Int[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Sim p[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a*e*(e*f *(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*( m + p + 1) + 2*c^2*d*f*(m + 2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2* p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, b, c, d, e, f, g]) | | !ILtQ[m + 2*p + 3, 0])
Leaf count of result is larger than twice the leaf count of optimal. \(973\) vs. \(2(378)=756\).
Time = 17.48 (sec) , antiderivative size = 974, normalized size of antiderivative = 2.51
method | result | size |
default | \(\frac {\frac {\left (25 a^{2} b \,c^{2} e^{5}-50 a^{2} c^{3} d \,e^{4}-15 a \,b^{3} c \,e^{5}+40 a \,b^{2} c^{2} d \,e^{4}-30 a b \,c^{3} d^{2} e^{3}+20 a \,c^{4} d^{3} e^{2}+2 b^{5} e^{5}-5 b^{4} c d \,e^{4}+10 b^{2} c^{3} d^{3} e^{2}-15 b \,c^{4} d^{4} e +6 c^{5} d^{5}\right ) x^{3}}{c^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}+\frac {\left (32 a^{3} c^{3} e^{5}+11 a^{2} b^{2} c^{2} e^{5}+10 a^{2} b \,c^{3} d \,e^{4}-160 a^{2} c^{4} d^{2} e^{3}-19 a \,b^{4} c \,e^{5}+40 a \,b^{3} c^{2} d \,e^{4}-10 a \,b^{2} c^{3} d^{2} e^{3}+60 a b \,c^{4} d^{3} e^{2}+3 b^{6} e^{5}-5 b^{5} c d \,e^{4}-10 b^{4} c^{2} d^{2} e^{3}+30 b^{3} c^{3} d^{3} e^{2}-45 b^{2} c^{4} d^{4} e +18 b \,c^{5} d^{5}\right ) x^{2}}{2 \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) c^{3}}+\frac {\left (31 a^{3} b \,c^{2} e^{5}-30 a^{3} c^{3} d \,e^{4}-22 a^{2} b^{3} c \,e^{5}+50 a^{2} b^{2} c^{2} d \,e^{4}-50 a^{2} b \,c^{3} d^{2} e^{3}-20 a^{2} c^{4} d^{3} e^{2}+3 a \,b^{5} e^{5}-5 a \,b^{4} c d \,e^{4}-10 a \,b^{3} c^{2} d^{2} e^{3}+50 a \,b^{2} c^{3} d^{3} e^{2}-25 a b \,c^{4} d^{4} e +10 a \,c^{5} d^{5}-5 b^{3} c^{3} d^{4} e +2 c^{4} b^{2} d^{5}\right ) x}{\left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right ) c^{3}}+\frac {24 a^{4} c^{2} e^{5}-21 a^{3} b^{2} c \,e^{5}+50 a^{3} b \,c^{2} d \,e^{4}-80 a^{3} c^{3} d^{2} e^{3}+3 a^{2} b^{4} e^{5}-5 a^{2} b^{3} c d \,e^{4}-10 a^{2} b^{2} c^{2} d^{2} e^{3}+60 a^{2} b \,c^{3} d^{3} e^{2}-40 a^{2} c^{4} d^{4} e -5 a \,b^{2} c^{3} d^{4} e +10 a b \,c^{4} d^{5}-b^{3} c^{3} d^{5}}{2 c^{3} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}}{\left (c \,x^{2}+b x +a \right )^{2}}+\frac {\frac {\left (16 e^{5} a^{2} c^{2}-8 a \,b^{2} e^{5} c +b^{4} e^{5}\right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (-7 a^{2} b c \,e^{5}+30 d \,e^{4} a^{2} c^{2}+a \,b^{3} e^{5}-30 a b \,c^{2} d^{2} e^{3}+20 d^{3} e^{2} c^{3} a +10 b^{2} c^{2} d^{3} e^{2}-15 b \,c^{3} d^{4} e +6 c^{4} d^{5}-\frac {\left (16 e^{5} a^{2} c^{2}-8 a \,b^{2} e^{5} c +b^{4} e^{5}\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}}}}{c^{2} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}\) | \(974\) |
risch | \(\text {Expression too large to display}\) | \(5551\) |
((25*a^2*b*c^2*e^5-50*a^2*c^3*d*e^4-15*a*b^3*c*e^5+40*a*b^2*c^2*d*e^4-30*a *b*c^3*d^2*e^3+20*a*c^4*d^3*e^2+2*b^5*e^5-5*b^4*c*d*e^4+10*b^2*c^3*d^3*e^2 -15*b*c^4*d^4*e+6*c^5*d^5)/c^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^3+1/2*(32*a^3* c^3*e^5+11*a^2*b^2*c^2*e^5+10*a^2*b*c^3*d*e^4-160*a^2*c^4*d^2*e^3-19*a*b^4 *c*e^5+40*a*b^3*c^2*d*e^4-10*a*b^2*c^3*d^2*e^3+60*a*b*c^4*d^3*e^2+3*b^6*e^ 5-5*b^5*c*d*e^4-10*b^4*c^2*d^2*e^3+30*b^3*c^3*d^3*e^2-45*b^2*c^4*d^4*e+18* b*c^5*d^5)/(16*a^2*c^2-8*a*b^2*c+b^4)/c^3*x^2+(31*a^3*b*c^2*e^5-30*a^3*c^3 *d*e^4-22*a^2*b^3*c*e^5+50*a^2*b^2*c^2*d*e^4-50*a^2*b*c^3*d^2*e^3-20*a^2*c ^4*d^3*e^2+3*a*b^5*e^5-5*a*b^4*c*d*e^4-10*a*b^3*c^2*d^2*e^3+50*a*b^2*c^3*d ^3*e^2-25*a*b*c^4*d^4*e+10*a*c^5*d^5-5*b^3*c^3*d^4*e+2*b^2*c^4*d^5)/(16*a^ 2*c^2-8*a*b^2*c+b^4)/c^3*x+1/2/c^3*(24*a^4*c^2*e^5-21*a^3*b^2*c*e^5+50*a^3 *b*c^2*d*e^4-80*a^3*c^3*d^2*e^3+3*a^2*b^4*e^5-5*a^2*b^3*c*d*e^4-10*a^2*b^2 *c^2*d^2*e^3+60*a^2*b*c^3*d^3*e^2-40*a^2*c^4*d^4*e-5*a*b^2*c^3*d^4*e+10*a* b*c^4*d^5-b^3*c^3*d^5)/(16*a^2*c^2-8*a*b^2*c+b^4))/(c*x^2+b*x+a)^2+1/c^2/( 16*a^2*c^2-8*a*b^2*c+b^4)*(1/2*(16*a^2*c^2*e^5-8*a*b^2*c*e^5+b^4*e^5)/c*ln (c*x^2+b*x+a)+2*(-7*a^2*b*c*e^5+30*d*e^4*a^2*c^2+a*b^3*e^5-30*a*b*c^2*d^2* e^3+20*d^3*e^2*c^3*a+10*b^2*c^2*d^3*e^2-15*b*c^3*d^4*e+6*c^4*d^5-1/2*(16*a ^2*c^2*e^5-8*a*b^2*c*e^5+b^4*e^5)*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. 1923 vs. \(2 (378) = 756\).
Time = 0.37 (sec) , antiderivative size = 3865, normalized size of antiderivative = 9.96 \[ \int \frac {(d+e x)^5}{\left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \]
[-1/2*((b^5*c^3 - 14*a*b^3*c^4 + 40*a^2*b*c^5)*d^5 + 5*(a*b^4*c^3 + 4*a^2* b^2*c^4 - 32*a^3*c^5)*d^4*e - 60*(a^2*b^3*c^3 - 4*a^3*b*c^4)*d^3*e^2 + 10* (a^2*b^4*c^2 + 4*a^3*b^2*c^3 - 32*a^4*c^4)*d^2*e^3 + 5*(a^2*b^5*c - 14*a^3 *b^3*c^2 + 40*a^4*b*c^3)*d*e^4 - 3*(a^2*b^6 - 11*a^3*b^4*c + 36*a^4*b^2*c^ 2 - 32*a^5*c^3)*e^5 - 2*(6*(b^2*c^6 - 4*a*c^7)*d^5 - 15*(b^3*c^5 - 4*a*b*c ^6)*d^4*e + 10*(b^4*c^4 - 2*a*b^2*c^5 - 8*a^2*c^6)*d^3*e^2 - 30*(a*b^3*c^4 - 4*a^2*b*c^5)*d^2*e^3 - 5*(b^6*c^2 - 12*a*b^4*c^3 + 42*a^2*b^2*c^4 - 40* a^3*c^5)*d*e^4 + (2*b^7*c - 23*a*b^5*c^2 + 85*a^2*b^3*c^3 - 100*a^3*b*c^4) *e^5)*x^3 - (18*(b^3*c^5 - 4*a*b*c^6)*d^5 - 45*(b^4*c^4 - 4*a*b^2*c^5)*d^4 *e + 30*(b^5*c^3 - 2*a*b^3*c^4 - 8*a^2*b*c^5)*d^3*e^2 - 10*(b^6*c^2 - 3*a* b^4*c^3 + 12*a^2*b^2*c^4 - 64*a^3*c^5)*d^2*e^3 - 5*(b^7*c - 12*a*b^5*c^2 + 30*a^2*b^3*c^3 + 8*a^3*b*c^4)*d*e^4 + (3*b^8 - 31*a*b^6*c + 87*a^2*b^4*c^ 2 - 12*a^3*b^2*c^3 - 128*a^4*c^4)*e^5)*x^2 + (12*a^2*c^5*d^5 - 30*a^2*b*c^ 4*d^4*e - 60*a^3*b*c^3*d^2*e^3 + 60*a^4*c^3*d*e^4 + 20*(a^2*b^2*c^3 + 2*a^ 3*c^4)*d^3*e^2 - (a^2*b^5 - 10*a^3*b^3*c + 30*a^4*b*c^2)*e^5 + (12*c^7*d^5 - 30*b*c^6*d^4*e - 60*a*b*c^5*d^2*e^3 + 60*a^2*c^5*d*e^4 + 20*(b^2*c^5 + 2*a*c^6)*d^3*e^2 - (b^5*c^2 - 10*a*b^3*c^3 + 30*a^2*b*c^4)*e^5)*x^4 + 2*(1 2*b*c^6*d^5 - 30*b^2*c^5*d^4*e - 60*a*b^2*c^4*d^2*e^3 + 60*a^2*b*c^4*d*e^4 + 20*(b^3*c^4 + 2*a*b*c^5)*d^3*e^2 - (b^6*c - 10*a*b^4*c^2 + 30*a^2*b^2*c ^3)*e^5)*x^3 + (12*(b^2*c^5 + 2*a*c^6)*d^5 - 30*(b^3*c^4 + 2*a*b*c^5)*d...
Timed out. \[ \int \frac {(d+e x)^5}{\left (a+b x+c x^2\right )^3} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {(d+e x)^5}{\left (a+b x+c x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]
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*a*c-b^2>0)', see `assume?` for more deta
Leaf count of result is larger than twice the leaf count of optimal. 845 vs. \(2 (378) = 756\).
Time = 0.29 (sec) , antiderivative size = 845, normalized size of antiderivative = 2.18 \[ \int \frac {(d+e x)^5}{\left (a+b x+c x^2\right )^3} \, dx=\frac {e^{5} \log \left (c x^{2} + b x + a\right )}{2 \, c^{3}} + \frac {{\left (12 \, c^{5} d^{5} - 30 \, b c^{4} d^{4} e + 20 \, b^{2} c^{3} d^{3} e^{2} + 40 \, a c^{4} d^{3} e^{2} - 60 \, a b c^{3} d^{2} e^{3} + 60 \, a^{2} c^{3} d e^{4} - b^{5} e^{5} + 10 \, a b^{3} c e^{5} - 30 \, a^{2} b c^{2} e^{5}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {b^{3} c^{3} d^{5} - 10 \, a b c^{4} d^{5} + 5 \, a b^{2} c^{3} d^{4} e + 40 \, a^{2} c^{4} d^{4} e - 60 \, a^{2} b c^{3} d^{3} e^{2} + 10 \, a^{2} b^{2} c^{2} d^{2} e^{3} + 80 \, a^{3} c^{3} d^{2} e^{3} + 5 \, a^{2} b^{3} c d e^{4} - 50 \, a^{3} b c^{2} d e^{4} - 3 \, a^{2} b^{4} e^{5} + 21 \, a^{3} b^{2} c e^{5} - 24 \, a^{4} c^{2} e^{5} - 2 \, {\left (6 \, c^{6} d^{5} - 15 \, b c^{5} d^{4} e + 10 \, b^{2} c^{4} d^{3} e^{2} + 20 \, a c^{5} d^{3} e^{2} - 30 \, a b c^{4} d^{2} e^{3} - 5 \, b^{4} c^{2} d e^{4} + 40 \, a b^{2} c^{3} d e^{4} - 50 \, a^{2} c^{4} d e^{4} + 2 \, b^{5} c e^{5} - 15 \, a b^{3} c^{2} e^{5} + 25 \, a^{2} b c^{3} e^{5}\right )} x^{3} - {\left (18 \, b c^{5} d^{5} - 45 \, b^{2} c^{4} d^{4} e + 30 \, b^{3} c^{3} d^{3} e^{2} + 60 \, a b c^{4} d^{3} e^{2} - 10 \, b^{4} c^{2} d^{2} e^{3} - 10 \, a b^{2} c^{3} d^{2} e^{3} - 160 \, a^{2} c^{4} d^{2} e^{3} - 5 \, b^{5} c d e^{4} + 40 \, a b^{3} c^{2} d e^{4} + 10 \, a^{2} b c^{3} d e^{4} + 3 \, b^{6} e^{5} - 19 \, a b^{4} c e^{5} + 11 \, a^{2} b^{2} c^{2} e^{5} + 32 \, a^{3} c^{3} e^{5}\right )} x^{2} - 2 \, {\left (2 \, b^{2} c^{4} d^{5} + 10 \, a c^{5} d^{5} - 5 \, b^{3} c^{3} d^{4} e - 25 \, a b c^{4} d^{4} e + 50 \, a b^{2} c^{3} d^{3} e^{2} - 20 \, a^{2} c^{4} d^{3} e^{2} - 10 \, a b^{3} c^{2} d^{2} e^{3} - 50 \, a^{2} b c^{3} d^{2} e^{3} - 5 \, a b^{4} c d e^{4} + 50 \, a^{2} b^{2} c^{2} d e^{4} - 30 \, a^{3} c^{3} d e^{4} + 3 \, a b^{5} e^{5} - 22 \, a^{2} b^{3} c e^{5} + 31 \, a^{3} b c^{2} e^{5}\right )} x}{2 \, {\left (c x^{2} + b x + a\right )}^{2} {\left (b^{2} - 4 \, a c\right )}^{2} c^{3}} \]
1/2*e^5*log(c*x^2 + b*x + a)/c^3 + (12*c^5*d^5 - 30*b*c^4*d^4*e + 20*b^2*c ^3*d^3*e^2 + 40*a*c^4*d^3*e^2 - 60*a*b*c^3*d^2*e^3 + 60*a^2*c^3*d*e^4 - b^ 5*e^5 + 10*a*b^3*c*e^5 - 30*a^2*b*c^2*e^5)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*sqrt(-b^2 + 4*a*c)) - 1/2*(b ^3*c^3*d^5 - 10*a*b*c^4*d^5 + 5*a*b^2*c^3*d^4*e + 40*a^2*c^4*d^4*e - 60*a^ 2*b*c^3*d^3*e^2 + 10*a^2*b^2*c^2*d^2*e^3 + 80*a^3*c^3*d^2*e^3 + 5*a^2*b^3* c*d*e^4 - 50*a^3*b*c^2*d*e^4 - 3*a^2*b^4*e^5 + 21*a^3*b^2*c*e^5 - 24*a^4*c ^2*e^5 - 2*(6*c^6*d^5 - 15*b*c^5*d^4*e + 10*b^2*c^4*d^3*e^2 + 20*a*c^5*d^3 *e^2 - 30*a*b*c^4*d^2*e^3 - 5*b^4*c^2*d*e^4 + 40*a*b^2*c^3*d*e^4 - 50*a^2* c^4*d*e^4 + 2*b^5*c*e^5 - 15*a*b^3*c^2*e^5 + 25*a^2*b*c^3*e^5)*x^3 - (18*b *c^5*d^5 - 45*b^2*c^4*d^4*e + 30*b^3*c^3*d^3*e^2 + 60*a*b*c^4*d^3*e^2 - 10 *b^4*c^2*d^2*e^3 - 10*a*b^2*c^3*d^2*e^3 - 160*a^2*c^4*d^2*e^3 - 5*b^5*c*d* e^4 + 40*a*b^3*c^2*d*e^4 + 10*a^2*b*c^3*d*e^4 + 3*b^6*e^5 - 19*a*b^4*c*e^5 + 11*a^2*b^2*c^2*e^5 + 32*a^3*c^3*e^5)*x^2 - 2*(2*b^2*c^4*d^5 + 10*a*c^5* d^5 - 5*b^3*c^3*d^4*e - 25*a*b*c^4*d^4*e + 50*a*b^2*c^3*d^3*e^2 - 20*a^2*c ^4*d^3*e^2 - 10*a*b^3*c^2*d^2*e^3 - 50*a^2*b*c^3*d^2*e^3 - 5*a*b^4*c*d*e^4 + 50*a^2*b^2*c^2*d*e^4 - 30*a^3*c^3*d*e^4 + 3*a*b^5*e^5 - 22*a^2*b^3*c*e^ 5 + 31*a^3*b*c^2*e^5)*x)/((c*x^2 + b*x + a)^2*(b^2 - 4*a*c)^2*c^3)
Time = 12.18 (sec) , antiderivative size = 1486, normalized size of antiderivative = 3.83 \[ \int \frac {(d+e x)^5}{\left (a+b x+c x^2\right )^3} \, dx=\frac {\mathrm {atan}\left (\frac {\left (\frac {x\,\left (b\,e-2\,c\,d\right )\,\left (30\,a^2\,c^2\,e^4-10\,a\,b^2\,c\,e^4-20\,a\,b\,c^2\,d\,e^3+20\,a\,c^3\,d^2\,e^2+b^4\,e^4+2\,b^3\,c\,d\,e^3+4\,b^2\,c^2\,d^2\,e^2-12\,b\,c^3\,d^3\,e+6\,c^4\,d^4\right )}{c^2\,{\left (4\,a\,c-b^2\right )}^5}+\frac {\left (b\,e-2\,c\,d\right )\,\left (16\,a^2\,b\,c^4-8\,a\,b^3\,c^3+b^5\,c^2\right )\,\left (30\,a^2\,c^2\,e^4-10\,a\,b^2\,c\,e^4-20\,a\,b\,c^2\,d\,e^3+20\,a\,c^3\,d^2\,e^2+b^4\,e^4+2\,b^3\,c\,d\,e^3+4\,b^2\,c^2\,d^2\,e^2-12\,b\,c^3\,d^3\,e+6\,c^4\,d^4\right )}{2\,c^5\,{\left (4\,a\,c-b^2\right )}^5\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}\right )\,\left (32\,a^2\,c^5\,{\left (4\,a\,c-b^2\right )}^{5/2}+2\,b^4\,c^3\,{\left (4\,a\,c-b^2\right )}^{5/2}-16\,a\,b^2\,c^4\,{\left (4\,a\,c-b^2\right )}^{5/2}\right )}{-30\,a^2\,b\,c^2\,e^5+60\,a^2\,c^3\,d\,e^4+10\,a\,b^3\,c\,e^5-60\,a\,b\,c^3\,d^2\,e^3+40\,a\,c^4\,d^3\,e^2-b^5\,e^5+20\,b^2\,c^3\,d^3\,e^2-30\,b\,c^4\,d^4\,e+12\,c^5\,d^5}\right )\,\left (b\,e-2\,c\,d\right )\,\left (30\,a^2\,c^2\,e^4-10\,a\,b^2\,c\,e^4-20\,a\,b\,c^2\,d\,e^3+20\,a\,c^3\,d^2\,e^2+b^4\,e^4+2\,b^3\,c\,d\,e^3+4\,b^2\,c^2\,d^2\,e^2-12\,b\,c^3\,d^3\,e+6\,c^4\,d^4\right )}{c^3\,{\left (4\,a\,c-b^2\right )}^{5/2}}-\frac {\ln \left (c\,x^2+b\,x+a\right )\,\left (-1024\,a^5\,c^5\,e^5+1280\,a^4\,b^2\,c^4\,e^5-640\,a^3\,b^4\,c^3\,e^5+160\,a^2\,b^6\,c^2\,e^5-20\,a\,b^8\,c\,e^5+b^{10}\,e^5\right )}{2\,\left (1024\,a^5\,c^8-1280\,a^4\,b^2\,c^7+640\,a^3\,b^4\,c^6-160\,a^2\,b^6\,c^5+20\,a\,b^8\,c^4-b^{10}\,c^3\right )}-\frac {\frac {-24\,a^4\,c^2\,e^5+21\,a^3\,b^2\,c\,e^5-50\,a^3\,b\,c^2\,d\,e^4+80\,a^3\,c^3\,d^2\,e^3-3\,a^2\,b^4\,e^5+5\,a^2\,b^3\,c\,d\,e^4+10\,a^2\,b^2\,c^2\,d^2\,e^3-60\,a^2\,b\,c^3\,d^3\,e^2+40\,a^2\,c^4\,d^4\,e+5\,a\,b^2\,c^3\,d^4\,e-10\,a\,b\,c^4\,d^5+b^3\,c^3\,d^5}{2\,c^3\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}+\frac {x\,\left (-31\,a^3\,b\,c^2\,e^5+30\,a^3\,c^3\,d\,e^4+22\,a^2\,b^3\,c\,e^5-50\,a^2\,b^2\,c^2\,d\,e^4+50\,a^2\,b\,c^3\,d^2\,e^3+20\,a^2\,c^4\,d^3\,e^2-3\,a\,b^5\,e^5+5\,a\,b^4\,c\,d\,e^4+10\,a\,b^3\,c^2\,d^2\,e^3-50\,a\,b^2\,c^3\,d^3\,e^2+25\,a\,b\,c^4\,d^4\,e-10\,a\,c^5\,d^5+5\,b^3\,c^3\,d^4\,e-2\,b^2\,c^4\,d^5\right )}{c^3\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}-\frac {x^2\,\left (32\,a^3\,c^3\,e^5+11\,a^2\,b^2\,c^2\,e^5+10\,a^2\,b\,c^3\,d\,e^4-160\,a^2\,c^4\,d^2\,e^3-19\,a\,b^4\,c\,e^5+40\,a\,b^3\,c^2\,d\,e^4-10\,a\,b^2\,c^3\,d^2\,e^3+60\,a\,b\,c^4\,d^3\,e^2+3\,b^6\,e^5-5\,b^5\,c\,d\,e^4-10\,b^4\,c^2\,d^2\,e^3+30\,b^3\,c^3\,d^3\,e^2-45\,b^2\,c^4\,d^4\,e+18\,b\,c^5\,d^5\right )}{2\,c^3\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}-\frac {x^3\,\left (25\,a^2\,b\,c^2\,e^5-50\,a^2\,c^3\,d\,e^4-15\,a\,b^3\,c\,e^5+40\,a\,b^2\,c^2\,d\,e^4-30\,a\,b\,c^3\,d^2\,e^3+20\,a\,c^4\,d^3\,e^2+2\,b^5\,e^5-5\,b^4\,c\,d\,e^4+10\,b^2\,c^3\,d^3\,e^2-15\,b\,c^4\,d^4\,e+6\,c^5\,d^5\right )}{c^2\,\left (16\,a^2\,c^2-8\,a\,b^2\,c+b^4\right )}}{x^2\,\left (b^2+2\,a\,c\right )+a^2+c^2\,x^4+2\,a\,b\,x+2\,b\,c\,x^3} \]
(atan((((x*(b*e - 2*c*d)*(b^4*e^4 + 6*c^4*d^4 + 30*a^2*c^2*e^4 + 20*a*c^3* d^2*e^2 + 4*b^2*c^2*d^2*e^2 - 10*a*b^2*c*e^4 - 12*b*c^3*d^3*e + 2*b^3*c*d* e^3 - 20*a*b*c^2*d*e^3))/(c^2*(4*a*c - b^2)^5) + ((b*e - 2*c*d)*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*(b^4*e^4 + 6*c^4*d^4 + 30*a^2*c^2*e^4 + 20*a* c^3*d^2*e^2 + 4*b^2*c^2*d^2*e^2 - 10*a*b^2*c*e^4 - 12*b*c^3*d^3*e + 2*b^3* c*d*e^3 - 20*a*b*c^2*d*e^3))/(2*c^5*(4*a*c - b^2)^5*(b^4 + 16*a^2*c^2 - 8* a*b^2*c)))*(32*a^2*c^5*(4*a*c - b^2)^(5/2) + 2*b^4*c^3*(4*a*c - b^2)^(5/2) - 16*a*b^2*c^4*(4*a*c - b^2)^(5/2)))/(12*c^5*d^5 - b^5*e^5 - 30*a^2*b*c^2 *e^5 + 40*a*c^4*d^3*e^2 + 60*a^2*c^3*d*e^4 + 20*b^2*c^3*d^3*e^2 + 10*a*b^3 *c*e^5 - 30*b*c^4*d^4*e - 60*a*b*c^3*d^2*e^3))*(b*e - 2*c*d)*(b^4*e^4 + 6* c^4*d^4 + 30*a^2*c^2*e^4 + 20*a*c^3*d^2*e^2 + 4*b^2*c^2*d^2*e^2 - 10*a*b^2 *c*e^4 - 12*b*c^3*d^3*e + 2*b^3*c*d*e^3 - 20*a*b*c^2*d*e^3))/(c^3*(4*a*c - b^2)^(5/2)) - (log(a + b*x + c*x^2)*(b^10*e^5 - 1024*a^5*c^5*e^5 + 160*a^ 2*b^6*c^2*e^5 - 640*a^3*b^4*c^3*e^5 + 1280*a^4*b^2*c^4*e^5 - 20*a*b^8*c*e^ 5))/(2*(1024*a^5*c^8 - b^10*c^3 + 20*a*b^8*c^4 - 160*a^2*b^6*c^5 + 640*a^3 *b^4*c^6 - 1280*a^4*b^2*c^7)) - ((b^3*c^3*d^5 - 24*a^4*c^2*e^5 - 3*a^2*b^4 *e^5 + 21*a^3*b^2*c*e^5 + 40*a^2*c^4*d^4*e + 80*a^3*c^3*d^2*e^3 - 10*a*b*c ^4*d^5 + 10*a^2*b^2*c^2*d^2*e^3 + 5*a*b^2*c^3*d^4*e + 5*a^2*b^3*c*d*e^4 - 50*a^3*b*c^2*d*e^4 - 60*a^2*b*c^3*d^3*e^2)/(2*c^3*(b^4 + 16*a^2*c^2 - 8*a* b^2*c)) + (x*(22*a^2*b^3*c*e^5 - 10*a*c^5*d^5 - 2*b^2*c^4*d^5 - 3*a*b^5...