Integrand size = 23, antiderivative size = 134 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )}{(d+e x)^7} \, dx=\frac {(B d-A e) \left (c d^2-b d e+a e^2\right )}{6 e^4 (d+e x)^6}+\frac {A e (2 c d-b e)-B \left (3 c d^2-e (2 b d-a e)\right )}{5 e^4 (d+e x)^5}+\frac {3 B c d-b B e-A c e}{4 e^4 (d+e x)^4}-\frac {B c}{3 e^4 (d+e x)^3} \]
1/6*(-A*e+B*d)*(a*e^2-b*d*e+c*d^2)/e^4/(e*x+d)^6+1/5*(A*e*(-b*e+2*c*d)-B*( 3*c*d^2-e*(-a*e+2*b*d)))/e^4/(e*x+d)^5+1/4*(-A*c*e-B*b*e+3*B*c*d)/e^4/(e*x +d)^4-1/3*B*c/e^4/(e*x+d)^3
Time = 0.04 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.89 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )}{(d+e x)^7} \, dx=-\frac {A e \left (2 e (b d+5 a e+6 b e x)+c \left (d^2+6 d e x+15 e^2 x^2\right )\right )+B \left (c \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )+e \left (2 a e (d+6 e x)+b \left (d^2+6 d e x+15 e^2 x^2\right )\right )\right )}{60 e^4 (d+e x)^6} \]
-1/60*(A*e*(2*e*(b*d + 5*a*e + 6*b*e*x) + c*(d^2 + 6*d*e*x + 15*e^2*x^2)) + B*(c*(d^3 + 6*d^2*e*x + 15*d*e^2*x^2 + 20*e^3*x^3) + e*(2*a*e*(d + 6*e*x ) + b*(d^2 + 6*d*e*x + 15*e^2*x^2))))/(e^4*(d + e*x)^6)
Time = 0.30 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {1195, 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.
\(\displaystyle \int \frac {(A+B x) \left (a+b x+c x^2\right )}{(d+e x)^7} \, dx\) |
\(\Big \downarrow \) 1195 |
\(\displaystyle \int \left (\frac {-B e (2 b d-a e)-A e (2 c d-b e)+3 B c d^2}{e^3 (d+e x)^6}+\frac {(A e-B d) \left (a e^2-b d e+c d^2\right )}{e^3 (d+e x)^7}+\frac {A c e+b B e-3 B c d}{e^3 (d+e x)^5}+\frac {B c}{e^3 (d+e x)^4}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {-B e (2 b d-a e)-A e (2 c d-b e)+3 B c d^2}{5 e^4 (d+e x)^5}+\frac {(B d-A e) \left (a e^2-b d e+c d^2\right )}{6 e^4 (d+e x)^6}+\frac {-A c e-b B e+3 B c d}{4 e^4 (d+e x)^4}-\frac {B c}{3 e^4 (d+e x)^3}\) |
((B*d - A*e)*(c*d^2 - b*d*e + a*e^2))/(6*e^4*(d + e*x)^6) - (3*B*c*d^2 - B *e*(2*b*d - a*e) - A*e*(2*c*d - b*e))/(5*e^4*(d + e*x)^5) + (3*B*c*d - b*B *e - A*c*e)/(4*e^4*(d + e*x)^4) - (B*c)/(3*e^4*(d + e*x)^3)
3.24.17.3.1 Defintions of rubi rules used
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && IGtQ[p, 0]
Time = 0.28 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.95
method | result | size |
risch | \(\frac {-\frac {B c \,x^{3}}{3 e}-\frac {\left (A c e +B b e +B c d \right ) x^{2}}{4 e^{2}}-\frac {\left (2 A b \,e^{2}+A c d e +2 B a \,e^{2}+B b d e +B c \,d^{2}\right ) x}{10 e^{3}}-\frac {10 A a \,e^{3}+2 A b d \,e^{2}+A c \,d^{2} e +2 B a d \,e^{2}+B b \,d^{2} e +B c \,d^{3}}{60 e^{4}}}{\left (e x +d \right )^{6}}\) | \(127\) |
gosper | \(-\frac {20 B c \,x^{3} e^{3}+15 A c \,e^{3} x^{2}+15 B \,x^{2} b \,e^{3}+15 B \,x^{2} c d \,e^{2}+12 A b \,e^{3} x +6 A c d \,e^{2} x +12 B x a \,e^{3}+6 B x b d \,e^{2}+6 B c \,d^{2} e x +10 A a \,e^{3}+2 A b d \,e^{2}+A c \,d^{2} e +2 B a d \,e^{2}+B b \,d^{2} e +B c \,d^{3}}{60 e^{4} \left (e x +d \right )^{6}}\) | \(141\) |
default | \(-\frac {A a \,e^{3}-A b d \,e^{2}+A c \,d^{2} e -B a d \,e^{2}+B b \,d^{2} e -B c \,d^{3}}{6 e^{4} \left (e x +d \right )^{6}}-\frac {A b \,e^{2}-2 A c d e +B a \,e^{2}-2 B b d e +3 B c \,d^{2}}{5 e^{4} \left (e x +d \right )^{5}}-\frac {B c}{3 e^{4} \left (e x +d \right )^{3}}-\frac {A c e +B b e -3 B c d}{4 e^{4} \left (e x +d \right )^{4}}\) | \(142\) |
norman | \(\frac {-\frac {B c \,x^{3}}{3 e}-\frac {\left (A c \,e^{3}+B \,e^{3} b +B c d \,e^{2}\right ) x^{2}}{4 e^{4}}-\frac {\left (2 A b \,e^{4}+A c d \,e^{3}+2 B \,e^{4} a +B b d \,e^{3}+B c \,d^{2} e^{2}\right ) x}{10 e^{5}}-\frac {10 A a \,e^{5}+2 A b d \,e^{4}+A c \,d^{2} e^{3}+2 B a d \,e^{4}+B b \,d^{2} e^{3}+B c \,d^{3} e^{2}}{60 e^{6}}}{\left (e x +d \right )^{6}}\) | \(148\) |
parallelrisch | \(-\frac {20 B c \,x^{3} e^{5}+15 A c \,e^{5} x^{2}+15 B b \,e^{5} x^{2}+15 B c d \,e^{4} x^{2}+12 A b \,e^{5} x +6 A c d \,e^{4} x +12 B a \,e^{5} x +6 B b d \,e^{4} x +6 B c \,d^{2} e^{3} x +10 A a \,e^{5}+2 A b d \,e^{4}+A c \,d^{2} e^{3}+2 B a d \,e^{4}+B b \,d^{2} e^{3}+B c \,d^{3} e^{2}}{60 e^{6} \left (e x +d \right )^{6}}\) | \(150\) |
(-1/3*B*c*x^3/e-1/4/e^2*(A*c*e+B*b*e+B*c*d)*x^2-1/10/e^3*(2*A*b*e^2+A*c*d* e+2*B*a*e^2+B*b*d*e+B*c*d^2)*x-1/60/e^4*(10*A*a*e^3+2*A*b*d*e^2+A*c*d^2*e+ 2*B*a*d*e^2+B*b*d^2*e+B*c*d^3))/(e*x+d)^6
Time = 0.30 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.33 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )}{(d+e x)^7} \, dx=-\frac {20 \, B c e^{3} x^{3} + B c d^{3} + 10 \, A a e^{3} + {\left (B b + A c\right )} d^{2} e + 2 \, {\left (B a + A b\right )} d e^{2} + 15 \, {\left (B c d e^{2} + {\left (B b + A c\right )} e^{3}\right )} x^{2} + 6 \, {\left (B c d^{2} e + {\left (B b + A c\right )} d e^{2} + 2 \, {\left (B a + A b\right )} e^{3}\right )} x}{60 \, {\left (e^{10} x^{6} + 6 \, d e^{9} x^{5} + 15 \, d^{2} e^{8} x^{4} + 20 \, d^{3} e^{7} x^{3} + 15 \, d^{4} e^{6} x^{2} + 6 \, d^{5} e^{5} x + d^{6} e^{4}\right )}} \]
-1/60*(20*B*c*e^3*x^3 + B*c*d^3 + 10*A*a*e^3 + (B*b + A*c)*d^2*e + 2*(B*a + A*b)*d*e^2 + 15*(B*c*d*e^2 + (B*b + A*c)*e^3)*x^2 + 6*(B*c*d^2*e + (B*b + A*c)*d*e^2 + 2*(B*a + A*b)*e^3)*x)/(e^10*x^6 + 6*d*e^9*x^5 + 15*d^2*e^8* x^4 + 20*d^3*e^7*x^3 + 15*d^4*e^6*x^2 + 6*d^5*e^5*x + d^6*e^4)
Timed out. \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )}{(d+e x)^7} \, dx=\text {Timed out} \]
Time = 0.19 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.33 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )}{(d+e x)^7} \, dx=-\frac {20 \, B c e^{3} x^{3} + B c d^{3} + 10 \, A a e^{3} + {\left (B b + A c\right )} d^{2} e + 2 \, {\left (B a + A b\right )} d e^{2} + 15 \, {\left (B c d e^{2} + {\left (B b + A c\right )} e^{3}\right )} x^{2} + 6 \, {\left (B c d^{2} e + {\left (B b + A c\right )} d e^{2} + 2 \, {\left (B a + A b\right )} e^{3}\right )} x}{60 \, {\left (e^{10} x^{6} + 6 \, d e^{9} x^{5} + 15 \, d^{2} e^{8} x^{4} + 20 \, d^{3} e^{7} x^{3} + 15 \, d^{4} e^{6} x^{2} + 6 \, d^{5} e^{5} x + d^{6} e^{4}\right )}} \]
-1/60*(20*B*c*e^3*x^3 + B*c*d^3 + 10*A*a*e^3 + (B*b + A*c)*d^2*e + 2*(B*a + A*b)*d*e^2 + 15*(B*c*d*e^2 + (B*b + A*c)*e^3)*x^2 + 6*(B*c*d^2*e + (B*b + A*c)*d*e^2 + 2*(B*a + A*b)*e^3)*x)/(e^10*x^6 + 6*d*e^9*x^5 + 15*d^2*e^8* x^4 + 20*d^3*e^7*x^3 + 15*d^4*e^6*x^2 + 6*d^5*e^5*x + d^6*e^4)
Time = 0.27 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.04 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )}{(d+e x)^7} \, dx=-\frac {20 \, B c e^{3} x^{3} + 15 \, B c d e^{2} x^{2} + 15 \, B b e^{3} x^{2} + 15 \, A c e^{3} x^{2} + 6 \, B c d^{2} e x + 6 \, B b d e^{2} x + 6 \, A c d e^{2} x + 12 \, B a e^{3} x + 12 \, A b e^{3} x + B c d^{3} + B b d^{2} e + A c d^{2} e + 2 \, B a d e^{2} + 2 \, A b d e^{2} + 10 \, A a e^{3}}{60 \, {\left (e x + d\right )}^{6} e^{4}} \]
-1/60*(20*B*c*e^3*x^3 + 15*B*c*d*e^2*x^2 + 15*B*b*e^3*x^2 + 15*A*c*e^3*x^2 + 6*B*c*d^2*e*x + 6*B*b*d*e^2*x + 6*A*c*d*e^2*x + 12*B*a*e^3*x + 12*A*b*e ^3*x + B*c*d^3 + B*b*d^2*e + A*c*d^2*e + 2*B*a*d*e^2 + 2*A*b*d*e^2 + 10*A* a*e^3)/((e*x + d)^6*e^4)
Time = 11.03 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.36 \[ \int \frac {(A+B x) \left (a+b x+c x^2\right )}{(d+e x)^7} \, dx=-\frac {\frac {10\,A\,a\,e^3+B\,c\,d^3+2\,A\,b\,d\,e^2+2\,B\,a\,d\,e^2+A\,c\,d^2\,e+B\,b\,d^2\,e}{60\,e^4}+\frac {x^2\,\left (A\,c\,e+B\,b\,e+B\,c\,d\right )}{4\,e^2}+\frac {x\,\left (2\,A\,b\,e^2+2\,B\,a\,e^2+B\,c\,d^2+A\,c\,d\,e+B\,b\,d\,e\right )}{10\,e^3}+\frac {B\,c\,x^3}{3\,e}}{d^6+6\,d^5\,e\,x+15\,d^4\,e^2\,x^2+20\,d^3\,e^3\,x^3+15\,d^2\,e^4\,x^4+6\,d\,e^5\,x^5+e^6\,x^6} \]
-((10*A*a*e^3 + B*c*d^3 + 2*A*b*d*e^2 + 2*B*a*d*e^2 + A*c*d^2*e + B*b*d^2* e)/(60*e^4) + (x^2*(A*c*e + B*b*e + B*c*d))/(4*e^2) + (x*(2*A*b*e^2 + 2*B* a*e^2 + B*c*d^2 + A*c*d*e + B*b*d*e))/(10*e^3) + (B*c*x^3)/(3*e))/(d^6 + e ^6*x^6 + 6*d*e^5*x^5 + 15*d^4*e^2*x^2 + 20*d^3*e^3*x^3 + 15*d^2*e^4*x^4 + 6*d^5*e*x)