Integrand size = 20, antiderivative size = 108 \[ \int \frac {(A+B x) \left (a+c x^2\right )}{(d+e x)^7} \, dx=\frac {(B d-A e) \left (c d^2+a e^2\right )}{6 e^4 (d+e x)^6}-\frac {3 B c d^2-2 A c d e+a B e^2}{5 e^4 (d+e x)^5}+\frac {c (3 B d-A e)}{4 e^4 (d+e x)^4}-\frac {B c}{3 e^4 (d+e x)^3} \] Output:
1/6*(-A*e+B*d)*(a*e^2+c*d^2)/e^4/(e*x+d)^6-1/5*(-2*A*c*d*e+B*a*e^2+3*B*c*d ^2)/e^4/(e*x+d)^5+1/4*c*(-A*e+3*B*d)/e^4/(e*x+d)^4-1/3*B*c/e^4/(e*x+d)^3
Time = 0.04 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.81 \[ \int \frac {(A+B x) \left (a+c x^2\right )}{(d+e x)^7} \, dx=-\frac {10 a A e^3+2 a B e^2 (d+6 e x)+A c e \left (d^2+6 d e x+15 e^2 x^2\right )+B c \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )}{60 e^4 (d+e x)^6} \] Input:
Integrate[((A + B*x)*(a + c*x^2))/(d + e*x)^7,x]
Output:
-1/60*(10*a*A*e^3 + 2*a*B*e^2*(d + 6*e*x) + A*c*e*(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^4*(d + e*x)^6 )
Time = 0.26 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {652, 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.
\(\displaystyle \int \frac {\left (a+c x^2\right ) (A+B x)}{(d+e x)^7} \, dx\) |
\(\Big \downarrow \) 652 |
\(\displaystyle \int \left (\frac {a B e^2-2 A c d e+3 B c d^2}{e^3 (d+e x)^6}+\frac {\left (a e^2+c d^2\right ) (A e-B d)}{e^3 (d+e x)^7}+\frac {c (A e-3 B d)}{e^3 (d+e x)^5}+\frac {B c}{e^3 (d+e x)^4}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a B e^2-2 A c d e+3 B c d^2}{5 e^4 (d+e x)^5}+\frac {\left (a e^2+c d^2\right ) (B d-A e)}{6 e^4 (d+e x)^6}+\frac {c (3 B d-A e)}{4 e^4 (d+e x)^4}-\frac {B c}{3 e^4 (d+e x)^3}\) |
Input:
Int[((A + B*x)*(a + c*x^2))/(d + e*x)^7,x]
Output:
((B*d - A*e)*(c*d^2 + a*e^2))/(6*e^4*(d + e*x)^6) - (3*B*c*d^2 - 2*A*c*d*e + a*B*e^2)/(5*e^4*(d + e*x)^5) + (c*(3*B*d - A*e))/(4*e^4*(d + e*x)^4) - (B*c)/(3*e^4*(d + e*x)^3)
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (c_.)*(x_ )^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + c *x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && IGtQ[p, 0]
Time = 0.61 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.88
method | result | size |
risch | \(\frac {-\frac {B c \,x^{3}}{3 e}-\frac {c \left (A e +B d \right ) x^{2}}{4 e^{2}}-\frac {\left (A c d e +2 B a \,e^{2}+B c \,d^{2}\right ) x}{10 e^{3}}-\frac {10 A a \,e^{3}+A c \,d^{2} e +2 B a d \,e^{2}+B c \,d^{3}}{60 e^{4}}}{\left (e x +d \right )^{6}}\) | \(95\) |
gosper | \(-\frac {20 B c \,x^{3} e^{3}+15 A \,x^{2} c \,e^{3}+15 B \,x^{2} c d \,e^{2}+6 A x c d \,e^{2}+12 B x a \,e^{3}+6 B x c \,d^{2} e +10 A a \,e^{3}+A c \,d^{2} e +2 B a d \,e^{2}+B c \,d^{3}}{60 e^{4} \left (e x +d \right )^{6}}\) | \(99\) |
orering | \(-\frac {20 B c \,x^{3} e^{3}+15 A \,x^{2} c \,e^{3}+15 B \,x^{2} c d \,e^{2}+6 A x c d \,e^{2}+12 B x a \,e^{3}+6 B x c \,d^{2} e +10 A a \,e^{3}+A c \,d^{2} e +2 B a d \,e^{2}+B c \,d^{3}}{60 e^{4} \left (e x +d \right )^{6}}\) | \(99\) |
parallelrisch | \(-\frac {20 B c \,x^{3} e^{5}+15 A c \,e^{5} x^{2}+15 B c d \,e^{4} x^{2}+6 A c d \,e^{4} x +12 B a \,e^{5} x +6 B c \,d^{2} e^{3} x +10 A a \,e^{5}+A c \,d^{2} e^{3}+2 B a d \,e^{4}+B c \,d^{3} e^{2}}{60 e^{6} \left (e x +d \right )^{6}}\) | \(106\) |
default | \(-\frac {B c}{3 e^{4} \left (e x +d \right )^{3}}-\frac {c \left (A e -3 B d \right )}{4 e^{4} \left (e x +d \right )^{4}}-\frac {-2 A c d e +B a \,e^{2}+3 B c \,d^{2}}{5 e^{4} \left (e x +d \right )^{5}}-\frac {A a \,e^{3}+A c \,d^{2} e -B a d \,e^{2}-B c \,d^{3}}{6 e^{4} \left (e x +d \right )^{6}}\) | \(110\) |
norman | \(\frac {-\frac {B c \,x^{3}}{3 e}-\frac {\left (A c \,e^{3}+B c d \,e^{2}\right ) x^{2}}{4 e^{4}}-\frac {\left (A c d \,e^{3}+2 B \,e^{4} a +B c \,d^{2} e^{2}\right ) x}{10 e^{5}}-\frac {10 A a \,e^{5}+A c \,d^{2} e^{3}+2 B a d \,e^{4}+B c \,d^{3} e^{2}}{60 e^{6}}}{\left (e x +d \right )^{6}}\) | \(111\) |
Input:
int((B*x+A)*(c*x^2+a)/(e*x+d)^7,x,method=_RETURNVERBOSE)
Output:
(-1/3*B*c*x^3/e-1/4*c/e^2*(A*e+B*d)*x^2-1/10/e^3*(A*c*d*e+2*B*a*e^2+B*c*d^ 2)*x-1/60/e^4*(10*A*a*e^3+A*c*d^2*e+2*B*a*d*e^2+B*c*d^3))/(e*x+d)^6
Time = 0.08 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.42 \[ \int \frac {(A+B x) \left (a+c x^2\right )}{(d+e x)^7} \, dx=-\frac {20 \, B c e^{3} x^{3} + B c d^{3} + A c d^{2} e + 2 \, B a d e^{2} + 10 \, A a e^{3} + 15 \, {\left (B c d e^{2} + A c e^{3}\right )} x^{2} + 6 \, {\left (B c d^{2} e + A c d e^{2} + 2 \, B a e^{3}\right )} x}{60 \, {\left (e^{10} x^{6} + 6 \, d e^{9} x^{5} + 15 \, d^{2} e^{8} x^{4} + 20 \, d^{3} e^{7} x^{3} + 15 \, d^{4} e^{6} x^{2} + 6 \, d^{5} e^{5} x + d^{6} e^{4}\right )}} \] Input:
integrate((B*x+A)*(c*x^2+a)/(e*x+d)^7,x, algorithm="fricas")
Output:
-1/60*(20*B*c*e^3*x^3 + B*c*d^3 + A*c*d^2*e + 2*B*a*d*e^2 + 10*A*a*e^3 + 1 5*(B*c*d*e^2 + A*c*e^3)*x^2 + 6*(B*c*d^2*e + A*c*d*e^2 + 2*B*a*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 = 5.40 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.60 \[ \int \frac {(A+B x) \left (a+c x^2\right )}{(d+e x)^7} \, dx=\frac {- 10 A a e^{3} - A c d^{2} e - 2 B a d e^{2} - B c d^{3} - 20 B c e^{3} x^{3} + x^{2} \left (- 15 A c e^{3} - 15 B c d e^{2}\right ) + x \left (- 6 A c d e^{2} - 12 B a e^{3} - 6 B c d^{2} e\right )}{60 d^{6} e^{4} + 360 d^{5} e^{5} x + 900 d^{4} e^{6} x^{2} + 1200 d^{3} e^{7} x^{3} + 900 d^{2} e^{8} x^{4} + 360 d e^{9} x^{5} + 60 e^{10} x^{6}} \] Input:
integrate((B*x+A)*(c*x**2+a)/(e*x+d)**7,x)
Output:
(-10*A*a*e**3 - A*c*d**2*e - 2*B*a*d*e**2 - B*c*d**3 - 20*B*c*e**3*x**3 + x**2*(-15*A*c*e**3 - 15*B*c*d*e**2) + x*(-6*A*c*d*e**2 - 12*B*a*e**3 - 6*B *c*d**2*e))/(60*d**6*e**4 + 360*d**5*e**5*x + 900*d**4*e**6*x**2 + 1200*d* *3*e**7*x**3 + 900*d**2*e**8*x**4 + 360*d*e**9*x**5 + 60*e**10*x**6)
Time = 0.04 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.42 \[ \int \frac {(A+B x) \left (a+c x^2\right )}{(d+e x)^7} \, dx=-\frac {20 \, B c e^{3} x^{3} + B c d^{3} + A c d^{2} e + 2 \, B a d e^{2} + 10 \, A a e^{3} + 15 \, {\left (B c d e^{2} + A c e^{3}\right )} x^{2} + 6 \, {\left (B c d^{2} e + A c d e^{2} + 2 \, B a e^{3}\right )} x}{60 \, {\left (e^{10} x^{6} + 6 \, d e^{9} x^{5} + 15 \, d^{2} e^{8} x^{4} + 20 \, d^{3} e^{7} x^{3} + 15 \, d^{4} e^{6} x^{2} + 6 \, d^{5} e^{5} x + d^{6} e^{4}\right )}} \] Input:
integrate((B*x+A)*(c*x^2+a)/(e*x+d)^7,x, algorithm="maxima")
Output:
-1/60*(20*B*c*e^3*x^3 + B*c*d^3 + A*c*d^2*e + 2*B*a*d*e^2 + 10*A*a*e^3 + 1 5*(B*c*d*e^2 + A*c*e^3)*x^2 + 6*(B*c*d^2*e + A*c*d*e^2 + 2*B*a*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 = 0.13 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.91 \[ \int \frac {(A+B x) \left (a+c x^2\right )}{(d+e x)^7} \, dx=-\frac {20 \, B c e^{3} x^{3} + 15 \, B c d e^{2} x^{2} + 15 \, A c e^{3} x^{2} + 6 \, B c d^{2} e x + 6 \, A c d e^{2} x + 12 \, B a e^{3} x + B c d^{3} + A c d^{2} e + 2 \, B a d e^{2} + 10 \, A a e^{3}}{60 \, {\left (e x + d\right )}^{6} e^{4}} \] Input:
integrate((B*x+A)*(c*x^2+a)/(e*x+d)^7,x, algorithm="giac")
Output:
-1/60*(20*B*c*e^3*x^3 + 15*B*c*d*e^2*x^2 + 15*A*c*e^3*x^2 + 6*B*c*d^2*e*x + 6*A*c*d*e^2*x + 12*B*a*e^3*x + B*c*d^3 + A*c*d^2*e + 2*B*a*d*e^2 + 10*A* a*e^3)/((e*x + d)^6*e^4)
Time = 6.87 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.39 \[ \int \frac {(A+B x) \left (a+c x^2\right )}{(d+e x)^7} \, dx=-\frac {\frac {B\,c\,d^3+A\,c\,d^2\,e+2\,B\,a\,d\,e^2+10\,A\,a\,e^3}{60\,e^4}+\frac {x\,\left (B\,c\,d^2+A\,c\,d\,e+2\,B\,a\,e^2\right )}{10\,e^3}+\frac {B\,c\,x^3}{3\,e}+\frac {c\,x^2\,\left (A\,e+B\,d\right )}{4\,e^2}}{d^6+6\,d^5\,e\,x+15\,d^4\,e^2\,x^2+20\,d^3\,e^3\,x^3+15\,d^2\,e^4\,x^4+6\,d\,e^5\,x^5+e^6\,x^6} \] Input:
int(((a + c*x^2)*(A + B*x))/(d + e*x)^7,x)
Output:
-((10*A*a*e^3 + B*c*d^3 + 2*B*a*d*e^2 + A*c*d^2*e)/(60*e^4) + (x*(2*B*a*e^ 2 + B*c*d^2 + A*c*d*e))/(10*e^3) + (B*c*x^3)/(3*e) + (c*x^2*(A*e + B*d))/( 4*e^2))/(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 + 1 5*d^2*e^4*x^4 + 6*d^5*e*x)
Time = 0.25 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.44 \[ \int \frac {(A+B x) \left (a+c x^2\right )}{(d+e x)^7} \, dx=\frac {-20 b c \,e^{3} x^{3}-15 a c \,e^{3} x^{2}-15 b c d \,e^{2} x^{2}-12 a b \,e^{3} x -6 a c d \,e^{2} x -6 b c \,d^{2} e x -10 a^{2} e^{3}-2 a b d \,e^{2}-a c \,d^{2} e -b c \,d^{3}}{60 e^{4} \left (e^{6} x^{6}+6 d \,e^{5} x^{5}+15 d^{2} e^{4} x^{4}+20 d^{3} e^{3} x^{3}+15 d^{4} e^{2} x^{2}+6 d^{5} e x +d^{6}\right )} \] Input:
int((B*x+A)*(c*x^2+a)/(e*x+d)^7,x)
Output:
( - 10*a**2*e**3 - 2*a*b*d*e**2 - 12*a*b*e**3*x - a*c*d**2*e - 6*a*c*d*e** 2*x - 15*a*c*e**3*x**2 - b*c*d**3 - 6*b*c*d**2*e*x - 15*b*c*d*e**2*x**2 - 20*b*c*e**3*x**3)/(60*e**4*(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))