\(\int \frac {(A+B x) (a+c x^2)^2}{(d+e x)^7} \, dx\) [58]

Optimal result

Integrand size = 22, antiderivative size = 204 \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{(d+e x)^7} \, dx=\frac {(B d-A e) \left (c d^2+a e^2\right )^2}{6 e^6 (d+e x)^6}-\frac {\left (c d^2+a e^2\right ) \left (5 B c d^2-4 A c d e+a B e^2\right )}{5 e^6 (d+e x)^5}+\frac {c \left (5 B c d^3-3 A c d^2 e+3 a B d e^2-a A e^3\right )}{2 e^6 (d+e x)^4}-\frac {2 c \left (5 B c d^2-2 A c d e+a B e^2\right )}{3 e^6 (d+e x)^3}+\frac {c^2 (5 B d-A e)}{2 e^6 (d+e x)^2}-\frac {B c^2}{e^6 (d+e x)} \] Output:

1/6*(-A*e+B*d)*(a*e^2+c*d^2)^2/e^6/(e*x+d)^6-1/5*(a*e^2+c*d^2)*(-4*A*c*d*e 
+B*a*e^2+5*B*c*d^2)/e^6/(e*x+d)^5+1/2*c*(-A*a*e^3-3*A*c*d^2*e+3*B*a*d*e^2+ 
5*B*c*d^3)/e^6/(e*x+d)^4-2/3*c*(-2*A*c*d*e+B*a*e^2+5*B*c*d^2)/e^6/(e*x+d)^ 
3+1/2*c^2*(-A*e+5*B*d)/e^6/(e*x+d)^2-B*c^2/e^6/(e*x+d)
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.97 \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{(d+e x)^7} \, dx=-\frac {A e \left (5 a^2 e^4+a c e^2 \left (d^2+6 d e x+15 e^2 x^2\right )+c^2 \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )\right )+B \left (a^2 e^4 (d+6 e x)+a c e^2 \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )+5 c^2 \left (d^5+6 d^4 e x+15 d^3 e^2 x^2+20 d^2 e^3 x^3+15 d e^4 x^4+6 e^5 x^5\right )\right )}{30 e^6 (d+e x)^6} \] Input:

Integrate[((A + B*x)*(a + c*x^2)^2)/(d + e*x)^7,x]
 

Output:

-1/30*(A*e*(5*a^2*e^4 + a*c*e^2*(d^2 + 6*d*e*x + 15*e^2*x^2) + c^2*(d^4 + 
6*d^3*e*x + 15*d^2*e^2*x^2 + 20*d*e^3*x^3 + 15*e^4*x^4)) + B*(a^2*e^4*(d + 
 6*e*x) + a*c*e^2*(d^3 + 6*d^2*e*x + 15*d*e^2*x^2 + 20*e^3*x^3) + 5*c^2*(d 
^5 + 6*d^4*e*x + 15*d^3*e^2*x^2 + 20*d^2*e^3*x^3 + 15*d*e^4*x^4 + 6*e^5*x^ 
5)))/(e^6*(d + e*x)^6)
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, 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.

\(\displaystyle \int \frac {\left (a+c x^2\right )^2 (A+B x)}{(d+e x)^7} \, dx\)

\(\Big \downarrow \) 652

\(\displaystyle \int \left (-\frac {2 c \left (-a B e^2+2 A c d e-5 B c d^2\right )}{e^5 (d+e x)^4}+\frac {\left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{e^5 (d+e x)^6}+\frac {\left (a e^2+c d^2\right )^2 (A e-B d)}{e^5 (d+e x)^7}+\frac {2 c \left (a A e^3-3 a B d e^2+3 A c d^2 e-5 B c d^3\right )}{e^5 (d+e x)^5}+\frac {c^2 (A e-5 B d)}{e^5 (d+e x)^3}+\frac {B c^2}{e^5 (d+e x)^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 c \left (a B e^2-2 A c d e+5 B c d^2\right )}{3 e^6 (d+e x)^3}-\frac {\left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{5 e^6 (d+e x)^5}+\frac {\left (a e^2+c d^2\right )^2 (B d-A e)}{6 e^6 (d+e x)^6}+\frac {c \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{2 e^6 (d+e x)^4}+\frac {c^2 (5 B d-A e)}{2 e^6 (d+e x)^2}-\frac {B c^2}{e^6 (d+e x)}\)

Input:

Int[((A + B*x)*(a + c*x^2)^2)/(d + e*x)^7,x]
 

Output:

((B*d - A*e)*(c*d^2 + a*e^2)^2)/(6*e^6*(d + e*x)^6) - ((c*d^2 + a*e^2)*(5* 
B*c*d^2 - 4*A*c*d*e + a*B*e^2))/(5*e^6*(d + e*x)^5) + (c*(5*B*c*d^3 - 3*A* 
c*d^2*e + 3*a*B*d*e^2 - a*A*e^3))/(2*e^6*(d + e*x)^4) - (2*c*(5*B*c*d^2 - 
2*A*c*d*e + a*B*e^2))/(3*e^6*(d + e*x)^3) + (c^2*(5*B*d - A*e))/(2*e^6*(d 
+ e*x)^2) - (B*c^2)/(e^6*(d + e*x))
 

Defintions of rubi rules used

rule 652

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]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 0.67 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.07

method	result	size
risch	\(\frac {-\frac {B \,c^{2} x^{5}}{e}-\frac {c^{2} \left (A e +5 B d \right ) x^{4}}{2 e^{2}}-\frac {2 c \left (A c d e +B a \,e^{2}+5 B c \,d^{2}\right ) x^{3}}{3 e^{3}}-\frac {c \left (A a \,e^{3}+A c \,d^{2} e +B a d \,e^{2}+5 B c \,d^{3}\right ) x^{2}}{2 e^{4}}-\frac {\left (A a c d \,e^{3}+A \,c^{2} d^{3} e +B \,e^{4} a^{2}+B a c \,d^{2} e^{2}+5 B \,c^{2} d^{4}\right ) x}{5 e^{5}}-\frac {5 A \,a^{2} e^{5}+A a c \,d^{2} e^{3}+A \,c^{2} d^{4} e +B \,a^{2} d \,e^{4}+B a c \,d^{3} e^{2}+5 B \,c^{2} d^{5}}{30 e^{6}}}{\left (e x +d \right )^{6}}\)	\(219\)
norman	\(\frac {-\frac {B \,c^{2} x^{5}}{e}-\frac {\left (A \,c^{2} e +5 B \,c^{2} d \right ) x^{4}}{2 e^{2}}-\frac {2 \left (A \,c^{2} d e +B \,e^{2} a c +5 B \,c^{2} d^{2}\right ) x^{3}}{3 e^{3}}-\frac {\left (A a c \,e^{3}+A \,c^{2} d^{2} e +B a c d \,e^{2}+5 B \,c^{2} d^{3}\right ) x^{2}}{2 e^{4}}-\frac {\left (A a c d \,e^{3}+A \,c^{2} d^{3} e +B \,e^{4} a^{2}+B a c \,d^{2} e^{2}+5 B \,c^{2} d^{4}\right ) x}{5 e^{5}}-\frac {5 A \,a^{2} e^{5}+A a c \,d^{2} e^{3}+A \,c^{2} d^{4} e +B \,a^{2} d \,e^{4}+B a c \,d^{3} e^{2}+5 B \,c^{2} d^{5}}{30 e^{6}}}{\left (e x +d \right )^{6}}\)	\(231\)
default	\(\frac {2 c \left (2 A c d e -B a \,e^{2}-5 B c \,d^{2}\right )}{3 e^{6} \left (e x +d \right )^{3}}-\frac {c \left (A a \,e^{3}+3 A c \,d^{2} e -3 B a d \,e^{2}-5 B c \,d^{3}\right )}{2 e^{6} \left (e x +d \right )^{4}}-\frac {-4 A a c d \,e^{3}-4 A \,c^{2} d^{3} e +B \,e^{4} a^{2}+6 B a c \,d^{2} e^{2}+5 B \,c^{2} d^{4}}{5 e^{6} \left (e x +d \right )^{5}}-\frac {B \,c^{2}}{e^{6} \left (e x +d \right )}-\frac {c^{2} \left (A e -5 B d \right )}{2 e^{6} \left (e x +d \right )^{2}}-\frac {A \,a^{2} e^{5}+2 A a c \,d^{2} e^{3}+A \,c^{2} d^{4} e -B \,a^{2} d \,e^{4}-2 B a c \,d^{3} e^{2}-B \,c^{2} d^{5}}{6 e^{6} \left (e x +d \right )^{6}}\)	\(249\)
gosper	\(-\frac {30 B \,x^{5} c^{2} e^{5}+15 A \,x^{4} c^{2} e^{5}+75 B \,x^{4} c^{2} d \,e^{4}+20 A \,x^{3} c^{2} d \,e^{4}+20 B \,x^{3} a c \,e^{5}+100 B \,x^{3} c^{2} d^{2} e^{3}+15 A \,x^{2} a c \,e^{5}+15 A \,x^{2} c^{2} d^{2} e^{3}+15 B \,x^{2} a c d \,e^{4}+75 B \,x^{2} c^{2} d^{3} e^{2}+6 A x a c d \,e^{4}+6 A x \,c^{2} d^{3} e^{2}+6 B x \,a^{2} e^{5}+6 B x a c \,d^{2} e^{3}+30 B x \,c^{2} d^{4} e +5 A \,a^{2} e^{5}+A a c \,d^{2} e^{3}+A \,c^{2} d^{4} e +B \,a^{2} d \,e^{4}+B a c \,d^{3} e^{2}+5 B \,c^{2} d^{5}}{30 \left (e x +d \right )^{6} e^{6}}\)	\(255\)
parallelrisch	\(-\frac {30 B \,x^{5} c^{2} e^{5}+15 A \,x^{4} c^{2} e^{5}+75 B \,x^{4} c^{2} d \,e^{4}+20 A \,x^{3} c^{2} d \,e^{4}+20 B \,x^{3} a c \,e^{5}+100 B \,x^{3} c^{2} d^{2} e^{3}+15 A \,x^{2} a c \,e^{5}+15 A \,x^{2} c^{2} d^{2} e^{3}+15 B \,x^{2} a c d \,e^{4}+75 B \,x^{2} c^{2} d^{3} e^{2}+6 A x a c d \,e^{4}+6 A x \,c^{2} d^{3} e^{2}+6 B x \,a^{2} e^{5}+6 B x a c \,d^{2} e^{3}+30 B x \,c^{2} d^{4} e +5 A \,a^{2} e^{5}+A a c \,d^{2} e^{3}+A \,c^{2} d^{4} e +B \,a^{2} d \,e^{4}+B a c \,d^{3} e^{2}+5 B \,c^{2} d^{5}}{30 \left (e x +d \right )^{6} e^{6}}\)	\(255\)
orering	\(-\frac {30 B \,x^{5} c^{2} e^{5}+15 A \,x^{4} c^{2} e^{5}+75 B \,x^{4} c^{2} d \,e^{4}+20 A \,x^{3} c^{2} d \,e^{4}+20 B \,x^{3} a c \,e^{5}+100 B \,x^{3} c^{2} d^{2} e^{3}+15 A \,x^{2} a c \,e^{5}+15 A \,x^{2} c^{2} d^{2} e^{3}+15 B \,x^{2} a c d \,e^{4}+75 B \,x^{2} c^{2} d^{3} e^{2}+6 A x a c d \,e^{4}+6 A x \,c^{2} d^{3} e^{2}+6 B x \,a^{2} e^{5}+6 B x a c \,d^{2} e^{3}+30 B x \,c^{2} d^{4} e +5 A \,a^{2} e^{5}+A a c \,d^{2} e^{3}+A \,c^{2} d^{4} e +B \,a^{2} d \,e^{4}+B a c \,d^{3} e^{2}+5 B \,c^{2} d^{5}}{30 \left (e x +d \right )^{6} e^{6}}\)	\(255\)

Input:

int((B*x+A)*(c*x^2+a)^2/(e*x+d)^7,x,method=_RETURNVERBOSE)
 

Output:

(-B*c^2*x^5/e-1/2*c^2*(A*e+5*B*d)/e^2*x^4-2/3*c*(A*c*d*e+B*a*e^2+5*B*c*d^2 
)/e^3*x^3-1/2*c*(A*a*e^3+A*c*d^2*e+B*a*d*e^2+5*B*c*d^3)/e^4*x^2-1/5*(A*a*c 
*d*e^3+A*c^2*d^3*e+B*a^2*e^4+B*a*c*d^2*e^2+5*B*c^2*d^4)/e^5*x-1/30*(5*A*a^ 
2*e^5+A*a*c*d^2*e^3+A*c^2*d^4*e+B*a^2*d*e^4+B*a*c*d^3*e^2+5*B*c^2*d^5)/e^6 
)/(e*x+d)^6
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.43 \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{(d+e x)^7} \, dx=-\frac {30 \, B c^{2} e^{5} x^{5} + 5 \, B c^{2} d^{5} + A c^{2} d^{4} e + B a c d^{3} e^{2} + A a c d^{2} e^{3} + B a^{2} d e^{4} + 5 \, A a^{2} e^{5} + 15 \, {\left (5 \, B c^{2} d e^{4} + A c^{2} e^{5}\right )} x^{4} + 20 \, {\left (5 \, B c^{2} d^{2} e^{3} + A c^{2} d e^{4} + B a c e^{5}\right )} x^{3} + 15 \, {\left (5 \, B c^{2} d^{3} e^{2} + A c^{2} d^{2} e^{3} + B a c d e^{4} + A a c e^{5}\right )} x^{2} + 6 \, {\left (5 \, B c^{2} d^{4} e + A c^{2} d^{3} e^{2} + B a c d^{2} e^{3} + A a c d e^{4} + B a^{2} e^{5}\right )} x}{30 \, {\left (e^{12} x^{6} + 6 \, d e^{11} x^{5} + 15 \, d^{2} e^{10} x^{4} + 20 \, d^{3} e^{9} x^{3} + 15 \, d^{4} e^{8} x^{2} + 6 \, d^{5} e^{7} x + d^{6} e^{6}\right )}} \] Input:

integrate((B*x+A)*(c*x^2+a)^2/(e*x+d)^7,x, algorithm="fricas")
 

Output:

-1/30*(30*B*c^2*e^5*x^5 + 5*B*c^2*d^5 + A*c^2*d^4*e + B*a*c*d^3*e^2 + A*a* 
c*d^2*e^3 + B*a^2*d*e^4 + 5*A*a^2*e^5 + 15*(5*B*c^2*d*e^4 + A*c^2*e^5)*x^4 
 + 20*(5*B*c^2*d^2*e^3 + A*c^2*d*e^4 + B*a*c*e^5)*x^3 + 15*(5*B*c^2*d^3*e^ 
2 + A*c^2*d^2*e^3 + B*a*c*d*e^4 + A*a*c*e^5)*x^2 + 6*(5*B*c^2*d^4*e + A*c^ 
2*d^3*e^2 + B*a*c*d^2*e^3 + A*a*c*d*e^4 + B*a^2*e^5)*x)/(e^12*x^6 + 6*d*e^ 
11*x^5 + 15*d^2*e^10*x^4 + 20*d^3*e^9*x^3 + 15*d^4*e^8*x^2 + 6*d^5*e^7*x + 
 d^6*e^6)
 

Sympy [A] (verification not implemented)

Time = 70.95 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.65 \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{(d+e x)^7} \, dx=\frac {- 5 A a^{2} e^{5} - A a c d^{2} e^{3} - A c^{2} d^{4} e - B a^{2} d e^{4} - B a c d^{3} e^{2} - 5 B c^{2} d^{5} - 30 B c^{2} e^{5} x^{5} + x^{4} \left (- 15 A c^{2} e^{5} - 75 B c^{2} d e^{4}\right ) + x^{3} \left (- 20 A c^{2} d e^{4} - 20 B a c e^{5} - 100 B c^{2} d^{2} e^{3}\right ) + x^{2} \left (- 15 A a c e^{5} - 15 A c^{2} d^{2} e^{3} - 15 B a c d e^{4} - 75 B c^{2} d^{3} e^{2}\right ) + x \left (- 6 A a c d e^{4} - 6 A c^{2} d^{3} e^{2} - 6 B a^{2} e^{5} - 6 B a c d^{2} e^{3} - 30 B c^{2} d^{4} e\right )}{30 d^{6} e^{6} + 180 d^{5} e^{7} x + 450 d^{4} e^{8} x^{2} + 600 d^{3} e^{9} x^{3} + 450 d^{2} e^{10} x^{4} + 180 d e^{11} x^{5} + 30 e^{12} x^{6}} \] Input:

integrate((B*x+A)*(c*x**2+a)**2/(e*x+d)**7,x)
 

Output:

(-5*A*a**2*e**5 - A*a*c*d**2*e**3 - A*c**2*d**4*e - B*a**2*d*e**4 - B*a*c* 
d**3*e**2 - 5*B*c**2*d**5 - 30*B*c**2*e**5*x**5 + x**4*(-15*A*c**2*e**5 - 
75*B*c**2*d*e**4) + x**3*(-20*A*c**2*d*e**4 - 20*B*a*c*e**5 - 100*B*c**2*d 
**2*e**3) + x**2*(-15*A*a*c*e**5 - 15*A*c**2*d**2*e**3 - 15*B*a*c*d*e**4 - 
 75*B*c**2*d**3*e**2) + x*(-6*A*a*c*d*e**4 - 6*A*c**2*d**3*e**2 - 6*B*a**2 
*e**5 - 6*B*a*c*d**2*e**3 - 30*B*c**2*d**4*e))/(30*d**6*e**6 + 180*d**5*e* 
*7*x + 450*d**4*e**8*x**2 + 600*d**3*e**9*x**3 + 450*d**2*e**10*x**4 + 180 
*d*e**11*x**5 + 30*e**12*x**6)
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.43 \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{(d+e x)^7} \, dx=-\frac {30 \, B c^{2} e^{5} x^{5} + 5 \, B c^{2} d^{5} + A c^{2} d^{4} e + B a c d^{3} e^{2} + A a c d^{2} e^{3} + B a^{2} d e^{4} + 5 \, A a^{2} e^{5} + 15 \, {\left (5 \, B c^{2} d e^{4} + A c^{2} e^{5}\right )} x^{4} + 20 \, {\left (5 \, B c^{2} d^{2} e^{3} + A c^{2} d e^{4} + B a c e^{5}\right )} x^{3} + 15 \, {\left (5 \, B c^{2} d^{3} e^{2} + A c^{2} d^{2} e^{3} + B a c d e^{4} + A a c e^{5}\right )} x^{2} + 6 \, {\left (5 \, B c^{2} d^{4} e + A c^{2} d^{3} e^{2} + B a c d^{2} e^{3} + A a c d e^{4} + B a^{2} e^{5}\right )} x}{30 \, {\left (e^{12} x^{6} + 6 \, d e^{11} x^{5} + 15 \, d^{2} e^{10} x^{4} + 20 \, d^{3} e^{9} x^{3} + 15 \, d^{4} e^{8} x^{2} + 6 \, d^{5} e^{7} x + d^{6} e^{6}\right )}} \] Input:

integrate((B*x+A)*(c*x^2+a)^2/(e*x+d)^7,x, algorithm="maxima")
 

Output:

-1/30*(30*B*c^2*e^5*x^5 + 5*B*c^2*d^5 + A*c^2*d^4*e + B*a*c*d^3*e^2 + A*a* 
c*d^2*e^3 + B*a^2*d*e^4 + 5*A*a^2*e^5 + 15*(5*B*c^2*d*e^4 + A*c^2*e^5)*x^4 
 + 20*(5*B*c^2*d^2*e^3 + A*c^2*d*e^4 + B*a*c*e^5)*x^3 + 15*(5*B*c^2*d^3*e^ 
2 + A*c^2*d^2*e^3 + B*a*c*d*e^4 + A*a*c*e^5)*x^2 + 6*(5*B*c^2*d^4*e + A*c^ 
2*d^3*e^2 + B*a*c*d^2*e^3 + A*a*c*d*e^4 + B*a^2*e^5)*x)/(e^12*x^6 + 6*d*e^ 
11*x^5 + 15*d^2*e^10*x^4 + 20*d^3*e^9*x^3 + 15*d^4*e^8*x^2 + 6*d^5*e^7*x + 
 d^6*e^6)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.25 \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{(d+e x)^7} \, dx=-\frac {30 \, B c^{2} e^{5} x^{5} + 75 \, B c^{2} d e^{4} x^{4} + 15 \, A c^{2} e^{5} x^{4} + 100 \, B c^{2} d^{2} e^{3} x^{3} + 20 \, A c^{2} d e^{4} x^{3} + 20 \, B a c e^{5} x^{3} + 75 \, B c^{2} d^{3} e^{2} x^{2} + 15 \, A c^{2} d^{2} e^{3} x^{2} + 15 \, B a c d e^{4} x^{2} + 15 \, A a c e^{5} x^{2} + 30 \, B c^{2} d^{4} e x + 6 \, A c^{2} d^{3} e^{2} x + 6 \, B a c d^{2} e^{3} x + 6 \, A a c d e^{4} x + 6 \, B a^{2} e^{5} x + 5 \, B c^{2} d^{5} + A c^{2} d^{4} e + B a c d^{3} e^{2} + A a c d^{2} e^{3} + B a^{2} d e^{4} + 5 \, A a^{2} e^{5}}{30 \, {\left (e x + d\right )}^{6} e^{6}} \] Input:

integrate((B*x+A)*(c*x^2+a)^2/(e*x+d)^7,x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

-1/30*(30*B*c^2*e^5*x^5 + 75*B*c^2*d*e^4*x^4 + 15*A*c^2*e^5*x^4 + 100*B*c^ 
2*d^2*e^3*x^3 + 20*A*c^2*d*e^4*x^3 + 20*B*a*c*e^5*x^3 + 75*B*c^2*d^3*e^2*x 
^2 + 15*A*c^2*d^2*e^3*x^2 + 15*B*a*c*d*e^4*x^2 + 15*A*a*c*e^5*x^2 + 30*B*c 
^2*d^4*e*x + 6*A*c^2*d^3*e^2*x + 6*B*a*c*d^2*e^3*x + 6*A*a*c*d*e^4*x + 6*B 
*a^2*e^5*x + 5*B*c^2*d^5 + A*c^2*d^4*e + B*a*c*d^3*e^2 + A*a*c*d^2*e^3 + B 
*a^2*d*e^4 + 5*A*a^2*e^5)/((e*x + d)^6*e^6)
 

Mupad [B] (verification not implemented)

Time = 6.26 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.34 \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{(d+e x)^7} \, dx=-\frac {\frac {B\,a^2\,d\,e^4+5\,A\,a^2\,e^5+B\,a\,c\,d^3\,e^2+A\,a\,c\,d^2\,e^3+5\,B\,c^2\,d^5+A\,c^2\,d^4\,e}{30\,e^6}+\frac {x\,\left (B\,a^2\,e^4+B\,a\,c\,d^2\,e^2+A\,a\,c\,d\,e^3+5\,B\,c^2\,d^4+A\,c^2\,d^3\,e\right )}{5\,e^5}+\frac {2\,c\,x^3\,\left (5\,B\,c\,d^2+A\,c\,d\,e+B\,a\,e^2\right )}{3\,e^3}+\frac {c^2\,x^4\,\left (A\,e+5\,B\,d\right )}{2\,e^2}+\frac {c\,x^2\,\left (5\,B\,c\,d^3+A\,c\,d^2\,e+B\,a\,d\,e^2+A\,a\,e^3\right )}{2\,e^4}+\frac {B\,c^2\,x^5}{e}}{d^6+6\,d^5\,e\,x+15\,d^4\,e^2\,x^2+20\,d^3\,e^3\,x^3+15\,d^2\,e^4\,x^4+6\,d\,e^5\,x^5+e^6\,x^6} \] Input:

int(((a + c*x^2)^2*(A + B*x))/(d + e*x)^7,x)
 

Output:

-((5*A*a^2*e^5 + 5*B*c^2*d^5 + B*a^2*d*e^4 + A*c^2*d^4*e + A*a*c*d^2*e^3 + 
 B*a*c*d^3*e^2)/(30*e^6) + (x*(B*a^2*e^4 + 5*B*c^2*d^4 + A*c^2*d^3*e + A*a 
*c*d*e^3 + B*a*c*d^2*e^2))/(5*e^5) + (2*c*x^3*(B*a*e^2 + 5*B*c*d^2 + A*c*d 
*e))/(3*e^3) + (c^2*x^4*(A*e + 5*B*d))/(2*e^2) + (c*x^2*(A*a*e^3 + 5*B*c*d 
^3 + B*a*d*e^2 + A*c*d^2*e))/(2*e^4) + (B*c^2*x^5)/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)
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.29 \[ \int \frac {(A+B x) \left (a+c x^2\right )^2}{(d+e x)^7} \, dx=\frac {5 b \,c^{2} e^{5} x^{6}-15 a \,c^{2} d \,e^{4} x^{4}-20 a b c d \,e^{4} x^{3}-20 a \,c^{2} d^{2} e^{3} x^{3}-15 a^{2} c d \,e^{4} x^{2}-15 a b c \,d^{2} e^{3} x^{2}-15 a \,c^{2} d^{3} e^{2} x^{2}-6 a^{2} b d \,e^{4} x -6 a^{2} c \,d^{2} e^{3} x -6 a b c \,d^{3} e^{2} x -6 a \,c^{2} d^{4} e x -5 a^{3} d \,e^{4}-a^{2} b \,d^{2} e^{3}-a^{2} c \,d^{3} e^{2}-a b c \,d^{4} e -a \,c^{2} d^{5}}{30 d \,e^{5} \left (e^{6} x^{6}+6 d \,e^{5} x^{5}+15 d^{2} e^{4} x^{4}+20 d^{3} e^{3} x^{3}+15 d^{4} e^{2} x^{2}+6 d^{5} e x +d^{6}\right )} \] Input:

int((B*x+A)*(c*x^2+a)^2/(e*x+d)^7,x)
 

Output:

( - 5*a**3*d*e**4 - a**2*b*d**2*e**3 - 6*a**2*b*d*e**4*x - a**2*c*d**3*e** 
2 - 6*a**2*c*d**2*e**3*x - 15*a**2*c*d*e**4*x**2 - a*b*c*d**4*e - 6*a*b*c* 
d**3*e**2*x - 15*a*b*c*d**2*e**3*x**2 - 20*a*b*c*d*e**4*x**3 - a*c**2*d**5 
 - 6*a*c**2*d**4*e*x - 15*a*c**2*d**3*e**2*x**2 - 20*a*c**2*d**2*e**3*x**3 
 - 15*a*c**2*d*e**4*x**4 + 5*b*c**2*e**5*x**6)/(30*d*e**5*(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))