\(\int (A+B x) (d+e x)^5 (a+c x^2) \, dx\) [33]

Optimal result

Integrand size = 20, antiderivative size = 108 \[ \int (A+B x) (d+e x)^5 \left (a+c x^2\right ) \, dx=-\frac {(B d-A e) \left (c d^2+a e^2\right ) (d+e x)^6}{6 e^4}+\frac {\left (3 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^7}{7 e^4}-\frac {c (3 B d-A e) (d+e x)^8}{8 e^4}+\frac {B c (d+e x)^9}{9 e^4} \] Output:

-1/6*(-A*e+B*d)*(a*e^2+c*d^2)*(e*x+d)^6/e^4+1/7*(-2*A*c*d*e+B*a*e^2+3*B*c* 
d^2)*(e*x+d)^7/e^4-1/8*c*(-A*e+3*B*d)*(e*x+d)^8/e^4+1/9*B*c*(e*x+d)^9/e^4
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(233\) vs. \(2(108)=216\).

Time = 0.06 (sec) , antiderivative size = 233, normalized size of antiderivative = 2.16 \[ \int (A+B x) (d+e x)^5 \left (a+c x^2\right ) \, dx=a A d^5 x+\frac {1}{2} a d^4 (B d+5 A e) x^2+\frac {1}{3} d^3 \left (A c d^2+5 a B d e+10 a A e^2\right ) x^3+\frac {1}{4} d^2 \left (B c d^3+5 A c d^2 e+10 a B d e^2+10 a A e^3\right ) x^4+d e \left (B c d^3+2 A c d^2 e+2 a B d e^2+a A e^3\right ) x^5+\frac {1}{6} e^2 \left (10 B c d^3+10 A c d^2 e+5 a B d e^2+a A e^3\right ) x^6+\frac {1}{7} e^3 \left (10 B c d^2+5 A c d e+a B e^2\right ) x^7+\frac {1}{8} c e^4 (5 B d+A e) x^8+\frac {1}{9} B c e^5 x^9 \] Input:

Integrate[(A + B*x)*(d + e*x)^5*(a + c*x^2),x]
 

Output:

a*A*d^5*x + (a*d^4*(B*d + 5*A*e)*x^2)/2 + (d^3*(A*c*d^2 + 5*a*B*d*e + 10*a 
*A*e^2)*x^3)/3 + (d^2*(B*c*d^3 + 5*A*c*d^2*e + 10*a*B*d*e^2 + 10*a*A*e^3)* 
x^4)/4 + d*e*(B*c*d^3 + 2*A*c*d^2*e + 2*a*B*d*e^2 + a*A*e^3)*x^5 + (e^2*(1 
0*B*c*d^3 + 10*A*c*d^2*e + 5*a*B*d*e^2 + a*A*e^3)*x^6)/6 + (e^3*(10*B*c*d^ 
2 + 5*A*c*d*e + a*B*e^2)*x^7)/7 + (c*e^4*(5*B*d + A*e)*x^8)/8 + (B*c*e^5*x 
^9)/9
 

Rubi [A] (verified)

Time = 0.35 (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.

\(\displaystyle \int \left (a+c x^2\right ) (A+B x) (d+e x)^5 \, dx\)

\(\Big \downarrow \) 652

\(\displaystyle \int \left (\frac {(d+e x)^6 \left (a B e^2-2 A c d e+3 B c d^2\right )}{e^3}+\frac {(d+e x)^5 \left (a e^2+c d^2\right ) (A e-B d)}{e^3}+\frac {c (d+e x)^7 (A e-3 B d)}{e^3}+\frac {B c (d+e x)^8}{e^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {(d+e x)^7 \left (a B e^2-2 A c d e+3 B c d^2\right )}{7 e^4}-\frac {(d+e x)^6 \left (a e^2+c d^2\right ) (B d-A e)}{6 e^4}-\frac {c (d+e x)^8 (3 B d-A e)}{8 e^4}+\frac {B c (d+e x)^9}{9 e^4}\)

Input:

Int[(A + B*x)*(d + e*x)^5*(a + c*x^2),x]
 

Output:

-1/6*((B*d - A*e)*(c*d^2 + a*e^2)*(d + e*x)^6)/e^4 + ((3*B*c*d^2 - 2*A*c*d 
*e + a*B*e^2)*(d + e*x)^7)/(7*e^4) - (c*(3*B*d - A*e)*(d + e*x)^8)/(8*e^4) 
 + (B*c*(d + e*x)^9)/(9*e^4)
 

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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(237\) vs. \(2(100)=200\).

Time = 0.50 (sec) , antiderivative size = 238, normalized size of antiderivative = 2.20

method	result	size
norman	\(\frac {B \,e^{5} c \,x^{9}}{9}+\left (\frac {1}{8} A c \,e^{5}+\frac {5}{8} B c d \,e^{4}\right ) x^{8}+\left (\frac {5}{7} A c d \,e^{4}+\frac {1}{7} B \,e^{5} a +\frac {10}{7} B c \,d^{2} e^{3}\right ) x^{7}+\left (\frac {1}{6} A a \,e^{5}+\frac {5}{3} A c \,d^{2} e^{3}+\frac {5}{6} B a d \,e^{4}+\frac {5}{3} B c \,d^{3} e^{2}\right ) x^{6}+\left (A a d \,e^{4}+2 A c \,d^{3} e^{2}+2 B a \,d^{2} e^{3}+B c \,d^{4} e \right ) x^{5}+\left (\frac {5}{2} A a \,d^{2} e^{3}+\frac {5}{4} A c \,d^{4} e +\frac {5}{2} B a \,d^{3} e^{2}+\frac {1}{4} B c \,d^{5}\right ) x^{4}+\left (\frac {10}{3} A a \,d^{3} e^{2}+\frac {1}{3} A \,d^{5} c +\frac {5}{3} B a \,d^{4} e \right ) x^{3}+\left (\frac {5}{2} A a \,d^{4} e +\frac {1}{2} B a \,d^{5}\right ) x^{2}+A \,d^{5} a x\)	\(238\)
default	\(\frac {B \,e^{5} c \,x^{9}}{9}+\frac {\left (A \,e^{5}+5 B d \,e^{4}\right ) c \,x^{8}}{8}+\frac {\left (\left (5 A d \,e^{4}+10 B \,d^{2} e^{3}\right ) c +B \,e^{5} a \right ) x^{7}}{7}+\frac {\left (\left (10 A \,d^{2} e^{3}+10 B \,d^{3} e^{2}\right ) c +\left (A \,e^{5}+5 B d \,e^{4}\right ) a \right ) x^{6}}{6}+\frac {\left (\left (10 A \,d^{3} e^{2}+5 B \,d^{4} e \right ) c +\left (5 A d \,e^{4}+10 B \,d^{2} e^{3}\right ) a \right ) x^{5}}{5}+\frac {\left (\left (5 A \,d^{4} e +B \,d^{5}\right ) c +\left (10 A \,d^{2} e^{3}+10 B \,d^{3} e^{2}\right ) a \right ) x^{4}}{4}+\frac {\left (A \,d^{5} c +\left (10 A \,d^{3} e^{2}+5 B \,d^{4} e \right ) a \right ) x^{3}}{3}+\frac {\left (5 A \,d^{4} e +B \,d^{5}\right ) a \,x^{2}}{2}+A \,d^{5} a x\)	\(247\)
gosper	\(\frac {1}{9} B \,e^{5} c \,x^{9}+\frac {1}{8} x^{8} A c \,e^{5}+\frac {5}{8} x^{8} B c d \,e^{4}+\frac {5}{7} x^{7} A c d \,e^{4}+\frac {1}{7} x^{7} B \,e^{5} a +\frac {10}{7} x^{7} B c \,d^{2} e^{3}+\frac {1}{6} x^{6} A a \,e^{5}+\frac {5}{3} x^{6} A c \,d^{2} e^{3}+\frac {5}{6} x^{6} B a d \,e^{4}+\frac {5}{3} x^{6} B c \,d^{3} e^{2}+A a d \,e^{4} x^{5}+2 A c \,d^{3} e^{2} x^{5}+2 B a \,d^{2} e^{3} x^{5}+B c \,d^{4} e \,x^{5}+\frac {5}{2} x^{4} A a \,d^{2} e^{3}+\frac {5}{4} x^{4} A c \,d^{4} e +\frac {5}{2} x^{4} B a \,d^{3} e^{2}+\frac {1}{4} B c \,d^{5} x^{4}+\frac {10}{3} x^{3} A a \,d^{3} e^{2}+\frac {1}{3} A c \,d^{5} x^{3}+\frac {5}{3} x^{3} B a \,d^{4} e +\frac {5}{2} x^{2} A a \,d^{4} e +\frac {1}{2} x^{2} B a \,d^{5}+A \,d^{5} a x\)	\(269\)
risch	\(\frac {1}{9} B \,e^{5} c \,x^{9}+\frac {1}{8} x^{8} A c \,e^{5}+\frac {5}{8} x^{8} B c d \,e^{4}+\frac {5}{7} x^{7} A c d \,e^{4}+\frac {1}{7} x^{7} B \,e^{5} a +\frac {10}{7} x^{7} B c \,d^{2} e^{3}+\frac {1}{6} x^{6} A a \,e^{5}+\frac {5}{3} x^{6} A c \,d^{2} e^{3}+\frac {5}{6} x^{6} B a d \,e^{4}+\frac {5}{3} x^{6} B c \,d^{3} e^{2}+A a d \,e^{4} x^{5}+2 A c \,d^{3} e^{2} x^{5}+2 B a \,d^{2} e^{3} x^{5}+B c \,d^{4} e \,x^{5}+\frac {5}{2} x^{4} A a \,d^{2} e^{3}+\frac {5}{4} x^{4} A c \,d^{4} e +\frac {5}{2} x^{4} B a \,d^{3} e^{2}+\frac {1}{4} B c \,d^{5} x^{4}+\frac {10}{3} x^{3} A a \,d^{3} e^{2}+\frac {1}{3} A c \,d^{5} x^{3}+\frac {5}{3} x^{3} B a \,d^{4} e +\frac {5}{2} x^{2} A a \,d^{4} e +\frac {1}{2} x^{2} B a \,d^{5}+A \,d^{5} a x\)	\(269\)
parallelrisch	\(\frac {1}{9} B \,e^{5} c \,x^{9}+\frac {1}{8} x^{8} A c \,e^{5}+\frac {5}{8} x^{8} B c d \,e^{4}+\frac {5}{7} x^{7} A c d \,e^{4}+\frac {1}{7} x^{7} B \,e^{5} a +\frac {10}{7} x^{7} B c \,d^{2} e^{3}+\frac {1}{6} x^{6} A a \,e^{5}+\frac {5}{3} x^{6} A c \,d^{2} e^{3}+\frac {5}{6} x^{6} B a d \,e^{4}+\frac {5}{3} x^{6} B c \,d^{3} e^{2}+A a d \,e^{4} x^{5}+2 A c \,d^{3} e^{2} x^{5}+2 B a \,d^{2} e^{3} x^{5}+B c \,d^{4} e \,x^{5}+\frac {5}{2} x^{4} A a \,d^{2} e^{3}+\frac {5}{4} x^{4} A c \,d^{4} e +\frac {5}{2} x^{4} B a \,d^{3} e^{2}+\frac {1}{4} B c \,d^{5} x^{4}+\frac {10}{3} x^{3} A a \,d^{3} e^{2}+\frac {1}{3} A c \,d^{5} x^{3}+\frac {5}{3} x^{3} B a \,d^{4} e +\frac {5}{2} x^{2} A a \,d^{4} e +\frac {1}{2} x^{2} B a \,d^{5}+A \,d^{5} a x\)	\(269\)
orering	\(\frac {x \left (56 B \,e^{5} c \,x^{8}+63 A c \,e^{5} x^{7}+315 B c d \,e^{4} x^{7}+360 A c d \,e^{4} x^{6}+72 B a \,e^{5} x^{6}+720 B c \,d^{2} e^{3} x^{6}+84 A a \,e^{5} x^{5}+840 A c \,d^{2} e^{3} x^{5}+420 B a d \,e^{4} x^{5}+840 B c \,d^{3} e^{2} x^{5}+504 A a d \,e^{4} x^{4}+1008 A c \,d^{3} e^{2} x^{4}+1008 B a \,d^{2} e^{3} x^{4}+504 B c \,d^{4} e \,x^{4}+1260 A a \,d^{2} e^{3} x^{3}+630 A c \,d^{4} e \,x^{3}+1260 B a \,d^{3} e^{2} x^{3}+126 B c \,d^{5} x^{3}+1680 A a \,d^{3} e^{2} x^{2}+168 A c \,d^{5} x^{2}+840 B a \,d^{4} e \,x^{2}+1260 A a \,d^{4} e x +252 B a \,d^{5} x +504 A \,d^{5} a \right )}{504}\)	\(270\)

Input:

int((B*x+A)*(e*x+d)^5*(c*x^2+a),x,method=_RETURNVERBOSE)
 

Output:

1/9*B*e^5*c*x^9+(1/8*A*c*e^5+5/8*B*c*d*e^4)*x^8+(5/7*A*c*d*e^4+1/7*B*e^5*a 
+10/7*B*c*d^2*e^3)*x^7+(1/6*A*a*e^5+5/3*A*c*d^2*e^3+5/6*B*a*d*e^4+5/3*B*c* 
d^3*e^2)*x^6+(A*a*d*e^4+2*A*c*d^3*e^2+2*B*a*d^2*e^3+B*c*d^4*e)*x^5+(5/2*A* 
a*d^2*e^3+5/4*A*c*d^4*e+5/2*B*a*d^3*e^2+1/4*B*c*d^5)*x^4+(10/3*A*a*d^3*e^2 
+1/3*A*d^5*c+5/3*B*a*d^4*e)*x^3+(5/2*A*a*d^4*e+1/2*B*a*d^5)*x^2+A*d^5*a*x
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 237 vs. \(2 (100) = 200\).

Time = 0.07 (sec) , antiderivative size = 237, normalized size of antiderivative = 2.19 \[ \int (A+B x) (d+e x)^5 \left (a+c x^2\right ) \, dx=\frac {1}{9} \, B c e^{5} x^{9} + \frac {1}{8} \, {\left (5 \, B c d e^{4} + A c e^{5}\right )} x^{8} + A a d^{5} x + \frac {1}{7} \, {\left (10 \, B c d^{2} e^{3} + 5 \, A c d e^{4} + B a e^{5}\right )} x^{7} + \frac {1}{6} \, {\left (10 \, B c d^{3} e^{2} + 10 \, A c d^{2} e^{3} + 5 \, B a d e^{4} + A a e^{5}\right )} x^{6} + {\left (B c d^{4} e + 2 \, A c d^{3} e^{2} + 2 \, B a d^{2} e^{3} + A a d e^{4}\right )} x^{5} + \frac {1}{4} \, {\left (B c d^{5} + 5 \, A c d^{4} e + 10 \, B a d^{3} e^{2} + 10 \, A a d^{2} e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (A c d^{5} + 5 \, B a d^{4} e + 10 \, A a d^{3} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B a d^{5} + 5 \, A a d^{4} e\right )} x^{2} \] Input:

integrate((B*x+A)*(e*x+d)^5*(c*x^2+a),x, algorithm="fricas")
 

Output:

1/9*B*c*e^5*x^9 + 1/8*(5*B*c*d*e^4 + A*c*e^5)*x^8 + A*a*d^5*x + 1/7*(10*B* 
c*d^2*e^3 + 5*A*c*d*e^4 + B*a*e^5)*x^7 + 1/6*(10*B*c*d^3*e^2 + 10*A*c*d^2* 
e^3 + 5*B*a*d*e^4 + A*a*e^5)*x^6 + (B*c*d^4*e + 2*A*c*d^3*e^2 + 2*B*a*d^2* 
e^3 + A*a*d*e^4)*x^5 + 1/4*(B*c*d^5 + 5*A*c*d^4*e + 10*B*a*d^3*e^2 + 10*A* 
a*d^2*e^3)*x^4 + 1/3*(A*c*d^5 + 5*B*a*d^4*e + 10*A*a*d^3*e^2)*x^3 + 1/2*(B 
*a*d^5 + 5*A*a*d^4*e)*x^2
                                                                                    
                                                                                    
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 287 vs. \(2 (105) = 210\).

Time = 0.04 (sec) , antiderivative size = 287, normalized size of antiderivative = 2.66 \[ \int (A+B x) (d+e x)^5 \left (a+c x^2\right ) \, dx=A a d^{5} x + \frac {B c e^{5} x^{9}}{9} + x^{8} \left (\frac {A c e^{5}}{8} + \frac {5 B c d e^{4}}{8}\right ) + x^{7} \cdot \left (\frac {5 A c d e^{4}}{7} + \frac {B a e^{5}}{7} + \frac {10 B c d^{2} e^{3}}{7}\right ) + x^{6} \left (\frac {A a e^{5}}{6} + \frac {5 A c d^{2} e^{3}}{3} + \frac {5 B a d e^{4}}{6} + \frac {5 B c d^{3} e^{2}}{3}\right ) + x^{5} \left (A a d e^{4} + 2 A c d^{3} e^{2} + 2 B a d^{2} e^{3} + B c d^{4} e\right ) + x^{4} \cdot \left (\frac {5 A a d^{2} e^{3}}{2} + \frac {5 A c d^{4} e}{4} + \frac {5 B a d^{3} e^{2}}{2} + \frac {B c d^{5}}{4}\right ) + x^{3} \cdot \left (\frac {10 A a d^{3} e^{2}}{3} + \frac {A c d^{5}}{3} + \frac {5 B a d^{4} e}{3}\right ) + x^{2} \cdot \left (\frac {5 A a d^{4} e}{2} + \frac {B a d^{5}}{2}\right ) \] Input:

integrate((B*x+A)*(e*x+d)**5*(c*x**2+a),x)
 

Output:

A*a*d**5*x + B*c*e**5*x**9/9 + x**8*(A*c*e**5/8 + 5*B*c*d*e**4/8) + x**7*( 
5*A*c*d*e**4/7 + B*a*e**5/7 + 10*B*c*d**2*e**3/7) + x**6*(A*a*e**5/6 + 5*A 
*c*d**2*e**3/3 + 5*B*a*d*e**4/6 + 5*B*c*d**3*e**2/3) + x**5*(A*a*d*e**4 + 
2*A*c*d**3*e**2 + 2*B*a*d**2*e**3 + B*c*d**4*e) + x**4*(5*A*a*d**2*e**3/2 
+ 5*A*c*d**4*e/4 + 5*B*a*d**3*e**2/2 + B*c*d**5/4) + x**3*(10*A*a*d**3*e** 
2/3 + A*c*d**5/3 + 5*B*a*d**4*e/3) + x**2*(5*A*a*d**4*e/2 + B*a*d**5/2)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 237 vs. \(2 (100) = 200\).

Time = 0.03 (sec) , antiderivative size = 237, normalized size of antiderivative = 2.19 \[ \int (A+B x) (d+e x)^5 \left (a+c x^2\right ) \, dx=\frac {1}{9} \, B c e^{5} x^{9} + \frac {1}{8} \, {\left (5 \, B c d e^{4} + A c e^{5}\right )} x^{8} + A a d^{5} x + \frac {1}{7} \, {\left (10 \, B c d^{2} e^{3} + 5 \, A c d e^{4} + B a e^{5}\right )} x^{7} + \frac {1}{6} \, {\left (10 \, B c d^{3} e^{2} + 10 \, A c d^{2} e^{3} + 5 \, B a d e^{4} + A a e^{5}\right )} x^{6} + {\left (B c d^{4} e + 2 \, A c d^{3} e^{2} + 2 \, B a d^{2} e^{3} + A a d e^{4}\right )} x^{5} + \frac {1}{4} \, {\left (B c d^{5} + 5 \, A c d^{4} e + 10 \, B a d^{3} e^{2} + 10 \, A a d^{2} e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (A c d^{5} + 5 \, B a d^{4} e + 10 \, A a d^{3} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B a d^{5} + 5 \, A a d^{4} e\right )} x^{2} \] Input:

integrate((B*x+A)*(e*x+d)^5*(c*x^2+a),x, algorithm="maxima")
 

Output:

1/9*B*c*e^5*x^9 + 1/8*(5*B*c*d*e^4 + A*c*e^5)*x^8 + A*a*d^5*x + 1/7*(10*B* 
c*d^2*e^3 + 5*A*c*d*e^4 + B*a*e^5)*x^7 + 1/6*(10*B*c*d^3*e^2 + 10*A*c*d^2* 
e^3 + 5*B*a*d*e^4 + A*a*e^5)*x^6 + (B*c*d^4*e + 2*A*c*d^3*e^2 + 2*B*a*d^2* 
e^3 + A*a*d*e^4)*x^5 + 1/4*(B*c*d^5 + 5*A*c*d^4*e + 10*B*a*d^3*e^2 + 10*A* 
a*d^2*e^3)*x^4 + 1/3*(A*c*d^5 + 5*B*a*d^4*e + 10*A*a*d^3*e^2)*x^3 + 1/2*(B 
*a*d^5 + 5*A*a*d^4*e)*x^2
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 268 vs. \(2 (100) = 200\).

Time = 0.11 (sec) , antiderivative size = 268, normalized size of antiderivative = 2.48 \[ \int (A+B x) (d+e x)^5 \left (a+c x^2\right ) \, dx=\frac {1}{9} \, B c e^{5} x^{9} + \frac {5}{8} \, B c d e^{4} x^{8} + \frac {1}{8} \, A c e^{5} x^{8} + \frac {10}{7} \, B c d^{2} e^{3} x^{7} + \frac {5}{7} \, A c d e^{4} x^{7} + \frac {1}{7} \, B a e^{5} x^{7} + \frac {5}{3} \, B c d^{3} e^{2} x^{6} + \frac {5}{3} \, A c d^{2} e^{3} x^{6} + \frac {5}{6} \, B a d e^{4} x^{6} + \frac {1}{6} \, A a e^{5} x^{6} + B c d^{4} e x^{5} + 2 \, A c d^{3} e^{2} x^{5} + 2 \, B a d^{2} e^{3} x^{5} + A a d e^{4} x^{5} + \frac {1}{4} \, B c d^{5} x^{4} + \frac {5}{4} \, A c d^{4} e x^{4} + \frac {5}{2} \, B a d^{3} e^{2} x^{4} + \frac {5}{2} \, A a d^{2} e^{3} x^{4} + \frac {1}{3} \, A c d^{5} x^{3} + \frac {5}{3} \, B a d^{4} e x^{3} + \frac {10}{3} \, A a d^{3} e^{2} x^{3} + \frac {1}{2} \, B a d^{5} x^{2} + \frac {5}{2} \, A a d^{4} e x^{2} + A a d^{5} x \] Input:

integrate((B*x+A)*(e*x+d)^5*(c*x^2+a),x, algorithm="giac")
 

Output:

1/9*B*c*e^5*x^9 + 5/8*B*c*d*e^4*x^8 + 1/8*A*c*e^5*x^8 + 10/7*B*c*d^2*e^3*x 
^7 + 5/7*A*c*d*e^4*x^7 + 1/7*B*a*e^5*x^7 + 5/3*B*c*d^3*e^2*x^6 + 5/3*A*c*d 
^2*e^3*x^6 + 5/6*B*a*d*e^4*x^6 + 1/6*A*a*e^5*x^6 + B*c*d^4*e*x^5 + 2*A*c*d 
^3*e^2*x^5 + 2*B*a*d^2*e^3*x^5 + A*a*d*e^4*x^5 + 1/4*B*c*d^5*x^4 + 5/4*A*c 
*d^4*e*x^4 + 5/2*B*a*d^3*e^2*x^4 + 5/2*A*a*d^2*e^3*x^4 + 1/3*A*c*d^5*x^3 + 
 5/3*B*a*d^4*e*x^3 + 10/3*A*a*d^3*e^2*x^3 + 1/2*B*a*d^5*x^2 + 5/2*A*a*d^4* 
e*x^2 + A*a*d^5*x
 

Mupad [B] (verification not implemented)

Time = 5.68 (sec) , antiderivative size = 231, normalized size of antiderivative = 2.14 \[ \int (A+B x) (d+e x)^5 \left (a+c x^2\right ) \, dx=x^5\,\left (B\,c\,d^4\,e+2\,A\,c\,d^3\,e^2+2\,B\,a\,d^2\,e^3+A\,a\,d\,e^4\right )+x^3\,\left (\frac {A\,c\,d^5}{3}+\frac {5\,B\,a\,d^4\,e}{3}+\frac {10\,A\,a\,d^3\,e^2}{3}\right )+x^7\,\left (\frac {10\,B\,c\,d^2\,e^3}{7}+\frac {5\,A\,c\,d\,e^4}{7}+\frac {B\,a\,e^5}{7}\right )+x^4\,\left (\frac {B\,c\,d^5}{4}+\frac {5\,A\,c\,d^4\,e}{4}+\frac {5\,B\,a\,d^3\,e^2}{2}+\frac {5\,A\,a\,d^2\,e^3}{2}\right )+x^6\,\left (\frac {5\,B\,c\,d^3\,e^2}{3}+\frac {5\,A\,c\,d^2\,e^3}{3}+\frac {5\,B\,a\,d\,e^4}{6}+\frac {A\,a\,e^5}{6}\right )+A\,a\,d^5\,x+\frac {B\,c\,e^5\,x^9}{9}+\frac {a\,d^4\,x^2\,\left (5\,A\,e+B\,d\right )}{2}+\frac {c\,e^4\,x^8\,\left (A\,e+5\,B\,d\right )}{8} \] Input:

int((a + c*x^2)*(A + B*x)*(d + e*x)^5,x)
 

Output:

x^5*(A*a*d*e^4 + B*c*d^4*e + 2*B*a*d^2*e^3 + 2*A*c*d^3*e^2) + x^3*((A*c*d^ 
5)/3 + (5*B*a*d^4*e)/3 + (10*A*a*d^3*e^2)/3) + x^7*((B*a*e^5)/7 + (5*A*c*d 
*e^4)/7 + (10*B*c*d^2*e^3)/7) + x^4*((B*c*d^5)/4 + (5*A*c*d^4*e)/4 + (5*A* 
a*d^2*e^3)/2 + (5*B*a*d^3*e^2)/2) + x^6*((A*a*e^5)/6 + (5*B*a*d*e^4)/6 + ( 
5*A*c*d^2*e^3)/3 + (5*B*c*d^3*e^2)/3) + A*a*d^5*x + (B*c*e^5*x^9)/9 + (a*d 
^4*x^2*(5*A*e + B*d))/2 + (c*e^4*x^8*(A*e + 5*B*d))/8
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 275, normalized size of antiderivative = 2.55 \[ \int (A+B x) (d+e x)^5 \left (a+c x^2\right ) \, dx=\frac {x \left (56 b c \,e^{5} x^{8}+63 a c \,e^{5} x^{7}+315 b c d \,e^{4} x^{7}+72 a b \,e^{5} x^{6}+360 a c d \,e^{4} x^{6}+720 b c \,d^{2} e^{3} x^{6}+84 a^{2} e^{5} x^{5}+420 a b d \,e^{4} x^{5}+840 a c \,d^{2} e^{3} x^{5}+840 b c \,d^{3} e^{2} x^{5}+504 a^{2} d \,e^{4} x^{4}+1008 a b \,d^{2} e^{3} x^{4}+1008 a c \,d^{3} e^{2} x^{4}+504 b c \,d^{4} e \,x^{4}+1260 a^{2} d^{2} e^{3} x^{3}+1260 a b \,d^{3} e^{2} x^{3}+630 a c \,d^{4} e \,x^{3}+126 b c \,d^{5} x^{3}+1680 a^{2} d^{3} e^{2} x^{2}+840 a b \,d^{4} e \,x^{2}+168 a c \,d^{5} x^{2}+1260 a^{2} d^{4} e x +252 a b \,d^{5} x +504 a^{2} d^{5}\right )}{504} \] Input:

int((B*x+A)*(e*x+d)^5*(c*x^2+a),x)
 

Output:

(x*(504*a**2*d**5 + 1260*a**2*d**4*e*x + 1680*a**2*d**3*e**2*x**2 + 1260*a 
**2*d**2*e**3*x**3 + 504*a**2*d*e**4*x**4 + 84*a**2*e**5*x**5 + 252*a*b*d* 
*5*x + 840*a*b*d**4*e*x**2 + 1260*a*b*d**3*e**2*x**3 + 1008*a*b*d**2*e**3* 
x**4 + 420*a*b*d*e**4*x**5 + 72*a*b*e**5*x**6 + 168*a*c*d**5*x**2 + 630*a* 
c*d**4*e*x**3 + 1008*a*c*d**3*e**2*x**4 + 840*a*c*d**2*e**3*x**5 + 360*a*c 
*d*e**4*x**6 + 63*a*c*e**5*x**7 + 126*b*c*d**5*x**3 + 504*b*c*d**4*e*x**4 
+ 840*b*c*d**3*e**2*x**5 + 720*b*c*d**2*e**3*x**6 + 315*b*c*d*e**4*x**7 + 
56*b*c*e**5*x**8))/504