\(\int (A+B x) (d+e x) (a+c x^2)^3 \, dx\) [66]

Optimal result

Integrand size = 20, antiderivative size = 148 \[ \int (A+B x) (d+e x) \left (a+c x^2\right )^3 \, dx=a^3 A d x+\frac {1}{2} a^3 (B d+A e) x^2+\frac {1}{3} a^2 (3 A c d+a B e) x^3+\frac {3}{4} a^2 c (B d+A e) x^4+\frac {3}{5} a c (A c d+a B e) x^5+\frac {1}{2} a c^2 (B d+A e) x^6+\frac {1}{7} c^2 (A c d+3 a B e) x^7+\frac {1}{8} c^3 (B d+A e) x^8+\frac {1}{9} B c^3 e x^9 \] Output:

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.91 \[ \int (A+B x) (d+e x) \left (a+c x^2\right )^3 \, dx=\frac {1}{6} a^3 x (3 A (2 d+e x)+B x (3 d+2 e x))+\frac {1}{20} a^2 c x^3 (5 A (4 d+3 e x)+3 B x (5 d+4 e x))+\frac {1}{70} a c^2 x^5 (7 A (6 d+5 e x)+5 B x (7 d+6 e x))+\frac {1}{504} c^3 x^7 (9 A (8 d+7 e x)+7 B x (9 d+8 e x)) \] Input:

Integrate[(A + B*x)*(d + e*x)*(a + c*x^2)^3,x]
 

Output:

(a^3*x*(3*A*(2*d + e*x) + B*x*(3*d + 2*e*x)))/6 + (a^2*c*x^3*(5*A*(4*d + 3 
*e*x) + 3*B*x*(5*d + 4*e*x)))/20 + (a*c^2*x^5*(7*A*(6*d + 5*e*x) + 5*B*x*( 
7*d + 6*e*x)))/70 + (c^3*x^7*(9*A*(8*d + 7*e*x) + 7*B*x*(9*d + 8*e*x)))/50 
4
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 148, 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.

\(\displaystyle \int \left (a+c x^2\right )^3 (A+B x) (d+e x) \, dx\)

\(\Big \downarrow \) 652

\(\displaystyle \int \left (a^3 x (A e+B d)+a^3 A d+3 a^2 c x^3 (A e+B d)+a^2 x^2 (a B e+3 A c d)+c^2 x^6 (3 a B e+A c d)+3 a c^2 x^5 (A e+B d)+3 a c x^4 (a B e+A c d)+c^3 x^7 (A e+B d)+B c^3 e x^8\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} a^3 x^2 (A e+B d)+a^3 A d x+\frac {3}{4} a^2 c x^4 (A e+B d)+\frac {1}{3} a^2 x^3 (a B e+3 A c d)+\frac {1}{7} c^2 x^7 (3 a B e+A c d)+\frac {1}{2} a c^2 x^6 (A e+B d)+\frac {3}{5} a c x^5 (a B e+A c d)+\frac {1}{8} c^3 x^8 (A e+B d)+\frac {1}{9} B c^3 e x^9\)

Input:

Int[(A + B*x)*(d + e*x)*(a + c*x^2)^3,x]
 

Output:

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

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 = 0.51 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.97

method	result	size
default	\(\frac {B \,c^{3} e \,x^{9}}{9}+\frac {c^{3} \left (A e +B d \right ) x^{8}}{8}+\frac {\left (A \,c^{3} d +3 B e a \,c^{2}\right ) x^{7}}{7}+\frac {a \,c^{2} \left (A e +B d \right ) x^{6}}{2}+\frac {\left (3 A d a \,c^{2}+3 B e \,a^{2} c \right ) x^{5}}{5}+\frac {3 a^{2} c \left (A e +B d \right ) x^{4}}{4}+\frac {\left (3 A \,a^{2} c d +B e \,a^{3}\right ) x^{3}}{3}+\frac {a^{3} \left (A e +B d \right ) x^{2}}{2}+a^{3} A d x\)	\(143\)
norman	\(\frac {B \,c^{3} e \,x^{9}}{9}+\left (\frac {1}{8} A \,c^{3} e +\frac {1}{8} B \,c^{3} d \right ) x^{8}+\left (\frac {1}{7} A \,c^{3} d +\frac {3}{7} B e a \,c^{2}\right ) x^{7}+\left (\frac {1}{2} A a \,c^{2} e +\frac {1}{2} B a \,c^{2} d \right ) x^{6}+\left (\frac {3}{5} A d a \,c^{2}+\frac {3}{5} B e \,a^{2} c \right ) x^{5}+\left (\frac {3}{4} A \,a^{2} c e +\frac {3}{4} B \,a^{2} c d \right ) x^{4}+\left (A \,a^{2} c d +\frac {1}{3} B e \,a^{3}\right ) x^{3}+\left (\frac {1}{2} A \,a^{3} e +\frac {1}{2} B \,a^{3} d \right ) x^{2}+a^{3} A d x\)	\(159\)
gosper	\(\frac {1}{9} B \,c^{3} e \,x^{9}+\frac {1}{8} x^{8} A \,c^{3} e +\frac {1}{8} x^{8} B \,c^{3} d +\frac {1}{7} x^{7} A \,c^{3} d +\frac {3}{7} x^{7} B e a \,c^{2}+\frac {1}{2} x^{6} A a \,c^{2} e +\frac {1}{2} x^{6} B a \,c^{2} d +\frac {3}{5} x^{5} A d a \,c^{2}+\frac {3}{5} x^{5} B e \,a^{2} c +\frac {3}{4} x^{4} A \,a^{2} c e +\frac {3}{4} x^{4} B \,a^{2} c d +x^{3} A \,a^{2} c d +\frac {1}{3} x^{3} B e \,a^{3}+\frac {1}{2} x^{2} A \,a^{3} e +\frac {1}{2} x^{2} B \,a^{3} d +a^{3} A d x\)	\(166\)
risch	\(\frac {1}{9} B \,c^{3} e \,x^{9}+\frac {1}{8} x^{8} A \,c^{3} e +\frac {1}{8} x^{8} B \,c^{3} d +\frac {1}{7} x^{7} A \,c^{3} d +\frac {3}{7} x^{7} B e a \,c^{2}+\frac {1}{2} x^{6} A a \,c^{2} e +\frac {1}{2} x^{6} B a \,c^{2} d +\frac {3}{5} x^{5} A d a \,c^{2}+\frac {3}{5} x^{5} B e \,a^{2} c +\frac {3}{4} x^{4} A \,a^{2} c e +\frac {3}{4} x^{4} B \,a^{2} c d +x^{3} A \,a^{2} c d +\frac {1}{3} x^{3} B e \,a^{3}+\frac {1}{2} x^{2} A \,a^{3} e +\frac {1}{2} x^{2} B \,a^{3} d +a^{3} A d x\)	\(166\)
parallelrisch	\(\frac {1}{9} B \,c^{3} e \,x^{9}+\frac {1}{8} x^{8} A \,c^{3} e +\frac {1}{8} x^{8} B \,c^{3} d +\frac {1}{7} x^{7} A \,c^{3} d +\frac {3}{7} x^{7} B e a \,c^{2}+\frac {1}{2} x^{6} A a \,c^{2} e +\frac {1}{2} x^{6} B a \,c^{2} d +\frac {3}{5} x^{5} A d a \,c^{2}+\frac {3}{5} x^{5} B e \,a^{2} c +\frac {3}{4} x^{4} A \,a^{2} c e +\frac {3}{4} x^{4} B \,a^{2} c d +x^{3} A \,a^{2} c d +\frac {1}{3} x^{3} B e \,a^{3}+\frac {1}{2} x^{2} A \,a^{3} e +\frac {1}{2} x^{2} B \,a^{3} d +a^{3} A d x\)	\(166\)
orering	\(\frac {x \left (280 B e \,c^{3} x^{8}+315 A \,c^{3} e \,x^{7}+315 B \,c^{3} d \,x^{7}+360 A \,c^{3} d \,x^{6}+1080 B a \,c^{2} e \,x^{6}+1260 A a \,c^{2} e \,x^{5}+1260 B a \,c^{2} d \,x^{5}+1512 A a \,c^{2} d \,x^{4}+1512 B \,a^{2} c e \,x^{4}+1890 A \,a^{2} c e \,x^{3}+1890 B \,a^{2} c d \,x^{3}+2520 A \,a^{2} c d \,x^{2}+840 B \,a^{3} e \,x^{2}+1260 A \,a^{3} e x +1260 B \,a^{3} d x +2520 A d \,a^{3}\right )}{2520}\)	\(166\)

Input:

int((B*x+A)*(e*x+d)*(c*x^2+a)^3,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.04 \[ \int (A+B x) (d+e x) \left (a+c x^2\right )^3 \, dx=\frac {1}{9} \, B c^{3} e x^{9} + \frac {1}{8} \, {\left (B c^{3} d + A c^{3} e\right )} x^{8} + \frac {1}{7} \, {\left (A c^{3} d + 3 \, B a c^{2} e\right )} x^{7} + \frac {1}{2} \, {\left (B a c^{2} d + A a c^{2} e\right )} x^{6} + A a^{3} d x + \frac {3}{5} \, {\left (A a c^{2} d + B a^{2} c e\right )} x^{5} + \frac {3}{4} \, {\left (B a^{2} c d + A a^{2} c e\right )} x^{4} + \frac {1}{3} \, {\left (3 \, A a^{2} c d + B a^{3} e\right )} x^{3} + \frac {1}{2} \, {\left (B a^{3} d + A a^{3} e\right )} x^{2} \] Input:

integrate((B*x+A)*(e*x+d)*(c*x^2+a)^3,x, algorithm="fricas")
 

Output:

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

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.23 \[ \int (A+B x) (d+e x) \left (a+c x^2\right )^3 \, dx=A a^{3} d x + \frac {B c^{3} e x^{9}}{9} + x^{8} \left (\frac {A c^{3} e}{8} + \frac {B c^{3} d}{8}\right ) + x^{7} \left (\frac {A c^{3} d}{7} + \frac {3 B a c^{2} e}{7}\right ) + x^{6} \left (\frac {A a c^{2} e}{2} + \frac {B a c^{2} d}{2}\right ) + x^{5} \cdot \left (\frac {3 A a c^{2} d}{5} + \frac {3 B a^{2} c e}{5}\right ) + x^{4} \cdot \left (\frac {3 A a^{2} c e}{4} + \frac {3 B a^{2} c d}{4}\right ) + x^{3} \left (A a^{2} c d + \frac {B a^{3} e}{3}\right ) + x^{2} \left (\frac {A a^{3} e}{2} + \frac {B a^{3} d}{2}\right ) \] Input:

integrate((B*x+A)*(e*x+d)*(c*x**2+a)**3,x)
 

Output:

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

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.04 \[ \int (A+B x) (d+e x) \left (a+c x^2\right )^3 \, dx=\frac {1}{9} \, B c^{3} e x^{9} + \frac {1}{8} \, {\left (B c^{3} d + A c^{3} e\right )} x^{8} + \frac {1}{7} \, {\left (A c^{3} d + 3 \, B a c^{2} e\right )} x^{7} + \frac {1}{2} \, {\left (B a c^{2} d + A a c^{2} e\right )} x^{6} + A a^{3} d x + \frac {3}{5} \, {\left (A a c^{2} d + B a^{2} c e\right )} x^{5} + \frac {3}{4} \, {\left (B a^{2} c d + A a^{2} c e\right )} x^{4} + \frac {1}{3} \, {\left (3 \, A a^{2} c d + B a^{3} e\right )} x^{3} + \frac {1}{2} \, {\left (B a^{3} d + A a^{3} e\right )} x^{2} \] Input:

integrate((B*x+A)*(e*x+d)*(c*x^2+a)^3,x, algorithm="maxima")
 

Output:

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

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.11 \[ \int (A+B x) (d+e x) \left (a+c x^2\right )^3 \, dx=\frac {1}{9} \, B c^{3} e x^{9} + \frac {1}{8} \, B c^{3} d x^{8} + \frac {1}{8} \, A c^{3} e x^{8} + \frac {1}{7} \, A c^{3} d x^{7} + \frac {3}{7} \, B a c^{2} e x^{7} + \frac {1}{2} \, B a c^{2} d x^{6} + \frac {1}{2} \, A a c^{2} e x^{6} + \frac {3}{5} \, A a c^{2} d x^{5} + \frac {3}{5} \, B a^{2} c e x^{5} + \frac {3}{4} \, B a^{2} c d x^{4} + \frac {3}{4} \, A a^{2} c e x^{4} + A a^{2} c d x^{3} + \frac {1}{3} \, B a^{3} e x^{3} + \frac {1}{2} \, B a^{3} d x^{2} + \frac {1}{2} \, A a^{3} e x^{2} + A a^{3} d x \] Input:

integrate((B*x+A)*(e*x+d)*(c*x^2+a)^3,x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

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

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.95 \[ \int (A+B x) (d+e x) \left (a+c x^2\right )^3 \, dx=x^3\,\left (\frac {B\,e\,a^3}{3}+A\,c\,d\,a^2\right )+x^7\,\left (\frac {A\,d\,c^3}{7}+\frac {3\,B\,a\,e\,c^2}{7}\right )+x^5\,\left (\frac {3\,B\,e\,a^2\,c}{5}+\frac {3\,A\,d\,a\,c^2}{5}\right )+\frac {a^3\,x^2\,\left (A\,e+B\,d\right )}{2}+\frac {c^3\,x^8\,\left (A\,e+B\,d\right )}{8}+A\,a^3\,d\,x+\frac {B\,c^3\,e\,x^9}{9}+\frac {3\,a^2\,c\,x^4\,\left (A\,e+B\,d\right )}{4}+\frac {a\,c^2\,x^6\,\left (A\,e+B\,d\right )}{2} \] Input:

int((a + c*x^2)^3*(A + B*x)*(d + e*x),x)
 

Output:

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

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.10 \[ \int (A+B x) (d+e x) \left (a+c x^2\right )^3 \, dx=\frac {x \left (280 b \,c^{3} e \,x^{8}+315 a \,c^{3} e \,x^{7}+315 b \,c^{3} d \,x^{7}+1080 a b \,c^{2} e \,x^{6}+360 a \,c^{3} d \,x^{6}+1260 a^{2} c^{2} e \,x^{5}+1260 a b \,c^{2} d \,x^{5}+1512 a^{2} b c e \,x^{4}+1512 a^{2} c^{2} d \,x^{4}+1890 a^{3} c e \,x^{3}+1890 a^{2} b c d \,x^{3}+840 a^{3} b e \,x^{2}+2520 a^{3} c d \,x^{2}+1260 a^{4} e x +1260 a^{3} b d x +2520 a^{4} d \right )}{2520} \] Input:

int((B*x+A)*(e*x+d)*(c*x^2+a)^3,x)
 

Output:

(x*(2520*a**4*d + 1260*a**4*e*x + 1260*a**3*b*d*x + 840*a**3*b*e*x**2 + 25 
20*a**3*c*d*x**2 + 1890*a**3*c*e*x**3 + 1890*a**2*b*c*d*x**3 + 1512*a**2*b 
*c*e*x**4 + 1512*a**2*c**2*d*x**4 + 1260*a**2*c**2*e*x**5 + 1260*a*b*c**2* 
d*x**5 + 1080*a*b*c**2*e*x**6 + 360*a*c**3*d*x**6 + 315*a*c**3*e*x**7 + 31 
5*b*c**3*d*x**7 + 280*b*c**3*e*x**8))/2520