Integrand size = 20, antiderivative size = 108 \[ \int (A+B x) (d+e x)^3 \left (a+c x^2\right ) \, dx=-\frac {(B d-A e) \left (c d^2+a e^2\right ) (d+e x)^4}{4 e^4}+\frac {\left (3 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^5}{5 e^4}-\frac {c (3 B d-A e) (d+e x)^6}{6 e^4}+\frac {B c (d+e x)^7}{7 e^4} \] Output:
-1/4*(-A*e+B*d)*(a*e^2+c*d^2)*(e*x+d)^4/e^4+1/5*(-2*A*c*d*e+B*a*e^2+3*B*c* d^2)*(e*x+d)^5/e^4-1/6*c*(-A*e+3*B*d)*(e*x+d)^6/e^4+1/7*B*c*(e*x+d)^7/e^4
Time = 0.04 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.39 \[ \int (A+B x) (d+e x)^3 \left (a+c x^2\right ) \, dx=a A d^3 x+\frac {1}{2} a d^2 (B d+3 A e) x^2+\frac {1}{3} d \left (A c d^2+3 a B d e+3 a A e^2\right ) x^3+\frac {1}{4} \left (B c d^3+3 A c d^2 e+3 a B d e^2+a A e^3\right ) x^4+\frac {1}{5} e \left (3 B c d^2+3 A c d e+a B e^2\right ) x^5+\frac {1}{6} c e^2 (3 B d+A e) x^6+\frac {1}{7} B c e^3 x^7 \] Input:
Integrate[(A + B*x)*(d + e*x)^3*(a + c*x^2),x]
Output:
a*A*d^3*x + (a*d^2*(B*d + 3*A*e)*x^2)/2 + (d*(A*c*d^2 + 3*a*B*d*e + 3*a*A* e^2)*x^3)/3 + ((B*c*d^3 + 3*A*c*d^2*e + 3*a*B*d*e^2 + a*A*e^3)*x^4)/4 + (e *(3*B*c*d^2 + 3*A*c*d*e + a*B*e^2)*x^5)/5 + (c*e^2*(3*B*d + A*e)*x^6)/6 + (B*c*e^3*x^7)/7
Time = 0.30 (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.
| \(\displaystyle \int \left (a+c x^2\right ) (A+B x) (d+e x)^3 \, dx\) |
| \(\Big \downarrow \) 652 |
| \(\displaystyle \int \left (\frac {(d+e x)^4 \left (a B e^2-2 A c d e+3 B c d^2\right )}{e^3}+\frac {(d+e x)^3 \left (a e^2+c d^2\right ) (A e-B d)}{e^3}+\frac {c (d+e x)^5 (A e-3 B d)}{e^3}+\frac {B c (d+e x)^6}{e^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(d+e x)^5 \left (a B e^2-2 A c d e+3 B c d^2\right )}{5 e^4}-\frac {(d+e x)^4 \left (a e^2+c d^2\right ) (B d-A e)}{4 e^4}-\frac {c (d+e x)^6 (3 B d-A e)}{6 e^4}+\frac {B c (d+e x)^7}{7 e^4}\) |
Input:
Int[(A + B*x)*(d + e*x)^3*(a + c*x^2),x]
Output:
-1/4*((B*d - A*e)*(c*d^2 + a*e^2)*(d + e*x)^4)/e^4 + ((3*B*c*d^2 - 2*A*c*d *e + a*B*e^2)*(d + e*x)^5)/(5*e^4) - (c*(3*B*d - A*e)*(d + e*x)^6)/(6*e^4) + (B*c*(d + e*x)^7)/(7*e^4)
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 = 0.52 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.37
| method | result | size |
| norman | \(\frac {B \,e^{3} c \,x^{7}}{7}+\left (\frac {1}{6} A c \,e^{3}+\frac {1}{2} B c d \,e^{2}\right ) x^{6}+\left (\frac {3}{5} A c d \,e^{2}+\frac {1}{5} B \,e^{3} a +\frac {3}{5} B c \,d^{2} e \right ) x^{5}+\left (\frac {1}{4} A a \,e^{3}+\frac {3}{4} A c \,d^{2} e +\frac {3}{4} B a d \,e^{2}+\frac {1}{4} B c \,d^{3}\right ) x^{4}+\left (A a d \,e^{2}+\frac {1}{3} A c \,d^{3}+B a \,d^{2} e \right ) x^{3}+\left (\frac {3}{2} A a \,d^{2} e +\frac {1}{2} B a \,d^{3}\right ) x^{2}+A \,d^{3} a x\) | \(148\) |
| default | \(\frac {B \,e^{3} c \,x^{7}}{7}+\frac {\left (A \,e^{3}+3 B d \,e^{2}\right ) c \,x^{6}}{6}+\frac {\left (\left (3 A d \,e^{2}+3 B \,d^{2} e \right ) c +B \,e^{3} a \right ) x^{5}}{5}+\frac {\left (\left (3 A \,d^{2} e +B \,d^{3}\right ) c +\left (A \,e^{3}+3 B d \,e^{2}\right ) a \right ) x^{4}}{4}+\frac {\left (A c \,d^{3}+\left (3 A d \,e^{2}+3 B \,d^{2} e \right ) a \right ) x^{3}}{3}+\frac {\left (3 A \,d^{2} e +B \,d^{3}\right ) a \,x^{2}}{2}+A \,d^{3} a x\) | \(151\) |
| gosper | \(\frac {1}{7} B \,e^{3} c \,x^{7}+\frac {1}{6} x^{6} A c \,e^{3}+\frac {1}{2} x^{6} B c d \,e^{2}+\frac {3}{5} x^{5} A c d \,e^{2}+\frac {1}{5} x^{5} B \,e^{3} a +\frac {3}{5} x^{5} B c \,d^{2} e +\frac {1}{4} x^{4} A a \,e^{3}+\frac {3}{4} x^{4} A c \,d^{2} e +\frac {3}{4} x^{4} B a d \,e^{2}+\frac {1}{4} x^{4} B c \,d^{3}+x^{3} A a d \,e^{2}+\frac {1}{3} x^{3} A c \,d^{3}+x^{3} B a \,d^{2} e +\frac {3}{2} x^{2} A a \,d^{2} e +\frac {1}{2} x^{2} B a \,d^{3}+A \,d^{3} a x\) | \(165\) |
| risch | \(\frac {1}{7} B \,e^{3} c \,x^{7}+\frac {1}{6} x^{6} A c \,e^{3}+\frac {1}{2} x^{6} B c d \,e^{2}+\frac {3}{5} x^{5} A c d \,e^{2}+\frac {1}{5} x^{5} B \,e^{3} a +\frac {3}{5} x^{5} B c \,d^{2} e +\frac {1}{4} x^{4} A a \,e^{3}+\frac {3}{4} x^{4} A c \,d^{2} e +\frac {3}{4} x^{4} B a d \,e^{2}+\frac {1}{4} x^{4} B c \,d^{3}+x^{3} A a d \,e^{2}+\frac {1}{3} x^{3} A c \,d^{3}+x^{3} B a \,d^{2} e +\frac {3}{2} x^{2} A a \,d^{2} e +\frac {1}{2} x^{2} B a \,d^{3}+A \,d^{3} a x\) | \(165\) |
| parallelrisch | \(\frac {1}{7} B \,e^{3} c \,x^{7}+\frac {1}{6} x^{6} A c \,e^{3}+\frac {1}{2} x^{6} B c d \,e^{2}+\frac {3}{5} x^{5} A c d \,e^{2}+\frac {1}{5} x^{5} B \,e^{3} a +\frac {3}{5} x^{5} B c \,d^{2} e +\frac {1}{4} x^{4} A a \,e^{3}+\frac {3}{4} x^{4} A c \,d^{2} e +\frac {3}{4} x^{4} B a d \,e^{2}+\frac {1}{4} x^{4} B c \,d^{3}+x^{3} A a d \,e^{2}+\frac {1}{3} x^{3} A c \,d^{3}+x^{3} B a \,d^{2} e +\frac {3}{2} x^{2} A a \,d^{2} e +\frac {1}{2} x^{2} B a \,d^{3}+A \,d^{3} a x\) | \(165\) |
| orering | \(\frac {x \left (60 B \,e^{3} c \,x^{6}+70 A c \,e^{3} x^{5}+210 B c d \,e^{2} x^{5}+252 A c d \,e^{2} x^{4}+84 B a \,e^{3} x^{4}+252 B c \,d^{2} e \,x^{4}+105 A a \,e^{3} x^{3}+315 A c \,d^{2} e \,x^{3}+315 B a d \,e^{2} x^{3}+105 B c \,d^{3} x^{3}+420 A a d \,e^{2} x^{2}+140 A c \,d^{3} x^{2}+420 B a \,d^{2} e \,x^{2}+630 A a \,d^{2} e x +210 B a \,d^{3} x +420 A \,d^{3} a \right )}{420}\) | \(166\) |
Input:
int((B*x+A)*(e*x+d)^3*(c*x^2+a),x,method=_RETURNVERBOSE)
Output:
1/7*B*e^3*c*x^7+(1/6*A*c*e^3+1/2*B*c*d*e^2)*x^6+(3/5*A*c*d*e^2+1/5*B*e^3*a +3/5*B*c*d^2*e)*x^5+(1/4*A*a*e^3+3/4*A*c*d^2*e+3/4*B*a*d*e^2+1/4*B*c*d^3)* x^4+(A*a*d*e^2+1/3*A*c*d^3+B*a*d^2*e)*x^3+(3/2*A*a*d^2*e+1/2*B*a*d^3)*x^2+ A*d^3*a*x
Time = 0.07 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.37 \[ \int (A+B x) (d+e x)^3 \left (a+c x^2\right ) \, dx=\frac {1}{7} \, B c e^{3} x^{7} + \frac {1}{6} \, {\left (3 \, B c d e^{2} + A c e^{3}\right )} x^{6} + A a d^{3} x + \frac {1}{5} \, {\left (3 \, B c d^{2} e + 3 \, A c d e^{2} + B a e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (B c d^{3} + 3 \, A c d^{2} e + 3 \, B a d e^{2} + A a e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (A c d^{3} + 3 \, B a d^{2} e + 3 \, A a d e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B a d^{3} + 3 \, A a d^{2} e\right )} x^{2} \] Input:
integrate((B*x+A)*(e*x+d)^3*(c*x^2+a),x, algorithm="fricas")
Output:
1/7*B*c*e^3*x^7 + 1/6*(3*B*c*d*e^2 + A*c*e^3)*x^6 + A*a*d^3*x + 1/5*(3*B*c *d^2*e + 3*A*c*d*e^2 + B*a*e^3)*x^5 + 1/4*(B*c*d^3 + 3*A*c*d^2*e + 3*B*a*d *e^2 + A*a*e^3)*x^4 + 1/3*(A*c*d^3 + 3*B*a*d^2*e + 3*A*a*d*e^2)*x^3 + 1/2* (B*a*d^3 + 3*A*a*d^2*e)*x^2
Time = 0.03 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.60 \[ \int (A+B x) (d+e x)^3 \left (a+c x^2\right ) \, dx=A a d^{3} x + \frac {B c e^{3} x^{7}}{7} + x^{6} \left (\frac {A c e^{3}}{6} + \frac {B c d e^{2}}{2}\right ) + x^{5} \cdot \left (\frac {3 A c d e^{2}}{5} + \frac {B a e^{3}}{5} + \frac {3 B c d^{2} e}{5}\right ) + x^{4} \left (\frac {A a e^{3}}{4} + \frac {3 A c d^{2} e}{4} + \frac {3 B a d e^{2}}{4} + \frac {B c d^{3}}{4}\right ) + x^{3} \left (A a d e^{2} + \frac {A c d^{3}}{3} + B a d^{2} e\right ) + x^{2} \cdot \left (\frac {3 A a d^{2} e}{2} + \frac {B a d^{3}}{2}\right ) \] Input:
integrate((B*x+A)*(e*x+d)**3*(c*x**2+a),x)
Output:
A*a*d**3*x + B*c*e**3*x**7/7 + x**6*(A*c*e**3/6 + B*c*d*e**2/2) + x**5*(3* A*c*d*e**2/5 + B*a*e**3/5 + 3*B*c*d**2*e/5) + x**4*(A*a*e**3/4 + 3*A*c*d** 2*e/4 + 3*B*a*d*e**2/4 + B*c*d**3/4) + x**3*(A*a*d*e**2 + A*c*d**3/3 + B*a *d**2*e) + x**2*(3*A*a*d**2*e/2 + B*a*d**3/2)
Time = 0.04 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.37 \[ \int (A+B x) (d+e x)^3 \left (a+c x^2\right ) \, dx=\frac {1}{7} \, B c e^{3} x^{7} + \frac {1}{6} \, {\left (3 \, B c d e^{2} + A c e^{3}\right )} x^{6} + A a d^{3} x + \frac {1}{5} \, {\left (3 \, B c d^{2} e + 3 \, A c d e^{2} + B a e^{3}\right )} x^{5} + \frac {1}{4} \, {\left (B c d^{3} + 3 \, A c d^{2} e + 3 \, B a d e^{2} + A a e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (A c d^{3} + 3 \, B a d^{2} e + 3 \, A a d e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B a d^{3} + 3 \, A a d^{2} e\right )} x^{2} \] Input:
integrate((B*x+A)*(e*x+d)^3*(c*x^2+a),x, algorithm="maxima")
Output:
1/7*B*c*e^3*x^7 + 1/6*(3*B*c*d*e^2 + A*c*e^3)*x^6 + A*a*d^3*x + 1/5*(3*B*c *d^2*e + 3*A*c*d*e^2 + B*a*e^3)*x^5 + 1/4*(B*c*d^3 + 3*A*c*d^2*e + 3*B*a*d *e^2 + A*a*e^3)*x^4 + 1/3*(A*c*d^3 + 3*B*a*d^2*e + 3*A*a*d*e^2)*x^3 + 1/2* (B*a*d^3 + 3*A*a*d^2*e)*x^2
Time = 0.13 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.52 \[ \int (A+B x) (d+e x)^3 \left (a+c x^2\right ) \, dx=\frac {1}{7} \, B c e^{3} x^{7} + \frac {1}{2} \, B c d e^{2} x^{6} + \frac {1}{6} \, A c e^{3} x^{6} + \frac {3}{5} \, B c d^{2} e x^{5} + \frac {3}{5} \, A c d e^{2} x^{5} + \frac {1}{5} \, B a e^{3} x^{5} + \frac {1}{4} \, B c d^{3} x^{4} + \frac {3}{4} \, A c d^{2} e x^{4} + \frac {3}{4} \, B a d e^{2} x^{4} + \frac {1}{4} \, A a e^{3} x^{4} + \frac {1}{3} \, A c d^{3} x^{3} + B a d^{2} e x^{3} + A a d e^{2} x^{3} + \frac {1}{2} \, B a d^{3} x^{2} + \frac {3}{2} \, A a d^{2} e x^{2} + A a d^{3} x \] Input:
integrate((B*x+A)*(e*x+d)^3*(c*x^2+a),x, algorithm="giac")
Output:
1/7*B*c*e^3*x^7 + 1/2*B*c*d*e^2*x^6 + 1/6*A*c*e^3*x^6 + 3/5*B*c*d^2*e*x^5 + 3/5*A*c*d*e^2*x^5 + 1/5*B*a*e^3*x^5 + 1/4*B*c*d^3*x^4 + 3/4*A*c*d^2*e*x^ 4 + 3/4*B*a*d*e^2*x^4 + 1/4*A*a*e^3*x^4 + 1/3*A*c*d^3*x^3 + B*a*d^2*e*x^3 + A*a*d*e^2*x^3 + 1/2*B*a*d^3*x^2 + 3/2*A*a*d^2*e*x^2 + A*a*d^3*x
Time = 0.03 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.31 \[ \int (A+B x) (d+e x)^3 \left (a+c x^2\right ) \, dx=x^4\,\left (\frac {B\,c\,d^3}{4}+\frac {3\,A\,c\,d^2\,e}{4}+\frac {3\,B\,a\,d\,e^2}{4}+\frac {A\,a\,e^3}{4}\right )+x^3\,\left (\frac {A\,c\,d^3}{3}+B\,a\,d^2\,e+A\,a\,d\,e^2\right )+x^5\,\left (\frac {3\,B\,c\,d^2\,e}{5}+\frac {3\,A\,c\,d\,e^2}{5}+\frac {B\,a\,e^3}{5}\right )+A\,a\,d^3\,x+\frac {B\,c\,e^3\,x^7}{7}+\frac {a\,d^2\,x^2\,\left (3\,A\,e+B\,d\right )}{2}+\frac {c\,e^2\,x^6\,\left (A\,e+3\,B\,d\right )}{6} \] Input:
int((a + c*x^2)*(A + B*x)*(d + e*x)^3,x)
Output:
x^4*((A*a*e^3)/4 + (B*c*d^3)/4 + (3*B*a*d*e^2)/4 + (3*A*c*d^2*e)/4) + x^3* ((A*c*d^3)/3 + A*a*d*e^2 + B*a*d^2*e) + x^5*((B*a*e^3)/5 + (3*A*c*d*e^2)/5 + (3*B*c*d^2*e)/5) + A*a*d^3*x + (B*c*e^3*x^7)/7 + (a*d^2*x^2*(3*A*e + B* d))/2 + (c*e^2*x^6*(A*e + 3*B*d))/6
Time = 0.26 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.56 \[ \int (A+B x) (d+e x)^3 \left (a+c x^2\right ) \, dx=\frac {x \left (60 b c \,e^{3} x^{6}+70 a c \,e^{3} x^{5}+210 b c d \,e^{2} x^{5}+84 a b \,e^{3} x^{4}+252 a c d \,e^{2} x^{4}+252 b c \,d^{2} e \,x^{4}+105 a^{2} e^{3} x^{3}+315 a b d \,e^{2} x^{3}+315 a c \,d^{2} e \,x^{3}+105 b c \,d^{3} x^{3}+420 a^{2} d \,e^{2} x^{2}+420 a b \,d^{2} e \,x^{2}+140 a c \,d^{3} x^{2}+630 a^{2} d^{2} e x +210 a b \,d^{3} x +420 a^{2} d^{3}\right )}{420} \] Input:
int((B*x+A)*(e*x+d)^3*(c*x^2+a),x)
Output:
(x*(420*a**2*d**3 + 630*a**2*d**2*e*x + 420*a**2*d*e**2*x**2 + 105*a**2*e* *3*x**3 + 210*a*b*d**3*x + 420*a*b*d**2*e*x**2 + 315*a*b*d*e**2*x**3 + 84* a*b*e**3*x**4 + 140*a*c*d**3*x**2 + 315*a*c*d**2*e*x**3 + 252*a*c*d*e**2*x **4 + 70*a*c*e**3*x**5 + 105*b*c*d**3*x**3 + 252*b*c*d**2*e*x**4 + 210*b*c *d*e**2*x**5 + 60*b*c*e**3*x**6))/420