Integrand size = 28, antiderivative size = 110 \[ \int (c+d x) \left (a+b x^2\right ) \left (A+B x+C x^2+D x^3\right ) \, dx=a A c x+\frac {1}{2} a (B c+A d) x^2+\frac {1}{3} (A b c+a c C+a B d) x^3+\frac {1}{4} (b B c+A b d+a C d+a c D) x^4+\frac {1}{5} (b c C+b B d+a d D) x^5+\frac {1}{6} b (C d+c D) x^6+\frac {1}{7} b d D x^7 \] Output:
a*A*c*x+1/2*a*(A*d+B*c)*x^2+1/3*(A*b*c+B*a*d+C*a*c)*x^3+1/4*(A*b*d+B*b*c+C *a*d+D*a*c)*x^4+1/5*(B*b*d+C*b*c+D*a*d)*x^5+1/6*b*(C*d+D*c)*x^6+1/7*b*d*D* x^7
Time = 0.06 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00 \[ \int (c+d x) \left (a+b x^2\right ) \left (A+B x+C x^2+D x^3\right ) \, dx=a A c x+\frac {1}{2} a (B c+A d) x^2+\frac {1}{3} (A b c+a c C+a B d) x^3+\frac {1}{4} (b B c+A b d+a C d+a c D) x^4+\frac {1}{5} (b c C+b B d+a d D) x^5+\frac {1}{6} b (C d+c D) x^6+\frac {1}{7} b d D x^7 \] Input:
Integrate[(c + d*x)*(a + b*x^2)*(A + B*x + C*x^2 + D*x^3),x]
Output:
a*A*c*x + (a*(B*c + A*d)*x^2)/2 + ((A*b*c + a*c*C + a*B*d)*x^3)/3 + ((b*B* c + A*b*d + a*C*d + a*c*D)*x^4)/4 + ((b*c*C + b*B*d + a*d*D)*x^5)/5 + (b*( C*d + c*D)*x^6)/6 + (b*d*D*x^7)/7
Time = 0.36 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2160, 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.
\(\displaystyle \int \left (a+b x^2\right ) (c+d x) \left (A+B x+C x^2+D x^3\right ) \, dx\) |
\(\Big \downarrow \) 2160 |
\(\displaystyle \int \left (x^3 (a c D+a C d+A b d+b B c)+x^2 (a B d+a c C+A b c)+a x (A d+B c)+a A c+x^4 (a d D+b B d+b c C)+b x^5 (c D+C d)+b d D x^6\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{4} x^4 (a c D+a C d+A b d+b B c)+\frac {1}{3} x^3 (a B d+a c C+A b c)+\frac {1}{2} a x^2 (A d+B c)+a A c x+\frac {1}{5} x^5 (a d D+b B d+b c C)+\frac {1}{6} b x^6 (c D+C d)+\frac {1}{7} b d D x^7\) |
Input:
Int[(c + d*x)*(a + b*x^2)*(A + B*x + C*x^2 + D*x^3),x]
Output:
a*A*c*x + (a*(B*c + A*d)*x^2)/2 + ((A*b*c + a*c*C + a*B*d)*x^3)/3 + ((b*B* c + A*b*d + a*C*d + a*c*D)*x^4)/4 + ((b*c*C + b*B*d + a*d*D)*x^5)/5 + (b*( C*d + c*D)*x^6)/6 + (b*d*D*x^7)/7
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
Time = 0.22 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.92
method | result | size |
default | \(\frac {b d D x^{7}}{7}+\frac {\left (b d C +b c D\right ) x^{6}}{6}+\frac {\left (B b d +b c C +D a d \right ) x^{5}}{5}+\frac {\left (A b d +B b c +C a d +D a c \right ) x^{4}}{4}+\frac {\left (A b c +B a d +a c C \right ) x^{3}}{3}+\frac {\left (A a d +a B c \right ) x^{2}}{2}+A a c x\) | \(101\) |
norman | \(\frac {b d D x^{7}}{7}+\left (\frac {1}{6} b d C +\frac {1}{6} b c D\right ) x^{6}+\left (\frac {1}{5} B b d +\frac {1}{5} b c C +\frac {1}{5} D a d \right ) x^{5}+\left (\frac {1}{4} A b d +\frac {1}{4} B b c +\frac {1}{4} C a d +\frac {1}{4} D a c \right ) x^{4}+\left (\frac {1}{3} A b c +\frac {1}{3} B a d +\frac {1}{3} a c C \right ) x^{3}+\left (\frac {1}{2} A a d +\frac {1}{2} a B c \right ) x^{2}+A a c x\) | \(110\) |
orering | \(\frac {x \left (60 D b d \,x^{6}+70 C b d \,x^{5}+70 D b c \,x^{5}+84 B \,x^{4} b d +84 C b c \,x^{4}+84 D a d \,x^{4}+105 A b d \,x^{3}+105 B b c \,x^{3}+105 C a d \,x^{3}+105 D a c \,x^{3}+140 A b c \,x^{2}+140 x^{2} B a d +140 C a c \,x^{2}+210 A a d x +210 B a c x +420 A a c \right )}{420}\) | \(126\) |
gosper | \(\frac {1}{7} b d D x^{7}+\frac {1}{6} x^{6} b d C +\frac {1}{6} x^{6} b c D+\frac {1}{5} B \,x^{5} b d +\frac {1}{5} x^{5} b c C +\frac {1}{5} x^{5} D a d +\frac {1}{4} x^{4} A b d +\frac {1}{4} x^{4} B b c +\frac {1}{4} x^{4} C a d +\frac {1}{4} x^{4} D a c +\frac {1}{3} x^{3} A b c +\frac {1}{3} x^{3} B a d +\frac {1}{3} x^{3} a c C +\frac {1}{2} x^{2} A a d +\frac {1}{2} B a c \,x^{2}+A a c x\) | \(127\) |
parallelrisch | \(\frac {1}{7} b d D x^{7}+\frac {1}{6} x^{6} b d C +\frac {1}{6} x^{6} b c D+\frac {1}{5} B \,x^{5} b d +\frac {1}{5} x^{5} b c C +\frac {1}{5} x^{5} D a d +\frac {1}{4} x^{4} A b d +\frac {1}{4} x^{4} B b c +\frac {1}{4} x^{4} C a d +\frac {1}{4} x^{4} D a c +\frac {1}{3} x^{3} A b c +\frac {1}{3} x^{3} B a d +\frac {1}{3} x^{3} a c C +\frac {1}{2} x^{2} A a d +\frac {1}{2} B a c \,x^{2}+A a c x\) | \(127\) |
Input:
int((d*x+c)*(b*x^2+a)*(D*x^3+C*x^2+B*x+A),x,method=_RETURNVERBOSE)
Output:
1/7*b*d*D*x^7+1/6*(C*b*d+D*b*c)*x^6+1/5*(B*b*d+C*b*c+D*a*d)*x^5+1/4*(A*b*d +B*b*c+C*a*d+D*a*c)*x^4+1/3*(A*b*c+B*a*d+C*a*c)*x^3+1/2*(A*a*d+B*a*c)*x^2+ A*a*c*x
Time = 0.06 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.95 \[ \int (c+d x) \left (a+b x^2\right ) \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {1}{7} \, D b d x^{7} + \frac {1}{6} \, {\left (D b c + C b d\right )} x^{6} + \frac {1}{5} \, {\left (C b c + {\left (D a + B b\right )} d\right )} x^{5} + \frac {1}{4} \, {\left ({\left (D a + B b\right )} c + {\left (C a + A b\right )} d\right )} x^{4} + A a c x + \frac {1}{3} \, {\left (B a d + {\left (C a + A b\right )} c\right )} x^{3} + \frac {1}{2} \, {\left (B a c + A a d\right )} x^{2} \] Input:
integrate((d*x+c)*(b*x^2+a)*(D*x^3+C*x^2+B*x+A),x, algorithm="fricas")
Output:
1/7*D*b*d*x^7 + 1/6*(D*b*c + C*b*d)*x^6 + 1/5*(C*b*c + (D*a + B*b)*d)*x^5 + 1/4*((D*a + B*b)*c + (C*a + A*b)*d)*x^4 + A*a*c*x + 1/3*(B*a*d + (C*a + A*b)*c)*x^3 + 1/2*(B*a*c + A*a*d)*x^2
Time = 0.02 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.16 \[ \int (c+d x) \left (a+b x^2\right ) \left (A+B x+C x^2+D x^3\right ) \, dx=A a c x + \frac {D b d x^{7}}{7} + x^{6} \left (\frac {C b d}{6} + \frac {D b c}{6}\right ) + x^{5} \left (\frac {B b d}{5} + \frac {C b c}{5} + \frac {D a d}{5}\right ) + x^{4} \left (\frac {A b d}{4} + \frac {B b c}{4} + \frac {C a d}{4} + \frac {D a c}{4}\right ) + x^{3} \left (\frac {A b c}{3} + \frac {B a d}{3} + \frac {C a c}{3}\right ) + x^{2} \left (\frac {A a d}{2} + \frac {B a c}{2}\right ) \] Input:
integrate((d*x+c)*(b*x**2+a)*(D*x**3+C*x**2+B*x+A),x)
Output:
A*a*c*x + D*b*d*x**7/7 + x**6*(C*b*d/6 + D*b*c/6) + x**5*(B*b*d/5 + C*b*c/ 5 + D*a*d/5) + x**4*(A*b*d/4 + B*b*c/4 + C*a*d/4 + D*a*c/4) + x**3*(A*b*c/ 3 + B*a*d/3 + C*a*c/3) + x**2*(A*a*d/2 + B*a*c/2)
Time = 0.03 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.95 \[ \int (c+d x) \left (a+b x^2\right ) \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {1}{7} \, D b d x^{7} + \frac {1}{6} \, {\left (D b c + C b d\right )} x^{6} + \frac {1}{5} \, {\left (C b c + {\left (D a + B b\right )} d\right )} x^{5} + \frac {1}{4} \, {\left ({\left (D a + B b\right )} c + {\left (C a + A b\right )} d\right )} x^{4} + A a c x + \frac {1}{3} \, {\left (B a d + {\left (C a + A b\right )} c\right )} x^{3} + \frac {1}{2} \, {\left (B a c + A a d\right )} x^{2} \] Input:
integrate((d*x+c)*(b*x^2+a)*(D*x^3+C*x^2+B*x+A),x, algorithm="maxima")
Output:
1/7*D*b*d*x^7 + 1/6*(D*b*c + C*b*d)*x^6 + 1/5*(C*b*c + (D*a + B*b)*d)*x^5 + 1/4*((D*a + B*b)*c + (C*a + A*b)*d)*x^4 + A*a*c*x + 1/3*(B*a*d + (C*a + A*b)*c)*x^3 + 1/2*(B*a*c + A*a*d)*x^2
Time = 0.34 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.15 \[ \int (c+d x) \left (a+b x^2\right ) \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {1}{7} \, D b d x^{7} + \frac {1}{6} \, D b c x^{6} + \frac {1}{6} \, C b d x^{6} + \frac {1}{5} \, C b c x^{5} + \frac {1}{5} \, D a d x^{5} + \frac {1}{5} \, B b d x^{5} + \frac {1}{4} \, D a c x^{4} + \frac {1}{4} \, B b c x^{4} + \frac {1}{4} \, C a d x^{4} + \frac {1}{4} \, A b d x^{4} + \frac {1}{3} \, C a c x^{3} + \frac {1}{3} \, A b c x^{3} + \frac {1}{3} \, B a d x^{3} + \frac {1}{2} \, B a c x^{2} + \frac {1}{2} \, A a d x^{2} + A a c x \] Input:
integrate((d*x+c)*(b*x^2+a)*(D*x^3+C*x^2+B*x+A),x, algorithm="giac")
Output:
1/7*D*b*d*x^7 + 1/6*D*b*c*x^6 + 1/6*C*b*d*x^6 + 1/5*C*b*c*x^5 + 1/5*D*a*d* x^5 + 1/5*B*b*d*x^5 + 1/4*D*a*c*x^4 + 1/4*B*b*c*x^4 + 1/4*C*a*d*x^4 + 1/4* A*b*d*x^4 + 1/3*C*a*c*x^3 + 1/3*A*b*c*x^3 + 1/3*B*a*d*x^3 + 1/2*B*a*c*x^2 + 1/2*A*a*d*x^2 + A*a*c*x
Time = 16.69 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.15 \[ \int (c+d x) \left (a+b x^2\right ) \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {A\,a\,d\,x^2}{2}+\frac {B\,a\,c\,x^2}{2}+\frac {A\,b\,c\,x^3}{3}+\frac {B\,a\,d\,x^3}{3}+\frac {C\,a\,c\,x^3}{3}+\frac {A\,b\,d\,x^4}{4}+\frac {B\,b\,c\,x^4}{4}+\frac {C\,a\,d\,x^4}{4}+\frac {B\,b\,d\,x^5}{5}+\frac {C\,b\,c\,x^5}{5}+\frac {a\,c\,x^4\,D}{4}+\frac {C\,b\,d\,x^6}{6}+\frac {a\,d\,x^5\,D}{5}+\frac {b\,c\,x^6\,D}{6}+\frac {b\,d\,x^7\,D}{7}+A\,a\,c\,x \] Input:
int((a + b*x^2)*(c + d*x)*(A + B*x + C*x^2 + x^3*D),x)
Output:
(A*a*d*x^2)/2 + (B*a*c*x^2)/2 + (A*b*c*x^3)/3 + (B*a*d*x^3)/3 + (C*a*c*x^3 )/3 + (A*b*d*x^4)/4 + (B*b*c*x^4)/4 + (C*a*d*x^4)/4 + (B*b*d*x^5)/5 + (C*b *c*x^5)/5 + (a*c*x^4*D)/4 + (C*b*d*x^6)/6 + (a*d*x^5*D)/5 + (b*c*x^6*D)/6 + (b*d*x^7*D)/7 + A*a*c*x
Time = 0.16 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.06 \[ \int (c+d x) \left (a+b x^2\right ) \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {x \left (60 b \,d^{2} x^{6}+140 b c d \,x^{5}+84 a \,d^{2} x^{4}+84 b^{2} d \,x^{4}+84 b \,c^{2} x^{4}+105 a b d \,x^{3}+210 a c d \,x^{3}+105 b^{2} c \,x^{3}+140 a b c \,x^{2}+140 a b d \,x^{2}+140 a \,c^{2} x^{2}+210 a^{2} d x +210 a b c x +420 a^{2} c \right )}{420} \] Input:
int((d*x+c)*(b*x^2+a)*(D*x^3+C*x^2+B*x+A),x)
Output:
(x*(420*a**2*c + 210*a**2*d*x + 140*a*b*c*x**2 + 210*a*b*c*x + 105*a*b*d*x **3 + 140*a*b*d*x**2 + 140*a*c**2*x**2 + 210*a*c*d*x**3 + 84*a*d**2*x**4 + 105*b**2*c*x**3 + 84*b**2*d*x**4 + 84*b*c**2*x**4 + 140*b*c*d*x**5 + 60*b *d**2*x**6))/420