Integrand size = 55, antiderivative size = 65 \[ \int \left (c^2+2 c d x+d^2 x^2\right ) \left (a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4\right ) \, dx=\frac {(b c-a d)^2 (a+b x)^5}{5 b^3}+\frac {d (b c-a d) (a+b x)^6}{3 b^3}+\frac {d^2 (a+b x)^7}{7 b^3} \] Output:
1/5*(-a*d+b*c)^2*(b*x+a)^5/b^3+1/3*d*(-a*d+b*c)*(b*x+a)^6/b^3+1/7*d^2*(b*x +a)^7/b^3
Leaf count is larger than twice the leaf count of optimal. \(148\) vs. \(2(65)=130\).
Time = 0.00 (sec) , antiderivative size = 148, normalized size of antiderivative = 2.28 \[ \int \left (c^2+2 c d x+d^2 x^2\right ) \left (a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4\right ) \, dx=a^4 c^2 x+a^3 c (2 b c+a d) x^2+\frac {1}{3} a^2 \left (6 b^2 c^2+8 a b c d+a^2 d^2\right ) x^3+a b \left (b^2 c^2+3 a b c d+a^2 d^2\right ) x^4+\frac {1}{5} b^2 \left (b^2 c^2+8 a b c d+6 a^2 d^2\right ) x^5+\frac {1}{3} b^3 d (b c+2 a d) x^6+\frac {1}{7} b^4 d^2 x^7 \] Input:
Integrate[(c^2 + 2*c*d*x + d^2*x^2)*(a^4 + 4*a^3*b*x + 6*a^2*b^2*x^2 + 4*a *b^3*x^3 + b^4*x^4),x]
Output:
a^4*c^2*x + a^3*c*(2*b*c + a*d)*x^2 + (a^2*(6*b^2*c^2 + 8*a*b*c*d + a^2*d^ 2)*x^3)/3 + a*b*(b^2*c^2 + 3*a*b*c*d + a^2*d^2)*x^4 + (b^2*(b^2*c^2 + 8*a* b*c*d + 6*a^2*d^2)*x^5)/5 + (b^3*d*(b*c + 2*a*d)*x^6)/3 + (b^4*d^2*x^7)/7
Time = 0.25 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2006, 1098, 27, 49, 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.
\(\displaystyle \int \left (a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4\right ) \left (c^2+2 c d x+d^2 x^2\right ) \, dx\) |
\(\Big \downarrow \) 2006 |
\(\displaystyle \int (a+b x)^4 \left (c^2+2 c d x+d^2 x^2\right )dx\) |
\(\Big \downarrow \) 1098 |
\(\displaystyle \frac {\int d^2 (a+b x)^4 (c+d x)^2dx}{d^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int (a+b x)^4 (c+d x)^2dx\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \int \left (\frac {2 d (a+b x)^5 (b c-a d)}{b^2}+\frac {(a+b x)^4 (b c-a d)^2}{b^2}+\frac {d^2 (a+b x)^6}{b^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {d (a+b x)^6 (b c-a d)}{3 b^3}+\frac {(a+b x)^5 (b c-a d)^2}{5 b^3}+\frac {d^2 (a+b x)^7}{7 b^3}\) |
Input:
Int[(c^2 + 2*c*d*x + d^2*x^2)*(a^4 + 4*a^3*b*x + 6*a^2*b^2*x^2 + 4*a*b^3*x ^3 + b^4*x^4),x]
Output:
((b*c - a*d)^2*(a + b*x)^5)/(5*b^3) + (d*(b*c - a*d)*(a + b*x)^6)/(3*b^3) + (d^2*(a + b*x)^7)/(7*b^3)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[((d_.) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_ Symbol] :> Simp[1/c^p Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[ {a, b, c, d, e, m}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Int[(u_.)*(Px_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^Expon[Px , x], x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; PolyQ[Px, x] && GtQ[Expon[P x, x], 1] && NeQ[Coeff[Px, x, 0], 0] && !MatchQ[Px, (a_.)*(v_)^Expon[Px, x ] /; FreeQ[a, x] && LinearQ[v, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(156\) vs. \(2(59)=118\).
Time = 0.08 (sec) , antiderivative size = 157, normalized size of antiderivative = 2.42
method | result | size |
norman | \(\frac {b^{4} d^{2} x^{7}}{7}+\left (\frac {2}{3} a \,b^{3} d^{2}+\frac {1}{3} b^{4} c d \right ) x^{6}+\left (\frac {6}{5} a^{2} b^{2} d^{2}+\frac {8}{5} a \,b^{3} c d +\frac {1}{5} b^{4} c^{2}\right ) x^{5}+\left (a^{3} b \,d^{2}+3 a^{2} b^{2} c d +a \,b^{3} c^{2}\right ) x^{4}+\left (\frac {1}{3} a^{4} d^{2}+\frac {8}{3} a^{3} b c d +2 a^{2} b^{2} c^{2}\right ) x^{3}+\left (a^{4} c d +2 a^{3} b \,c^{2}\right ) x^{2}+a^{4} c^{2} x\) | \(157\) |
default | \(\frac {b^{4} d^{2} x^{7}}{7}+\frac {\left (4 a \,b^{3} d^{2}+2 b^{4} c d \right ) x^{6}}{6}+\frac {\left (6 a^{2} b^{2} d^{2}+8 a \,b^{3} c d +b^{4} c^{2}\right ) x^{5}}{5}+\frac {\left (4 a^{3} b \,d^{2}+12 a^{2} b^{2} c d +4 a \,b^{3} c^{2}\right ) x^{4}}{4}+\frac {\left (a^{4} d^{2}+8 a^{3} b c d +6 a^{2} b^{2} c^{2}\right ) x^{3}}{3}+\frac {\left (2 a^{4} c d +4 a^{3} b \,c^{2}\right ) x^{2}}{2}+a^{4} c^{2} x\) | \(163\) |
risch | \(\frac {1}{7} b^{4} d^{2} x^{7}+\frac {2}{3} x^{6} a \,b^{3} d^{2}+\frac {1}{3} x^{6} b^{4} c d +\frac {6}{5} x^{5} a^{2} b^{2} d^{2}+\frac {8}{5} x^{5} a \,b^{3} c d +\frac {1}{5} x^{5} b^{4} c^{2}+a^{3} b \,d^{2} x^{4}+3 a^{2} b^{2} c d \,x^{4}+a \,b^{3} c^{2} x^{4}+\frac {1}{3} x^{3} a^{4} d^{2}+\frac {8}{3} x^{3} a^{3} b c d +2 x^{3} a^{2} b^{2} c^{2}+a^{4} c d \,x^{2}+2 a^{3} b \,c^{2} x^{2}+a^{4} c^{2} x\) | \(171\) |
parallelrisch | \(\frac {1}{7} b^{4} d^{2} x^{7}+\frac {2}{3} x^{6} a \,b^{3} d^{2}+\frac {1}{3} x^{6} b^{4} c d +\frac {6}{5} x^{5} a^{2} b^{2} d^{2}+\frac {8}{5} x^{5} a \,b^{3} c d +\frac {1}{5} x^{5} b^{4} c^{2}+a^{3} b \,d^{2} x^{4}+3 a^{2} b^{2} c d \,x^{4}+a \,b^{3} c^{2} x^{4}+\frac {1}{3} x^{3} a^{4} d^{2}+\frac {8}{3} x^{3} a^{3} b c d +2 x^{3} a^{2} b^{2} c^{2}+a^{4} c d \,x^{2}+2 a^{3} b \,c^{2} x^{2}+a^{4} c^{2} x\) | \(171\) |
gosper | \(\frac {x \left (15 b^{4} d^{2} x^{6}+70 a \,b^{3} d^{2} x^{5}+35 b^{4} c d \,x^{5}+126 a^{2} b^{2} d^{2} x^{4}+168 a \,b^{3} c d \,x^{4}+21 b^{4} c^{2} x^{4}+105 a^{3} b \,d^{2} x^{3}+315 a^{2} b^{2} c d \,x^{3}+105 a \,b^{3} c^{2} x^{3}+35 a^{4} d^{2} x^{2}+280 x^{2} a^{3} b c d +210 a^{2} b^{2} c^{2} x^{2}+105 a^{4} c d x +210 a^{3} b \,c^{2} x +105 a^{4} c^{2}\right )}{105}\) | \(173\) |
orering | \(\frac {x \left (15 b^{4} d^{2} x^{6}+70 a \,b^{3} d^{2} x^{5}+35 b^{4} c d \,x^{5}+126 a^{2} b^{2} d^{2} x^{4}+168 a \,b^{3} c d \,x^{4}+21 b^{4} c^{2} x^{4}+105 a^{3} b \,d^{2} x^{3}+315 a^{2} b^{2} c d \,x^{3}+105 a \,b^{3} c^{2} x^{3}+35 a^{4} d^{2} x^{2}+280 x^{2} a^{3} b c d +210 a^{2} b^{2} c^{2} x^{2}+105 a^{4} c d x +210 a^{3} b \,c^{2} x +105 a^{4} c^{2}\right ) \left (d^{2} x^{2}+2 c d x +c^{2}\right ) \left (b^{4} x^{4}+4 a \,x^{3} b^{3}+6 a^{2} b^{2} x^{2}+4 a^{3} b x +a^{4}\right )}{105 \left (b x +a \right )^{4} \left (x d +c \right )^{2}}\) | \(241\) |
Input:
int((d^2*x^2+2*c*d*x+c^2)*(b^4*x^4+4*a*b^3*x^3+6*a^2*b^2*x^2+4*a^3*b*x+a^4 ),x,method=_RETURNVERBOSE)
Output:
1/7*b^4*d^2*x^7+(2/3*a*b^3*d^2+1/3*b^4*c*d)*x^6+(6/5*a^2*b^2*d^2+8/5*a*b^3 *c*d+1/5*b^4*c^2)*x^5+(a^3*b*d^2+3*a^2*b^2*c*d+a*b^3*c^2)*x^4+(1/3*a^4*d^2 +8/3*a^3*b*c*d+2*a^2*b^2*c^2)*x^3+(a^4*c*d+2*a^3*b*c^2)*x^2+a^4*c^2*x
Leaf count of result is larger than twice the leaf count of optimal. 156 vs. \(2 (59) = 118\).
Time = 0.06 (sec) , antiderivative size = 156, normalized size of antiderivative = 2.40 \[ \int \left (c^2+2 c d x+d^2 x^2\right ) \left (a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4\right ) \, dx=\frac {1}{7} \, b^{4} d^{2} x^{7} + a^{4} c^{2} x + \frac {1}{3} \, {\left (b^{4} c d + 2 \, a b^{3} d^{2}\right )} x^{6} + \frac {1}{5} \, {\left (b^{4} c^{2} + 8 \, a b^{3} c d + 6 \, a^{2} b^{2} d^{2}\right )} x^{5} + {\left (a b^{3} c^{2} + 3 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{4} + \frac {1}{3} \, {\left (6 \, a^{2} b^{2} c^{2} + 8 \, a^{3} b c d + a^{4} d^{2}\right )} x^{3} + {\left (2 \, a^{3} b c^{2} + a^{4} c d\right )} x^{2} \] Input:
integrate((d^2*x^2+2*c*d*x+c^2)*(b^4*x^4+4*a*b^3*x^3+6*a^2*b^2*x^2+4*a^3*b *x+a^4),x, algorithm="fricas")
Output:
1/7*b^4*d^2*x^7 + a^4*c^2*x + 1/3*(b^4*c*d + 2*a*b^3*d^2)*x^6 + 1/5*(b^4*c ^2 + 8*a*b^3*c*d + 6*a^2*b^2*d^2)*x^5 + (a*b^3*c^2 + 3*a^2*b^2*c*d + a^3*b *d^2)*x^4 + 1/3*(6*a^2*b^2*c^2 + 8*a^3*b*c*d + a^4*d^2)*x^3 + (2*a^3*b*c^2 + a^4*c*d)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 168 vs. \(2 (54) = 108\).
Time = 0.03 (sec) , antiderivative size = 168, normalized size of antiderivative = 2.58 \[ \int \left (c^2+2 c d x+d^2 x^2\right ) \left (a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4\right ) \, dx=a^{4} c^{2} x + \frac {b^{4} d^{2} x^{7}}{7} + x^{6} \cdot \left (\frac {2 a b^{3} d^{2}}{3} + \frac {b^{4} c d}{3}\right ) + x^{5} \cdot \left (\frac {6 a^{2} b^{2} d^{2}}{5} + \frac {8 a b^{3} c d}{5} + \frac {b^{4} c^{2}}{5}\right ) + x^{4} \left (a^{3} b d^{2} + 3 a^{2} b^{2} c d + a b^{3} c^{2}\right ) + x^{3} \left (\frac {a^{4} d^{2}}{3} + \frac {8 a^{3} b c d}{3} + 2 a^{2} b^{2} c^{2}\right ) + x^{2} \left (a^{4} c d + 2 a^{3} b c^{2}\right ) \] Input:
integrate((d**2*x**2+2*c*d*x+c**2)*(b**4*x**4+4*a*b**3*x**3+6*a**2*b**2*x* *2+4*a**3*b*x+a**4),x)
Output:
a**4*c**2*x + b**4*d**2*x**7/7 + x**6*(2*a*b**3*d**2/3 + b**4*c*d/3) + x** 5*(6*a**2*b**2*d**2/5 + 8*a*b**3*c*d/5 + b**4*c**2/5) + x**4*(a**3*b*d**2 + 3*a**2*b**2*c*d + a*b**3*c**2) + x**3*(a**4*d**2/3 + 8*a**3*b*c*d/3 + 2* a**2*b**2*c**2) + x**2*(a**4*c*d + 2*a**3*b*c**2)
Leaf count of result is larger than twice the leaf count of optimal. 156 vs. \(2 (59) = 118\).
Time = 0.04 (sec) , antiderivative size = 156, normalized size of antiderivative = 2.40 \[ \int \left (c^2+2 c d x+d^2 x^2\right ) \left (a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4\right ) \, dx=\frac {1}{7} \, b^{4} d^{2} x^{7} + a^{4} c^{2} x + \frac {1}{3} \, {\left (b^{4} c d + 2 \, a b^{3} d^{2}\right )} x^{6} + \frac {1}{5} \, {\left (b^{4} c^{2} + 8 \, a b^{3} c d + 6 \, a^{2} b^{2} d^{2}\right )} x^{5} + {\left (a b^{3} c^{2} + 3 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{4} + \frac {1}{3} \, {\left (6 \, a^{2} b^{2} c^{2} + 8 \, a^{3} b c d + a^{4} d^{2}\right )} x^{3} + {\left (2 \, a^{3} b c^{2} + a^{4} c d\right )} x^{2} \] Input:
integrate((d^2*x^2+2*c*d*x+c^2)*(b^4*x^4+4*a*b^3*x^3+6*a^2*b^2*x^2+4*a^3*b *x+a^4),x, algorithm="maxima")
Output:
1/7*b^4*d^2*x^7 + a^4*c^2*x + 1/3*(b^4*c*d + 2*a*b^3*d^2)*x^6 + 1/5*(b^4*c ^2 + 8*a*b^3*c*d + 6*a^2*b^2*d^2)*x^5 + (a*b^3*c^2 + 3*a^2*b^2*c*d + a^3*b *d^2)*x^4 + 1/3*(6*a^2*b^2*c^2 + 8*a^3*b*c*d + a^4*d^2)*x^3 + (2*a^3*b*c^2 + a^4*c*d)*x^2
Leaf count of result is larger than twice the leaf count of optimal. 170 vs. \(2 (59) = 118\).
Time = 0.12 (sec) , antiderivative size = 170, normalized size of antiderivative = 2.62 \[ \int \left (c^2+2 c d x+d^2 x^2\right ) \left (a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4\right ) \, dx=\frac {1}{7} \, b^{4} d^{2} x^{7} + \frac {1}{3} \, b^{4} c d x^{6} + \frac {2}{3} \, a b^{3} d^{2} x^{6} + \frac {1}{5} \, b^{4} c^{2} x^{5} + \frac {8}{5} \, a b^{3} c d x^{5} + \frac {6}{5} \, a^{2} b^{2} d^{2} x^{5} + a b^{3} c^{2} x^{4} + 3 \, a^{2} b^{2} c d x^{4} + a^{3} b d^{2} x^{4} + 2 \, a^{2} b^{2} c^{2} x^{3} + \frac {8}{3} \, a^{3} b c d x^{3} + \frac {1}{3} \, a^{4} d^{2} x^{3} + 2 \, a^{3} b c^{2} x^{2} + a^{4} c d x^{2} + a^{4} c^{2} x \] Input:
integrate((d^2*x^2+2*c*d*x+c^2)*(b^4*x^4+4*a*b^3*x^3+6*a^2*b^2*x^2+4*a^3*b *x+a^4),x, algorithm="giac")
Output:
1/7*b^4*d^2*x^7 + 1/3*b^4*c*d*x^6 + 2/3*a*b^3*d^2*x^6 + 1/5*b^4*c^2*x^5 + 8/5*a*b^3*c*d*x^5 + 6/5*a^2*b^2*d^2*x^5 + a*b^3*c^2*x^4 + 3*a^2*b^2*c*d*x^ 4 + a^3*b*d^2*x^4 + 2*a^2*b^2*c^2*x^3 + 8/3*a^3*b*c*d*x^3 + 1/3*a^4*d^2*x^ 3 + 2*a^3*b*c^2*x^2 + a^4*c*d*x^2 + a^4*c^2*x
Time = 0.04 (sec) , antiderivative size = 144, normalized size of antiderivative = 2.22 \[ \int \left (c^2+2 c d x+d^2 x^2\right ) \left (a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4\right ) \, dx=x^3\,\left (\frac {a^4\,d^2}{3}+\frac {8\,a^3\,b\,c\,d}{3}+2\,a^2\,b^2\,c^2\right )+x^5\,\left (\frac {6\,a^2\,b^2\,d^2}{5}+\frac {8\,a\,b^3\,c\,d}{5}+\frac {b^4\,c^2}{5}\right )+a^4\,c^2\,x+\frac {b^4\,d^2\,x^7}{7}+a^3\,c\,x^2\,\left (a\,d+2\,b\,c\right )+\frac {b^3\,d\,x^6\,\left (2\,a\,d+b\,c\right )}{3}+a\,b\,x^4\,\left (a^2\,d^2+3\,a\,b\,c\,d+b^2\,c^2\right ) \] Input:
int((c^2 + d^2*x^2 + 2*c*d*x)*(a^4 + b^4*x^4 + 4*a*b^3*x^3 + 6*a^2*b^2*x^2 + 4*a^3*b*x),x)
Output:
x^3*((a^4*d^2)/3 + 2*a^2*b^2*c^2 + (8*a^3*b*c*d)/3) + x^5*((b^4*c^2)/5 + ( 6*a^2*b^2*d^2)/5 + (8*a*b^3*c*d)/5) + a^4*c^2*x + (b^4*d^2*x^7)/7 + a^3*c* x^2*(a*d + 2*b*c) + (b^3*d*x^6*(2*a*d + b*c))/3 + a*b*x^4*(a^2*d^2 + b^2*c ^2 + 3*a*b*c*d)
Time = 0.17 (sec) , antiderivative size = 172, normalized size of antiderivative = 2.65 \[ \int \left (c^2+2 c d x+d^2 x^2\right ) \left (a^4+4 a^3 b x+6 a^2 b^2 x^2+4 a b^3 x^3+b^4 x^4\right ) \, dx=\frac {x \left (15 b^{4} d^{2} x^{6}+70 a \,b^{3} d^{2} x^{5}+35 b^{4} c d \,x^{5}+126 a^{2} b^{2} d^{2} x^{4}+168 a \,b^{3} c d \,x^{4}+21 b^{4} c^{2} x^{4}+105 a^{3} b \,d^{2} x^{3}+315 a^{2} b^{2} c d \,x^{3}+105 a \,b^{3} c^{2} x^{3}+35 a^{4} d^{2} x^{2}+280 a^{3} b c d \,x^{2}+210 a^{2} b^{2} c^{2} x^{2}+105 a^{4} c d x +210 a^{3} b \,c^{2} x +105 a^{4} c^{2}\right )}{105} \] Input:
int((d^2*x^2+2*c*d*x+c^2)*(b^4*x^4+4*a*b^3*x^3+6*a^2*b^2*x^2+4*a^3*b*x+a^4 ),x)
Output:
(x*(105*a**4*c**2 + 105*a**4*c*d*x + 35*a**4*d**2*x**2 + 210*a**3*b*c**2*x + 280*a**3*b*c*d*x**2 + 105*a**3*b*d**2*x**3 + 210*a**2*b**2*c**2*x**2 + 315*a**2*b**2*c*d*x**3 + 126*a**2*b**2*d**2*x**4 + 105*a*b**3*c**2*x**3 + 168*a*b**3*c*d*x**4 + 70*a*b**3*d**2*x**5 + 21*b**4*c**2*x**4 + 35*b**4*c* d*x**5 + 15*b**4*d**2*x**6))/105