Integrand size = 21, antiderivative size = 51 \[ \int (c+d x)^3 \left (a+b (c+d x)^2\right )^2 \, dx=\frac {a^2 (c+d x)^4}{4 d}+\frac {a b (c+d x)^6}{3 d}+\frac {b^2 (c+d x)^8}{8 d} \] Output:
1/4*a^2*(d*x+c)^4/d+1/3*a*b*(d*x+c)^6/d+1/8*b^2*(d*x+c)^8/d
Leaf count is larger than twice the leaf count of optimal. \(172\) vs. \(2(51)=102\).
Time = 0.02 (sec) , antiderivative size = 172, normalized size of antiderivative = 3.37 \[ \int (c+d x)^3 \left (a+b (c+d x)^2\right )^2 \, dx=c^3 \left (a+b c^2\right )^2 x+\frac {1}{2} c^2 \left (3 a^2+10 a b c^2+7 b^2 c^4\right ) d x^2+\frac {1}{3} c \left (3 a^2+20 a b c^2+21 b^2 c^4\right ) d^2 x^3+\frac {1}{4} \left (a^2+20 a b c^2+35 b^2 c^4\right ) d^3 x^4+b c \left (2 a+7 b c^2\right ) d^4 x^5+\frac {1}{6} b \left (2 a+21 b c^2\right ) d^5 x^6+b^2 c d^6 x^7+\frac {1}{8} b^2 d^7 x^8 \] Input:
Integrate[(c + d*x)^3*(a + b*(c + d*x)^2)^2,x]
Output:
c^3*(a + b*c^2)^2*x + (c^2*(3*a^2 + 10*a*b*c^2 + 7*b^2*c^4)*d*x^2)/2 + (c* (3*a^2 + 20*a*b*c^2 + 21*b^2*c^4)*d^2*x^3)/3 + ((a^2 + 20*a*b*c^2 + 35*b^2 *c^4)*d^3*x^4)/4 + b*c*(2*a + 7*b*c^2)*d^4*x^5 + (b*(2*a + 21*b*c^2)*d^5*x ^6)/6 + b^2*c*d^6*x^7 + (b^2*d^7*x^8)/8
Time = 0.30 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.96, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {895, 243, 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.
\(\displaystyle \int (c+d x)^3 \left (a+b (c+d x)^2\right )^2 \, dx\) |
\(\Big \downarrow \) 895 |
\(\displaystyle \frac {\int (c+d x)^3 \left (b (c+d x)^2+a\right )^2d(c+d x)}{d}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {\int (c+d x)^2 \left (b (c+d x)^2+a\right )^2d(c+d x)^2}{2 d}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {\int \left (b^2 (c+d x)^6+2 a b (c+d x)^4+a^2 (c+d x)^2\right )d(c+d x)^2}{2 d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {1}{2} a^2 (c+d x)^4+\frac {2}{3} a b (c+d x)^6+\frac {1}{4} b^2 (c+d x)^8}{2 d}\) |
Input:
Int[(c + d*x)^3*(a + b*(c + d*x)^2)^2,x]
Output:
((a^2*(c + d*x)^4)/2 + (2*a*b*(c + d*x)^6)/3 + (b^2*(c + d*x)^8)/4)/(2*d)
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[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[(u_)^(m_.)*((a_) + (b_.)*(v_)^(n_))^(p_.), x_Symbol] :> Simp[u^m/(Coeff icient[v, x, 1]*v^m) Subst[Int[x^m*(a + b*x^n)^p, x], x, v], x] /; FreeQ[ {a, b, m, n, p}, x] && LinearPairQ[u, v, x]
Time = 1.45 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.80
method | result | size |
default | \(-\frac {\frac {a \left (a +b \left (x d +c \right )^{2}\right )^{3}}{3}-\frac {\left (a +b \left (x d +c \right )^{2}\right )^{4}}{4}}{2 d \,b^{2}}\) | \(41\) |
norman | \(\frac {d^{7} b^{2} x^{8}}{8}+c \,d^{6} b^{2} x^{7}+\left (\frac {7}{2} b^{2} c^{2} d^{5}+\frac {1}{3} a b \,d^{5}\right ) x^{6}+\left (7 c^{3} b^{2} d^{4}+2 a b c \,d^{4}\right ) x^{5}+\left (\frac {35}{4} c^{4} b^{2} d^{3}+5 a b \,c^{2} d^{3}+\frac {1}{4} a^{2} d^{3}\right ) x^{4}+\left (7 b^{2} c^{5} d^{2}+\frac {20}{3} a b \,c^{3} d^{2}+c \,a^{2} d^{2}\right ) x^{3}+\left (\frac {7}{2} b^{2} c^{6} d +5 a b \,c^{4} d +\frac {3}{2} a^{2} c^{2} d \right ) x^{2}+\left (b^{2} c^{7}+2 a b \,c^{5}+a^{2} c^{3}\right ) x\) | \(194\) |
gosper | \(\frac {x \left (3 d^{7} b^{2} x^{7}+24 c \,d^{6} b^{2} x^{6}+84 x^{5} b^{2} c^{2} d^{5}+168 b^{2} c^{3} d^{4} x^{4}+8 x^{5} a b \,d^{5}+210 x^{3} c^{4} b^{2} d^{3}+48 a b c \,d^{4} x^{4}+168 x^{2} b^{2} c^{5} d^{2}+120 x^{3} a b \,c^{2} d^{3}+84 x \,b^{2} c^{6} d +160 x^{2} a b \,c^{3} d^{2}+24 b^{2} c^{7}+6 a^{2} d^{3} x^{3}+120 x a b \,c^{4} d +24 a^{2} c \,d^{2} x^{2}+48 a b \,c^{5}+36 a^{2} c^{2} d x +24 a^{2} c^{3}\right )}{24}\) | \(206\) |
parallelrisch | \(b^{2} c^{7} x +2 b a \,c^{5} x +a^{2} c^{3} x +\frac {7}{2} d \,b^{2} c^{6} x^{2}+5 d b a \,c^{4} x^{2}+\frac {3}{2} a^{2} c^{2} d \,x^{2}+7 d^{2} b^{2} c^{5} x^{3}+\frac {20}{3} d^{2} b a \,c^{3} x^{3}+a^{2} c \,d^{2} x^{3}+\frac {35}{4} d^{3} b^{2} c^{4} x^{4}+5 d^{3} b a \,c^{2} x^{4}+\frac {1}{4} a^{2} d^{3} x^{4}+7 d^{4} b^{2} c^{3} x^{5}+2 d^{4} b a c \,x^{5}+\frac {7}{2} d^{5} b^{2} c^{2} x^{6}+\frac {1}{3} d^{5} b a \,x^{6}+c \,d^{6} b^{2} x^{7}+\frac {1}{8} d^{7} b^{2} x^{8}\) | \(208\) |
orering | \(\frac {x \left (3 d^{7} b^{2} x^{7}+24 c \,d^{6} b^{2} x^{6}+84 x^{5} b^{2} c^{2} d^{5}+168 b^{2} c^{3} d^{4} x^{4}+8 x^{5} a b \,d^{5}+210 x^{3} c^{4} b^{2} d^{3}+48 a b c \,d^{4} x^{4}+168 x^{2} b^{2} c^{5} d^{2}+120 x^{3} a b \,c^{2} d^{3}+84 x \,b^{2} c^{6} d +160 x^{2} a b \,c^{3} d^{2}+24 b^{2} c^{7}+6 a^{2} d^{3} x^{3}+120 x a b \,c^{4} d +24 a^{2} c \,d^{2} x^{2}+48 a b \,c^{5}+36 a^{2} c^{2} d x +24 a^{2} c^{3}\right ) \left (a +b \left (x d +c \right )^{2}\right )^{2}}{24 \left (b \,d^{2} x^{2}+2 b c d x +b \,c^{2}+a \right )^{2}}\) | \(242\) |
risch | \(\frac {3 a^{2} c^{2} d \,x^{2}}{2}-\frac {a^{4}}{24 d \,b^{2}}+\frac {a^{2} d^{3} x^{4}}{4}+2 d^{4} b a c \,x^{5}+5 d^{3} b a \,c^{2} x^{4}+\frac {20 d^{2} b a \,c^{3} x^{3}}{3}+5 d b a \,c^{4} x^{2}+a^{2} c \,d^{2} x^{3}+2 b a \,c^{5} x +a^{2} c^{3} x +\frac {d^{5} b a \,x^{6}}{3}+c \,d^{6} b^{2} x^{7}+\frac {7 d^{5} b^{2} c^{2} x^{6}}{2}+7 d^{4} b^{2} c^{3} x^{5}+\frac {35 d^{3} b^{2} c^{4} x^{4}}{4}+7 d^{2} b^{2} c^{5} x^{3}+\frac {7 d \,b^{2} c^{6} x^{2}}{2}+b^{2} c^{7} x +\frac {b a \,c^{6}}{3 d}+\frac {a^{2} c^{4}}{4 d}+\frac {b^{2} c^{8}}{8 d}+\frac {d^{7} b^{2} x^{8}}{8}\) | \(251\) |
Input:
int((d*x+c)^3*(a+b*(d*x+c)^2)^2,x,method=_RETURNVERBOSE)
Output:
-1/2/d/b^2*(1/3*a*(a+b*(d*x+c)^2)^3-1/4*(a+b*(d*x+c)^2)^4)
Leaf count of result is larger than twice the leaf count of optimal. 176 vs. \(2 (45) = 90\).
Time = 0.07 (sec) , antiderivative size = 176, normalized size of antiderivative = 3.45 \[ \int (c+d x)^3 \left (a+b (c+d x)^2\right )^2 \, dx=\frac {1}{8} \, b^{2} d^{7} x^{8} + b^{2} c d^{6} x^{7} + \frac {1}{6} \, {\left (21 \, b^{2} c^{2} + 2 \, a b\right )} d^{5} x^{6} + {\left (7 \, b^{2} c^{3} + 2 \, a b c\right )} d^{4} x^{5} + \frac {1}{4} \, {\left (35 \, b^{2} c^{4} + 20 \, a b c^{2} + a^{2}\right )} d^{3} x^{4} + \frac {1}{3} \, {\left (21 \, b^{2} c^{5} + 20 \, a b c^{3} + 3 \, a^{2} c\right )} d^{2} x^{3} + \frac {1}{2} \, {\left (7 \, b^{2} c^{6} + 10 \, a b c^{4} + 3 \, a^{2} c^{2}\right )} d x^{2} + {\left (b^{2} c^{7} + 2 \, a b c^{5} + a^{2} c^{3}\right )} x \] Input:
integrate((d*x+c)^3*(a+b*(d*x+c)^2)^2,x, algorithm="fricas")
Output:
1/8*b^2*d^7*x^8 + b^2*c*d^6*x^7 + 1/6*(21*b^2*c^2 + 2*a*b)*d^5*x^6 + (7*b^ 2*c^3 + 2*a*b*c)*d^4*x^5 + 1/4*(35*b^2*c^4 + 20*a*b*c^2 + a^2)*d^3*x^4 + 1 /3*(21*b^2*c^5 + 20*a*b*c^3 + 3*a^2*c)*d^2*x^3 + 1/2*(7*b^2*c^6 + 10*a*b*c ^4 + 3*a^2*c^2)*d*x^2 + (b^2*c^7 + 2*a*b*c^5 + a^2*c^3)*x
Leaf count of result is larger than twice the leaf count of optimal. 209 vs. \(2 (39) = 78\).
Time = 0.06 (sec) , antiderivative size = 209, normalized size of antiderivative = 4.10 \[ \int (c+d x)^3 \left (a+b (c+d x)^2\right )^2 \, dx=b^{2} c d^{6} x^{7} + \frac {b^{2} d^{7} x^{8}}{8} + x^{6} \left (\frac {a b d^{5}}{3} + \frac {7 b^{2} c^{2} d^{5}}{2}\right ) + x^{5} \cdot \left (2 a b c d^{4} + 7 b^{2} c^{3} d^{4}\right ) + x^{4} \left (\frac {a^{2} d^{3}}{4} + 5 a b c^{2} d^{3} + \frac {35 b^{2} c^{4} d^{3}}{4}\right ) + x^{3} \left (a^{2} c d^{2} + \frac {20 a b c^{3} d^{2}}{3} + 7 b^{2} c^{5} d^{2}\right ) + x^{2} \cdot \left (\frac {3 a^{2} c^{2} d}{2} + 5 a b c^{4} d + \frac {7 b^{2} c^{6} d}{2}\right ) + x \left (a^{2} c^{3} + 2 a b c^{5} + b^{2} c^{7}\right ) \] Input:
integrate((d*x+c)**3*(a+b*(d*x+c)**2)**2,x)
Output:
b**2*c*d**6*x**7 + b**2*d**7*x**8/8 + x**6*(a*b*d**5/3 + 7*b**2*c**2*d**5/ 2) + x**5*(2*a*b*c*d**4 + 7*b**2*c**3*d**4) + x**4*(a**2*d**3/4 + 5*a*b*c* *2*d**3 + 35*b**2*c**4*d**3/4) + x**3*(a**2*c*d**2 + 20*a*b*c**3*d**2/3 + 7*b**2*c**5*d**2) + x**2*(3*a**2*c**2*d/2 + 5*a*b*c**4*d + 7*b**2*c**6*d/2 ) + x*(a**2*c**3 + 2*a*b*c**5 + b**2*c**7)
Leaf count of result is larger than twice the leaf count of optimal. 176 vs. \(2 (45) = 90\).
Time = 0.03 (sec) , antiderivative size = 176, normalized size of antiderivative = 3.45 \[ \int (c+d x)^3 \left (a+b (c+d x)^2\right )^2 \, dx=\frac {1}{8} \, b^{2} d^{7} x^{8} + b^{2} c d^{6} x^{7} + \frac {1}{6} \, {\left (21 \, b^{2} c^{2} + 2 \, a b\right )} d^{5} x^{6} + {\left (7 \, b^{2} c^{3} + 2 \, a b c\right )} d^{4} x^{5} + \frac {1}{4} \, {\left (35 \, b^{2} c^{4} + 20 \, a b c^{2} + a^{2}\right )} d^{3} x^{4} + \frac {1}{3} \, {\left (21 \, b^{2} c^{5} + 20 \, a b c^{3} + 3 \, a^{2} c\right )} d^{2} x^{3} + \frac {1}{2} \, {\left (7 \, b^{2} c^{6} + 10 \, a b c^{4} + 3 \, a^{2} c^{2}\right )} d x^{2} + {\left (b^{2} c^{7} + 2 \, a b c^{5} + a^{2} c^{3}\right )} x \] Input:
integrate((d*x+c)^3*(a+b*(d*x+c)^2)^2,x, algorithm="maxima")
Output:
1/8*b^2*d^7*x^8 + b^2*c*d^6*x^7 + 1/6*(21*b^2*c^2 + 2*a*b)*d^5*x^6 + (7*b^ 2*c^3 + 2*a*b*c)*d^4*x^5 + 1/4*(35*b^2*c^4 + 20*a*b*c^2 + a^2)*d^3*x^4 + 1 /3*(21*b^2*c^5 + 20*a*b*c^3 + 3*a^2*c)*d^2*x^3 + 1/2*(7*b^2*c^6 + 10*a*b*c ^4 + 3*a^2*c^2)*d*x^2 + (b^2*c^7 + 2*a*b*c^5 + a^2*c^3)*x
Leaf count of result is larger than twice the leaf count of optimal. 173 vs. \(2 (45) = 90\).
Time = 0.12 (sec) , antiderivative size = 173, normalized size of antiderivative = 3.39 \[ \int (c+d x)^3 \left (a+b (c+d x)^2\right )^2 \, dx=\frac {1}{2} \, {\left (d x^{2} + 2 \, c x\right )} b^{2} c^{6} + \frac {3}{4} \, {\left (d x^{2} + 2 \, c x\right )}^{2} b^{2} c^{4} d + \frac {1}{2} \, {\left (d x^{2} + 2 \, c x\right )}^{3} b^{2} c^{2} d^{2} + \frac {1}{8} \, {\left (d x^{2} + 2 \, c x\right )}^{4} b^{2} d^{3} + {\left (d x^{2} + 2 \, c x\right )} a b c^{4} + {\left (d x^{2} + 2 \, c x\right )}^{2} a b c^{2} d + \frac {1}{3} \, {\left (d x^{2} + 2 \, c x\right )}^{3} a b d^{2} + \frac {1}{2} \, {\left (d x^{2} + 2 \, c x\right )} a^{2} c^{2} + \frac {1}{4} \, {\left (d x^{2} + 2 \, c x\right )}^{2} a^{2} d \] Input:
integrate((d*x+c)^3*(a+b*(d*x+c)^2)^2,x, algorithm="giac")
Output:
1/2*(d*x^2 + 2*c*x)*b^2*c^6 + 3/4*(d*x^2 + 2*c*x)^2*b^2*c^4*d + 1/2*(d*x^2 + 2*c*x)^3*b^2*c^2*d^2 + 1/8*(d*x^2 + 2*c*x)^4*b^2*d^3 + (d*x^2 + 2*c*x)* a*b*c^4 + (d*x^2 + 2*c*x)^2*a*b*c^2*d + 1/3*(d*x^2 + 2*c*x)^3*a*b*d^2 + 1/ 2*(d*x^2 + 2*c*x)*a^2*c^2 + 1/4*(d*x^2 + 2*c*x)^2*a^2*d
Time = 0.45 (sec) , antiderivative size = 162, normalized size of antiderivative = 3.18 \[ \int (c+d x)^3 \left (a+b (c+d x)^2\right )^2 \, dx=c^3\,x\,{\left (b\,c^2+a\right )}^2+\frac {b^2\,d^7\,x^8}{8}+\frac {d^3\,x^4\,\left (a^2+20\,a\,b\,c^2+35\,b^2\,c^4\right )}{4}+\frac {b\,d^5\,x^6\,\left (21\,b\,c^2+2\,a\right )}{6}+b^2\,c\,d^6\,x^7+\frac {c^2\,d\,x^2\,\left (3\,a^2+10\,a\,b\,c^2+7\,b^2\,c^4\right )}{2}+\frac {c\,d^2\,x^3\,\left (3\,a^2+20\,a\,b\,c^2+21\,b^2\,c^4\right )}{3}+b\,c\,d^4\,x^5\,\left (7\,b\,c^2+2\,a\right ) \] Input:
int((a + b*(c + d*x)^2)^2*(c + d*x)^3,x)
Output:
c^3*x*(a + b*c^2)^2 + (b^2*d^7*x^8)/8 + (d^3*x^4*(a^2 + 35*b^2*c^4 + 20*a* b*c^2))/4 + (b*d^5*x^6*(2*a + 21*b*c^2))/6 + b^2*c*d^6*x^7 + (c^2*d*x^2*(3 *a^2 + 7*b^2*c^4 + 10*a*b*c^2))/2 + (c*d^2*x^3*(3*a^2 + 21*b^2*c^4 + 20*a* b*c^2))/3 + b*c*d^4*x^5*(2*a + 7*b*c^2)
Time = 0.27 (sec) , antiderivative size = 205, normalized size of antiderivative = 4.02 \[ \int (c+d x)^3 \left (a+b (c+d x)^2\right )^2 \, dx=\frac {x \left (3 b^{2} d^{7} x^{7}+24 b^{2} c \,d^{6} x^{6}+84 b^{2} c^{2} d^{5} x^{5}+168 b^{2} c^{3} d^{4} x^{4}+8 a b \,d^{5} x^{5}+210 b^{2} c^{4} d^{3} x^{3}+48 a b c \,d^{4} x^{4}+168 b^{2} c^{5} d^{2} x^{2}+120 a b \,c^{2} d^{3} x^{3}+84 b^{2} c^{6} d x +160 a b \,c^{3} d^{2} x^{2}+24 b^{2} c^{7}+6 a^{2} d^{3} x^{3}+120 a b \,c^{4} d x +24 a^{2} c \,d^{2} x^{2}+48 a b \,c^{5}+36 a^{2} c^{2} d x +24 a^{2} c^{3}\right )}{24} \] Input:
int((d*x+c)^3*(a+b*(d*x+c)^2)^2,x)
Output:
(x*(24*a**2*c**3 + 36*a**2*c**2*d*x + 24*a**2*c*d**2*x**2 + 6*a**2*d**3*x* *3 + 48*a*b*c**5 + 120*a*b*c**4*d*x + 160*a*b*c**3*d**2*x**2 + 120*a*b*c** 2*d**3*x**3 + 48*a*b*c*d**4*x**4 + 8*a*b*d**5*x**5 + 24*b**2*c**7 + 84*b** 2*c**6*d*x + 168*b**2*c**5*d**2*x**2 + 210*b**2*c**4*d**3*x**3 + 168*b**2* c**3*d**4*x**4 + 84*b**2*c**2*d**5*x**5 + 24*b**2*c*d**6*x**6 + 3*b**2*d** 7*x**7))/24