Integrand size = 28, antiderivative size = 103 \[ \int \frac {(c+d x)^2 \left (b c^2-b d^2 x^2\right )^2}{x^9} \, dx=-\frac {b^2 c^6}{8 x^8}-\frac {2 b^2 c^5 d}{7 x^7}+\frac {b^2 c^4 d^2}{6 x^6}+\frac {4 b^2 c^3 d^3}{5 x^5}+\frac {b^2 c^2 d^4}{4 x^4}-\frac {2 b^2 c d^5}{3 x^3}-\frac {b^2 d^6}{2 x^2} \] Output:
-1/8*b^2*c^6/x^8-2/7*b^2*c^5*d/x^7+1/6*b^2*c^4*d^2/x^6+4/5*b^2*c^3*d^3/x^5 +1/4*b^2*c^2*d^4/x^4-2/3*b^2*c*d^5/x^3-1/2*b^2*d^6/x^2
Time = 0.01 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.83 \[ \int \frac {(c+d x)^2 \left (b c^2-b d^2 x^2\right )^2}{x^9} \, dx=b^2 \left (-\frac {c^6}{8 x^8}-\frac {2 c^5 d}{7 x^7}+\frac {c^4 d^2}{6 x^6}+\frac {4 c^3 d^3}{5 x^5}+\frac {c^2 d^4}{4 x^4}-\frac {2 c d^5}{3 x^3}-\frac {d^6}{2 x^2}\right ) \] Input:
Integrate[((c + d*x)^2*(b*c^2 - b*d^2*x^2)^2)/x^9,x]
Output:
b^2*(-1/8*c^6/x^8 - (2*c^5*d)/(7*x^7) + (c^4*d^2)/(6*x^6) + (4*c^3*d^3)/(5 *x^5) + (c^2*d^4)/(4*x^4) - (2*c*d^5)/(3*x^3) - d^6/(2*x^2))
Time = 0.40 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {516, 99, 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.
\(\displaystyle \int \frac {(c+d x)^2 \left (b c^2-b d^2 x^2\right )^2}{x^9} \, dx\) |
\(\Big \downarrow \) 516 |
\(\displaystyle \int \frac {(c+d x)^4 (b c-b d x)^2}{x^9}dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (\frac {b^2 c^6}{x^9}+\frac {2 b^2 c^5 d}{x^8}-\frac {b^2 c^4 d^2}{x^7}-\frac {4 b^2 c^3 d^3}{x^6}-\frac {b^2 c^2 d^4}{x^5}+\frac {2 b^2 c d^5}{x^4}+\frac {b^2 d^6}{x^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {b^2 c^6}{8 x^8}-\frac {2 b^2 c^5 d}{7 x^7}+\frac {b^2 c^4 d^2}{6 x^6}+\frac {4 b^2 c^3 d^3}{5 x^5}+\frac {b^2 c^2 d^4}{4 x^4}-\frac {2 b^2 c d^5}{3 x^3}-\frac {b^2 d^6}{2 x^2}\) |
Input:
Int[((c + d*x)^2*(b*c^2 - b*d^2*x^2)^2)/x^9,x]
Output:
-1/8*(b^2*c^6)/x^8 - (2*b^2*c^5*d)/(7*x^7) + (b^2*c^4*d^2)/(6*x^6) + (4*b^ 2*c^3*d^3)/(5*x^5) + (b^2*c^2*d^4)/(4*x^4) - (2*b^2*c*d^5)/(3*x^3) - (b^2* d^6)/(2*x^2)
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. ), x_Symbol] :> Int[(e*x)^m*(c + d*x)^(n + p)*(a/c + (b/d)*x)^p, x] /; Free Q[{a, b, c, d, e, m, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[c, 0] && !IntegerQ[n]))
Time = 0.13 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.70
method | result | size |
gosper | \(-\frac {b^{2} \left (420 d^{6} x^{6}+560 c \,d^{5} x^{5}-210 c^{2} d^{4} x^{4}-672 d^{3} c^{3} x^{3}-140 c^{4} d^{2} x^{2}+240 c^{5} d x +105 c^{6}\right )}{840 x^{8}}\) | \(72\) |
default | \(b^{2} \left (\frac {4 d^{3} c^{3}}{5 x^{5}}+\frac {c^{4} d^{2}}{6 x^{6}}-\frac {2 c \,d^{5}}{3 x^{3}}-\frac {d^{6}}{2 x^{2}}-\frac {c^{6}}{8 x^{8}}+\frac {c^{2} d^{4}}{4 x^{4}}-\frac {2 c^{5} d}{7 x^{7}}\right )\) | \(73\) |
norman | \(\frac {-\frac {1}{8} c^{6} b^{2}-\frac {1}{2} b^{2} d^{6} x^{6}-\frac {2}{3} b^{2} c \,d^{5} x^{5}+\frac {1}{4} b^{2} c^{2} d^{4} x^{4}+\frac {4}{5} b^{2} c^{3} d^{3} x^{3}+\frac {1}{6} b^{2} c^{4} d^{2} x^{2}-\frac {2}{7} b^{2} c^{5} d x}{x^{8}}\) | \(89\) |
risch | \(\frac {-\frac {1}{8} c^{6} b^{2}-\frac {1}{2} b^{2} d^{6} x^{6}-\frac {2}{3} b^{2} c \,d^{5} x^{5}+\frac {1}{4} b^{2} c^{2} d^{4} x^{4}+\frac {4}{5} b^{2} c^{3} d^{3} x^{3}+\frac {1}{6} b^{2} c^{4} d^{2} x^{2}-\frac {2}{7} b^{2} c^{5} d x}{x^{8}}\) | \(89\) |
parallelrisch | \(\frac {-420 b^{2} d^{6} x^{6}-560 b^{2} c \,d^{5} x^{5}+210 b^{2} c^{2} d^{4} x^{4}+672 b^{2} c^{3} d^{3} x^{3}+140 b^{2} c^{4} d^{2} x^{2}-240 b^{2} c^{5} d x -105 c^{6} b^{2}}{840 x^{8}}\) | \(90\) |
orering | \(-\frac {\left (420 d^{6} x^{6}+560 c \,d^{5} x^{5}-210 c^{2} d^{4} x^{4}-672 d^{3} c^{3} x^{3}-140 c^{4} d^{2} x^{2}+240 c^{5} d x +105 c^{6}\right ) \left (-b \,x^{2} d^{2}+b \,c^{2}\right )^{2}}{840 x^{8} \left (-d x +c \right )^{2} \left (d x +c \right )^{2}}\) | \(101\) |
Input:
int((d*x+c)^2*(-b*d^2*x^2+b*c^2)^2/x^9,x,method=_RETURNVERBOSE)
Output:
-1/840*b^2*(420*d^6*x^6+560*c*d^5*x^5-210*c^2*d^4*x^4-672*c^3*d^3*x^3-140* c^4*d^2*x^2+240*c^5*d*x+105*c^6)/x^8
Time = 0.08 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.86 \[ \int \frac {(c+d x)^2 \left (b c^2-b d^2 x^2\right )^2}{x^9} \, dx=-\frac {420 \, b^{2} d^{6} x^{6} + 560 \, b^{2} c d^{5} x^{5} - 210 \, b^{2} c^{2} d^{4} x^{4} - 672 \, b^{2} c^{3} d^{3} x^{3} - 140 \, b^{2} c^{4} d^{2} x^{2} + 240 \, b^{2} c^{5} d x + 105 \, b^{2} c^{6}}{840 \, x^{8}} \] Input:
integrate((d*x+c)^2*(-b*d^2*x^2+b*c^2)^2/x^9,x, algorithm="fricas")
Output:
-1/840*(420*b^2*d^6*x^6 + 560*b^2*c*d^5*x^5 - 210*b^2*c^2*d^4*x^4 - 672*b^ 2*c^3*d^3*x^3 - 140*b^2*c^4*d^2*x^2 + 240*b^2*c^5*d*x + 105*b^2*c^6)/x^8
Time = 0.23 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.92 \[ \int \frac {(c+d x)^2 \left (b c^2-b d^2 x^2\right )^2}{x^9} \, dx=\frac {- 105 b^{2} c^{6} - 240 b^{2} c^{5} d x + 140 b^{2} c^{4} d^{2} x^{2} + 672 b^{2} c^{3} d^{3} x^{3} + 210 b^{2} c^{2} d^{4} x^{4} - 560 b^{2} c d^{5} x^{5} - 420 b^{2} d^{6} x^{6}}{840 x^{8}} \] Input:
integrate((d*x+c)**2*(-b*d**2*x**2+b*c**2)**2/x**9,x)
Output:
(-105*b**2*c**6 - 240*b**2*c**5*d*x + 140*b**2*c**4*d**2*x**2 + 672*b**2*c **3*d**3*x**3 + 210*b**2*c**2*d**4*x**4 - 560*b**2*c*d**5*x**5 - 420*b**2* d**6*x**6)/(840*x**8)
Time = 0.03 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.86 \[ \int \frac {(c+d x)^2 \left (b c^2-b d^2 x^2\right )^2}{x^9} \, dx=-\frac {420 \, b^{2} d^{6} x^{6} + 560 \, b^{2} c d^{5} x^{5} - 210 \, b^{2} c^{2} d^{4} x^{4} - 672 \, b^{2} c^{3} d^{3} x^{3} - 140 \, b^{2} c^{4} d^{2} x^{2} + 240 \, b^{2} c^{5} d x + 105 \, b^{2} c^{6}}{840 \, x^{8}} \] Input:
integrate((d*x+c)^2*(-b*d^2*x^2+b*c^2)^2/x^9,x, algorithm="maxima")
Output:
-1/840*(420*b^2*d^6*x^6 + 560*b^2*c*d^5*x^5 - 210*b^2*c^2*d^4*x^4 - 672*b^ 2*c^3*d^3*x^3 - 140*b^2*c^4*d^2*x^2 + 240*b^2*c^5*d*x + 105*b^2*c^6)/x^8
Time = 0.12 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.86 \[ \int \frac {(c+d x)^2 \left (b c^2-b d^2 x^2\right )^2}{x^9} \, dx=-\frac {420 \, b^{2} d^{6} x^{6} + 560 \, b^{2} c d^{5} x^{5} - 210 \, b^{2} c^{2} d^{4} x^{4} - 672 \, b^{2} c^{3} d^{3} x^{3} - 140 \, b^{2} c^{4} d^{2} x^{2} + 240 \, b^{2} c^{5} d x + 105 \, b^{2} c^{6}}{840 \, x^{8}} \] Input:
integrate((d*x+c)^2*(-b*d^2*x^2+b*c^2)^2/x^9,x, algorithm="giac")
Output:
-1/840*(420*b^2*d^6*x^6 + 560*b^2*c*d^5*x^5 - 210*b^2*c^2*d^4*x^4 - 672*b^ 2*c^3*d^3*x^3 - 140*b^2*c^4*d^2*x^2 + 240*b^2*c^5*d*x + 105*b^2*c^6)/x^8
Time = 6.14 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.86 \[ \int \frac {(c+d x)^2 \left (b c^2-b d^2 x^2\right )^2}{x^9} \, dx=-\frac {\frac {b^2\,c^6}{8}+\frac {2\,b^2\,c^5\,d\,x}{7}-\frac {b^2\,c^4\,d^2\,x^2}{6}-\frac {4\,b^2\,c^3\,d^3\,x^3}{5}-\frac {b^2\,c^2\,d^4\,x^4}{4}+\frac {2\,b^2\,c\,d^5\,x^5}{3}+\frac {b^2\,d^6\,x^6}{2}}{x^8} \] Input:
int(((b*c^2 - b*d^2*x^2)^2*(c + d*x)^2)/x^9,x)
Output:
-((b^2*c^6)/8 + (b^2*d^6*x^6)/2 + (2*b^2*c*d^5*x^5)/3 - (b^2*c^4*d^2*x^2)/ 6 - (4*b^2*c^3*d^3*x^3)/5 - (b^2*c^2*d^4*x^4)/4 + (2*b^2*c^5*d*x)/7)/x^8
Time = 0.19 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.69 \[ \int \frac {(c+d x)^2 \left (b c^2-b d^2 x^2\right )^2}{x^9} \, dx=\frac {b^{2} \left (-420 d^{6} x^{6}-560 c \,d^{5} x^{5}+210 c^{2} d^{4} x^{4}+672 c^{3} d^{3} x^{3}+140 c^{4} d^{2} x^{2}-240 c^{5} d x -105 c^{6}\right )}{840 x^{8}} \] Input:
int((d*x+c)^2*(-b*d^2*x^2+b*c^2)^2/x^9,x)
Output:
(b**2*( - 105*c**6 - 240*c**5*d*x + 140*c**4*d**2*x**2 + 672*c**3*d**3*x** 3 + 210*c**2*d**4*x**4 - 560*c*d**5*x**5 - 420*d**6*x**6))/(840*x**8)