Integrand size = 24, antiderivative size = 73 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^{10}} \, dx=-\frac {\left (b^2-4 a c\right )^2}{288 c^3 d^{10} (b+2 c x)^9}+\frac {b^2-4 a c}{112 c^3 d^{10} (b+2 c x)^7}-\frac {1}{160 c^3 d^{10} (b+2 c x)^5} \] Output:
-1/288*(-4*a*c+b^2)^2/c^3/d^10/(2*c*x+b)^9+1/112*(-4*a*c+b^2)/c^3/d^10/(2* c*x+b)^7-1/160/c^3/d^10/(2*c*x+b)^5
Time = 0.03 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.81 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^{10}} \, dx=\frac {-35 \left (b^2-4 a c\right )^2+90 \left (b^2-4 a c\right ) (b+2 c x)^2-63 (b+2 c x)^4}{10080 c^3 d^{10} (b+2 c x)^9} \] Input:
Integrate[(a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^10,x]
Output:
(-35*(b^2 - 4*a*c)^2 + 90*(b^2 - 4*a*c)*(b + 2*c*x)^2 - 63*(b + 2*c*x)^4)/ (10080*c^3*d^10*(b + 2*c*x)^9)
Time = 0.24 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1107, 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.
\(\displaystyle \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^{10}} \, dx\) |
\(\Big \downarrow \) 1107 |
\(\displaystyle \int \left (\frac {\left (4 a c-b^2\right )^2}{16 c^2 d^{10} (b+2 c x)^{10}}+\frac {4 a c-b^2}{8 c^2 d^{10} (b+2 c x)^8}+\frac {1}{16 c^2 d^{10} (b+2 c x)^6}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\left (b^2-4 a c\right )^2}{288 c^3 d^{10} (b+2 c x)^9}+\frac {b^2-4 a c}{112 c^3 d^{10} (b+2 c x)^7}-\frac {1}{160 c^3 d^{10} (b+2 c x)^5}\) |
Input:
Int[(a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^10,x]
Output:
-1/288*(b^2 - 4*a*c)^2/(c^3*d^10*(b + 2*c*x)^9) + (b^2 - 4*a*c)/(112*c^3*d ^10*(b + 2*c*x)^7) - 1/(160*c^3*d^10*(b + 2*c*x)^5)
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_ Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; F reeQ[{a, b, c, d, e, m}, x] && EqQ[2*c*d - b*e, 0] && IGtQ[p, 0] && !(EqQ[ m, 3] && NeQ[p, 1])
Time = 0.83 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.01
method | result | size |
default | \(\frac {-\frac {4 a c -b^{2}}{112 c^{3} \left (2 c x +b \right )^{7}}-\frac {1}{160 c^{3} \left (2 c x +b \right )^{5}}-\frac {16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}}{288 c^{3} \left (2 c x +b \right )^{9}}}{d^{10}}\) | \(74\) |
risch | \(\frac {-\frac {c \,x^{4}}{10}-\frac {b \,x^{3}}{5}-\frac {\left (5 a c +4 b^{2}\right ) x^{2}}{35 c}-\frac {b \left (10 a c +b^{2}\right ) x}{70 c^{2}}-\frac {70 a^{2} c^{2}+10 c a \,b^{2}+b^{4}}{1260 c^{3}}}{d^{10} \left (2 c x +b \right )^{9}}\) | \(83\) |
gosper | \(-\frac {126 c^{4} x^{4}+252 b \,c^{3} x^{3}+180 a \,c^{3} x^{2}+144 b^{2} c^{2} x^{2}+180 a b \,c^{2} x +18 b^{3} c x +70 a^{2} c^{2}+10 c a \,b^{2}+b^{4}}{1260 \left (2 c x +b \right )^{9} d^{10} c^{3}}\) | \(88\) |
norman | \(\frac {\frac {a^{2} x}{b d}+\frac {\left (8 a^{2} c +a \,b^{2}\right ) x^{2}}{b^{2} d}+\frac {\left (112 a^{2} c^{2}+16 c a \,b^{2}+b^{4}\right ) x^{3}}{3 b^{3} d}+\frac {64 c^{5} \left (70 a^{2} c^{2}+10 c a \,b^{2}+b^{4}\right ) x^{8}}{35 b^{8} d}+\frac {128 c^{4} \left (70 a^{2} c^{2}+10 c a \,b^{2}+b^{4}\right ) x^{7}}{35 b^{7} d}+\frac {64 c^{3} \left (70 a^{2} c^{2}+10 c a \,b^{2}+b^{4}\right ) x^{6}}{15 b^{6} d}+\frac {16 c^{2} \left (70 a^{2} c^{2}+10 c a \,b^{2}+b^{4}\right ) x^{5}}{5 b^{5} d}+\frac {c \left (224 a^{2} c^{2}+32 c a \,b^{2}+3 b^{4}\right ) x^{4}}{2 b^{4} d}+\frac {128 c^{6} \left (70 a^{2} c^{2}+10 c a \,b^{2}+b^{4}\right ) x^{9}}{315 b^{9} d}}{d^{9} \left (2 c x +b \right )^{9}}\) | \(275\) |
parallelrisch | \(\frac {17920 x^{9} a^{2} c^{8}+2560 x^{9} a \,b^{2} c^{7}+256 x^{9} b^{4} c^{6}+80640 x^{8} a^{2} b \,c^{7}+11520 x^{8} a \,b^{3} c^{6}+1152 x^{8} b^{5} c^{5}+161280 x^{7} a^{2} b^{2} c^{6}+23040 x^{7} a \,b^{4} c^{5}+2304 x^{7} b^{6} c^{4}+188160 x^{6} a^{2} b^{3} c^{5}+26880 x^{6} a \,b^{5} c^{4}+2688 x^{6} b^{7} c^{3}+141120 x^{5} a^{2} b^{4} c^{4}+20160 x^{5} a \,b^{6} c^{3}+2016 x^{5} b^{8} c^{2}+70560 x^{4} a^{2} b^{5} c^{3}+10080 x^{4} a \,b^{7} c^{2}+945 x^{4} b^{9} c +23520 x^{3} a^{2} b^{6} c^{2}+3360 x^{3} a \,b^{8} c +210 x^{3} b^{10}+5040 x^{2} a^{2} b^{7} c +630 x^{2} a \,b^{9}+630 x \,a^{2} b^{8}}{630 b^{9} d^{10} \left (2 c x +b \right )^{9}}\) | \(295\) |
orering | \(\frac {x \left (17920 a^{2} c^{8} x^{8}+2560 a \,b^{2} c^{7} x^{8}+256 b^{4} c^{6} x^{8}+80640 a^{2} b \,c^{7} x^{7}+11520 a \,b^{3} c^{6} x^{7}+1152 b^{5} c^{5} x^{7}+161280 a^{2} b^{2} c^{6} x^{6}+23040 a \,b^{4} c^{5} x^{6}+2304 b^{6} c^{4} x^{6}+188160 a^{2} b^{3} c^{5} x^{5}+26880 a \,b^{5} c^{4} x^{5}+2688 b^{7} c^{3} x^{5}+141120 a^{2} b^{4} c^{4} x^{4}+20160 a \,b^{6} c^{3} x^{4}+2016 b^{8} c^{2} x^{4}+70560 a^{2} b^{5} c^{3} x^{3}+10080 a \,b^{7} c^{2} x^{3}+945 b^{9} c \,x^{3}+23520 a^{2} b^{6} c^{2} x^{2}+3360 a \,b^{8} c \,x^{2}+210 b^{10} x^{2}+5040 a^{2} b^{7} c x +630 a \,b^{9} x +630 a^{2} b^{8}\right ) \left (2 c x +b \right )}{630 b^{9} \left (2 c d x +b d \right )^{10}}\) | \(297\) |
Input:
int((c*x^2+b*x+a)^2/(2*c*d*x+b*d)^10,x,method=_RETURNVERBOSE)
Output:
1/d^10*(-1/112*(4*a*c-b^2)/c^3/(2*c*x+b)^7-1/160/c^3/(2*c*x+b)^5-1/288*(16 *a^2*c^2-8*a*b^2*c+b^4)/c^3/(2*c*x+b)^9)
Leaf count of result is larger than twice the leaf count of optimal. 206 vs. \(2 (67) = 134\).
Time = 0.08 (sec) , antiderivative size = 206, normalized size of antiderivative = 2.82 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^{10}} \, dx=-\frac {126 \, c^{4} x^{4} + 252 \, b c^{3} x^{3} + b^{4} + 10 \, a b^{2} c + 70 \, a^{2} c^{2} + 36 \, {\left (4 \, b^{2} c^{2} + 5 \, a c^{3}\right )} x^{2} + 18 \, {\left (b^{3} c + 10 \, a b c^{2}\right )} x}{1260 \, {\left (512 \, c^{12} d^{10} x^{9} + 2304 \, b c^{11} d^{10} x^{8} + 4608 \, b^{2} c^{10} d^{10} x^{7} + 5376 \, b^{3} c^{9} d^{10} x^{6} + 4032 \, b^{4} c^{8} d^{10} x^{5} + 2016 \, b^{5} c^{7} d^{10} x^{4} + 672 \, b^{6} c^{6} d^{10} x^{3} + 144 \, b^{7} c^{5} d^{10} x^{2} + 18 \, b^{8} c^{4} d^{10} x + b^{9} c^{3} d^{10}\right )}} \] Input:
integrate((c*x^2+b*x+a)^2/(2*c*d*x+b*d)^10,x, algorithm="fricas")
Output:
-1/1260*(126*c^4*x^4 + 252*b*c^3*x^3 + b^4 + 10*a*b^2*c + 70*a^2*c^2 + 36* (4*b^2*c^2 + 5*a*c^3)*x^2 + 18*(b^3*c + 10*a*b*c^2)*x)/(512*c^12*d^10*x^9 + 2304*b*c^11*d^10*x^8 + 4608*b^2*c^10*d^10*x^7 + 5376*b^3*c^9*d^10*x^6 + 4032*b^4*c^8*d^10*x^5 + 2016*b^5*c^7*d^10*x^4 + 672*b^6*c^6*d^10*x^3 + 144 *b^7*c^5*d^10*x^2 + 18*b^8*c^4*d^10*x + b^9*c^3*d^10)
Leaf count of result is larger than twice the leaf count of optimal. 219 vs. \(2 (70) = 140\).
Time = 1.29 (sec) , antiderivative size = 219, normalized size of antiderivative = 3.00 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^{10}} \, dx=\frac {- 70 a^{2} c^{2} - 10 a b^{2} c - b^{4} - 252 b c^{3} x^{3} - 126 c^{4} x^{4} + x^{2} \left (- 180 a c^{3} - 144 b^{2} c^{2}\right ) + x \left (- 180 a b c^{2} - 18 b^{3} c\right )}{1260 b^{9} c^{3} d^{10} + 22680 b^{8} c^{4} d^{10} x + 181440 b^{7} c^{5} d^{10} x^{2} + 846720 b^{6} c^{6} d^{10} x^{3} + 2540160 b^{5} c^{7} d^{10} x^{4} + 5080320 b^{4} c^{8} d^{10} x^{5} + 6773760 b^{3} c^{9} d^{10} x^{6} + 5806080 b^{2} c^{10} d^{10} x^{7} + 2903040 b c^{11} d^{10} x^{8} + 645120 c^{12} d^{10} x^{9}} \] Input:
integrate((c*x**2+b*x+a)**2/(2*c*d*x+b*d)**10,x)
Output:
(-70*a**2*c**2 - 10*a*b**2*c - b**4 - 252*b*c**3*x**3 - 126*c**4*x**4 + x* *2*(-180*a*c**3 - 144*b**2*c**2) + x*(-180*a*b*c**2 - 18*b**3*c))/(1260*b* *9*c**3*d**10 + 22680*b**8*c**4*d**10*x + 181440*b**7*c**5*d**10*x**2 + 84 6720*b**6*c**6*d**10*x**3 + 2540160*b**5*c**7*d**10*x**4 + 5080320*b**4*c* *8*d**10*x**5 + 6773760*b**3*c**9*d**10*x**6 + 5806080*b**2*c**10*d**10*x* *7 + 2903040*b*c**11*d**10*x**8 + 645120*c**12*d**10*x**9)
Leaf count of result is larger than twice the leaf count of optimal. 206 vs. \(2 (67) = 134\).
Time = 0.06 (sec) , antiderivative size = 206, normalized size of antiderivative = 2.82 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^{10}} \, dx=-\frac {126 \, c^{4} x^{4} + 252 \, b c^{3} x^{3} + b^{4} + 10 \, a b^{2} c + 70 \, a^{2} c^{2} + 36 \, {\left (4 \, b^{2} c^{2} + 5 \, a c^{3}\right )} x^{2} + 18 \, {\left (b^{3} c + 10 \, a b c^{2}\right )} x}{1260 \, {\left (512 \, c^{12} d^{10} x^{9} + 2304 \, b c^{11} d^{10} x^{8} + 4608 \, b^{2} c^{10} d^{10} x^{7} + 5376 \, b^{3} c^{9} d^{10} x^{6} + 4032 \, b^{4} c^{8} d^{10} x^{5} + 2016 \, b^{5} c^{7} d^{10} x^{4} + 672 \, b^{6} c^{6} d^{10} x^{3} + 144 \, b^{7} c^{5} d^{10} x^{2} + 18 \, b^{8} c^{4} d^{10} x + b^{9} c^{3} d^{10}\right )}} \] Input:
integrate((c*x^2+b*x+a)^2/(2*c*d*x+b*d)^10,x, algorithm="maxima")
Output:
-1/1260*(126*c^4*x^4 + 252*b*c^3*x^3 + b^4 + 10*a*b^2*c + 70*a^2*c^2 + 36* (4*b^2*c^2 + 5*a*c^3)*x^2 + 18*(b^3*c + 10*a*b*c^2)*x)/(512*c^12*d^10*x^9 + 2304*b*c^11*d^10*x^8 + 4608*b^2*c^10*d^10*x^7 + 5376*b^3*c^9*d^10*x^6 + 4032*b^4*c^8*d^10*x^5 + 2016*b^5*c^7*d^10*x^4 + 672*b^6*c^6*d^10*x^3 + 144 *b^7*c^5*d^10*x^2 + 18*b^8*c^4*d^10*x + b^9*c^3*d^10)
Time = 0.13 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.19 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^{10}} \, dx=-\frac {126 \, c^{4} x^{4} + 252 \, b c^{3} x^{3} + 144 \, b^{2} c^{2} x^{2} + 180 \, a c^{3} x^{2} + 18 \, b^{3} c x + 180 \, a b c^{2} x + b^{4} + 10 \, a b^{2} c + 70 \, a^{2} c^{2}}{1260 \, {\left (2 \, c x + b\right )}^{9} c^{3} d^{10}} \] Input:
integrate((c*x^2+b*x+a)^2/(2*c*d*x+b*d)^10,x, algorithm="giac")
Output:
-1/1260*(126*c^4*x^4 + 252*b*c^3*x^3 + 144*b^2*c^2*x^2 + 180*a*c^3*x^2 + 1 8*b^3*c*x + 180*a*b*c^2*x + b^4 + 10*a*b^2*c + 70*a^2*c^2)/((2*c*x + b)^9* c^3*d^10)
Time = 5.34 (sec) , antiderivative size = 199, normalized size of antiderivative = 2.73 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^{10}} \, dx=-\frac {\frac {70\,a^2\,c^2+10\,a\,b^2\,c+b^4}{1260\,c^3}+\frac {b\,x^3}{5}+\frac {c\,x^4}{10}+\frac {x^2\,\left (4\,b^2+5\,a\,c\right )}{35\,c}+\frac {b\,x\,\left (b^2+10\,a\,c\right )}{70\,c^2}}{b^9\,d^{10}+18\,b^8\,c\,d^{10}\,x+144\,b^7\,c^2\,d^{10}\,x^2+672\,b^6\,c^3\,d^{10}\,x^3+2016\,b^5\,c^4\,d^{10}\,x^4+4032\,b^4\,c^5\,d^{10}\,x^5+5376\,b^3\,c^6\,d^{10}\,x^6+4608\,b^2\,c^7\,d^{10}\,x^7+2304\,b\,c^8\,d^{10}\,x^8+512\,c^9\,d^{10}\,x^9} \] Input:
int((a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^10,x)
Output:
-((b^4 + 70*a^2*c^2 + 10*a*b^2*c)/(1260*c^3) + (b*x^3)/5 + (c*x^4)/10 + (x ^2*(5*a*c + 4*b^2))/(35*c) + (b*x*(10*a*c + b^2))/(70*c^2))/(b^9*d^10 + 51 2*c^9*d^10*x^9 + 2304*b*c^8*d^10*x^8 + 144*b^7*c^2*d^10*x^2 + 672*b^6*c^3* d^10*x^3 + 2016*b^5*c^4*d^10*x^4 + 4032*b^4*c^5*d^10*x^5 + 5376*b^3*c^6*d^ 10*x^6 + 4608*b^2*c^7*d^10*x^7 + 18*b^8*c*d^10*x)
Time = 0.18 (sec) , antiderivative size = 177, normalized size of antiderivative = 2.42 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^{10}} \, dx=\frac {-126 c^{4} x^{4}-252 b \,c^{3} x^{3}-180 a \,c^{3} x^{2}-144 b^{2} c^{2} x^{2}-180 a b \,c^{2} x -18 b^{3} c x -70 a^{2} c^{2}-10 a \,b^{2} c -b^{4}}{1260 c^{3} d^{10} \left (512 c^{9} x^{9}+2304 b \,c^{8} x^{8}+4608 b^{2} c^{7} x^{7}+5376 b^{3} c^{6} x^{6}+4032 b^{4} c^{5} x^{5}+2016 b^{5} c^{4} x^{4}+672 b^{6} c^{3} x^{3}+144 b^{7} c^{2} x^{2}+18 b^{8} c x +b^{9}\right )} \] Input:
int((c*x^2+b*x+a)^2/(2*c*d*x+b*d)^10,x)
Output:
( - 70*a**2*c**2 - 10*a*b**2*c - 180*a*b*c**2*x - 180*a*c**3*x**2 - b**4 - 18*b**3*c*x - 144*b**2*c**2*x**2 - 252*b*c**3*x**3 - 126*c**4*x**4)/(1260 *c**3*d**10*(b**9 + 18*b**8*c*x + 144*b**7*c**2*x**2 + 672*b**6*c**3*x**3 + 2016*b**5*c**4*x**4 + 4032*b**4*c**5*x**5 + 5376*b**3*c**6*x**6 + 4608*b **2*c**7*x**7 + 2304*b*c**8*x**8 + 512*c**9*x**9))