Integrand size = 24, antiderivative size = 37 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^9} \, dx=\frac {\left (a+b x+c x^2\right )^4}{4 \left (b^2-4 a c\right ) d^9 (b+2 c x)^8} \] Output:
1/4*(c*x^2+b*x+a)^4/(-4*a*c+b^2)/d^9/(2*c*x+b)^8
Leaf count is larger than twice the leaf count of optimal. \(96\) vs. \(2(37)=74\).
Time = 0.07 (sec) , antiderivative size = 96, normalized size of antiderivative = 2.59 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^9} \, dx=\frac {b^6-12 a b^4 c+48 a^2 b^2 c^2-64 a^3 c^3-4 \left (b^2-4 a c\right )^2 (b+2 c x)^2+6 \left (b^2-4 a c\right ) (b+2 c x)^4-4 (b+2 c x)^6}{1024 c^4 d^9 (b+2 c x)^8} \] Input:
Integrate[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^9,x]
Output:
(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3 - 4*(b^2 - 4*a*c)^2*(b + 2 *c*x)^2 + 6*(b^2 - 4*a*c)*(b + 2*c*x)^4 - 4*(b + 2*c*x)^6)/(1024*c^4*d^9*( b + 2*c*x)^8)
Time = 0.17 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {1106}
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.
\(\displaystyle \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^9} \, dx\) |
\(\Big \downarrow \) 1106 |
\(\displaystyle \frac {\left (a+b x+c x^2\right )^4}{4 d^9 \left (b^2-4 a c\right ) (b+2 c x)^8}\) |
Input:
Int[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^9,x]
Output:
(a + b*x + c*x^2)^4/(4*(b^2 - 4*a*c)*d^9*(b + 2*c*x)^8)
Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_S ymbol] :> Simp[2*c*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(e*(p + 1)* (b^2 - 4*a*c))), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[2*c*d - b*e, 0] && EqQ[m + 2*p + 3, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(120\) vs. \(2(35)=70\).
Time = 0.83 (sec) , antiderivative size = 121, normalized size of antiderivative = 3.27
method | result | size |
default | \(\frac {-\frac {12 a c -3 b^{2}}{512 c^{4} \left (2 c x +b \right )^{4}}-\frac {48 a^{2} c^{2}-24 c a \,b^{2}+3 b^{4}}{768 c^{4} \left (2 c x +b \right )^{6}}-\frac {64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}{1024 c^{4} \left (2 c x +b \right )^{8}}-\frac {1}{256 c^{4} \left (2 c x +b \right )^{2}}}{d^{9}}\) | \(121\) |
risch | \(\frac {-\frac {c^{2} x^{6}}{4}-\frac {3 x^{5} b c}{4}+\left (-\frac {3 a c}{8}-\frac {27 b^{2}}{32}\right ) x^{4}-\frac {b \left (12 a c +7 b^{2}\right ) x^{3}}{16 c}-\frac {\left (16 a^{2} c^{2}+28 c a \,b^{2}+7 b^{4}\right ) x^{2}}{64 c^{2}}-\frac {b \left (16 a^{2} c^{2}+4 c a \,b^{2}+b^{4}\right ) x}{64 c^{3}}-\frac {64 a^{3} c^{3}+16 a^{2} b^{2} c^{2}+4 a \,b^{4} c +b^{6}}{1024 c^{4}}}{d^{9} \left (2 c x +b \right )^{8}}\) | \(152\) |
gosper | \(-\frac {256 x^{6} c^{6}+768 x^{5} b \,c^{5}+384 a \,c^{5} x^{4}+864 x^{4} b^{2} c^{4}+768 a b \,c^{4} x^{3}+448 b^{3} c^{3} x^{3}+256 a^{2} c^{4} x^{2}+448 a \,b^{2} c^{3} x^{2}+112 c^{2} x^{2} b^{4}+256 a^{2} b \,c^{3} x +64 x a \,b^{3} c^{2}+16 x c \,b^{5}+64 a^{3} c^{3}+16 a^{2} b^{2} c^{2}+4 a \,b^{4} c +b^{6}}{1024 \left (2 c x +b \right )^{8} d^{9} c^{4}}\) | \(166\) |
norman | \(\frac {\frac {a^{3} x}{b d}+\frac {\left (28 c^{2} a^{3}+7 a^{2} b^{2} c +a \,b^{4}\right ) x^{3}}{d \,b^{3}}+\frac {c^{3} \left (64 a^{3} c^{3}+16 a^{2} b^{2} c^{2}+4 a \,b^{4} c +b^{6}\right ) x^{7}}{b^{7} d}+\frac {c \left (112 a^{3} c^{3}+28 a^{2} b^{2} c^{2}+7 a \,b^{4} c +b^{6}\right ) x^{5}}{b^{5} d}+\frac {\left (14 c \,a^{3}+3 a^{2} b^{2}\right ) x^{2}}{2 b^{2} d}+\frac {\left (280 a^{3} c^{3}+70 a^{2} b^{2} c^{2}+16 a \,b^{4} c +b^{6}\right ) x^{4}}{4 b^{4} d}+\frac {c^{2} \left (224 a^{3} c^{3}+56 a^{2} b^{2} c^{2}+14 a \,b^{4} c +3 b^{6}\right ) x^{6}}{2 d \,b^{6}}+\frac {c^{4} \left (64 a^{3} c^{3}+16 a^{2} b^{2} c^{2}+4 a \,b^{4} c +b^{6}\right ) x^{8}}{4 b^{8} d}}{d^{8} \left (2 c x +b \right )^{8}}\) | \(299\) |
parallelrisch | \(\frac {64 x^{8} a^{3} c^{7}+16 x^{8} a^{2} b^{2} c^{6}+4 x^{8} a \,b^{4} c^{5}+x^{8} b^{6} c^{4}+256 x^{7} a^{3} b \,c^{6}+64 x^{7} a^{2} b^{3} c^{5}+16 x^{7} a \,b^{5} c^{4}+4 x^{7} b^{7} c^{3}+448 x^{6} a^{3} b^{2} c^{5}+112 x^{6} a^{2} b^{4} c^{4}+28 x^{6} a \,b^{6} c^{3}+6 x^{6} b^{8} c^{2}+448 x^{5} a^{3} b^{3} c^{4}+112 x^{5} a^{2} b^{5} c^{3}+28 x^{5} a \,b^{7} c^{2}+4 x^{5} b^{9} c +280 x^{4} a^{3} b^{4} c^{3}+70 x^{4} a^{2} b^{6} c^{2}+16 x^{4} a \,b^{8} c +x^{4} b^{10}+112 x^{3} a^{3} b^{5} c^{2}+28 x^{3} a^{2} b^{7} c +4 x^{3} a \,b^{9}+28 x^{2} a^{3} b^{6} c +6 x^{2} a^{2} b^{8}+4 a^{3} b^{7} x}{4 b^{8} d^{9} \left (2 c x +b \right )^{8}}\) | \(326\) |
orering | \(\frac {x \left (64 a^{3} c^{7} x^{7}+16 a^{2} b^{2} c^{6} x^{7}+4 a \,b^{4} c^{5} x^{7}+b^{6} c^{4} x^{7}+256 a^{3} b \,c^{6} x^{6}+64 a^{2} b^{3} c^{5} x^{6}+16 a \,b^{5} c^{4} x^{6}+4 b^{7} c^{3} x^{6}+448 a^{3} b^{2} c^{5} x^{5}+112 a^{2} b^{4} c^{4} x^{5}+28 a \,b^{6} c^{3} x^{5}+6 b^{8} c^{2} x^{5}+448 a^{3} b^{3} c^{4} x^{4}+112 a^{2} b^{5} c^{3} x^{4}+28 a \,b^{7} c^{2} x^{4}+4 b^{9} c \,x^{4}+280 a^{3} b^{4} c^{3} x^{3}+70 a^{2} b^{6} c^{2} x^{3}+16 a \,b^{8} c \,x^{3}+b^{10} x^{3}+112 a^{3} b^{5} c^{2} x^{2}+28 a^{2} b^{7} c \,x^{2}+4 x^{2} a \,b^{9}+28 a^{3} b^{6} c x +6 a^{2} b^{8} x +4 a^{3} b^{7}\right ) \left (2 c x +b \right )}{4 b^{8} \left (2 c d x +b d \right )^{9}}\) | \(328\) |
Input:
int((c*x^2+b*x+a)^3/(2*c*d*x+b*d)^9,x,method=_RETURNVERBOSE)
Output:
1/d^9*(-1/512*(12*a*c-3*b^2)/c^4/(2*c*x+b)^4-1/768*(48*a^2*c^2-24*a*b^2*c+ 3*b^4)/c^4/(2*c*x+b)^6-1/1024*(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)/c ^4/(2*c*x+b)^8-1/256/c^4/(2*c*x+b)^2)
Leaf count of result is larger than twice the leaf count of optimal. 266 vs. \(2 (35) = 70\).
Time = 0.08 (sec) , antiderivative size = 266, normalized size of antiderivative = 7.19 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^9} \, dx=-\frac {256 \, c^{6} x^{6} + 768 \, b c^{5} x^{5} + b^{6} + 4 \, a b^{4} c + 16 \, a^{2} b^{2} c^{2} + 64 \, a^{3} c^{3} + 96 \, {\left (9 \, b^{2} c^{4} + 4 \, a c^{5}\right )} x^{4} + 64 \, {\left (7 \, b^{3} c^{3} + 12 \, a b c^{4}\right )} x^{3} + 16 \, {\left (7 \, b^{4} c^{2} + 28 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{2} + 16 \, {\left (b^{5} c + 4 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x}{1024 \, {\left (256 \, c^{12} d^{9} x^{8} + 1024 \, b c^{11} d^{9} x^{7} + 1792 \, b^{2} c^{10} d^{9} x^{6} + 1792 \, b^{3} c^{9} d^{9} x^{5} + 1120 \, b^{4} c^{8} d^{9} x^{4} + 448 \, b^{5} c^{7} d^{9} x^{3} + 112 \, b^{6} c^{6} d^{9} x^{2} + 16 \, b^{7} c^{5} d^{9} x + b^{8} c^{4} d^{9}\right )}} \] Input:
integrate((c*x^2+b*x+a)^3/(2*c*d*x+b*d)^9,x, algorithm="fricas")
Output:
-1/1024*(256*c^6*x^6 + 768*b*c^5*x^5 + b^6 + 4*a*b^4*c + 16*a^2*b^2*c^2 + 64*a^3*c^3 + 96*(9*b^2*c^4 + 4*a*c^5)*x^4 + 64*(7*b^3*c^3 + 12*a*b*c^4)*x^ 3 + 16*(7*b^4*c^2 + 28*a*b^2*c^3 + 16*a^2*c^4)*x^2 + 16*(b^5*c + 4*a*b^3*c ^2 + 16*a^2*b*c^3)*x)/(256*c^12*d^9*x^8 + 1024*b*c^11*d^9*x^7 + 1792*b^2*c ^10*d^9*x^6 + 1792*b^3*c^9*d^9*x^5 + 1120*b^4*c^8*d^9*x^4 + 448*b^5*c^7*d^ 9*x^3 + 112*b^6*c^6*d^9*x^2 + 16*b^7*c^5*d^9*x + b^8*c^4*d^9)
Leaf count of result is larger than twice the leaf count of optimal. 282 vs. \(2 (32) = 64\).
Time = 5.02 (sec) , antiderivative size = 282, normalized size of antiderivative = 7.62 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^9} \, dx=\frac {- 64 a^{3} c^{3} - 16 a^{2} b^{2} c^{2} - 4 a b^{4} c - b^{6} - 768 b c^{5} x^{5} - 256 c^{6} x^{6} + x^{4} \left (- 384 a c^{5} - 864 b^{2} c^{4}\right ) + x^{3} \left (- 768 a b c^{4} - 448 b^{3} c^{3}\right ) + x^{2} \left (- 256 a^{2} c^{4} - 448 a b^{2} c^{3} - 112 b^{4} c^{2}\right ) + x \left (- 256 a^{2} b c^{3} - 64 a b^{3} c^{2} - 16 b^{5} c\right )}{1024 b^{8} c^{4} d^{9} + 16384 b^{7} c^{5} d^{9} x + 114688 b^{6} c^{6} d^{9} x^{2} + 458752 b^{5} c^{7} d^{9} x^{3} + 1146880 b^{4} c^{8} d^{9} x^{4} + 1835008 b^{3} c^{9} d^{9} x^{5} + 1835008 b^{2} c^{10} d^{9} x^{6} + 1048576 b c^{11} d^{9} x^{7} + 262144 c^{12} d^{9} x^{8}} \] Input:
integrate((c*x**2+b*x+a)**3/(2*c*d*x+b*d)**9,x)
Output:
(-64*a**3*c**3 - 16*a**2*b**2*c**2 - 4*a*b**4*c - b**6 - 768*b*c**5*x**5 - 256*c**6*x**6 + x**4*(-384*a*c**5 - 864*b**2*c**4) + x**3*(-768*a*b*c**4 - 448*b**3*c**3) + x**2*(-256*a**2*c**4 - 448*a*b**2*c**3 - 112*b**4*c**2) + x*(-256*a**2*b*c**3 - 64*a*b**3*c**2 - 16*b**5*c))/(1024*b**8*c**4*d**9 + 16384*b**7*c**5*d**9*x + 114688*b**6*c**6*d**9*x**2 + 458752*b**5*c**7* d**9*x**3 + 1146880*b**4*c**8*d**9*x**4 + 1835008*b**3*c**9*d**9*x**5 + 18 35008*b**2*c**10*d**9*x**6 + 1048576*b*c**11*d**9*x**7 + 262144*c**12*d**9 *x**8)
Leaf count of result is larger than twice the leaf count of optimal. 266 vs. \(2 (35) = 70\).
Time = 0.05 (sec) , antiderivative size = 266, normalized size of antiderivative = 7.19 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^9} \, dx=-\frac {256 \, c^{6} x^{6} + 768 \, b c^{5} x^{5} + b^{6} + 4 \, a b^{4} c + 16 \, a^{2} b^{2} c^{2} + 64 \, a^{3} c^{3} + 96 \, {\left (9 \, b^{2} c^{4} + 4 \, a c^{5}\right )} x^{4} + 64 \, {\left (7 \, b^{3} c^{3} + 12 \, a b c^{4}\right )} x^{3} + 16 \, {\left (7 \, b^{4} c^{2} + 28 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{2} + 16 \, {\left (b^{5} c + 4 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x}{1024 \, {\left (256 \, c^{12} d^{9} x^{8} + 1024 \, b c^{11} d^{9} x^{7} + 1792 \, b^{2} c^{10} d^{9} x^{6} + 1792 \, b^{3} c^{9} d^{9} x^{5} + 1120 \, b^{4} c^{8} d^{9} x^{4} + 448 \, b^{5} c^{7} d^{9} x^{3} + 112 \, b^{6} c^{6} d^{9} x^{2} + 16 \, b^{7} c^{5} d^{9} x + b^{8} c^{4} d^{9}\right )}} \] Input:
integrate((c*x^2+b*x+a)^3/(2*c*d*x+b*d)^9,x, algorithm="maxima")
Output:
-1/1024*(256*c^6*x^6 + 768*b*c^5*x^5 + b^6 + 4*a*b^4*c + 16*a^2*b^2*c^2 + 64*a^3*c^3 + 96*(9*b^2*c^4 + 4*a*c^5)*x^4 + 64*(7*b^3*c^3 + 12*a*b*c^4)*x^ 3 + 16*(7*b^4*c^2 + 28*a*b^2*c^3 + 16*a^2*c^4)*x^2 + 16*(b^5*c + 4*a*b^3*c ^2 + 16*a^2*b*c^3)*x)/(256*c^12*d^9*x^8 + 1024*b*c^11*d^9*x^7 + 1792*b^2*c ^10*d^9*x^6 + 1792*b^3*c^9*d^9*x^5 + 1120*b^4*c^8*d^9*x^4 + 448*b^5*c^7*d^ 9*x^3 + 112*b^6*c^6*d^9*x^2 + 16*b^7*c^5*d^9*x + b^8*c^4*d^9)
Leaf count of result is larger than twice the leaf count of optimal. 165 vs. \(2 (35) = 70\).
Time = 0.12 (sec) , antiderivative size = 165, normalized size of antiderivative = 4.46 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^9} \, dx=-\frac {256 \, c^{6} x^{6} + 768 \, b c^{5} x^{5} + 864 \, b^{2} c^{4} x^{4} + 384 \, a c^{5} x^{4} + 448 \, b^{3} c^{3} x^{3} + 768 \, a b c^{4} x^{3} + 112 \, b^{4} c^{2} x^{2} + 448 \, a b^{2} c^{3} x^{2} + 256 \, a^{2} c^{4} x^{2} + 16 \, b^{5} c x + 64 \, a b^{3} c^{2} x + 256 \, a^{2} b c^{3} x + b^{6} + 4 \, a b^{4} c + 16 \, a^{2} b^{2} c^{2} + 64 \, a^{3} c^{3}}{1024 \, {\left (2 \, c x + b\right )}^{8} c^{4} d^{9}} \] Input:
integrate((c*x^2+b*x+a)^3/(2*c*d*x+b*d)^9,x, algorithm="giac")
Output:
-1/1024*(256*c^6*x^6 + 768*b*c^5*x^5 + 864*b^2*c^4*x^4 + 384*a*c^5*x^4 + 4 48*b^3*c^3*x^3 + 768*a*b*c^4*x^3 + 112*b^4*c^2*x^2 + 448*a*b^2*c^3*x^2 + 2 56*a^2*c^4*x^2 + 16*b^5*c*x + 64*a*b^3*c^2*x + 256*a^2*b*c^3*x + b^6 + 4*a *b^4*c + 16*a^2*b^2*c^2 + 64*a^3*c^3)/((2*c*x + b)^8*c^4*d^9)
Time = 5.41 (sec) , antiderivative size = 254, normalized size of antiderivative = 6.86 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^9} \, dx=-\frac {\frac {64\,a^3\,c^3+16\,a^2\,b^2\,c^2+4\,a\,b^4\,c+b^6}{1024\,c^4}+x^4\,\left (\frac {27\,b^2}{32}+\frac {3\,a\,c}{8}\right )+\frac {c^2\,x^6}{4}+\frac {x^2\,\left (16\,a^2\,c^2+28\,a\,b^2\,c+7\,b^4\right )}{64\,c^2}+\frac {3\,b\,c\,x^5}{4}+\frac {x^3\,\left (7\,b^3+12\,a\,c\,b\right )}{16\,c}+\frac {b\,x\,\left (16\,a^2\,c^2+4\,a\,b^2\,c+b^4\right )}{64\,c^3}}{b^8\,d^9+16\,b^7\,c\,d^9\,x+112\,b^6\,c^2\,d^9\,x^2+448\,b^5\,c^3\,d^9\,x^3+1120\,b^4\,c^4\,d^9\,x^4+1792\,b^3\,c^5\,d^9\,x^5+1792\,b^2\,c^6\,d^9\,x^6+1024\,b\,c^7\,d^9\,x^7+256\,c^8\,d^9\,x^8} \] Input:
int((a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^9,x)
Output:
-((b^6 + 64*a^3*c^3 + 16*a^2*b^2*c^2 + 4*a*b^4*c)/(1024*c^4) + x^4*((3*a*c )/8 + (27*b^2)/32) + (c^2*x^6)/4 + (x^2*(7*b^4 + 16*a^2*c^2 + 28*a*b^2*c)) /(64*c^2) + (3*b*c*x^5)/4 + (x^3*(7*b^3 + 12*a*b*c))/(16*c) + (b*x*(b^4 + 16*a^2*c^2 + 4*a*b^2*c))/(64*c^3))/(b^8*d^9 + 256*c^8*d^9*x^8 + 1024*b*c^7 *d^9*x^7 + 112*b^6*c^2*d^9*x^2 + 448*b^5*c^3*d^9*x^3 + 1120*b^4*c^4*d^9*x^ 4 + 1792*b^3*c^5*d^9*x^5 + 1792*b^2*c^6*d^9*x^6 + 16*b^7*c*d^9*x)
Time = 0.28 (sec) , antiderivative size = 244, normalized size of antiderivative = 6.59 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^9} \, dx=\frac {-256 c^{6} x^{6}-768 b \,c^{5} x^{5}-384 a \,c^{5} x^{4}-864 b^{2} c^{4} x^{4}-768 a b \,c^{4} x^{3}-448 b^{3} c^{3} x^{3}-256 a^{2} c^{4} x^{2}-448 a \,b^{2} c^{3} x^{2}-112 b^{4} c^{2} x^{2}-256 a^{2} b \,c^{3} x -64 a \,b^{3} c^{2} x -16 b^{5} c x -64 a^{3} c^{3}-16 a^{2} b^{2} c^{2}-4 a \,b^{4} c -b^{6}}{1024 c^{4} d^{9} \left (256 c^{8} x^{8}+1024 b \,c^{7} x^{7}+1792 b^{2} c^{6} x^{6}+1792 b^{3} c^{5} x^{5}+1120 b^{4} c^{4} x^{4}+448 b^{5} c^{3} x^{3}+112 b^{6} c^{2} x^{2}+16 b^{7} c x +b^{8}\right )} \] Input:
int((c*x^2+b*x+a)^3/(2*c*d*x+b*d)^9,x)
Output:
( - 64*a**3*c**3 - 16*a**2*b**2*c**2 - 256*a**2*b*c**3*x - 256*a**2*c**4*x **2 - 4*a*b**4*c - 64*a*b**3*c**2*x - 448*a*b**2*c**3*x**2 - 768*a*b*c**4* x**3 - 384*a*c**5*x**4 - b**6 - 16*b**5*c*x - 112*b**4*c**2*x**2 - 448*b** 3*c**3*x**3 - 864*b**2*c**4*x**4 - 768*b*c**5*x**5 - 256*c**6*x**6)/(1024* c**4*d**9*(b**8 + 16*b**7*c*x + 112*b**6*c**2*x**2 + 448*b**5*c**3*x**3 + 1120*b**4*c**4*x**4 + 1792*b**3*c**5*x**5 + 1792*b**2*c**6*x**6 + 1024*b*c **7*x**7 + 256*c**8*x**8))