\(\int \frac {(a+b x+c x^2)^2}{(b d+2 c d x)^{10}} \, dx\) [1135]

Optimal result

Integrand size = 24, antiderivative size = 73 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^{10}} \, dx=-\frac {\left (b^2-4 a c\right )^2}{288 c^3 d^{10} (b+2 c x)^9}+\frac {b^2-4 a c}{112 c^3 d^{10} (b+2 c x)^7}-\frac {1}{160 c^3 d^{10} (b+2 c x)^5} \]

[Out]

-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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {697} \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^{10}} \, dx=-\frac {\left (b^2-4 a c\right )^2}{288 c^3 d^{10} (b+2 c x)^9}+\frac {b^2-4 a c}{112 c^3 d^{10} (b+2 c x)^7}-\frac {1}{160 c^3 d^{10} (b+2 c x)^5} \]

[In]

Int[(a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^10,x]

[Out]

-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)

Rule 697

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] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e,
0] && IGtQ[p, 0] &&  !(EqQ[m, 3] && NeQ[p, 1])

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (-b^2+4 a c\right )^2}{16 c^2 d^{10} (b+2 c x)^{10}}+\frac {-b^2+4 a c}{8 c^2 d^{10} (b+2 c x)^8}+\frac {1}{16 c^2 d^{10} (b+2 c x)^6}\right ) \, 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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.81 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^{10}} \, dx=\frac {-35 \left (b^2-4 a c\right )^2+90 \left (b^2-4 a c\right ) (b+2 c x)^2-63 (b+2 c x)^4}{10080 c^3 d^{10} (b+2 c x)^9} \]

[In]

Integrate[(a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^10,x]

[Out]

(-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)

Maple [A] (verified)

Time = 2.33 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.01

method	result	size
default	\(\frac {-\frac {1}{160 c^{3} \left (2 c x +b \right )^{5}}-\frac {4 a c -b^{2}}{112 c^{3} \left (2 c x +b \right )^{7}}-\frac {16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}{288 c^{3} \left (2 c x +b \right )^{9}}}{d^{10}}\)	\(74\)
risch	\(\frac {-\frac {x^{4} c}{10}-\frac {b \,x^{3}}{5}-\frac {\left (5 a c +4 b^{2}\right ) x^{2}}{35 c}-\frac {b \left (10 a c +b^{2}\right ) x}{70 c^{2}}-\frac {70 a^{2} c^{2}+10 a \,b^{2} c +b^{4}}{1260 c^{3}}}{d^{10} \left (2 c x +b \right )^{9}}\)	\(83\)
gosper	\(-\frac {126 c^{4} x^{4}+252 b \,c^{3} x^{3}+180 x^{2} a \,c^{3}+144 b^{2} c^{2} x^{2}+180 a b \,c^{2} x +18 b^{3} c x +70 a^{2} c^{2}+10 a \,b^{2} c +b^{4}}{1260 \left (2 c x +b \right )^{9} d^{10} c^{3}}\)	\(88\)
norman	\(\frac {\frac {a^{2} x}{b d}+\frac {\left (8 a^{2} c +a \,b^{2}\right ) x^{2}}{b^{2} d}+\frac {\left (112 a^{2} c^{2}+16 a \,b^{2} c +b^{4}\right ) x^{3}}{3 b^{3} d}+\frac {64 c^{5} \left (70 a^{2} c^{2}+10 a \,b^{2} c +b^{4}\right ) x^{8}}{35 b^{8} d}+\frac {128 c^{4} \left (70 a^{2} c^{2}+10 a \,b^{2} c +b^{4}\right ) x^{7}}{35 b^{7} d}+\frac {64 c^{3} \left (70 a^{2} c^{2}+10 a \,b^{2} c +b^{4}\right ) x^{6}}{15 b^{6} d}+\frac {16 c^{2} \left (70 a^{2} c^{2}+10 a \,b^{2} c +b^{4}\right ) x^{5}}{5 b^{5} d}+\frac {c \left (224 a^{2} c^{2}+32 a \,b^{2} c +3 b^{4}\right ) x^{4}}{2 b^{4} d}+\frac {128 c^{6} \left (70 a^{2} c^{2}+10 a \,b^{2} c +b^{4}\right ) x^{9}}{315 b^{9} d}}{d^{9} \left (2 c x +b \right )^{9}}\)	\(275\)
parallelrisch	\(\frac {17920 x^{9} a^{2} c^{8}+2560 x^{9} a \,b^{2} c^{7}+256 x^{9} b^{4} c^{6}+80640 x^{8} a^{2} b \,c^{7}+11520 x^{8} a \,b^{3} c^{6}+1152 x^{8} b^{5} c^{5}+161280 x^{7} a^{2} b^{2} c^{6}+23040 x^{7} a \,b^{4} c^{5}+2304 x^{7} b^{6} c^{4}+188160 x^{6} a^{2} b^{3} c^{5}+26880 x^{6} a \,b^{5} c^{4}+2688 x^{6} b^{7} c^{3}+141120 x^{5} a^{2} b^{4} c^{4}+20160 x^{5} a \,b^{6} c^{3}+2016 x^{5} b^{8} c^{2}+70560 x^{4} a^{2} b^{5} c^{3}+10080 x^{4} a \,b^{7} c^{2}+945 x^{4} b^{9} c +23520 x^{3} a^{2} b^{6} c^{2}+3360 x^{3} a \,b^{8} c +210 b^{10} x^{3}+5040 x^{2} a^{2} b^{7} c +630 x^{2} b^{9} a +630 b^{8} x \,a^{2}}{630 b^{9} d^{10} \left (2 c x +b \right )^{9}}\)	\(295\)

[In]

int((c*x^2+b*x+a)^2/(2*c*d*x+b*d)^10,x,method=_RETURNVERBOSE)

[Out]

1/d^10*(-1/160/c^3/(2*c*x+b)^5-1/112*(4*a*c-b^2)/c^3/(2*c*x+b)^7-1/288*(16*a^2*c^2-8*a*b^2*c+b^4)/c^3/(2*c*x+b
)^9)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 206 vs. \(2 (67) = 134\).

Time = 0.27 (sec) , antiderivative size = 206, normalized size of antiderivative = 2.82 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^{10}} \, dx=-\frac {126 \, c^{4} x^{4} + 252 \, b c^{3} x^{3} + b^{4} + 10 \, a b^{2} c + 70 \, a^{2} c^{2} + 36 \, {\left (4 \, b^{2} c^{2} + 5 \, a c^{3}\right )} x^{2} + 18 \, {\left (b^{3} c + 10 \, a b c^{2}\right )} x}{1260 \, {\left (512 \, c^{12} d^{10} x^{9} + 2304 \, b c^{11} d^{10} x^{8} + 4608 \, b^{2} c^{10} d^{10} x^{7} + 5376 \, b^{3} c^{9} d^{10} x^{6} + 4032 \, b^{4} c^{8} d^{10} x^{5} + 2016 \, b^{5} c^{7} d^{10} x^{4} + 672 \, b^{6} c^{6} d^{10} x^{3} + 144 \, b^{7} c^{5} d^{10} x^{2} + 18 \, b^{8} c^{4} d^{10} x + b^{9} c^{3} d^{10}\right )}} \]

[In]

integrate((c*x^2+b*x+a)^2/(2*c*d*x+b*d)^10,x, algorithm="fricas")

[Out]

-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^1
0*x + b^9*c^3*d^10)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 219 vs. \(2 (70) = 140\).

Time = 1.43 (sec) , antiderivative size = 219, normalized size of antiderivative = 3.00 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^{10}} \, dx=\frac {- 70 a^{2} c^{2} - 10 a b^{2} c - b^{4} - 252 b c^{3} x^{3} - 126 c^{4} x^{4} + x^{2} \left (- 180 a c^{3} - 144 b^{2} c^{2}\right ) + x \left (- 180 a b c^{2} - 18 b^{3} c\right )}{1260 b^{9} c^{3} d^{10} + 22680 b^{8} c^{4} d^{10} x + 181440 b^{7} c^{5} d^{10} x^{2} + 846720 b^{6} c^{6} d^{10} x^{3} + 2540160 b^{5} c^{7} d^{10} x^{4} + 5080320 b^{4} c^{8} d^{10} x^{5} + 6773760 b^{3} c^{9} d^{10} x^{6} + 5806080 b^{2} c^{10} d^{10} x^{7} + 2903040 b c^{11} d^{10} x^{8} + 645120 c^{12} d^{10} x^{9}} \]

[In]

integrate((c*x**2+b*x+a)**2/(2*c*d*x+b*d)**10,x)

[Out]

(-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 +
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)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 206 vs. \(2 (67) = 134\).

Time = 0.20 (sec) , antiderivative size = 206, normalized size of antiderivative = 2.82 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^{10}} \, dx=-\frac {126 \, c^{4} x^{4} + 252 \, b c^{3} x^{3} + b^{4} + 10 \, a b^{2} c + 70 \, a^{2} c^{2} + 36 \, {\left (4 \, b^{2} c^{2} + 5 \, a c^{3}\right )} x^{2} + 18 \, {\left (b^{3} c + 10 \, a b c^{2}\right )} x}{1260 \, {\left (512 \, c^{12} d^{10} x^{9} + 2304 \, b c^{11} d^{10} x^{8} + 4608 \, b^{2} c^{10} d^{10} x^{7} + 5376 \, b^{3} c^{9} d^{10} x^{6} + 4032 \, b^{4} c^{8} d^{10} x^{5} + 2016 \, b^{5} c^{7} d^{10} x^{4} + 672 \, b^{6} c^{6} d^{10} x^{3} + 144 \, b^{7} c^{5} d^{10} x^{2} + 18 \, b^{8} c^{4} d^{10} x + b^{9} c^{3} d^{10}\right )}} \]

[In]

integrate((c*x^2+b*x+a)^2/(2*c*d*x+b*d)^10,x, algorithm="maxima")

[Out]

-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^1
0*x + b^9*c^3*d^10)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.19 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^{10}} \, dx=-\frac {126 \, c^{4} x^{4} + 252 \, b c^{3} x^{3} + 144 \, b^{2} c^{2} x^{2} + 180 \, a c^{3} x^{2} + 18 \, b^{3} c x + 180 \, a b c^{2} x + b^{4} + 10 \, a b^{2} c + 70 \, a^{2} c^{2}}{1260 \, {\left (2 \, c x + b\right )}^{9} c^{3} d^{10}} \]

[In]

integrate((c*x^2+b*x+a)^2/(2*c*d*x+b*d)^10,x, algorithm="giac")

[Out]

-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 + 18*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)

Mupad [B] (verification not implemented)

Time = 10.11 (sec) , antiderivative size = 199, normalized size of antiderivative = 2.73 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^{10}} \, dx=-\frac {\frac {70\,a^2\,c^2+10\,a\,b^2\,c+b^4}{1260\,c^3}+\frac {b\,x^3}{5}+\frac {c\,x^4}{10}+\frac {x^2\,\left (4\,b^2+5\,a\,c\right )}{35\,c}+\frac {b\,x\,\left (b^2+10\,a\,c\right )}{70\,c^2}}{b^9\,d^{10}+18\,b^8\,c\,d^{10}\,x+144\,b^7\,c^2\,d^{10}\,x^2+672\,b^6\,c^3\,d^{10}\,x^3+2016\,b^5\,c^4\,d^{10}\,x^4+4032\,b^4\,c^5\,d^{10}\,x^5+5376\,b^3\,c^6\,d^{10}\,x^6+4608\,b^2\,c^7\,d^{10}\,x^7+2304\,b\,c^8\,d^{10}\,x^8+512\,c^9\,d^{10}\,x^9} \]

[In]

int((a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^10,x)

[Out]

-((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*(1
0*a*c + b^2))/(70*c^2))/(b^9*d^10 + 512*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 + 1
8*b^8*c*d^10*x)