\(\int \frac {(a+b x+c x^2)^3}{(b d+2 c d x)^8} \, dx\) [1149]

Optimal result

Integrand size = 24, antiderivative size = 101 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^8} \, dx=\frac {\left (b^2-4 a c\right )^3}{896 c^4 d^8 (b+2 c x)^7}-\frac {3 \left (b^2-4 a c\right )^2}{640 c^4 d^8 (b+2 c x)^5}+\frac {b^2-4 a c}{128 c^4 d^8 (b+2 c x)^3}-\frac {1}{128 c^4 d^8 (b+2 c x)} \]

[Out]

1/896*(-4*a*c+b^2)^3/c^4/d^8/(2*c*x+b)^7-3/640*(-4*a*c+b^2)^2/c^4/d^8/(2*c*x+b)^5+1/128*(-4*a*c+b^2)/c^4/d^8/(
2*c*x+b)^3-1/128/c^4/d^8/(2*c*x+b)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 101, 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} \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^8} \, dx=\frac {\left (b^2-4 a c\right )^3}{896 c^4 d^8 (b+2 c x)^7}-\frac {3 \left (b^2-4 a c\right )^2}{640 c^4 d^8 (b+2 c x)^5}+\frac {b^2-4 a c}{128 c^4 d^8 (b+2 c x)^3}-\frac {1}{128 c^4 d^8 (b+2 c x)} \]

[In]

Int[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^8,x]

[Out]

(b^2 - 4*a*c)^3/(896*c^4*d^8*(b + 2*c*x)^7) - (3*(b^2 - 4*a*c)^2)/(640*c^4*d^8*(b + 2*c*x)^5) + (b^2 - 4*a*c)/
(128*c^4*d^8*(b + 2*c*x)^3) - 1/(128*c^4*d^8*(b + 2*c*x))

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 )^3}{64 c^3 d^8 (b+2 c x)^8}+\frac {3 \left (-b^2+4 a c\right )^2}{64 c^3 d^8 (b+2 c x)^6}+\frac {3 \left (-b^2+4 a c\right )}{64 c^3 d^8 (b+2 c x)^4}+\frac {1}{64 c^3 d^8 (b+2 c x)^2}\right ) \, dx \\ & = \frac {\left (b^2-4 a c\right )^3}{896 c^4 d^8 (b+2 c x)^7}-\frac {3 \left (b^2-4 a c\right )^2}{640 c^4 d^8 (b+2 c x)^5}+\frac {b^2-4 a c}{128 c^4 d^8 (b+2 c x)^3}-\frac {1}{128 c^4 d^8 (b+2 c x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.78 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^8} \, dx=\frac {5 \left (b^2-4 a c\right )^3-21 \left (b^2-4 a c\right )^2 (b+2 c x)^2+35 \left (b^2-4 a c\right ) (b+2 c x)^4-35 (b+2 c x)^6}{4480 c^4 d^8 (b+2 c x)^7} \]

[In]

Integrate[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^8,x]

[Out]

(5*(b^2 - 4*a*c)^3 - 21*(b^2 - 4*a*c)^2*(b + 2*c*x)^2 + 35*(b^2 - 4*a*c)*(b + 2*c*x)^4 - 35*(b + 2*c*x)^6)/(44
80*c^4*d^8*(b + 2*c*x)^7)

Maple [A] (verified)

Time = 2.87 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.20

method	result	size
default	\(\frac {-\frac {48 a^{2} c^{2}-24 a \,b^{2} c +3 b^{4}}{640 c^{4} \left (2 c x +b \right )^{5}}-\frac {12 a c -3 b^{2}}{384 c^{4} \left (2 c x +b \right )^{3}}-\frac {64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}{896 c^{4} \left (2 c x +b \right )^{7}}-\frac {1}{128 c^{4} \left (2 c x +b \right )}}{d^{8}}\)	\(121\)
risch	\(\frac {-\frac {c^{2} x^{6}}{2}-\frac {3 b c \,x^{5}}{2}+\left (-\frac {a c}{2}-\frac {7 b^{2}}{4}\right ) x^{4}-\frac {b \left (a c +b^{2}\right ) x^{3}}{c}-\frac {3 \left (a^{2} c^{2}+2 a \,b^{2} c +b^{4}\right ) x^{2}}{10 c^{2}}-\frac {b \left (6 a^{2} c^{2}+2 a \,b^{2} c +b^{4}\right ) x}{20 c^{3}}-\frac {20 c^{3} a^{3}+6 a^{2} b^{2} c^{2}+2 a \,b^{4} c +b^{6}}{280 c^{4}}}{d^{8} \left (2 c x +b \right )^{7}}\)	\(146\)
gosper	\(-\frac {140 c^{6} x^{6}+420 b \,c^{5} x^{5}+140 a \,c^{5} x^{4}+490 b^{2} c^{4} x^{4}+280 a b \,c^{4} x^{3}+280 x^{3} b^{3} c^{3}+84 a^{2} c^{4} x^{2}+168 a \,b^{2} c^{3} x^{2}+84 x^{2} b^{4} c^{2}+84 a^{2} b \,c^{3} x +28 x a \,b^{3} c^{2}+14 x \,b^{5} c +20 c^{3} a^{3}+6 a^{2} b^{2} c^{2}+2 a \,b^{4} c +b^{6}}{280 \left (2 c x +b \right )^{7} d^{8} c^{4}}\)	\(166\)
norman	\(\frac {\frac {a^{3} x}{b d}+\frac {\left (20 a^{3} c^{2}+6 a^{2} b^{2} c +a \,b^{4}\right ) x^{3}}{b^{3} d}+\frac {3 \left (4 c \,a^{3}+a^{2} b^{2}\right ) x^{2}}{2 b^{2} d}+\frac {\left (160 c^{3} a^{3}+48 a^{2} b^{2} c^{2}+14 a \,b^{4} c +b^{6}\right ) x^{4}}{4 b^{4} d}+\frac {3 c \left (160 c^{3} a^{3}+48 a^{2} b^{2} c^{2}+16 a \,b^{4} c +3 b^{6}\right ) x^{5}}{10 b^{5} d}+\frac {c^{2} \left (320 c^{3} a^{3}+96 a^{2} b^{2} c^{2}+32 a \,b^{4} c +11 b^{6}\right ) x^{6}}{10 b^{6} d}+\frac {16 c^{3} \left (20 c^{3} a^{3}+6 a^{2} b^{2} c^{2}+2 a \,b^{4} c +b^{6}\right ) x^{7}}{35 b^{7} d}}{d^{7} \left (2 c x +b \right )^{7}}\)	\(258\)
parallelrisch	\(\frac {1280 x^{7} a^{3} c^{6}+384 x^{7} a^{2} b^{2} c^{5}+128 x^{7} a \,b^{4} c^{4}+64 x^{7} b^{6} c^{3}+4480 x^{6} a^{3} b \,c^{5}+1344 x^{6} a^{2} b^{3} c^{4}+448 x^{6} a \,b^{5} c^{3}+154 x^{6} b^{7} c^{2}+6720 x^{5} a^{3} b^{2} c^{4}+2016 x^{5} a^{2} b^{4} c^{3}+672 x^{5} a \,b^{6} c^{2}+126 x^{5} b^{8} c +5600 x^{4} a^{3} b^{3} c^{3}+1680 x^{4} a^{2} b^{5} c^{2}+490 x^{4} a \,b^{7} c +35 x^{4} b^{9}+2800 x^{3} a^{3} b^{4} c^{2}+840 x^{3} a^{2} b^{6} c +140 x^{3} a \,b^{8}+840 x^{2} a^{3} b^{5} c +210 x^{2} a^{2} b^{7}+140 a^{3} b^{6} x}{140 b^{7} d^{8} \left (2 c x +b \right )^{7}}\)	\(277\)

[In]

int((c*x^2+b*x+a)^3/(2*c*d*x+b*d)^8,x,method=_RETURNVERBOSE)

[Out]

1/d^8*(-1/640*(48*a^2*c^2-24*a*b^2*c+3*b^4)/c^4/(2*c*x+b)^5-1/384*(12*a*c-3*b^2)/c^4/(2*c*x+b)^3-1/896*(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)^7-1/128/c^4/(2*c*x+b))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 248 vs. \(2 (93) = 186\).

Time = 0.26 (sec) , antiderivative size = 248, normalized size of antiderivative = 2.46 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^8} \, dx=-\frac {140 \, c^{6} x^{6} + 420 \, b c^{5} x^{5} + b^{6} + 2 \, a b^{4} c + 6 \, a^{2} b^{2} c^{2} + 20 \, a^{3} c^{3} + 70 \, {\left (7 \, b^{2} c^{4} + 2 \, a c^{5}\right )} x^{4} + 280 \, {\left (b^{3} c^{3} + a b c^{4}\right )} x^{3} + 84 \, {\left (b^{4} c^{2} + 2 \, a b^{2} c^{3} + a^{2} c^{4}\right )} x^{2} + 14 \, {\left (b^{5} c + 2 \, a b^{3} c^{2} + 6 \, a^{2} b c^{3}\right )} x}{280 \, {\left (128 \, c^{11} d^{8} x^{7} + 448 \, b c^{10} d^{8} x^{6} + 672 \, b^{2} c^{9} d^{8} x^{5} + 560 \, b^{3} c^{8} d^{8} x^{4} + 280 \, b^{4} c^{7} d^{8} x^{3} + 84 \, b^{5} c^{6} d^{8} x^{2} + 14 \, b^{6} c^{5} d^{8} x + b^{7} c^{4} d^{8}\right )}} \]

[In]

integrate((c*x^2+b*x+a)^3/(2*c*d*x+b*d)^8,x, algorithm="fricas")

[Out]

-1/280*(140*c^6*x^6 + 420*b*c^5*x^5 + b^6 + 2*a*b^4*c + 6*a^2*b^2*c^2 + 20*a^3*c^3 + 70*(7*b^2*c^4 + 2*a*c^5)*
x^4 + 280*(b^3*c^3 + a*b*c^4)*x^3 + 84*(b^4*c^2 + 2*a*b^2*c^3 + a^2*c^4)*x^2 + 14*(b^5*c + 2*a*b^3*c^2 + 6*a^2
*b*c^3)*x)/(128*c^11*d^8*x^7 + 448*b*c^10*d^8*x^6 + 672*b^2*c^9*d^8*x^5 + 560*b^3*c^8*d^8*x^4 + 280*b^4*c^7*d^
8*x^3 + 84*b^5*c^6*d^8*x^2 + 14*b^6*c^5*d^8*x + b^7*c^4*d^8)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 267 vs. \(2 (97) = 194\).

Time = 2.52 (sec) , antiderivative size = 267, normalized size of antiderivative = 2.64 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^8} \, dx=\frac {- 20 a^{3} c^{3} - 6 a^{2} b^{2} c^{2} - 2 a b^{4} c - b^{6} - 420 b c^{5} x^{5} - 140 c^{6} x^{6} + x^{4} \left (- 140 a c^{5} - 490 b^{2} c^{4}\right ) + x^{3} \left (- 280 a b c^{4} - 280 b^{3} c^{3}\right ) + x^{2} \left (- 84 a^{2} c^{4} - 168 a b^{2} c^{3} - 84 b^{4} c^{2}\right ) + x \left (- 84 a^{2} b c^{3} - 28 a b^{3} c^{2} - 14 b^{5} c\right )}{280 b^{7} c^{4} d^{8} + 3920 b^{6} c^{5} d^{8} x + 23520 b^{5} c^{6} d^{8} x^{2} + 78400 b^{4} c^{7} d^{8} x^{3} + 156800 b^{3} c^{8} d^{8} x^{4} + 188160 b^{2} c^{9} d^{8} x^{5} + 125440 b c^{10} d^{8} x^{6} + 35840 c^{11} d^{8} x^{7}} \]

[In]

integrate((c*x**2+b*x+a)**3/(2*c*d*x+b*d)**8,x)

[Out]

(-20*a**3*c**3 - 6*a**2*b**2*c**2 - 2*a*b**4*c - b**6 - 420*b*c**5*x**5 - 140*c**6*x**6 + x**4*(-140*a*c**5 -
490*b**2*c**4) + x**3*(-280*a*b*c**4 - 280*b**3*c**3) + x**2*(-84*a**2*c**4 - 168*a*b**2*c**3 - 84*b**4*c**2)
+ x*(-84*a**2*b*c**3 - 28*a*b**3*c**2 - 14*b**5*c))/(280*b**7*c**4*d**8 + 3920*b**6*c**5*d**8*x + 23520*b**5*c
**6*d**8*x**2 + 78400*b**4*c**7*d**8*x**3 + 156800*b**3*c**8*d**8*x**4 + 188160*b**2*c**9*d**8*x**5 + 125440*b
*c**10*d**8*x**6 + 35840*c**11*d**8*x**7)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 248 vs. \(2 (93) = 186\).

Time = 0.21 (sec) , antiderivative size = 248, normalized size of antiderivative = 2.46 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^8} \, dx=-\frac {140 \, c^{6} x^{6} + 420 \, b c^{5} x^{5} + b^{6} + 2 \, a b^{4} c + 6 \, a^{2} b^{2} c^{2} + 20 \, a^{3} c^{3} + 70 \, {\left (7 \, b^{2} c^{4} + 2 \, a c^{5}\right )} x^{4} + 280 \, {\left (b^{3} c^{3} + a b c^{4}\right )} x^{3} + 84 \, {\left (b^{4} c^{2} + 2 \, a b^{2} c^{3} + a^{2} c^{4}\right )} x^{2} + 14 \, {\left (b^{5} c + 2 \, a b^{3} c^{2} + 6 \, a^{2} b c^{3}\right )} x}{280 \, {\left (128 \, c^{11} d^{8} x^{7} + 448 \, b c^{10} d^{8} x^{6} + 672 \, b^{2} c^{9} d^{8} x^{5} + 560 \, b^{3} c^{8} d^{8} x^{4} + 280 \, b^{4} c^{7} d^{8} x^{3} + 84 \, b^{5} c^{6} d^{8} x^{2} + 14 \, b^{6} c^{5} d^{8} x + b^{7} c^{4} d^{8}\right )}} \]

[In]

integrate((c*x^2+b*x+a)^3/(2*c*d*x+b*d)^8,x, algorithm="maxima")

[Out]

-1/280*(140*c^6*x^6 + 420*b*c^5*x^5 + b^6 + 2*a*b^4*c + 6*a^2*b^2*c^2 + 20*a^3*c^3 + 70*(7*b^2*c^4 + 2*a*c^5)*
x^4 + 280*(b^3*c^3 + a*b*c^4)*x^3 + 84*(b^4*c^2 + 2*a*b^2*c^3 + a^2*c^4)*x^2 + 14*(b^5*c + 2*a*b^3*c^2 + 6*a^2
*b*c^3)*x)/(128*c^11*d^8*x^7 + 448*b*c^10*d^8*x^6 + 672*b^2*c^9*d^8*x^5 + 560*b^3*c^8*d^8*x^4 + 280*b^4*c^7*d^
8*x^3 + 84*b^5*c^6*d^8*x^2 + 14*b^6*c^5*d^8*x + b^7*c^4*d^8)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.63 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^8} \, dx=-\frac {140 \, c^{6} x^{6} + 420 \, b c^{5} x^{5} + 490 \, b^{2} c^{4} x^{4} + 140 \, a c^{5} x^{4} + 280 \, b^{3} c^{3} x^{3} + 280 \, a b c^{4} x^{3} + 84 \, b^{4} c^{2} x^{2} + 168 \, a b^{2} c^{3} x^{2} + 84 \, a^{2} c^{4} x^{2} + 14 \, b^{5} c x + 28 \, a b^{3} c^{2} x + 84 \, a^{2} b c^{3} x + b^{6} + 2 \, a b^{4} c + 6 \, a^{2} b^{2} c^{2} + 20 \, a^{3} c^{3}}{280 \, {\left (2 \, c x + b\right )}^{7} c^{4} d^{8}} \]

[In]

integrate((c*x^2+b*x+a)^3/(2*c*d*x+b*d)^8,x, algorithm="giac")

[Out]

-1/280*(140*c^6*x^6 + 420*b*c^5*x^5 + 490*b^2*c^4*x^4 + 140*a*c^5*x^4 + 280*b^3*c^3*x^3 + 280*a*b*c^4*x^3 + 84
*b^4*c^2*x^2 + 168*a*b^2*c^3*x^2 + 84*a^2*c^4*x^2 + 14*b^5*c*x + 28*a*b^3*c^2*x + 84*a^2*b*c^3*x + b^6 + 2*a*b
^4*c + 6*a^2*b^2*c^2 + 20*a^3*c^3)/((2*c*x + b)^7*c^4*d^8)

Mupad [B] (verification not implemented)

Time = 9.64 (sec) , antiderivative size = 233, normalized size of antiderivative = 2.31 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^8} \, dx=-\frac {\frac {20\,a^3\,c^3+6\,a^2\,b^2\,c^2+2\,a\,b^4\,c+b^6}{280\,c^4}+x^4\,\left (\frac {7\,b^2}{4}+\frac {a\,c}{2}\right )+\frac {c^2\,x^6}{2}+\frac {x^3\,\left (b^3+a\,c\,b\right )}{c}+\frac {3\,x^2\,\left (a^2\,c^2+2\,a\,b^2\,c+b^4\right )}{10\,c^2}+\frac {3\,b\,c\,x^5}{2}+\frac {b\,x\,\left (6\,a^2\,c^2+2\,a\,b^2\,c+b^4\right )}{20\,c^3}}{b^7\,d^8+14\,b^6\,c\,d^8\,x+84\,b^5\,c^2\,d^8\,x^2+280\,b^4\,c^3\,d^8\,x^3+560\,b^3\,c^4\,d^8\,x^4+672\,b^2\,c^5\,d^8\,x^5+448\,b\,c^6\,d^8\,x^6+128\,c^7\,d^8\,x^7} \]

[In]

int((a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^8,x)

[Out]

-((b^6 + 20*a^3*c^3 + 6*a^2*b^2*c^2 + 2*a*b^4*c)/(280*c^4) + x^4*((a*c)/2 + (7*b^2)/4) + (c^2*x^6)/2 + (x^3*(b
^3 + a*b*c))/c + (3*x^2*(b^4 + a^2*c^2 + 2*a*b^2*c))/(10*c^2) + (3*b*c*x^5)/2 + (b*x*(b^4 + 6*a^2*c^2 + 2*a*b^
2*c))/(20*c^3))/(b^7*d^8 + 128*c^7*d^8*x^7 + 448*b*c^6*d^8*x^6 + 84*b^5*c^2*d^8*x^2 + 280*b^4*c^3*d^8*x^3 + 56
0*b^3*c^4*d^8*x^4 + 672*b^2*c^5*d^8*x^5 + 14*b^6*c*d^8*x)