\(\int \frac {a+b x+c x^2}{(b d+2 c d x)^8} \, dx\) [1120]

Optimal result

Integrand size = 22, antiderivative size = 45 \[ \int \frac {a+b x+c x^2}{(b d+2 c d x)^8} \, dx=\frac {b^2-4 a c}{56 c^2 d^8 (b+2 c x)^7}-\frac {1}{40 c^2 d^8 (b+2 c x)^5} \]

[Out]

1/56*(-4*a*c+b^2)/c^2/d^8/(2*c*x+b)^7-1/40/c^2/d^8/(2*c*x+b)^5

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {697} \[ \int \frac {a+b x+c x^2}{(b d+2 c d x)^8} \, dx=\frac {b^2-4 a c}{56 c^2 d^8 (b+2 c x)^7}-\frac {1}{40 c^2 d^8 (b+2 c x)^5} \]

[In]

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

[Out]

(b^2 - 4*a*c)/(56*c^2*d^8*(b + 2*c*x)^7) - 1/(40*c^2*d^8*(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 {-b^2+4 a c}{4 c d^8 (b+2 c x)^8}+\frac {1}{4 c d^8 (b+2 c x)^6}\right ) \, dx \\ & = \frac {b^2-4 a c}{56 c^2 d^8 (b+2 c x)^7}-\frac {1}{40 c^2 d^8 (b+2 c x)^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.87 \[ \int \frac {a+b x+c x^2}{(b d+2 c d x)^8} \, dx=\frac {5 \left (b^2-4 a c\right )-7 (b+2 c x)^2}{280 c^2 d^8 (b+2 c x)^7} \]

[In]

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

[Out]

(5*(b^2 - 4*a*c) - 7*(b + 2*c*x)^2)/(280*c^2*d^8*(b + 2*c*x)^7)

Maple [A] (verified)

Time = 2.18 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.84

method	result	size
gosper	\(-\frac {14 c^{2} x^{2}+14 b c x +10 a c +b^{2}}{140 \left (2 c x +b \right )^{7} d^{8} c^{2}}\)	\(38\)
risch	\(\frac {-\frac {x^{2}}{10}-\frac {b x}{10 c}-\frac {10 a c +b^{2}}{140 c^{2}}}{d^{8} \left (2 c x +b \right )^{7}}\)	\(39\)
default	\(\frac {-\frac {4 a c -b^{2}}{56 c^{2} \left (2 c x +b \right )^{7}}-\frac {1}{40 c^{2} \left (2 c x +b \right )^{5}}}{d^{8}}\)	\(42\)
norman	\(\frac {\frac {a x}{b d}+\frac {\left (12 a c +b^{2}\right ) x^{2}}{2 b^{2} d}+\frac {16 c^{4} \left (10 a c +b^{2}\right ) x^{6}}{5 b^{6} d}+\frac {24 c^{3} \left (10 a c +b^{2}\right ) x^{5}}{5 b^{5} d}+\frac {4 c^{2} \left (10 a c +b^{2}\right ) x^{4}}{b^{4} d}+\frac {2 c \left (10 a c +b^{2}\right ) x^{3}}{b^{3} d}+\frac {32 c^{5} \left (10 a c +b^{2}\right ) x^{7}}{35 b^{7} d}}{d^{7} \left (2 c x +b \right )^{7}}\)	\(150\)
parallelrisch	\(\frac {640 x^{7} a \,c^{6}+64 x^{7} b^{2} c^{5}+2240 x^{6} a b \,c^{5}+224 x^{6} b^{3} c^{4}+3360 x^{5} a \,b^{2} c^{4}+336 x^{5} b^{4} c^{3}+2800 a \,b^{3} c^{3} x^{4}+280 x^{4} b^{5} c^{2}+1400 c^{2} x^{3} a \,b^{4}+140 x^{3} b^{6} c +420 x^{2} a \,b^{5} c +35 x^{2} b^{7}+70 x a \,b^{6}}{70 b^{7} d^{8} \left (2 c x +b \right )^{7}}\)	\(151\)

[In]

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

[Out]

-1/140*(14*c^2*x^2+14*b*c*x+10*a*c+b^2)/(2*c*x+b)^7/d^8/c^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 127 vs. \(2 (41) = 82\).

Time = 0.60 (sec) , antiderivative size = 127, normalized size of antiderivative = 2.82 \[ \int \frac {a+b x+c x^2}{(b d+2 c d x)^8} \, dx=-\frac {14 \, c^{2} x^{2} + 14 \, b c x + b^{2} + 10 \, a c}{140 \, {\left (128 \, c^{9} d^{8} x^{7} + 448 \, b c^{8} d^{8} x^{6} + 672 \, b^{2} c^{7} d^{8} x^{5} + 560 \, b^{3} c^{6} d^{8} x^{4} + 280 \, b^{4} c^{5} d^{8} x^{3} + 84 \, b^{5} c^{4} d^{8} x^{2} + 14 \, b^{6} c^{3} d^{8} x + b^{7} c^{2} d^{8}\right )}} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 136 vs. \(2 (42) = 84\).

Time = 0.48 (sec) , antiderivative size = 136, normalized size of antiderivative = 3.02 \[ \int \frac {a+b x+c x^2}{(b d+2 c d x)^8} \, dx=\frac {- 10 a c - b^{2} - 14 b c x - 14 c^{2} x^{2}}{140 b^{7} c^{2} d^{8} + 1960 b^{6} c^{3} d^{8} x + 11760 b^{5} c^{4} d^{8} x^{2} + 39200 b^{4} c^{5} d^{8} x^{3} + 78400 b^{3} c^{6} d^{8} x^{4} + 94080 b^{2} c^{7} d^{8} x^{5} + 62720 b c^{8} d^{8} x^{6} + 17920 c^{9} d^{8} x^{7}} \]

[In]

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

[Out]

(-10*a*c - b**2 - 14*b*c*x - 14*c**2*x**2)/(140*b**7*c**2*d**8 + 1960*b**6*c**3*d**8*x + 11760*b**5*c**4*d**8*
x**2 + 39200*b**4*c**5*d**8*x**3 + 78400*b**3*c**6*d**8*x**4 + 94080*b**2*c**7*d**8*x**5 + 62720*b*c**8*d**8*x
**6 + 17920*c**9*d**8*x**7)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 127 vs. \(2 (41) = 82\).

Time = 0.21 (sec) , antiderivative size = 127, normalized size of antiderivative = 2.82 \[ \int \frac {a+b x+c x^2}{(b d+2 c d x)^8} \, dx=-\frac {14 \, c^{2} x^{2} + 14 \, b c x + b^{2} + 10 \, a c}{140 \, {\left (128 \, c^{9} d^{8} x^{7} + 448 \, b c^{8} d^{8} x^{6} + 672 \, b^{2} c^{7} d^{8} x^{5} + 560 \, b^{3} c^{6} d^{8} x^{4} + 280 \, b^{4} c^{5} d^{8} x^{3} + 84 \, b^{5} c^{4} d^{8} x^{2} + 14 \, b^{6} c^{3} d^{8} x + b^{7} c^{2} d^{8}\right )}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.82 \[ \int \frac {a+b x+c x^2}{(b d+2 c d x)^8} \, dx=-\frac {14 \, c^{2} x^{2} + 14 \, b c x + b^{2} + 10 \, a c}{140 \, {\left (2 \, c x + b\right )}^{7} c^{2} d^{8}} \]

[In]

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

[Out]

-1/140*(14*c^2*x^2 + 14*b*c*x + b^2 + 10*a*c)/((2*c*x + b)^7*c^2*d^8)

Mupad [B] (verification not implemented)

Time = 10.08 (sec) , antiderivative size = 127, normalized size of antiderivative = 2.82 \[ \int \frac {a+b x+c x^2}{(b d+2 c d x)^8} \, dx=-\frac {\frac {b^2+10\,a\,c}{140\,c^2}+\frac {x^2}{10}+\frac {b\,x}{10\,c}}{b^7\,d^8+14\,b^6\,c\,d^8\,x+84\,b^5\,c^2\,d^8\,x^2+280\,b^4\,c^3\,d^8\,x^3+560\,b^3\,c^4\,d^8\,x^4+672\,b^2\,c^5\,d^8\,x^5+448\,b\,c^6\,d^8\,x^6+128\,c^7\,d^8\,x^7} \]

[In]

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

[Out]

-((10*a*c + b^2)/(140*c^2) + x^2/10 + (b*x)/(10*c))/(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 + 560*b^3*c^4*d^8*x^4 + 672*b^2*c^5*d^8*x^5 + 14*b^6*c*d^8*x)