\(\int \frac {a+b x+c x^2}{(b d+2 c d x)^6} \, dx\) [1118]

Optimal result

Integrand size = 22, antiderivative size = 45 \[ \int \frac {a+b x+c x^2}{(b d+2 c d x)^6} \, dx=\frac {b^2-4 a c}{40 c^2 d^6 (b+2 c x)^5}-\frac {1}{24 c^2 d^6 (b+2 c x)^3} \]

[Out]

1/40*(-4*a*c+b^2)/c^2/d^6/(2*c*x+b)^5-1/24/c^2/d^6/(2*c*x+b)^3

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} \[ \int \frac {a+b x+c x^2}{(b d+2 c d x)^6} \, dx=\frac {b^2-4 a c}{40 c^2 d^6 (b+2 c x)^5}-\frac {1}{24 c^2 d^6 (b+2 c x)^3} \]

[In]

Int[(a + b*x + c*x^2)/(b*d + 2*c*d*x)^6,x]

[Out]

(b^2 - 4*a*c)/(40*c^2*d^6*(b + 2*c*x)^5) - 1/(24*c^2*d^6*(b + 2*c*x)^3)

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^6 (b+2 c x)^6}+\frac {1}{4 c d^6 (b+2 c x)^4}\right ) \, dx \\ & = \frac {b^2-4 a c}{40 c^2 d^6 (b+2 c x)^5}-\frac {1}{24 c^2 d^6 (b+2 c x)^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.96 \[ \int \frac {a+b x+c x^2}{(b d+2 c d x)^6} \, dx=\frac {\frac {b^2-4 a c}{40 c^2 (b+2 c x)^5}-\frac {1}{24 c^2 (b+2 c x)^3}}{d^6} \]

[In]

Integrate[(a + b*x + c*x^2)/(b*d + 2*c*d*x)^6,x]

[Out]

((b^2 - 4*a*c)/(40*c^2*(b + 2*c*x)^5) - 1/(24*c^2*(b + 2*c*x)^3))/d^6

Maple [A] (verified)

Time = 2.45 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.84

method	result	size
gosper	\(-\frac {10 c^{2} x^{2}+10 b c x +6 a c +b^{2}}{60 \left (2 c x +b \right )^{5} d^{6} c^{2}}\)	\(38\)
risch	\(\frac {-\frac {x^{2}}{6}-\frac {b x}{6 c}-\frac {6 a c +b^{2}}{60 c^{2}}}{d^{6} \left (2 c x +b \right )^{5}}\)	\(39\)
default	\(\frac {-\frac {1}{24 c^{2} \left (2 c x +b \right )^{3}}-\frac {4 a c -b^{2}}{40 c^{2} \left (2 c x +b \right )^{5}}}{d^{6}}\)	\(42\)
parallelrisch	\(\frac {96 x^{5} a \,c^{4}+16 x^{5} b^{2} c^{3}+240 x^{4} a b \,c^{3}+40 c^{2} b^{3} x^{4}+240 a \,b^{2} c^{2} x^{3}+40 b^{4} c \,x^{3}+120 a \,b^{3} c \,x^{2}+15 b^{5} x^{2}+30 b^{4} x a}{30 b^{5} d^{6} \left (2 c x +b \right )^{5}}\)	\(105\)
norman	\(\frac {\frac {a x}{b d}+\frac {\left (8 a c +b^{2}\right ) x^{2}}{2 b^{2} d}+\frac {4 c^{2} \left (6 a c +b^{2}\right ) x^{4}}{3 b^{4} d}+\frac {4 c \left (6 a c +b^{2}\right ) x^{3}}{3 b^{3} d}+\frac {8 c^{3} \left (6 a c +b^{2}\right ) x^{5}}{15 b^{5} d}}{d^{5} \left (2 c x +b \right )^{5}}\)	\(106\)

[In]

int((c*x^2+b*x+a)/(2*c*d*x+b*d)^6,x,method=_RETURNVERBOSE)

[Out]

-1/60*(10*c^2*x^2+10*b*c*x+6*a*c+b^2)/(2*c*x+b)^5/d^6/c^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (41) = 82\).

Time = 0.35 (sec) , antiderivative size = 99, normalized size of antiderivative = 2.20 \[ \int \frac {a+b x+c x^2}{(b d+2 c d x)^6} \, dx=-\frac {10 \, c^{2} x^{2} + 10 \, b c x + b^{2} + 6 \, a c}{60 \, {\left (32 \, c^{7} d^{6} x^{5} + 80 \, b c^{6} d^{6} x^{4} + 80 \, b^{2} c^{5} d^{6} x^{3} + 40 \, b^{3} c^{4} d^{6} x^{2} + 10 \, b^{4} c^{3} d^{6} x + b^{5} c^{2} d^{6}\right )}} \]

[In]

integrate((c*x^2+b*x+a)/(2*c*d*x+b*d)^6,x, algorithm="fricas")

[Out]

-1/60*(10*c^2*x^2 + 10*b*c*x + b^2 + 6*a*c)/(32*c^7*d^6*x^5 + 80*b*c^6*d^6*x^4 + 80*b^2*c^5*d^6*x^3 + 40*b^3*c
^4*d^6*x^2 + 10*b^4*c^3*d^6*x + b^5*c^2*d^6)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (42) = 84\).

Time = 0.35 (sec) , antiderivative size = 105, normalized size of antiderivative = 2.33 \[ \int \frac {a+b x+c x^2}{(b d+2 c d x)^6} \, dx=\frac {- 6 a c - b^{2} - 10 b c x - 10 c^{2} x^{2}}{60 b^{5} c^{2} d^{6} + 600 b^{4} c^{3} d^{6} x + 2400 b^{3} c^{4} d^{6} x^{2} + 4800 b^{2} c^{5} d^{6} x^{3} + 4800 b c^{6} d^{6} x^{4} + 1920 c^{7} d^{6} x^{5}} \]

[In]

integrate((c*x**2+b*x+a)/(2*c*d*x+b*d)**6,x)

[Out]

(-6*a*c - b**2 - 10*b*c*x - 10*c**2*x**2)/(60*b**5*c**2*d**6 + 600*b**4*c**3*d**6*x + 2400*b**3*c**4*d**6*x**2
+ 4800*b**2*c**5*d**6*x**3 + 4800*b*c**6*d**6*x**4 + 1920*c**7*d**6*x**5)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (41) = 82\).

Time = 0.20 (sec) , antiderivative size = 99, normalized size of antiderivative = 2.20 \[ \int \frac {a+b x+c x^2}{(b d+2 c d x)^6} \, dx=-\frac {10 \, c^{2} x^{2} + 10 \, b c x + b^{2} + 6 \, a c}{60 \, {\left (32 \, c^{7} d^{6} x^{5} + 80 \, b c^{6} d^{6} x^{4} + 80 \, b^{2} c^{5} d^{6} x^{3} + 40 \, b^{3} c^{4} d^{6} x^{2} + 10 \, b^{4} c^{3} d^{6} x + b^{5} c^{2} d^{6}\right )}} \]

[In]

integrate((c*x^2+b*x+a)/(2*c*d*x+b*d)^6,x, algorithm="maxima")

[Out]

-1/60*(10*c^2*x^2 + 10*b*c*x + b^2 + 6*a*c)/(32*c^7*d^6*x^5 + 80*b*c^6*d^6*x^4 + 80*b^2*c^5*d^6*x^3 + 40*b^3*c
^4*d^6*x^2 + 10*b^4*c^3*d^6*x + b^5*c^2*d^6)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.82 \[ \int \frac {a+b x+c x^2}{(b d+2 c d x)^6} \, dx=-\frac {10 \, c^{2} x^{2} + 10 \, b c x + b^{2} + 6 \, a c}{60 \, {\left (2 \, c x + b\right )}^{5} c^{2} d^{6}} \]

[In]

integrate((c*x^2+b*x+a)/(2*c*d*x+b*d)^6,x, algorithm="giac")

[Out]

-1/60*(10*c^2*x^2 + 10*b*c*x + b^2 + 6*a*c)/((2*c*x + b)^5*c^2*d^6)

Mupad [B] (verification not implemented)

Time = 10.03 (sec) , antiderivative size = 99, normalized size of antiderivative = 2.20 \[ \int \frac {a+b x+c x^2}{(b d+2 c d x)^6} \, dx=-\frac {\frac {b^2+6\,a\,c}{60\,c^2}+\frac {x^2}{6}+\frac {b\,x}{6\,c}}{b^5\,d^6+10\,b^4\,c\,d^6\,x+40\,b^3\,c^2\,d^6\,x^2+80\,b^2\,c^3\,d^6\,x^3+80\,b\,c^4\,d^6\,x^4+32\,c^5\,d^6\,x^5} \]

[In]

int((a + b*x + c*x^2)/(b*d + 2*c*d*x)^6,x)

[Out]

-((6*a*c + b^2)/(60*c^2) + x^2/6 + (b*x)/(6*c))/(b^5*d^6 + 32*c^5*d^6*x^5 + 80*b*c^4*d^6*x^4 + 40*b^3*c^2*d^6*
x^2 + 80*b^2*c^3*d^6*x^3 + 10*b^4*c*d^6*x)