\(\int \frac {(a+b x+c x^2)^2}{(b d+2 c d x)^6} \, dx\) [1131]

Optimal result

Integrand size = 24, antiderivative size = 73 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^6} \, dx=-\frac {\left (b^2-4 a c\right )^2}{160 c^3 d^6 (b+2 c x)^5}+\frac {b^2-4 a c}{48 c^3 d^6 (b+2 c x)^3}-\frac {1}{32 c^3 d^6 (b+2 c x)} \]

[Out]

-1/160*(-4*a*c+b^2)^2/c^3/d^6/(2*c*x+b)^5+1/48*(-4*a*c+b^2)/c^3/d^6/(2*c*x+b)^3-1/32/c^3/d^6/(2*c*x+b)

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} \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^6} \, dx=-\frac {\left (b^2-4 a c\right )^2}{160 c^3 d^6 (b+2 c x)^5}+\frac {b^2-4 a c}{48 c^3 d^6 (b+2 c x)^3}-\frac {1}{32 c^3 d^6 (b+2 c x)} \]

[In]

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

[Out]

-1/160*(b^2 - 4*a*c)^2/(c^3*d^6*(b + 2*c*x)^5) + (b^2 - 4*a*c)/(48*c^3*d^6*(b + 2*c*x)^3) - 1/(32*c^3*d^6*(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 )^2}{16 c^2 d^6 (b+2 c x)^6}+\frac {-b^2+4 a c}{8 c^2 d^6 (b+2 c x)^4}+\frac {1}{16 c^2 d^6 (b+2 c x)^2}\right ) \, dx \\ & = -\frac {\left (b^2-4 a c\right )^2}{160 c^3 d^6 (b+2 c x)^5}+\frac {b^2-4 a c}{48 c^3 d^6 (b+2 c x)^3}-\frac {1}{32 c^3 d^6 (b+2 c x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.81 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^6} \, dx=\frac {-3 \left (b^2-4 a c\right )^2+10 \left (b^2-4 a c\right ) (b+2 c x)^2-15 (b+2 c x)^4}{480 c^3 d^6 (b+2 c x)^5} \]

[In]

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

[Out]

(-3*(b^2 - 4*a*c)^2 + 10*(b^2 - 4*a*c)*(b + 2*c*x)^2 - 15*(b + 2*c*x)^4)/(480*c^3*d^6*(b + 2*c*x)^5)

Maple [A] (verified)

Time = 2.93 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.01

method	result	size
default	\(\frac {-\frac {16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}{160 c^{3} \left (2 c x +b \right )^{5}}-\frac {4 a c -b^{2}}{48 c^{3} \left (2 c x +b \right )^{3}}-\frac {1}{32 c^{3} \left (2 c x +b \right )}}{d^{6}}\)	\(74\)
risch	\(\frac {-\frac {x^{4} c}{2}-b \,x^{3}-\frac {\left (a c +2 b^{2}\right ) x^{2}}{3 c}-\frac {\left (2 a c +b^{2}\right ) b x}{6 c^{2}}-\frac {6 a^{2} c^{2}+2 a \,b^{2} c +b^{4}}{60 c^{3}}}{d^{6} \left (2 c x +b \right )^{5}}\)	\(82\)
gosper	\(-\frac {30 c^{4} x^{4}+60 b \,c^{3} x^{3}+20 x^{2} a \,c^{3}+40 b^{2} c^{2} x^{2}+20 a b \,c^{2} x +10 b^{3} c x +6 a^{2} c^{2}+2 a \,b^{2} c +b^{4}}{60 \left (2 c x +b \right )^{5} c^{3} d^{6}}\)	\(88\)
norman	\(\frac {\frac {a^{2} x}{b d}+\frac {\left (4 a^{2} c +a \,b^{2}\right ) x^{2}}{b^{2} d}+\frac {\left (24 a^{2} c^{2}+8 a \,b^{2} c +b^{4}\right ) x^{3}}{3 b^{3} d}+\frac {c \left (48 a^{2} c^{2}+16 a \,b^{2} c +5 b^{4}\right ) x^{4}}{6 b^{4} d}+\frac {8 c^{2} \left (6 a^{2} c^{2}+2 a \,b^{2} c +b^{4}\right ) x^{5}}{15 b^{5} d}}{d^{5} \left (2 c x +b \right )^{5}}\)	\(143\)
parallelrisch	\(\frac {96 x^{5} a^{2} c^{4}+32 x^{5} a \,b^{2} c^{3}+16 b^{4} c^{2} x^{5}+240 a^{2} b \,c^{3} x^{4}+80 x^{4} a \,b^{3} c^{2}+25 x^{4} b^{5} c +240 x^{3} a^{2} b^{2} c^{2}+80 a \,b^{4} c \,x^{3}+10 x^{3} b^{6}+120 x^{2} a^{2} b^{3} c +30 a \,b^{5} x^{2}+30 a^{2} b^{4} x}{30 b^{5} d^{6} \left (2 c x +b \right )^{5}}\)	\(147\)

[In]

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

[Out]

1/d^6*(-1/160*(16*a^2*c^2-8*a*b^2*c+b^4)/c^3/(2*c*x+b)^5-1/48*(4*a*c-b^2)/c^3/(2*c*x+b)^3-1/32/c^3/(2*c*x+b))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 149 vs. \(2 (67) = 134\).

Time = 0.34 (sec) , antiderivative size = 149, normalized size of antiderivative = 2.04 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^6} \, dx=-\frac {30 \, c^{4} x^{4} + 60 \, b c^{3} x^{3} + b^{4} + 2 \, a b^{2} c + 6 \, a^{2} c^{2} + 20 \, {\left (2 \, b^{2} c^{2} + a c^{3}\right )} x^{2} + 10 \, {\left (b^{3} c + 2 \, a b c^{2}\right )} x}{60 \, {\left (32 \, c^{8} d^{6} x^{5} + 80 \, b c^{7} d^{6} x^{4} + 80 \, b^{2} c^{6} d^{6} x^{3} + 40 \, b^{3} c^{5} d^{6} x^{2} + 10 \, b^{4} c^{4} d^{6} x + b^{5} c^{3} d^{6}\right )}} \]

[In]

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

[Out]

-1/60*(30*c^4*x^4 + 60*b*c^3*x^3 + b^4 + 2*a*b^2*c + 6*a^2*c^2 + 20*(2*b^2*c^2 + a*c^3)*x^2 + 10*(b^3*c + 2*a*
b*c^2)*x)/(32*c^8*d^6*x^5 + 80*b*c^7*d^6*x^4 + 80*b^2*c^6*d^6*x^3 + 40*b^3*c^5*d^6*x^2 + 10*b^4*c^4*d^6*x + b^
5*c^3*d^6)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 158 vs. \(2 (68) = 136\).

Time = 0.73 (sec) , antiderivative size = 158, normalized size of antiderivative = 2.16 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^6} \, dx=\frac {- 6 a^{2} c^{2} - 2 a b^{2} c - b^{4} - 60 b c^{3} x^{3} - 30 c^{4} x^{4} + x^{2} \left (- 20 a c^{3} - 40 b^{2} c^{2}\right ) + x \left (- 20 a b c^{2} - 10 b^{3} c\right )}{60 b^{5} c^{3} d^{6} + 600 b^{4} c^{4} d^{6} x + 2400 b^{3} c^{5} d^{6} x^{2} + 4800 b^{2} c^{6} d^{6} x^{3} + 4800 b c^{7} d^{6} x^{4} + 1920 c^{8} d^{6} x^{5}} \]

[In]

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

[Out]

(-6*a**2*c**2 - 2*a*b**2*c - b**4 - 60*b*c**3*x**3 - 30*c**4*x**4 + x**2*(-20*a*c**3 - 40*b**2*c**2) + x*(-20*
a*b*c**2 - 10*b**3*c))/(60*b**5*c**3*d**6 + 600*b**4*c**4*d**6*x + 2400*b**3*c**5*d**6*x**2 + 4800*b**2*c**6*d
**6*x**3 + 4800*b*c**7*d**6*x**4 + 1920*c**8*d**6*x**5)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 149 vs. \(2 (67) = 134\).

Time = 0.21 (sec) , antiderivative size = 149, normalized size of antiderivative = 2.04 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^6} \, dx=-\frac {30 \, c^{4} x^{4} + 60 \, b c^{3} x^{3} + b^{4} + 2 \, a b^{2} c + 6 \, a^{2} c^{2} + 20 \, {\left (2 \, b^{2} c^{2} + a c^{3}\right )} x^{2} + 10 \, {\left (b^{3} c + 2 \, a b c^{2}\right )} x}{60 \, {\left (32 \, c^{8} d^{6} x^{5} + 80 \, b c^{7} d^{6} x^{4} + 80 \, b^{2} c^{6} d^{6} x^{3} + 40 \, b^{3} c^{5} d^{6} x^{2} + 10 \, b^{4} c^{4} d^{6} x + b^{5} c^{3} d^{6}\right )}} \]

[In]

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

[Out]

-1/60*(30*c^4*x^4 + 60*b*c^3*x^3 + b^4 + 2*a*b^2*c + 6*a^2*c^2 + 20*(2*b^2*c^2 + a*c^3)*x^2 + 10*(b^3*c + 2*a*
b*c^2)*x)/(32*c^8*d^6*x^5 + 80*b*c^7*d^6*x^4 + 80*b^2*c^6*d^6*x^3 + 40*b^3*c^5*d^6*x^2 + 10*b^4*c^4*d^6*x + b^
5*c^3*d^6)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.19 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^6} \, dx=-\frac {30 \, c^{4} x^{4} + 60 \, b c^{3} x^{3} + 40 \, b^{2} c^{2} x^{2} + 20 \, a c^{3} x^{2} + 10 \, b^{3} c x + 20 \, a b c^{2} x + b^{4} + 2 \, a b^{2} c + 6 \, a^{2} c^{2}}{60 \, {\left (2 \, c x + b\right )}^{5} c^{3} d^{6}} \]

[In]

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

[Out]

-1/60*(30*c^4*x^4 + 60*b*c^3*x^3 + 40*b^2*c^2*x^2 + 20*a*c^3*x^2 + 10*b^3*c*x + 20*a*b*c^2*x + b^4 + 2*a*b^2*c
+ 6*a^2*c^2)/((2*c*x + b)^5*c^3*d^6)

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.93 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^6} \, dx=-\frac {\frac {6\,a^2\,c^2+2\,a\,b^2\,c+b^4}{60\,c^3}+b\,x^3+\frac {c\,x^4}{2}+\frac {x^2\,\left (2\,b^2+a\,c\right )}{3\,c}+\frac {b\,x\,\left (b^2+2\,a\,c\right )}{6\,c^2}}{b^5\,d^6+10\,b^4\,c\,d^6\,x+40\,b^3\,c^2\,d^6\,x^2+80\,b^2\,c^3\,d^6\,x^3+80\,b\,c^4\,d^6\,x^4+32\,c^5\,d^6\,x^5} \]

[In]

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

[Out]

-((b^4 + 6*a^2*c^2 + 2*a*b^2*c)/(60*c^3) + b*x^3 + (c*x^4)/2 + (x^2*(a*c + 2*b^2))/(3*c) + (b*x*(2*a*c + b^2))
/(6*c^2))/(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)