3.1134 \(\int \frac{(a+b x+c x^2)^2}{(b d+2 c d x)^9} \, dx\)

Optimal. Leaf size=73 \[ -\frac{\left (b^2-4 a c\right )^2}{256 c^3 d^9 (b+2 c x)^8}+\frac{b^2-4 a c}{96 c^3 d^9 (b+2 c x)^6}-\frac{1}{128 c^3 d^9 (b+2 c x)^4} \]

[Out]

-(b^2 - 4*a*c)^2/(256*c^3*d^9*(b + 2*c*x)^8) + (b^2 - 4*a*c)/(96*c^3*d^9*(b + 2*c*x)^6) - 1/(128*c^3*d^9*(b +
2*c*x)^4)

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Rubi [A]  time = 0.0548289, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {683} \[ -\frac{\left (b^2-4 a c\right )^2}{256 c^3 d^9 (b+2 c x)^8}+\frac{b^2-4 a c}{96 c^3 d^9 (b+2 c x)^6}-\frac{1}{128 c^3 d^9 (b+2 c x)^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^9,x]

[Out]

-(b^2 - 4*a*c)^2/(256*c^3*d^9*(b + 2*c*x)^8) + (b^2 - 4*a*c)/(96*c^3*d^9*(b + 2*c*x)^6) - 1/(128*c^3*d^9*(b +
2*c*x)^4)

Rule 683

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*} \int \frac{\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^9} \, dx &=\int \left (\frac{\left (-b^2+4 a c\right )^2}{16 c^2 d^9 (b+2 c x)^9}+\frac{-b^2+4 a c}{8 c^2 d^9 (b+2 c x)^7}+\frac{1}{16 c^2 d^9 (b+2 c x)^5}\right ) \, dx\\ &=-\frac{\left (b^2-4 a c\right )^2}{256 c^3 d^9 (b+2 c x)^8}+\frac{b^2-4 a c}{96 c^3 d^9 (b+2 c x)^6}-\frac{1}{128 c^3 d^9 (b+2 c x)^4}\\ \end{align*}

Mathematica [A]  time = 0.0260712, size = 59, normalized size = 0.81 \[ \frac{8 \left (b^2-4 a c\right ) (b+2 c x)^2-3 \left (b^2-4 a c\right )^2-6 (b+2 c x)^4}{768 c^3 d^9 (b+2 c x)^8} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^9,x]

[Out]

(-3*(b^2 - 4*a*c)^2 + 8*(b^2 - 4*a*c)*(b + 2*c*x)^2 - 6*(b + 2*c*x)^4)/(768*c^3*d^9*(b + 2*c*x)^8)

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Maple [A]  time = 0.043, size = 74, normalized size = 1. \begin{align*}{\frac{1}{{d}^{9}} \left ( -{\frac{4\,ac-{b}^{2}}{96\,{c}^{3} \left ( 2\,cx+b \right ) ^{6}}}-{\frac{16\,{a}^{2}{c}^{2}-8\,ac{b}^{2}+{b}^{4}}{256\,{c}^{3} \left ( 2\,cx+b \right ) ^{8}}}-{\frac{1}{128\,{c}^{3} \left ( 2\,cx+b \right ) ^{4}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)^2/(2*c*d*x+b*d)^9,x)

[Out]

1/d^9*(-1/96*(4*a*c-b^2)/c^3/(2*c*x+b)^6-1/256*(16*a^2*c^2-8*a*b^2*c+b^4)/c^3/(2*c*x+b)^8-1/128/c^3/(2*c*x+b)^
4)

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Maxima [B]  time = 1.06347, size = 259, normalized size = 3.55 \begin{align*} -\frac{96 \, c^{4} x^{4} + 192 \, b c^{3} x^{3} + b^{4} + 8 \, a b^{2} c + 48 \, a^{2} c^{2} + 16 \,{\left (7 \, b^{2} c^{2} + 8 \, a c^{3}\right )} x^{2} + 16 \,{\left (b^{3} c + 8 \, a b c^{2}\right )} x}{768 \,{\left (256 \, c^{11} d^{9} x^{8} + 1024 \, b c^{10} d^{9} x^{7} + 1792 \, b^{2} c^{9} d^{9} x^{6} + 1792 \, b^{3} c^{8} d^{9} x^{5} + 1120 \, b^{4} c^{7} d^{9} x^{4} + 448 \, b^{5} c^{6} d^{9} x^{3} + 112 \, b^{6} c^{5} d^{9} x^{2} + 16 \, b^{7} c^{4} d^{9} x + b^{8} c^{3} d^{9}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^2/(2*c*d*x+b*d)^9,x, algorithm="maxima")

[Out]

-1/768*(96*c^4*x^4 + 192*b*c^3*x^3 + b^4 + 8*a*b^2*c + 48*a^2*c^2 + 16*(7*b^2*c^2 + 8*a*c^3)*x^2 + 16*(b^3*c +
8*a*b*c^2)*x)/(256*c^11*d^9*x^8 + 1024*b*c^10*d^9*x^7 + 1792*b^2*c^9*d^9*x^6 + 1792*b^3*c^8*d^9*x^5 + 1120*b^
4*c^7*d^9*x^4 + 448*b^5*c^6*d^9*x^3 + 112*b^6*c^5*d^9*x^2 + 16*b^7*c^4*d^9*x + b^8*c^3*d^9)

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Fricas [B]  time = 1.97067, size = 423, normalized size = 5.79 \begin{align*} -\frac{96 \, c^{4} x^{4} + 192 \, b c^{3} x^{3} + b^{4} + 8 \, a b^{2} c + 48 \, a^{2} c^{2} + 16 \,{\left (7 \, b^{2} c^{2} + 8 \, a c^{3}\right )} x^{2} + 16 \,{\left (b^{3} c + 8 \, a b c^{2}\right )} x}{768 \,{\left (256 \, c^{11} d^{9} x^{8} + 1024 \, b c^{10} d^{9} x^{7} + 1792 \, b^{2} c^{9} d^{9} x^{6} + 1792 \, b^{3} c^{8} d^{9} x^{5} + 1120 \, b^{4} c^{7} d^{9} x^{4} + 448 \, b^{5} c^{6} d^{9} x^{3} + 112 \, b^{6} c^{5} d^{9} x^{2} + 16 \, b^{7} c^{4} d^{9} x + b^{8} c^{3} d^{9}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^2/(2*c*d*x+b*d)^9,x, algorithm="fricas")

[Out]

-1/768*(96*c^4*x^4 + 192*b*c^3*x^3 + b^4 + 8*a*b^2*c + 48*a^2*c^2 + 16*(7*b^2*c^2 + 8*a*c^3)*x^2 + 16*(b^3*c +
8*a*b*c^2)*x)/(256*c^11*d^9*x^8 + 1024*b*c^10*d^9*x^7 + 1792*b^2*c^9*d^9*x^6 + 1792*b^3*c^8*d^9*x^5 + 1120*b^
4*c^7*d^9*x^4 + 448*b^5*c^6*d^9*x^3 + 112*b^6*c^5*d^9*x^2 + 16*b^7*c^4*d^9*x + b^8*c^3*d^9)

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Sympy [B]  time = 5.11816, size = 202, normalized size = 2.77 \begin{align*} - \frac{48 a^{2} c^{2} + 8 a b^{2} c + b^{4} + 192 b c^{3} x^{3} + 96 c^{4} x^{4} + x^{2} \left (128 a c^{3} + 112 b^{2} c^{2}\right ) + x \left (128 a b c^{2} + 16 b^{3} c\right )}{768 b^{8} c^{3} d^{9} + 12288 b^{7} c^{4} d^{9} x + 86016 b^{6} c^{5} d^{9} x^{2} + 344064 b^{5} c^{6} d^{9} x^{3} + 860160 b^{4} c^{7} d^{9} x^{4} + 1376256 b^{3} c^{8} d^{9} x^{5} + 1376256 b^{2} c^{9} d^{9} x^{6} + 786432 b c^{10} d^{9} x^{7} + 196608 c^{11} d^{9} x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)**2/(2*c*d*x+b*d)**9,x)

[Out]

-(48*a**2*c**2 + 8*a*b**2*c + b**4 + 192*b*c**3*x**3 + 96*c**4*x**4 + x**2*(128*a*c**3 + 112*b**2*c**2) + x*(1
28*a*b*c**2 + 16*b**3*c))/(768*b**8*c**3*d**9 + 12288*b**7*c**4*d**9*x + 86016*b**6*c**5*d**9*x**2 + 344064*b*
*5*c**6*d**9*x**3 + 860160*b**4*c**7*d**9*x**4 + 1376256*b**3*c**8*d**9*x**5 + 1376256*b**2*c**9*d**9*x**6 + 7
86432*b*c**10*d**9*x**7 + 196608*c**11*d**9*x**8)

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Giac [A]  time = 1.1501, size = 117, normalized size = 1.6 \begin{align*} -\frac{96 \, c^{4} x^{4} + 192 \, b c^{3} x^{3} + 112 \, b^{2} c^{2} x^{2} + 128 \, a c^{3} x^{2} + 16 \, b^{3} c x + 128 \, a b c^{2} x + b^{4} + 8 \, a b^{2} c + 48 \, a^{2} c^{2}}{768 \,{\left (2 \, c x + b\right )}^{8} c^{3} d^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^2/(2*c*d*x+b*d)^9,x, algorithm="giac")

[Out]

-1/768*(96*c^4*x^4 + 192*b*c^3*x^3 + 112*b^2*c^2*x^2 + 128*a*c^3*x^2 + 16*b^3*c*x + 128*a*b*c^2*x + b^4 + 8*a*
b^2*c + 48*a^2*c^2)/((2*c*x + b)^8*c^3*d^9)