### 3.1136 $$\int \frac{(a+b x+c x^2)^2}{(b d+2 c d x)^{11}} \, dx$$

Optimal. Leaf size=73 $-\frac{\left (b^2-4 a c\right )^2}{320 c^3 d^{11} (b+2 c x)^{10}}+\frac{b^2-4 a c}{128 c^3 d^{11} (b+2 c x)^8}-\frac{1}{192 c^3 d^{11} (b+2 c x)^6}$

[Out]

-(b^2 - 4*a*c)^2/(320*c^3*d^11*(b + 2*c*x)^10) + (b^2 - 4*a*c)/(128*c^3*d^11*(b + 2*c*x)^8) - 1/(192*c^3*d^11*
(b + 2*c*x)^6)

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Rubi [A]  time = 0.0542056, 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} $-\frac{\left (b^2-4 a c\right )^2}{320 c^3 d^{11} (b+2 c x)^{10}}+\frac{b^2-4 a c}{128 c^3 d^{11} (b+2 c x)^8}-\frac{1}{192 c^3 d^{11} (b+2 c x)^6}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^11,x]

[Out]

-(b^2 - 4*a*c)^2/(320*c^3*d^11*(b + 2*c*x)^10) + (b^2 - 4*a*c)/(128*c^3*d^11*(b + 2*c*x)^8) - 1/(192*c^3*d^11*
(b + 2*c*x)^6)

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)^{11}} \, dx &=\int \left (\frac{\left (-b^2+4 a c\right )^2}{16 c^2 d^{11} (b+2 c x)^{11}}+\frac{-b^2+4 a c}{8 c^2 d^{11} (b+2 c x)^9}+\frac{1}{16 c^2 d^{11} (b+2 c x)^7}\right ) \, dx\\ &=-\frac{\left (b^2-4 a c\right )^2}{320 c^3 d^{11} (b+2 c x)^{10}}+\frac{b^2-4 a c}{128 c^3 d^{11} (b+2 c x)^8}-\frac{1}{192 c^3 d^{11} (b+2 c x)^6}\\ \end{align*}

Mathematica [A]  time = 0.0324032, size = 59, normalized size = 0.81 $\frac{15 \left (b^2-4 a c\right ) (b+2 c x)^2-6 \left (b^2-4 a c\right )^2-10 (b+2 c x)^4}{1920 c^3 d^{11} (b+2 c x)^{10}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^11,x]

[Out]

(-6*(b^2 - 4*a*c)^2 + 15*(b^2 - 4*a*c)*(b + 2*c*x)^2 - 10*(b + 2*c*x)^4)/(1920*c^3*d^11*(b + 2*c*x)^10)

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Maple [A]  time = 0.046, size = 74, normalized size = 1. \begin{align*}{\frac{1}{{d}^{11}} \left ( -{\frac{1}{192\,{c}^{3} \left ( 2\,cx+b \right ) ^{6}}}-{\frac{16\,{a}^{2}{c}^{2}-8\,ac{b}^{2}+{b}^{4}}{320\,{c}^{3} \left ( 2\,cx+b \right ) ^{10}}}-{\frac{4\,ac-{b}^{2}}{128\,{c}^{3} \left ( 2\,cx+b \right ) ^{8}}} \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)^11,x)

[Out]

1/d^11*(-1/192/c^3/(2*c*x+b)^6-1/320*(16*a^2*c^2-8*a*b^2*c+b^4)/c^3/(2*c*x+b)^10-1/128*(4*a*c-b^2)/c^3/(2*c*x+
b)^8)

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Maxima [B]  time = 1.21255, size = 297, normalized size = 4.07 \begin{align*} -\frac{160 \, c^{4} x^{4} + 320 \, b c^{3} x^{3} + b^{4} + 12 \, a b^{2} c + 96 \, a^{2} c^{2} + 60 \,{\left (3 \, b^{2} c^{2} + 4 \, a c^{3}\right )} x^{2} + 20 \,{\left (b^{3} c + 12 \, a b c^{2}\right )} x}{1920 \,{\left (1024 \, c^{13} d^{11} x^{10} + 5120 \, b c^{12} d^{11} x^{9} + 11520 \, b^{2} c^{11} d^{11} x^{8} + 15360 \, b^{3} c^{10} d^{11} x^{7} + 13440 \, b^{4} c^{9} d^{11} x^{6} + 8064 \, b^{5} c^{8} d^{11} x^{5} + 3360 \, b^{6} c^{7} d^{11} x^{4} + 960 \, b^{7} c^{6} d^{11} x^{3} + 180 \, b^{8} c^{5} d^{11} x^{2} + 20 \, b^{9} c^{4} d^{11} x + b^{10} c^{3} d^{11}\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)^11,x, algorithm="maxima")

[Out]

-1/1920*(160*c^4*x^4 + 320*b*c^3*x^3 + b^4 + 12*a*b^2*c + 96*a^2*c^2 + 60*(3*b^2*c^2 + 4*a*c^3)*x^2 + 20*(b^3*
c + 12*a*b*c^2)*x)/(1024*c^13*d^11*x^10 + 5120*b*c^12*d^11*x^9 + 11520*b^2*c^11*d^11*x^8 + 15360*b^3*c^10*d^11
*x^7 + 13440*b^4*c^9*d^11*x^6 + 8064*b^5*c^8*d^11*x^5 + 3360*b^6*c^7*d^11*x^4 + 960*b^7*c^6*d^11*x^3 + 180*b^8
*c^5*d^11*x^2 + 20*b^9*c^4*d^11*x + b^10*c^3*d^11)

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Fricas [B]  time = 1.97898, size = 516, normalized size = 7.07 \begin{align*} -\frac{160 \, c^{4} x^{4} + 320 \, b c^{3} x^{3} + b^{4} + 12 \, a b^{2} c + 96 \, a^{2} c^{2} + 60 \,{\left (3 \, b^{2} c^{2} + 4 \, a c^{3}\right )} x^{2} + 20 \,{\left (b^{3} c + 12 \, a b c^{2}\right )} x}{1920 \,{\left (1024 \, c^{13} d^{11} x^{10} + 5120 \, b c^{12} d^{11} x^{9} + 11520 \, b^{2} c^{11} d^{11} x^{8} + 15360 \, b^{3} c^{10} d^{11} x^{7} + 13440 \, b^{4} c^{9} d^{11} x^{6} + 8064 \, b^{5} c^{8} d^{11} x^{5} + 3360 \, b^{6} c^{7} d^{11} x^{4} + 960 \, b^{7} c^{6} d^{11} x^{3} + 180 \, b^{8} c^{5} d^{11} x^{2} + 20 \, b^{9} c^{4} d^{11} x + b^{10} c^{3} d^{11}\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)^11,x, algorithm="fricas")

[Out]

-1/1920*(160*c^4*x^4 + 320*b*c^3*x^3 + b^4 + 12*a*b^2*c + 96*a^2*c^2 + 60*(3*b^2*c^2 + 4*a*c^3)*x^2 + 20*(b^3*
c + 12*a*b*c^2)*x)/(1024*c^13*d^11*x^10 + 5120*b*c^12*d^11*x^9 + 11520*b^2*c^11*d^11*x^8 + 15360*b^3*c^10*d^11
*x^7 + 13440*b^4*c^9*d^11*x^6 + 8064*b^5*c^8*d^11*x^5 + 3360*b^6*c^7*d^11*x^4 + 960*b^7*c^6*d^11*x^3 + 180*b^8
*c^5*d^11*x^2 + 20*b^9*c^4*d^11*x + b^10*c^3*d^11)

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Sympy [B]  time = 9.00974, size = 233, normalized size = 3.19 \begin{align*} - \frac{96 a^{2} c^{2} + 12 a b^{2} c + b^{4} + 320 b c^{3} x^{3} + 160 c^{4} x^{4} + x^{2} \left (240 a c^{3} + 180 b^{2} c^{2}\right ) + x \left (240 a b c^{2} + 20 b^{3} c\right )}{1920 b^{10} c^{3} d^{11} + 38400 b^{9} c^{4} d^{11} x + 345600 b^{8} c^{5} d^{11} x^{2} + 1843200 b^{7} c^{6} d^{11} x^{3} + 6451200 b^{6} c^{7} d^{11} x^{4} + 15482880 b^{5} c^{8} d^{11} x^{5} + 25804800 b^{4} c^{9} d^{11} x^{6} + 29491200 b^{3} c^{10} d^{11} x^{7} + 22118400 b^{2} c^{11} d^{11} x^{8} + 9830400 b c^{12} d^{11} x^{9} + 1966080 c^{13} d^{11} x^{10}} \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)**11,x)

[Out]

-(96*a**2*c**2 + 12*a*b**2*c + b**4 + 320*b*c**3*x**3 + 160*c**4*x**4 + x**2*(240*a*c**3 + 180*b**2*c**2) + x*
(240*a*b*c**2 + 20*b**3*c))/(1920*b**10*c**3*d**11 + 38400*b**9*c**4*d**11*x + 345600*b**8*c**5*d**11*x**2 + 1
843200*b**7*c**6*d**11*x**3 + 6451200*b**6*c**7*d**11*x**4 + 15482880*b**5*c**8*d**11*x**5 + 25804800*b**4*c**
9*d**11*x**6 + 29491200*b**3*c**10*d**11*x**7 + 22118400*b**2*c**11*d**11*x**8 + 9830400*b*c**12*d**11*x**9 +
1966080*c**13*d**11*x**10)

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Giac [A]  time = 1.17609, size = 117, normalized size = 1.6 \begin{align*} -\frac{160 \, c^{4} x^{4} + 320 \, b c^{3} x^{3} + 180 \, b^{2} c^{2} x^{2} + 240 \, a c^{3} x^{2} + 20 \, b^{3} c x + 240 \, a b c^{2} x + b^{4} + 12 \, a b^{2} c + 96 \, a^{2} c^{2}}{1920 \,{\left (2 \, c x + b\right )}^{10} c^{3} d^{11}} \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)^11,x, algorithm="giac")

[Out]

-1/1920*(160*c^4*x^4 + 320*b*c^3*x^3 + 180*b^2*c^2*x^2 + 240*a*c^3*x^2 + 20*b^3*c*x + 240*a*b*c^2*x + b^4 + 12
*a*b^2*c + 96*a^2*c^2)/((2*c*x + b)^10*c^3*d^11)