### 3.1131 $$\int \frac{(a+b x+c x^2)^2}{(b d+2 c d x)^6} \, dx$$

Optimal. Leaf size=73 $-\frac{\left (b^2-4 a c\right )^2}{160 c^3 d^6 (b+2 c x)^5}+\frac{b^2-4 a c}{48 c^3 d^6 (b+2 c x)^3}-\frac{1}{32 c^3 d^6 (b+2 c x)}$

[Out]

-(b^2 - 4*a*c)^2/(160*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))

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Rubi [A]  time = 0.0569732, 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} $-\frac{\left (b^2-4 a c\right )^2}{160 c^3 d^6 (b+2 c x)^5}+\frac{b^2-4 a c}{48 c^3 d^6 (b+2 c x)^3}-\frac{1}{32 c^3 d^6 (b+2 c x)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(b^2 - 4*a*c)^2/(160*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 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)^6} \, dx &=\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]  time = 0.034111, size = 59, normalized size = 0.81 $\frac{10 \left (b^2-4 a c\right ) (b+2 c x)^2-3 \left (b^2-4 a c\right )^2-15 (b+2 c x)^4}{480 c^3 d^6 (b+2 c x)^5}$

Antiderivative was successfully veriﬁed.

[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)

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Maple [A]  time = 0.045, size = 74, normalized size = 1. \begin{align*}{\frac{1}{{d}^{6}} \left ( -{\frac{4\,ac-{b}^{2}}{48\,{c}^{3} \left ( 2\,cx+b \right ) ^{3}}}-{\frac{1}{32\,{c}^{3} \left ( 2\,cx+b \right ) }}-{\frac{16\,{a}^{2}{c}^{2}-8\,ac{b}^{2}+{b}^{4}}{160\,{c}^{3} \left ( 2\,cx+b \right ) ^{5}}} \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)^6,x)

[Out]

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

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Maxima [B]  time = 1.28222, size = 201, normalized size = 2.75 \begin{align*} -\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 )}} \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)^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)

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Fricas [B]  time = 1.98415, size = 313, normalized size = 4.29 \begin{align*} -\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 )}} \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)^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)

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Sympy [B]  time = 1.89372, size = 156, normalized size = 2.14 \begin{align*} - \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}} \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)**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)

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Giac [A]  time = 1.19, size = 117, normalized size = 1.6 \begin{align*} -\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}} \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)^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)