### 3.1118 $$\int \frac{a+b x+c x^2}{(b d+2 c d x)^6} \, dx$$

Optimal. Leaf size=45 $\frac{b^2-4 a c}{40 c^2 d^6 (b+2 c x)^5}-\frac{1}{24 c^2 d^6 (b+2 c x)^3}$

[Out]

(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)

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Rubi [A]  time = 0.0328772, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.045, Rules used = {683} $\frac{b^2-4 a c}{40 c^2 d^6 (b+2 c x)^5}-\frac{1}{24 c^2 d^6 (b+2 c x)^3}$

Antiderivative was successfully veriﬁed.

[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 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{a+b x+c x^2}{(b d+2 c d x)^6} \, dx &=\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]  time = 0.0167315, size = 43, normalized size = 0.96 $\frac{\frac{b^2-4 a c}{40 c^2 (b+2 c x)^5}-\frac{1}{24 c^2 (b+2 c x)^3}}{d^6}$

Antiderivative was successfully veriﬁed.

[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

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Maple [A]  time = 0.044, size = 42, normalized size = 0.9 \begin{align*}{\frac{1}{{d}^{6}} \left ( -{\frac{1}{24\,{c}^{2} \left ( 2\,cx+b \right ) ^{3}}}-{\frac{4\,ac-{b}^{2}}{40\,{c}^{2} \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*c*d*x+b*d)^6,x)

[Out]

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

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Maxima [B]  time = 1.45589, size = 134, normalized size = 2.98 \begin{align*} -\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 )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [B]  time = 2.47657, size = 211, normalized size = 4.69 \begin{align*} -\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 )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [B]  time = 1.14445, size = 105, normalized size = 2.33 \begin{align*} - \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}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Giac [A]  time = 1.1946, size = 50, normalized size = 1.11 \begin{align*} -\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}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)