3.1120 \(\int \frac{a+b x+c x^2}{(b d+2 c d x)^8} \, dx\)

Optimal. Leaf size=45 \[ \frac{b^2-4 a c}{56 c^2 d^8 (b+2 c x)^7}-\frac{1}{40 c^2 d^8 (b+2 c x)^5} \]

[Out]

(b^2 - 4*a*c)/(56*c^2*d^8*(b + 2*c*x)^7) - 1/(40*c^2*d^8*(b + 2*c*x)^5)

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Rubi [A]  time = 0.0322843, 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} \[ \frac{b^2-4 a c}{56 c^2 d^8 (b+2 c x)^7}-\frac{1}{40 c^2 d^8 (b+2 c x)^5} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2)/(b*d + 2*c*d*x)^8,x]

[Out]

(b^2 - 4*a*c)/(56*c^2*d^8*(b + 2*c*x)^7) - 1/(40*c^2*d^8*(b + 2*c*x)^5)

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)^8} \, dx &=\int \left (\frac{-b^2+4 a c}{4 c d^8 (b+2 c x)^8}+\frac{1}{4 c d^8 (b+2 c x)^6}\right ) \, dx\\ &=\frac{b^2-4 a c}{56 c^2 d^8 (b+2 c x)^7}-\frac{1}{40 c^2 d^8 (b+2 c x)^5}\\ \end{align*}

Mathematica [A]  time = 0.0149663, size = 39, normalized size = 0.87 \[ \frac{5 \left (b^2-4 a c\right )-7 (b+2 c x)^2}{280 c^2 d^8 (b+2 c x)^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x + c*x^2)/(b*d + 2*c*d*x)^8,x]

[Out]

(5*(b^2 - 4*a*c) - 7*(b + 2*c*x)^2)/(280*c^2*d^8*(b + 2*c*x)^7)

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Maple [A]  time = 0.043, size = 42, normalized size = 0.9 \begin{align*}{\frac{1}{{d}^{8}} \left ( -{\frac{4\,ac-{b}^{2}}{56\,{c}^{2} \left ( 2\,cx+b \right ) ^{7}}}-{\frac{1}{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)^8,x)

[Out]

1/d^8*(-1/56*(4*a*c-b^2)/c^2/(2*c*x+b)^7-1/40/c^2/(2*c*x+b)^5)

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Maxima [B]  time = 1.15251, size = 171, normalized size = 3.8 \begin{align*} -\frac{14 \, c^{2} x^{2} + 14 \, b c x + b^{2} + 10 \, a c}{140 \,{\left (128 \, c^{9} d^{8} x^{7} + 448 \, b c^{8} d^{8} x^{6} + 672 \, b^{2} c^{7} d^{8} x^{5} + 560 \, b^{3} c^{6} d^{8} x^{4} + 280 \, b^{4} c^{5} d^{8} x^{3} + 84 \, b^{5} c^{4} d^{8} x^{2} + 14 \, b^{6} c^{3} d^{8} x + b^{7} c^{2} d^{8}\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)^8,x, algorithm="maxima")

[Out]

-1/140*(14*c^2*x^2 + 14*b*c*x + b^2 + 10*a*c)/(128*c^9*d^8*x^7 + 448*b*c^8*d^8*x^6 + 672*b^2*c^7*d^8*x^5 + 560
*b^3*c^6*d^8*x^4 + 280*b^4*c^5*d^8*x^3 + 84*b^5*c^4*d^8*x^2 + 14*b^6*c^3*d^8*x + b^7*c^2*d^8)

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Fricas [B]  time = 2.09167, size = 277, normalized size = 6.16 \begin{align*} -\frac{14 \, c^{2} x^{2} + 14 \, b c x + b^{2} + 10 \, a c}{140 \,{\left (128 \, c^{9} d^{8} x^{7} + 448 \, b c^{8} d^{8} x^{6} + 672 \, b^{2} c^{7} d^{8} x^{5} + 560 \, b^{3} c^{6} d^{8} x^{4} + 280 \, b^{4} c^{5} d^{8} x^{3} + 84 \, b^{5} c^{4} d^{8} x^{2} + 14 \, b^{6} c^{3} d^{8} x + b^{7} c^{2} d^{8}\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)^8,x, algorithm="fricas")

[Out]

-1/140*(14*c^2*x^2 + 14*b*c*x + b^2 + 10*a*c)/(128*c^9*d^8*x^7 + 448*b*c^8*d^8*x^6 + 672*b^2*c^7*d^8*x^5 + 560
*b^3*c^6*d^8*x^4 + 280*b^4*c^5*d^8*x^3 + 84*b^5*c^4*d^8*x^2 + 14*b^6*c^3*d^8*x + b^7*c^2*d^8)

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Sympy [B]  time = 1.40456, size = 136, normalized size = 3.02 \begin{align*} - \frac{10 a c + b^{2} + 14 b c x + 14 c^{2} x^{2}}{140 b^{7} c^{2} d^{8} + 1960 b^{6} c^{3} d^{8} x + 11760 b^{5} c^{4} d^{8} x^{2} + 39200 b^{4} c^{5} d^{8} x^{3} + 78400 b^{3} c^{6} d^{8} x^{4} + 94080 b^{2} c^{7} d^{8} x^{5} + 62720 b c^{8} d^{8} x^{6} + 17920 c^{9} d^{8} x^{7}} \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)**8,x)

[Out]

-(10*a*c + b**2 + 14*b*c*x + 14*c**2*x**2)/(140*b**7*c**2*d**8 + 1960*b**6*c**3*d**8*x + 11760*b**5*c**4*d**8*
x**2 + 39200*b**4*c**5*d**8*x**3 + 78400*b**3*c**6*d**8*x**4 + 94080*b**2*c**7*d**8*x**5 + 62720*b*c**8*d**8*x
**6 + 17920*c**9*d**8*x**7)

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Giac [A]  time = 1.20759, size = 50, normalized size = 1.11 \begin{align*} -\frac{14 \, c^{2} x^{2} + 14 \, b c x + b^{2} + 10 \, a c}{140 \,{\left (2 \, c x + b\right )}^{7} c^{2} d^{8}} \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)^8,x, algorithm="giac")

[Out]

-1/140*(14*c^2*x^2 + 14*b*c*x + b^2 + 10*a*c)/((2*c*x + b)^7*c^2*d^8)