3.1116 \(\int \frac {a+b x+c x^2}{(b d+2 c d x)^4} \, dx\)

Optimal. Leaf size=45 \[ \frac {b^2-4 a c}{24 c^2 d^4 (b+2 c x)^3}-\frac {1}{8 c^2 d^4 (b+2 c x)} \]

[Out]

1/24*(-4*a*c+b^2)/c^2/d^4/(2*c*x+b)^3-1/8/c^2/d^4/(2*c*x+b)

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Rubi [A]  time = 0.03, antiderivative size = 45, normalized size of antiderivative = 1.00, 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} \[ \frac {b^2-4 a c}{24 c^2 d^4 (b+2 c x)^3}-\frac {1}{8 c^2 d^4 (b+2 c x)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2)/(b*d + 2*c*d*x)^4,x]

[Out]

(b^2 - 4*a*c)/(24*c^2*d^4*(b + 2*c*x)^3) - 1/(8*c^2*d^4*(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 {a+b x+c x^2}{(b d+2 c d x)^4} \, dx &=\int \left (\frac {-b^2+4 a c}{4 c d^4 (b+2 c x)^4}+\frac {1}{4 c d^4 (b+2 c x)^2}\right ) \, dx\\ &=\frac {b^2-4 a c}{24 c^2 d^4 (b+2 c x)^3}-\frac {1}{8 c^2 d^4 (b+2 c x)}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 38, normalized size = 0.84 \[ -\frac {2 c \left (a+3 c x^2\right )+b^2+6 b c x}{12 c^2 d^4 (b+2 c x)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x + c*x^2)/(b*d + 2*c*d*x)^4,x]

[Out]

-1/12*(b^2 + 6*b*c*x + 2*c*(a + 3*c*x^2))/(c^2*d^4*(b + 2*c*x)^3)

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fricas [A]  time = 1.13, size = 71, normalized size = 1.58 \[ -\frac {6 \, c^{2} x^{2} + 6 \, b c x + b^{2} + 2 \, a c}{12 \, {\left (8 \, c^{5} d^{4} x^{3} + 12 \, b c^{4} d^{4} x^{2} + 6 \, b^{2} c^{3} d^{4} x + b^{3} c^{2} d^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)/(2*c*d*x+b*d)^4,x, algorithm="fricas")

[Out]

-1/12*(6*c^2*x^2 + 6*b*c*x + b^2 + 2*a*c)/(8*c^5*d^4*x^3 + 12*b*c^4*d^4*x^2 + 6*b^2*c^3*d^4*x + b^3*c^2*d^4)

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giac [A]  time = 0.15, size = 37, normalized size = 0.82 \[ -\frac {6 \, c^{2} x^{2} + 6 \, b c x + b^{2} + 2 \, a c}{12 \, {\left (2 \, c x + b\right )}^{3} c^{2} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)/(2*c*d*x+b*d)^4,x, algorithm="giac")

[Out]

-1/12*(6*c^2*x^2 + 6*b*c*x + b^2 + 2*a*c)/((2*c*x + b)^3*c^2*d^4)

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maple [A]  time = 0.05, size = 42, normalized size = 0.93 \[ \frac {-\frac {4 a c -b^{2}}{24 \left (2 c x +b \right )^{3} c^{2}}-\frac {1}{8 \left (2 c x +b \right ) c^{2}}}{d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)/(2*c*d*x+b*d)^4,x)

[Out]

1/d^4*(-1/24*(4*a*c-b^2)/c^2/(2*c*x+b)^3-1/8/c^2/(2*c*x+b))

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maxima [A]  time = 1.36, size = 71, normalized size = 1.58 \[ -\frac {6 \, c^{2} x^{2} + 6 \, b c x + b^{2} + 2 \, a c}{12 \, {\left (8 \, c^{5} d^{4} x^{3} + 12 \, b c^{4} d^{4} x^{2} + 6 \, b^{2} c^{3} d^{4} x + b^{3} c^{2} d^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)/(2*c*d*x+b*d)^4,x, algorithm="maxima")

[Out]

-1/12*(6*c^2*x^2 + 6*b*c*x + b^2 + 2*a*c)/(8*c^5*d^4*x^3 + 12*b*c^4*d^4*x^2 + 6*b^2*c^3*d^4*x + b^3*c^2*d^4)

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mupad [B]  time = 0.43, size = 71, normalized size = 1.58 \[ -\frac {\frac {b^2+2\,a\,c}{12\,c^2}+\frac {x^2}{2}+\frac {b\,x}{2\,c}}{b^3\,d^4+6\,b^2\,c\,d^4\,x+12\,b\,c^2\,d^4\,x^2+8\,c^3\,d^4\,x^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x + c*x^2)/(b*d + 2*c*d*x)^4,x)

[Out]

-((2*a*c + b^2)/(12*c^2) + x^2/2 + (b*x)/(2*c))/(b^3*d^4 + 8*c^3*d^4*x^3 + 12*b*c^2*d^4*x^2 + 6*b^2*c*d^4*x)

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sympy [A]  time = 0.44, size = 75, normalized size = 1.67 \[ \frac {- 2 a c - b^{2} - 6 b c x - 6 c^{2} x^{2}}{12 b^{3} c^{2} d^{4} + 72 b^{2} c^{3} d^{4} x + 144 b c^{4} d^{4} x^{2} + 96 c^{5} d^{4} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)/(2*c*d*x+b*d)**4,x)

[Out]

(-2*a*c - b**2 - 6*b*c*x - 6*c**2*x**2)/(12*b**3*c**2*d**4 + 72*b**2*c**3*d**4*x + 144*b*c**4*d**4*x**2 + 96*c
**5*d**4*x**3)

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