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

Optimal. Leaf size=38 \[ \frac {b^2-4 a c}{8 c^2 d^2 (b+2 c x)}+\frac {x}{4 c d^2} \]

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Rubi [A]  time = 0.03, antiderivative size = 38, 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} \begin {gather*} \frac {b^2-4 a c}{8 c^2 d^2 (b+2 c x)}+\frac {x}{4 c d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.01, size = 41, normalized size = 1.08 \begin {gather*} \frac {\frac {b^2-4 a c}{8 c^2 (b+2 c x)}+\frac {b+2 c x}{8 c^2}}{d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a+b x+c x^2}{(b d+2 c d x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.39, size = 43, normalized size = 1.13 \begin {gather*} \frac {4 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 4 \, a c}{8 \, {\left (2 \, c^{3} d^{2} x + b c^{2} d^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.16, size = 170, normalized size = 4.47 \begin {gather*} -\frac {1}{8} \, c {\left (\frac {b^{2}}{{\left (2 \, c d x + b d\right )} c^{3} d} - \frac {2 \, b \log \left (\frac {{\left | 2 \, c d x + b d \right |}}{2 \, {\left (2 \, c d x + b d\right )}^{2} {\left | c \right |} {\left | d \right |}}\right )}{c^{3} d^{2}} - \frac {2 \, c d x + b d}{c^{3} d^{3}}\right )} + \frac {b {\left (\frac {b}{{\left (2 \, c d x + b d\right )} c} - \frac {\log \left (\frac {{\left | 2 \, c d x + b d \right |}}{2 \, {\left (2 \, c d x + b d\right )}^{2} {\left | c \right |} {\left | d \right |}}\right )}{c d}\right )}}{4 \, c d} - \frac {a}{2 \, {\left (2 \, c d x + b d\right )} c d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.05, size = 35, normalized size = 0.92 \begin {gather*} \frac {\frac {x}{4 c}-\frac {4 a c -b^{2}}{8 \left (2 c x +b \right ) c^{2}}}{d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.37, size = 40, normalized size = 1.05 \begin {gather*} \frac {b^{2} - 4 \, a c}{8 \, {\left (2 \, c^{3} d^{2} x + b c^{2} d^{2}\right )}} + \frac {x}{4 \, c d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.43, size = 36, normalized size = 0.95 \begin {gather*} \frac {x}{4\,c\,d^2}-\frac {\frac {a\,c}{2}-\frac {b^2}{8}}{c^2\,d^2\,\left (b+2\,c\,x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.23, size = 36, normalized size = 0.95 \begin {gather*} \frac {- 4 a c + b^{2}}{8 b c^{2} d^{2} + 16 c^{3} d^{2} x} + \frac {x}{4 c d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-4*a*c + b**2)/(8*b*c**2*d**2 + 16*c**3*d**2*x) + x/(4*c*d**2)

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