3.14.52 \(\int \frac {b+2 c x}{(a+b x+c x^2)^3} \, dx\)

Optimal. Leaf size=16 \[ -\frac {1}{2 \left (a+b x+c x^2\right )^2} \]

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Rubi [A]  time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {629} \begin {gather*} -\frac {1}{2 \left (a+b x+c x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b + 2*c*x)/(a + b*x + c*x^2)^3,x]

[Out]

-1/(2*(a + b*x + c*x^2)^2)

Rule 629

Int[((d_) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d*(a + b*x + c*x^2)^(p +
1))/(b*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 15, normalized size = 0.94 \begin {gather*} -\frac {1}{2 (a+x (b+c x))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b + 2*c*x)/(a + b*x + c*x^2)^3,x]

[Out]

-1/2*1/(a + x*(b + c*x))^2

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {b+2 c x}{\left (a+b x+c x^2\right )^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(b + 2*c*x)/(a + b*x + c*x^2)^3,x]

[Out]

IntegrateAlgebraic[(b + 2*c*x)/(a + b*x + c*x^2)^3, x]

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fricas [B]  time = 0.41, size = 39, normalized size = 2.44 \begin {gather*} -\frac {1}{2 \, {\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)/(c*x^2+b*x+a)^3,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.16, size = 14, normalized size = 0.88 \begin {gather*} -\frac {1}{2 \, {\left (c x^{2} + b x + a\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)/(c*x^2+b*x+a)^3,x, algorithm="giac")

[Out]

-1/2/(c*x^2 + b*x + a)^2

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maple [A]  time = 0.04, size = 15, normalized size = 0.94 \begin {gather*} -\frac {1}{2 \left (c \,x^{2}+b x +a \right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*x+b)/(c*x^2+b*x+a)^3,x)

[Out]

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

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maxima [A]  time = 0.69, size = 14, normalized size = 0.88 \begin {gather*} -\frac {1}{2 \, {\left (c x^{2} + b x + a\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)/(c*x^2+b*x+a)^3,x, algorithm="maxima")

[Out]

-1/2/(c*x^2 + b*x + a)^2

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mupad [B]  time = 0.06, size = 43, normalized size = 2.69 \begin {gather*} -\frac {1}{2\,\left (x^2\,\left (b^2+2\,a\,c\right )+a^2+c^2\,x^4+2\,a\,b\,x+2\,b\,c\,x^3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b + 2*c*x)/(a + b*x + c*x^2)^3,x)

[Out]

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

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sympy [B]  time = 0.82, size = 44, normalized size = 2.75 \begin {gather*} - \frac {1}{2 a^{2} + 4 a b x + 4 b c x^{3} + 2 c^{2} x^{4} + x^{2} \left (4 a c + 2 b^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)/(c*x**2+b*x+a)**3,x)

[Out]

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

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