3.12.86 \(\int \frac {b d+2 c d x}{(a+b x+c x^2)^3} \, dx\) [1186]

Optimal. Leaf size=17 \[ -\frac {d}{2 \left (a+b x+c x^2\right )^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {643} \begin {gather*} -\frac {d}{2 \left (a+b x+c x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*d/(a + b*x + c*x^2)^2

Rule 643

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

________________________________________________________________________________________

Mathematica [A]
time = 0.00, size = 16, normalized size = 0.94 \begin {gather*} -\frac {d}{2 (a+x (b+c x))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*d/(a + x*(b + c*x))^2

________________________________________________________________________________________

Maple [A]
time = 0.64, size = 16, normalized size = 0.94

method	result	size
gosper	\(-\frac {d}{2 \left (c \,x^{2}+b x +a \right )^{2}}\)	\(16\)
default	\(-\frac {d}{2 \left (c \,x^{2}+b x +a \right )^{2}}\)	\(16\)
norman	\(-\frac {d}{2 \left (c \,x^{2}+b x +a \right )^{2}}\)	\(16\)
risch	\(-\frac {d}{2 \left (c \,x^{2}+b x +a \right )^{2}}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*d*x+b*d)/(c*x^2+b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.27, size = 15, normalized size = 0.88 \begin {gather*} -\frac {d}{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*d*x+b*d)/(c*x^2+b*x+a)^3,x, algorithm="maxima")

[Out]

-1/2*d/(c*x^2 + b*x + a)^2

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 40 vs. \(2 (15) = 30\).
time = 1.38, size = 40, normalized size = 2.35 \begin {gather*} -\frac {d}{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*d*x+b*d)/(c*x^2+b*x+a)^3,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (15) = 30\).
time = 0.46, size = 44, normalized size = 2.59 \begin {gather*} - \frac {d}{2 a^{2} + 4 a b x + 4 b c x^{3} + 2 c^{2} x^{4} + x^{2} \cdot \left (4 a c + 2 b^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-d/(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))

________________________________________________________________________________________

Giac [A]
time = 1.20, size = 22, normalized size = 1.29 \begin {gather*} -\frac {d}{2 \, {\left (a + \frac {c d x^{2} + b d x}{d}\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*d/(a + (c*d*x^2 + b*d*x)/d)^2

________________________________________________________________________________________

Mupad [B]
time = 0.05, size = 44, normalized size = 2.59 \begin {gather*} -\frac {d}{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*d + 2*c*d*x)/(a + b*x + c*x^2)^3,x)

[Out]

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

________________________________________________________________________________________