3.1243 \(\int \frac{c+d x}{(a+b x)^3} \, dx\)

Optimal. Leaf size=28 \[ -\frac{(c+d x)^2}{2 (a+b x)^2 (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.0043352, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {37} \[ -\frac{(c+d x)^2}{2 (a+b x)^2 (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)/(a + b*x)^3,x]

[Out]

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

Rule 37

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

Rubi steps

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

Mathematica [A]  time = 0.0093039, size = 26, normalized size = 0.93 \[ -\frac{a d+b (c+2 d x)}{2 b^2 (a+b x)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)/(a + b*x)^3,x]

[Out]

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

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Maple [A]  time = 0.006, size = 35, normalized size = 1.3 \begin{align*} -{\frac{d}{{b}^{2} \left ( bx+a \right ) }}-{\frac{-ad+bc}{2\,{b}^{2} \left ( bx+a \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.965442, size = 51, normalized size = 1.82 \begin{align*} -\frac{2 \, b d x + b c + a d}{2 \,{\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.96827, size = 81, normalized size = 2.89 \begin{align*} -\frac{2 \, b d x + b c + a d}{2 \,{\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.428885, size = 39, normalized size = 1.39 \begin{align*} - \frac{a d + b c + 2 b d x}{2 a^{2} b^{2} + 4 a b^{3} x + 2 b^{4} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.06438, size = 32, normalized size = 1.14 \begin{align*} -\frac{2 \, b d x + b c + a d}{2 \,{\left (b x + a\right )}^{2} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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