3.1357 \(\int \frac{a+b x}{(c+d x)^3} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0029068, 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} \[ \frac{(a+b x)^2}{2 (c+d x)^2 (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)/(c + d*x)^3,x]

[Out]

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

Mathematica [A]  time = 0.0086918, size = 26, normalized size = 0.93 \[ -\frac{a d+b (c+2 d x)}{2 d^2 (c+d x)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)/(c + d*x)^3,x]

[Out]

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

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Maple [A]  time = 0.004, size = 35, normalized size = 1.3 \begin{align*} -{\frac{b}{{d}^{2} \left ( dx+c \right ) }}-{\frac{ad-bc}{2\,{d}^{2} \left ( dx+c \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.958779, size = 51, normalized size = 1.82 \begin{align*} -\frac{2 \, b d x + b c + a d}{2 \,{\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.71051, size = 81, normalized size = 2.89 \begin{align*} -\frac{2 \, b d x + b c + a d}{2 \,{\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.407639, size = 39, normalized size = 1.39 \begin{align*} - \frac{a d + b c + 2 b d x}{2 c^{2} d^{2} + 4 c d^{3} x + 2 d^{4} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.06412, size = 32, normalized size = 1.14 \begin{align*} -\frac{2 \, b d x + b c + a d}{2 \,{\left (d x + c\right )}^{2} d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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