3.1011 \(\int \frac{\frac{b c}{d}+b x}{(c+d x)^3} \, dx\)

Optimal. Leaf size=13 \[ -\frac{b}{d^2 (c+d x)} \]

[Out]

-(b/(d^2*(c + d*x)))

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Rubi [A]  time = 0.0032366, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {21, 32} \[ -\frac{b}{d^2 (c+d x)} \]

Antiderivative was successfully verified.

[In]

Int[((b*c)/d + b*x)/(c + d*x)^3,x]

[Out]

-(b/(d^2*(c + d*x)))

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0040674, size = 13, normalized size = 1. \[ -\frac{b}{d^2 (c+d x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b/(d^2*(c + d*x)))

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Maple [A]  time = 0.001, size = 14, normalized size = 1.1 \begin{align*} -{\frac{b}{{d}^{2} \left ( dx+c \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.987419, size = 22, normalized size = 1.69 \begin{align*} -\frac{b}{d^{3} x + c d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b/(d^3*x + c*d^2)

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Fricas [A]  time = 1.45861, size = 27, normalized size = 2.08 \begin{align*} -\frac{b}{d^{3} x + c d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b/(d^3*x + c*d^2)

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Sympy [A]  time = 0.284583, size = 12, normalized size = 0.92 \begin{align*} - \frac{b}{c d^{2} + d^{3} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b/(c*d**2 + d**3*x)

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Giac [A]  time = 1.05929, size = 18, normalized size = 1.38 \begin{align*} -\frac{b}{{\left (d x + c\right )} d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b/((d*x + c)*d^2)