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

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

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Rubi [A]  time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {b}{d^2 (c+d x)} \end {gather*}

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*}

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Mathematica [A]  time = 0.00, size = 13, normalized size = 1.00 \begin {gather*} -\frac {b}{d^2 (c+d x)} \end {gather*}

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.20, size = 16, normalized size = 1.23 \begin {gather*} -\frac {b}{d^{3} x + c d^{2}} \end {gather*}

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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giac [A]  time = 1.00, size = 13, normalized size = 1.00 \begin {gather*} -\frac {b}{{\left (d x + c\right )} d^{2}} \end {gather*}

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)

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maple [A]  time = 0.00, size = 14, normalized size = 1.08 \begin {gather*} -\frac {b}{\left (d x +c \right ) d^{2}} \end {gather*}

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 = 1.38, size = 16, normalized size = 1.23 \begin {gather*} -\frac {b}{d^{3} x + c d^{2}} \end {gather*}

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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mupad [B]  time = 0.04, size = 13, normalized size = 1.00 \begin {gather*} -\frac {b}{d^2\,\left (c+d\,x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x + (b*c)/d)/(c + d*x)^3,x)

[Out]

-b/(d^2*(c + d*x))

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sympy [A]  time = 0.15, size = 12, normalized size = 0.92 \begin {gather*} - \frac {b}{c d^{2} + d^{3} x} \end {gather*}

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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