3.1003 \(\int \frac{a+b x}{(\frac{a d}{b}+d x)^3} \, dx\)

Optimal. Leaf size=15 \[ -\frac{b^2}{d^3 (a+b x)} \]

[Out]

-(b^2/(d^3*(a + b*x)))

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Rubi [A]  time = 0.0033476, antiderivative size = 15, 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} \[ -\frac{b^2}{d^3 (a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)/((a*d)/b + d*x)^3,x]

[Out]

-(b^2/(d^3*(a + b*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{a+b x}{\left (\frac{a d}{b}+d x\right )^3} \, dx &=\frac{b^3 \int \frac{1}{(a+b x)^2} \, dx}{d^3}\\ &=-\frac{b^2}{d^3 (a+b x)}\\ \end{align*}

Mathematica [A]  time = 0.0039192, size = 15, normalized size = 1. \[ -\frac{b^2}{d^3 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b^2/(d^3*(a + b*x)))

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Maple [A]  time = 0.001, size = 16, normalized size = 1.1 \begin{align*} -{\frac{{b}^{2}}{{d}^{3} \left ( bx+a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.979857, size = 26, normalized size = 1.73 \begin{align*} -\frac{b^{2}}{b d^{3} x + a d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b^2/(b*d^3*x + a*d^3)

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Fricas [A]  time = 1.5042, size = 32, normalized size = 2.13 \begin{align*} -\frac{b^{2}}{b d^{3} x + a d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b^2/(b*d^3*x + a*d^3)

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Sympy [A]  time = 0.309068, size = 19, normalized size = 1.27 \begin{align*} - \frac{b^{3}}{a b d^{3} + b^{2} d^{3} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b**3/(a*b*d**3 + b**2*d**3*x)

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Giac [A]  time = 1.06447, size = 20, normalized size = 1.33 \begin{align*} -\frac{b^{2}}{{\left (b x + a\right )} d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b^2/((b*x + a)*d^3)