3.1005 \(\int \frac{1}{(a+b x)^2 (\frac{a d}{b}+d x)^3} \, dx\)

Optimal. Leaf size=17 \[ -\frac{b^2}{4 d^3 (a+b x)^4} \]

[Out]

-b^2/(4*d^3*(a + b*x)^4)

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Rubi [A]  time = 0.0032718, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {21, 32} \[ -\frac{b^2}{4 d^3 (a+b x)^4} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x)^2*((a*d)/b + d*x)^3),x]

[Out]

-b^2/(4*d^3*(a + b*x)^4)

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{1}{(a+b x)^2 \left (\frac{a d}{b}+d x\right )^3} \, dx &=\frac{b^3 \int \frac{1}{(a+b x)^5} \, dx}{d^3}\\ &=-\frac{b^2}{4 d^3 (a+b x)^4}\\ \end{align*}

Mathematica [A]  time = 0.0054998, size = 17, normalized size = 1. \[ -\frac{b^2}{4 d^3 (a+b x)^4} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x)^2*((a*d)/b + d*x)^3),x]

[Out]

-b^2/(4*d^3*(a + b*x)^4)

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Maple [A]  time = 0., size = 16, normalized size = 0.9 \begin{align*} -{\frac{{b}^{2}}{4\,{d}^{3} \left ( bx+a \right ) ^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x+a)^2/(a*d/b+d*x)^3,x)

[Out]

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

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Maxima [B]  time = 1.0036, size = 82, normalized size = 4.82 \begin{align*} -\frac{b^{2}}{4 \,{\left (b^{4} d^{3} x^{4} + 4 \, a b^{3} d^{3} x^{3} + 6 \, a^{2} b^{2} d^{3} x^{2} + 4 \, a^{3} b d^{3} x + a^{4} d^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^2/(a*d/b+d*x)^3,x, algorithm="maxima")

[Out]

-1/4*b^2/(b^4*d^3*x^4 + 4*a*b^3*d^3*x^3 + 6*a^2*b^2*d^3*x^2 + 4*a^3*b*d^3*x + a^4*d^3)

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Fricas [B]  time = 1.63316, size = 119, normalized size = 7. \begin{align*} -\frac{b^{2}}{4 \,{\left (b^{4} d^{3} x^{4} + 4 \, a b^{3} d^{3} x^{3} + 6 \, a^{2} b^{2} d^{3} x^{2} + 4 \, a^{3} b d^{3} x + a^{4} d^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^2/(a*d/b+d*x)^3,x, algorithm="fricas")

[Out]

-1/4*b^2/(b^4*d^3*x^4 + 4*a*b^3*d^3*x^3 + 6*a^2*b^2*d^3*x^2 + 4*a^3*b*d^3*x + a^4*d^3)

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Sympy [B]  time = 0.474918, size = 68, normalized size = 4. \begin{align*} - \frac{b^{3}}{4 a^{4} b d^{3} + 16 a^{3} b^{2} d^{3} x + 24 a^{2} b^{3} d^{3} x^{2} + 16 a b^{4} d^{3} x^{3} + 4 b^{5} d^{3} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)**2/(a*d/b+d*x)**3,x)

[Out]

-b**3/(4*a**4*b*d**3 + 16*a**3*b**2*d**3*x + 24*a**2*b**3*d**3*x**2 + 16*a*b**4*d**3*x**3 + 4*b**5*d**3*x**4)

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Giac [A]  time = 1.0528, size = 20, normalized size = 1.18 \begin{align*} -\frac{b^{2}}{4 \,{\left (b x + a\right )}^{4} d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x+a)^2/(a*d/b+d*x)^3,x, algorithm="giac")

[Out]

-1/4*b^2/((b*x + a)^4*d^3)