3.1006 \(\int \frac{1}{(a+b x)^3 (\frac{a d}{b}+d x)^3} \, dx\)

Optimal. Leaf size=17 \[ -\frac{b^2}{5 d^3 (a+b x)^5} \]

[Out]

-b^2/(5*d^3*(a + b*x)^5)

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Rubi [A]  time = 0.003293, 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} \[ -\frac{b^2}{5 d^3 (a+b x)^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-b^2/(5*d^3*(a + b*x)^5)

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)^3 \left (\frac{a d}{b}+d x\right )^3} \, dx &=\frac{b^3 \int \frac{1}{(a+b x)^6} \, dx}{d^3}\\ &=-\frac{b^2}{5 d^3 (a+b x)^5}\\ \end{align*}

Mathematica [A]  time = 0.0064176, size = 17, normalized size = 1. \[ -\frac{b^2}{5 d^3 (a+b x)^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-b^2/(5*d^3*(a + b*x)^5)

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Maple [A]  time = 0.002, size = 16, normalized size = 0.9 \begin{align*} -{\frac{{b}^{2}}{5\,{d}^{3} \left ( bx+a \right ) ^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*b^2/d^3/(b*x+a)^5

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Maxima [B]  time = 1.02923, size = 101, normalized size = 5.94 \begin{align*} -\frac{b^{2}}{5 \,{\left (b^{5} d^{3} x^{5} + 5 \, a b^{4} d^{3} x^{4} + 10 \, a^{2} b^{3} d^{3} x^{3} + 10 \, a^{3} b^{2} d^{3} x^{2} + 5 \, a^{4} b d^{3} x + a^{5} d^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*b^2/(b^5*d^3*x^5 + 5*a*b^4*d^3*x^4 + 10*a^2*b^3*d^3*x^3 + 10*a^3*b^2*d^3*x^2 + 5*a^4*b*d^3*x + a^5*d^3)

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Fricas [B]  time = 1.69989, size = 149, normalized size = 8.76 \begin{align*} -\frac{b^{2}}{5 \,{\left (b^{5} d^{3} x^{5} + 5 \, a b^{4} d^{3} x^{4} + 10 \, a^{2} b^{3} d^{3} x^{3} + 10 \, a^{3} b^{2} d^{3} x^{2} + 5 \, a^{4} b d^{3} x + a^{5} d^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*b^2/(b^5*d^3*x^5 + 5*a*b^4*d^3*x^4 + 10*a^2*b^3*d^3*x^3 + 10*a^3*b^2*d^3*x^2 + 5*a^4*b*d^3*x + a^5*d^3)

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Sympy [B]  time = 0.546246, size = 83, normalized size = 4.88 \begin{align*} - \frac{b^{3}}{5 a^{5} b d^{3} + 25 a^{4} b^{2} d^{3} x + 50 a^{3} b^{3} d^{3} x^{2} + 50 a^{2} b^{4} d^{3} x^{3} + 25 a b^{5} d^{3} x^{4} + 5 b^{6} d^{3} x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b**3/(5*a**5*b*d**3 + 25*a**4*b**2*d**3*x + 50*a**3*b**3*d**3*x**2 + 50*a**2*b**4*d**3*x**3 + 25*a*b**5*d**3*
x**4 + 5*b**6*d**3*x**5)

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Giac [A]  time = 1.05078, size = 20, normalized size = 1.18 \begin{align*} -\frac{b^{2}}{5 \,{\left (b x + a\right )}^{5} d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*b^2/((b*x + a)^5*d^3)