3.1014 \(\int \frac{1}{(\frac{b c}{d}+b x)^3 (c+d x)^3} \, dx\)

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

[Out]

-d^2/(5*b^3*(c + d*x)^5)

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Rubi [A]  time = 0.0034984, 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} \[ -\frac{d^2}{5 b^3 (c+d x)^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0057995, size = 17, normalized size = 1. \[ -\frac{d^2}{5 b^3 (c+d x)^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-d^2/(5*b^3*(c + d*x)^5)

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Maple [A]  time = 0., size = 16, normalized size = 0.9 \begin{align*} -{\frac{{d}^{2}}{5\,{b}^{3} \left ( dx+c \right ) ^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.05315, size = 101, normalized size = 5.94 \begin{align*} -\frac{d^{2}}{5 \,{\left (b^{3} d^{5} x^{5} + 5 \, b^{3} c d^{4} x^{4} + 10 \, b^{3} c^{2} d^{3} x^{3} + 10 \, b^{3} c^{3} d^{2} x^{2} + 5 \, b^{3} c^{4} d x + b^{3} c^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.39587, size = 149, normalized size = 8.76 \begin{align*} -\frac{d^{2}}{5 \,{\left (b^{3} d^{5} x^{5} + 5 \, b^{3} c d^{4} x^{4} + 10 \, b^{3} c^{2} d^{3} x^{3} + 10 \, b^{3} c^{3} d^{2} x^{2} + 5 \, b^{3} c^{4} d x + b^{3} c^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.571301, size = 83, normalized size = 4.88 \begin{align*} - \frac{d^{3}}{5 b^{3} c^{5} d + 25 b^{3} c^{4} d^{2} x + 50 b^{3} c^{3} d^{3} x^{2} + 50 b^{3} c^{2} d^{4} x^{3} + 25 b^{3} c d^{5} x^{4} + 5 b^{3} d^{6} x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.05843, size = 20, normalized size = 1.18 \begin{align*} -\frac{d^{2}}{5 \,{\left (d x + c\right )}^{5} b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5*d^2/((d*x + c)^5*b^3)