3.1013 \(\int \frac{1}{(\frac{b c}{d}+b x)^2 (c+d x)^3} \, dx\)

Optimal. Leaf size=15 \[ -\frac{d}{4 b^2 (c+d x)^4} \]

[Out]

-d/(4*b^2*(c + d*x)^4)

________________________________________________________________________________________

Rubi [A]  time = 0.0032222, antiderivative size = 15, 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} \[ -\frac{d}{4 b^2 (c+d x)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0053454, size = 15, normalized size = 1. \[ -\frac{d}{4 b^2 (c+d x)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-d/(4*b^2*(c + d*x)^4)

________________________________________________________________________________________

Maple [A]  time = 0.001, size = 14, normalized size = 0.9 \begin{align*} -{\frac{d}{4\,{b}^{2} \left ( dx+c \right ) ^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 1.00136, size = 80, normalized size = 5.33 \begin{align*} -\frac{d}{4 \,{\left (b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} x^{2} + 4 \, b^{2} c^{3} d x + b^{2} c^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [B]  time = 1.44963, size = 116, normalized size = 7.73 \begin{align*} -\frac{d}{4 \,{\left (b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} x^{2} + 4 \, b^{2} c^{3} d x + b^{2} c^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [B]  time = 0.48291, size = 68, normalized size = 4.53 \begin{align*} - \frac{d^{2}}{4 b^{2} c^{4} d + 16 b^{2} c^{3} d^{2} x + 24 b^{2} c^{2} d^{3} x^{2} + 16 b^{2} c d^{4} x^{3} + 4 b^{2} d^{5} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.06731, size = 27, normalized size = 1.8 \begin{align*} -\frac{b^{2}}{4 \,{\left (b x + \frac{b c}{d}\right )}^{4} d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*b^2/((b*x + b*c/d)^4*d^3)