3.1012 \(\int \frac{1}{(\frac{b c}{d}+b x) (c+d x)^3} \, dx\)

Optimal. Leaf size=14 \[ -\frac{1}{3 b (c+d x)^3} \]

[Out]

-1/(3*b*(c + d*x)^3)

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Rubi [A]  time = 0.0029554, antiderivative size = 14, 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} \[ -\frac{1}{3 b (c+d x)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(3*b*(c + d*x)^3)

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

Mathematica [A]  time = 0.0052916, size = 14, normalized size = 1. \[ -\frac{1}{3 b (c+d x)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(3*b*(c + d*x)^3)

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Maple [A]  time = 0.001, size = 13, normalized size = 0.9 \begin{align*} -{\frac{1}{3\,b \left ( dx+c \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.00733, size = 49, normalized size = 3.5 \begin{align*} -\frac{1}{3 \,{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.44479, size = 76, normalized size = 5.43 \begin{align*} -\frac{1}{3 \,{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.399423, size = 44, normalized size = 3.14 \begin{align*} - \frac{d}{3 b c^{3} d + 9 b c^{2} d^{2} x + 9 b c d^{3} x^{2} + 3 b d^{4} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-d/(3*b*c**3*d + 9*b*c**2*d**2*x + 9*b*c*d**3*x**2 + 3*b*d**4*x**3)

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Giac [A]  time = 1.0682, size = 16, normalized size = 1.14 \begin{align*} -\frac{1}{3 \,{\left (d x + c\right )}^{3} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3/((d*x + c)^3*b)