\(\int \frac {1}{(\frac {b c}{d}+b x) (c+d x)^3} \, dx\) [1012]

Optimal result

Integrand size = 20, antiderivative size = 14 \[ \int \frac {1}{\left (\frac {b c}{d}+b x\right ) (c+d x)^3} \, dx=-\frac {1}{3 b (c+d x)^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {21, 32} \[ \int \frac {1}{\left (\frac {b c}{d}+b x\right ) (c+d x)^3} \, dx=-\frac {1}{3 b (c+d x)^3} \]

[In]

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

[Out]

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (\frac {b c}{d}+b x\right ) (c+d x)^3} \, dx=-\frac {1}{3 b (c+d x)^3} \]

[In]

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

[Out]

-1/3*1/(b*(c + d*x)^3)

Maple [A] (verified)

Time = 0.16 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93

method	result	size
gosper	\(-\frac {1}{3 b \left (d x +c \right )^{3}}\)	\(13\)
default	\(-\frac {1}{3 b \left (d x +c \right )^{3}}\)	\(13\)
norman	\(-\frac {1}{3 b \left (d x +c \right )^{3}}\)	\(13\)
risch	\(-\frac {1}{3 b \left (d x +c \right )^{3}}\)	\(13\)
parallelrisch	\(-\frac {1}{3 b \left (d x +c \right )^{3}}\)	\(13\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (12) = 24\).

Time = 0.21 (sec) , antiderivative size = 36, normalized size of antiderivative = 2.57 \[ \int \frac {1}{\left (\frac {b c}{d}+b x\right ) (c+d x)^3} \, dx=-\frac {1}{3 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )}} \]

[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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (12) = 24\).

Time = 0.40 (sec) , antiderivative size = 44, normalized size of antiderivative = 3.14 \[ \int \frac {1}{\left (\frac {b c}{d}+b x\right ) (c+d x)^3} \, dx=- \frac {d}{3 b c^{3} d + 9 b c^{2} d^{2} x + 9 b c d^{3} x^{2} + 3 b d^{4} x^{3}} \]

[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)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (12) = 24\).

Time = 0.20 (sec) , antiderivative size = 36, normalized size of antiderivative = 2.57 \[ \int \frac {1}{\left (\frac {b c}{d}+b x\right ) (c+d x)^3} \, dx=-\frac {1}{3 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )}} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {1}{\left (\frac {b c}{d}+b x\right ) (c+d x)^3} \, dx=-\frac {1}{3 \, {\left (d x + c\right )}^{3} b} \]

[In]

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

[Out]

-1/3/((d*x + c)^3*b)

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 38, normalized size of antiderivative = 2.71 \[ \int \frac {1}{\left (\frac {b c}{d}+b x\right ) (c+d x)^3} \, dx=-\frac {1}{3\,b\,c^3+9\,b\,c^2\,d\,x+9\,b\,c\,d^2\,x^2+3\,b\,d^3\,x^3} \]

[In]

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

[Out]

-1/(3*b*c^3 + 3*b*d^3*x^3 + 9*b*c^2*d*x + 9*b*c*d^2*x^2)