\(\int \frac {a+b x}{(a c-b c x)^3} \, dx\) [1034]

Optimal result

Integrand size = 17, antiderivative size = 13 \[ \int \frac {a+b x}{(a c-b c x)^3} \, dx=\frac {x}{c^3 (a-b x)^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {34} \[ \int \frac {a+b x}{(a c-b c x)^3} \, dx=\frac {x}{c^3 (a-b x)^2} \]

[In]

Int[(a + b*x)/(a*c - b*c*x)^3,x]

[Out]

x/(c^3*(a - b*x)^2)

Rule 34

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_)), x_Symbol] :> Simp[d*x*((a + b*x)^(m + 1)/(b*(m + 2))), x] /
; FreeQ[{a, b, c, d, m}, x] && EqQ[a*d - b*c*(m + 2), 0]

Rubi steps \begin{align*} \text {integral}& = \frac {x}{c^3 (a-b x)^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {a+b x}{(a c-b c x)^3} \, dx=\frac {x}{c^3 (a-b x)^2} \]

[In]

Integrate[(a + b*x)/(a*c - b*c*x)^3,x]

[Out]

x/(c^3*(a - b*x)^2)

Maple [A] (verified)

Time = 0.16 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08

method	result	size
gosper	\(\frac {x}{c^{3} \left (-b x +a \right )^{2}}\)	\(14\)
norman	\(\frac {x}{c^{3} \left (-b x +a \right )^{2}}\)	\(14\)
risch	\(\frac {x}{c^{3} \left (-b x +a \right )^{2}}\)	\(14\)
parallelrisch	\(\frac {x}{c^{3} \left (b x -a \right )^{2}}\)	\(15\)
default	\(\frac {-\frac {1}{b \left (-b x +a \right )}+\frac {a}{b \left (-b x +a \right )^{2}}}{c^{3}}\)	\(32\)

[In]

int((b*x+a)/(-b*c*x+a*c)^3,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 30 vs. \(2 (14) = 28\).

Time = 0.21 (sec) , antiderivative size = 30, normalized size of antiderivative = 2.31 \[ \int \frac {a+b x}{(a c-b c x)^3} \, dx=\frac {x}{b^{2} c^{3} x^{2} - 2 \, a b c^{3} x + a^{2} c^{3}} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 27 vs. \(2 (10) = 20\).

Time = 0.24 (sec) , antiderivative size = 27, normalized size of antiderivative = 2.08 \[ \int \frac {a+b x}{(a c-b c x)^3} \, dx=\frac {x}{a^{2} c^{3} - 2 a b c^{3} x + b^{2} c^{3} x^{2}} \]

[In]

integrate((b*x+a)/(-b*c*x+a*c)**3,x)

[Out]

x/(a**2*c**3 - 2*a*b*c**3*x + b**2*c**3*x**2)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 30 vs. \(2 (14) = 28\).

Time = 0.23 (sec) , antiderivative size = 30, normalized size of antiderivative = 2.31 \[ \int \frac {a+b x}{(a c-b c x)^3} \, dx=\frac {x}{b^{2} c^{3} x^{2} - 2 \, a b c^{3} x + a^{2} c^{3}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08 \[ \int \frac {a+b x}{(a c-b c x)^3} \, dx=\frac {x}{{\left (b x - a\right )}^{2} c^{3}} \]

[In]

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

[Out]

x/((b*x - a)^2*c^3)

Mupad [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {a+b x}{(a c-b c x)^3} \, dx=\frac {x}{c^3\,{\left (a-b\,x\right )}^2} \]

[In]

int((a + b*x)/(a*c - b*c*x)^3,x)

[Out]

x/(c^3*(a - b*x)^2)