\(\int \frac {a+b x}{(a c-b c x)^4} \, dx\) [1035]

Optimal result

Integrand size = 17, antiderivative size = 38 \[ \int \frac {a+b x}{(a c-b c x)^4} \, dx=\frac {2 a}{3 b c^4 (a-b x)^3}-\frac {1}{2 b c^4 (a-b x)^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {45} \[ \int \frac {a+b x}{(a c-b c x)^4} \, dx=\frac {2 a}{3 b c^4 (a-b x)^3}-\frac {1}{2 b c^4 (a-b x)^2} \]

[In]

Int[(a + b*x)/(a*c - b*c*x)^4,x]

[Out]

(2*a)/(3*b*c^4*(a - b*x)^3) - 1/(2*b*c^4*(a - b*x)^2)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {2 a}{c^4 (a-b x)^4}-\frac {1}{c^4 (a-b x)^3}\right ) \, dx \\ & = \frac {2 a}{3 b c^4 (a-b x)^3}-\frac {1}{2 b c^4 (a-b x)^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.66 \[ \int \frac {a+b x}{(a c-b c x)^4} \, dx=-\frac {a+3 b x}{6 b c^4 (-a+b x)^3} \]

[In]

Integrate[(a + b*x)/(a*c - b*c*x)^4,x]

[Out]

-1/6*(a + 3*b*x)/(b*c^4*(-a + b*x)^3)

Maple [A] (verified)

Time = 0.30 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.61

method	result	size
gosper	\(\frac {3 b x +a}{6 \left (-b x +a \right )^{3} c^{4} b}\)	\(23\)
risch	\(\frac {\frac {x}{2}+\frac {a}{6 b}}{c^{4} \left (-b x +a \right )^{3}}\)	\(23\)
norman	\(\frac {\frac {a}{6 b c}+\frac {x}{2 c}}{c^{3} \left (-b x +a \right )^{3}}\)	\(29\)
parallelrisch	\(\frac {-3 b^{3} x -a \,b^{2}}{6 b^{3} c^{4} \left (b x -a \right )^{3}}\)	\(31\)
default	\(\frac {-\frac {1}{2 b \left (-b x +a \right )^{2}}+\frac {2 a}{3 b \left (-b x +a \right )^{3}}}{c^{4}}\)	\(33\)

[In]

int((b*x+a)/(-b*c*x+a*c)^4,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.42 \[ \int \frac {a+b x}{(a c-b c x)^4} \, dx=-\frac {3 \, b x + a}{6 \, {\left (b^{4} c^{4} x^{3} - 3 \, a b^{3} c^{4} x^{2} + 3 \, a^{2} b^{2} c^{4} x - a^{3} b c^{4}\right )}} \]

[In]

integrate((b*x+a)/(-b*c*x+a*c)^4,x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.38 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.47 \[ \int \frac {a+b x}{(a c-b c x)^4} \, dx=\frac {- a - 3 b x}{- 6 a^{3} b c^{4} + 18 a^{2} b^{2} c^{4} x - 18 a b^{3} c^{4} x^{2} + 6 b^{4} c^{4} x^{3}} \]

[In]

integrate((b*x+a)/(-b*c*x+a*c)**4,x)

[Out]

(-a - 3*b*x)/(-6*a**3*b*c**4 + 18*a**2*b**2*c**4*x - 18*a*b**3*c**4*x**2 + 6*b**4*c**4*x**3)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.42 \[ \int \frac {a+b x}{(a c-b c x)^4} \, dx=-\frac {3 \, b x + a}{6 \, {\left (b^{4} c^{4} x^{3} - 3 \, a b^{3} c^{4} x^{2} + 3 \, a^{2} b^{2} c^{4} x - a^{3} b c^{4}\right )}} \]

[In]

integrate((b*x+a)/(-b*c*x+a*c)^4,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.61 \[ \int \frac {a+b x}{(a c-b c x)^4} \, dx=-\frac {3 \, b x + a}{6 \, {\left (b x - a\right )}^{3} b c^{4}} \]

[In]

integrate((b*x+a)/(-b*c*x+a*c)^4,x, algorithm="giac")

[Out]

-1/6*(3*b*x + a)/((b*x - a)^3*b*c^4)

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.42 \[ \int \frac {a+b x}{(a c-b c x)^4} \, dx=\frac {\frac {x}{2}+\frac {a}{6\,b}}{a^3\,c^4-3\,a^2\,b\,c^4\,x+3\,a\,b^2\,c^4\,x^2-b^3\,c^4\,x^3} \]

[In]

int((a + b*x)/(a*c - b*c*x)^4,x)

[Out]

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