\(\int \frac {a+b x}{(a c-b c x)^6} \, dx\) [1037]

Optimal result

Integrand size = 17, antiderivative size = 38 \[ \int \frac {a+b x}{(a c-b c x)^6} \, dx=\frac {2 a}{5 b c^6 (a-b x)^5}-\frac {1}{4 b c^6 (a-b x)^4} \]

[Out]

2/5*a/b/c^6/(-b*x+a)^5-1/4/b/c^6/(-b*x+a)^4

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} \[ \int \frac {a+b x}{(a c-b c x)^6} \, dx=\frac {2 a}{5 b c^6 (a-b x)^5}-\frac {1}{4 b c^6 (a-b x)^4} \]

[In]

Int[(a + b*x)/(a*c - b*c*x)^6,x]

[Out]

(2*a)/(5*b*c^6*(a - b*x)^5) - 1/(4*b*c^6*(a - b*x)^4)

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^6 (a-b x)^6}-\frac {1}{c^6 (a-b x)^5}\right ) \, dx \\ & = \frac {2 a}{5 b c^6 (a-b x)^5}-\frac {1}{4 b c^6 (a-b x)^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.71 \[ \int \frac {a+b x}{(a c-b c x)^6} \, dx=-\frac {3 a+5 b x}{20 b c^6 (-a+b x)^5} \]

[In]

Integrate[(a + b*x)/(a*c - b*c*x)^6,x]

[Out]

-1/20*(3*a + 5*b*x)/(b*c^6*(-a + b*x)^5)

Maple [A] (verified)

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

method	result	size
risch	\(\frac {\frac {x}{4}+\frac {3 a}{20 b}}{c^{6} \left (-b x +a \right )^{5}}\)	\(23\)
gosper	\(\frac {5 b x +3 a}{20 \left (-b x +a \right )^{5} c^{6} b}\)	\(25\)
norman	\(\frac {\frac {3 a}{20 b c}+\frac {x}{4 c}}{c^{5} \left (-b x +a \right )^{5}}\)	\(29\)
parallelrisch	\(\frac {-5 b^{5} x -3 a \,b^{4}}{20 b^{5} c^{6} \left (b x -a \right )^{5}}\)	\(31\)
default	\(\frac {-\frac {1}{4 b \left (-b x +a \right )^{4}}+\frac {2 a}{5 b \left (-b x +a \right )^{5}}}{c^{6}}\)	\(33\)

[In]

int((b*x+a)/(-b*c*x+a*c)^6,x,method=_RETURNVERBOSE)

[Out]

(1/4*x+3/20*a/b)/c^6/(-b*x+a)^5

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (36) = 72\).

Time = 0.21 (sec) , antiderivative size = 84, normalized size of antiderivative = 2.21 \[ \int \frac {a+b x}{(a c-b c x)^6} \, dx=-\frac {5 \, b x + 3 \, a}{20 \, {\left (b^{6} c^{6} x^{5} - 5 \, a b^{5} c^{6} x^{4} + 10 \, a^{2} b^{4} c^{6} x^{3} - 10 \, a^{3} b^{3} c^{6} x^{2} + 5 \, a^{4} b^{2} c^{6} x - a^{5} b c^{6}\right )}} \]

[In]

integrate((b*x+a)/(-b*c*x+a*c)^6,x, algorithm="fricas")

[Out]

-1/20*(5*b*x + 3*a)/(b^6*c^6*x^5 - 5*a*b^5*c^6*x^4 + 10*a^2*b^4*c^6*x^3 - 10*a^3*b^3*c^6*x^2 + 5*a^4*b^2*c^6*x
- a^5*b*c^6)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (31) = 62\).

Time = 0.25 (sec) , antiderivative size = 88, normalized size of antiderivative = 2.32 \[ \int \frac {a+b x}{(a c-b c x)^6} \, dx=\frac {- 3 a - 5 b x}{- 20 a^{5} b c^{6} + 100 a^{4} b^{2} c^{6} x - 200 a^{3} b^{3} c^{6} x^{2} + 200 a^{2} b^{4} c^{6} x^{3} - 100 a b^{5} c^{6} x^{4} + 20 b^{6} c^{6} x^{5}} \]

[In]

integrate((b*x+a)/(-b*c*x+a*c)**6,x)

[Out]

(-3*a - 5*b*x)/(-20*a**5*b*c**6 + 100*a**4*b**2*c**6*x - 200*a**3*b**3*c**6*x**2 + 200*a**2*b**4*c**6*x**3 - 1
00*a*b**5*c**6*x**4 + 20*b**6*c**6*x**5)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (36) = 72\).

Time = 0.21 (sec) , antiderivative size = 84, normalized size of antiderivative = 2.21 \[ \int \frac {a+b x}{(a c-b c x)^6} \, dx=-\frac {5 \, b x + 3 \, a}{20 \, {\left (b^{6} c^{6} x^{5} - 5 \, a b^{5} c^{6} x^{4} + 10 \, a^{2} b^{4} c^{6} x^{3} - 10 \, a^{3} b^{3} c^{6} x^{2} + 5 \, a^{4} b^{2} c^{6} x - a^{5} b c^{6}\right )}} \]

[In]

integrate((b*x+a)/(-b*c*x+a*c)^6,x, algorithm="maxima")

[Out]

-1/20*(5*b*x + 3*a)/(b^6*c^6*x^5 - 5*a*b^5*c^6*x^4 + 10*a^2*b^4*c^6*x^3 - 10*a^3*b^3*c^6*x^2 + 5*a^4*b^2*c^6*x
- a^5*b*c^6)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.66 \[ \int \frac {a+b x}{(a c-b c x)^6} \, dx=-\frac {5 \, b x + 3 \, a}{20 \, {\left (b x - a\right )}^{5} b c^{6}} \]

[In]

integrate((b*x+a)/(-b*c*x+a*c)^6,x, algorithm="giac")

[Out]

-1/20*(5*b*x + 3*a)/((b*x - a)^5*b*c^6)

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.16 \[ \int \frac {a+b x}{(a c-b c x)^6} \, dx=\frac {\frac {x}{4}+\frac {3\,a}{20\,b}}{a^5\,c^6-5\,a^4\,b\,c^6\,x+10\,a^3\,b^2\,c^6\,x^2-10\,a^2\,b^3\,c^6\,x^3+5\,a\,b^4\,c^6\,x^4-b^5\,c^6\,x^5} \]

[In]

int((a + b*x)/(a*c - b*c*x)^6,x)

[Out]

(x/4 + (3*a)/(20*b))/(a^5*c^6 - b^5*c^6*x^5 + 5*a*b^4*c^6*x^4 + 10*a^3*b^2*c^6*x^2 - 10*a^2*b^3*c^6*x^3 - 5*a^
4*b*c^6*x)