\(\int \frac {(a+b x)^5}{(a c+b c x)^8} \, dx\) [1026]

Optimal result

Integrand size = 18, antiderivative size = 17 \[ \int \frac {(a+b x)^5}{(a c+b c x)^8} \, dx=-\frac {1}{2 b c^8 (a+b x)^2} \]

[Out]

-1/2/b/c^8/(b*x+a)^2

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {21, 32} \[ \int \frac {(a+b x)^5}{(a c+b c x)^8} \, dx=-\frac {1}{2 b c^8 (a+b x)^2} \]

[In]

Int[(a + b*x)^5/(a*c + b*c*x)^8,x]

[Out]

-1/2*1/(b*c^8*(a + b*x)^2)

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 {\int \frac {1}{(a+b x)^3} \, dx}{c^8} \\ & = -\frac {1}{2 b c^8 (a+b x)^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^5}{(a c+b c x)^8} \, dx=-\frac {1}{2 b c^8 (a+b x)^2} \]

[In]

Integrate[(a + b*x)^5/(a*c + b*c*x)^8,x]

[Out]

-1/2*1/(b*c^8*(a + b*x)^2)

Maple [A] (verified)

Time = 0.16 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94

method	result	size
gosper	\(-\frac {1}{2 b \,c^{8} \left (b x +a \right )^{2}}\)	\(16\)
default	\(-\frac {1}{2 b \,c^{8} \left (b x +a \right )^{2}}\)	\(16\)
risch	\(-\frac {1}{2 b \,c^{8} \left (b x +a \right )^{2}}\)	\(16\)
parallelrisch	\(-\frac {1}{2 b \,c^{8} \left (b x +a \right )^{2}}\)	\(16\)
norman	\(\frac {-\frac {5 a^{3} b \,x^{2}}{c}-\frac {a^{5}}{2 b c}-\frac {b^{4} x^{5}}{2 c}-\frac {5 a \,b^{3} x^{4}}{2 c}-\frac {5 a^{2} b^{2} x^{3}}{c}-\frac {5 a^{4} x}{2 c}}{c^{7} \left (b x +a \right )^{7}}\)	\(82\)

[In]

int((b*x+a)^5/(b*c*x+a*c)^8,x,method=_RETURNVERBOSE)

[Out]

-1/2/b/c^8/(b*x+a)^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (15) = 30\).

Time = 0.21 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.94 \[ \int \frac {(a+b x)^5}{(a c+b c x)^8} \, dx=-\frac {1}{2 \, {\left (b^{3} c^{8} x^{2} + 2 \, a b^{2} c^{8} x + a^{2} b c^{8}\right )}} \]

[In]

integrate((b*x+a)^5/(b*c*x+a*c)^8,x, algorithm="fricas")

[Out]

-1/2/(b^3*c^8*x^2 + 2*a*b^2*c^8*x + a^2*b*c^8)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (15) = 30\).

Time = 0.31 (sec) , antiderivative size = 36, normalized size of antiderivative = 2.12 \[ \int \frac {(a+b x)^5}{(a c+b c x)^8} \, dx=- \frac {1}{2 a^{2} b c^{8} + 4 a b^{2} c^{8} x + 2 b^{3} c^{8} x^{2}} \]

[In]

integrate((b*x+a)**5/(b*c*x+a*c)**8,x)

[Out]

-1/(2*a**2*b*c**8 + 4*a*b**2*c**8*x + 2*b**3*c**8*x**2)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (15) = 30\).

Time = 0.21 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.94 \[ \int \frac {(a+b x)^5}{(a c+b c x)^8} \, dx=-\frac {1}{2 \, {\left (b^{3} c^{8} x^{2} + 2 \, a b^{2} c^{8} x + a^{2} b c^{8}\right )}} \]

[In]

integrate((b*x+a)^5/(b*c*x+a*c)^8,x, algorithm="maxima")

[Out]

-1/2/(b^3*c^8*x^2 + 2*a*b^2*c^8*x + a^2*b*c^8)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {(a+b x)^5}{(a c+b c x)^8} \, dx=-\frac {1}{2 \, {\left (b x + a\right )}^{2} b c^{8}} \]

[In]

integrate((b*x+a)^5/(b*c*x+a*c)^8,x, algorithm="giac")

[Out]

-1/2/((b*x + a)^2*b*c^8)

Mupad [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 35, normalized size of antiderivative = 2.06 \[ \int \frac {(a+b x)^5}{(a c+b c x)^8} \, dx=-\frac {1}{2\,a^2\,b\,c^8+4\,a\,b^2\,c^8\,x+2\,b^3\,c^8\,x^2} \]

[In]

int((a + b*x)^5/(a*c + b*c*x)^8,x)

[Out]

-1/(2*a^2*b*c^8 + 2*b^3*c^8*x^2 + 4*a*b^2*c^8*x)