\(\int \frac {(a+b x)^5}{(a c+b c x)^7} \, dx\) [1025]

Optimal result

Integrand size = 18, antiderivative size = 15 \[ \int \frac {(a+b x)^5}{(a c+b c x)^7} \, dx=-\frac {1}{b c^7 (a+b x)} \]

[Out]

-1/b/c^7/(b*x+a)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 15, 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} \[ \int \frac {(a+b x)^5}{(a c+b c x)^7} \, dx=-\frac {1}{b c^7 (a+b x)} \]

[In]

Int[(a + b*x)^5/(a*c + b*c*x)^7,x]

[Out]

-(1/(b*c^7*(a + b*x)))

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)^2} \, dx}{c^7} \\ & = -\frac {1}{b c^7 (a+b x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^5}{(a c+b c x)^7} \, dx=-\frac {1}{b c^7 (a+b x)} \]

[In]

Integrate[(a + b*x)^5/(a*c + b*c*x)^7,x]

[Out]

-(1/(b*c^7*(a + b*x)))

Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.07

method	result	size
gosper	\(-\frac {1}{b \,c^{7} \left (b x +a \right )}\)	\(16\)
default	\(-\frac {1}{b \,c^{7} \left (b x +a \right )}\)	\(16\)
risch	\(-\frac {1}{b \,c^{7} \left (b x +a \right )}\)	\(16\)
parallelrisch	\(\frac {x}{a \,c^{7} \left (b x +a \right )}\)	\(16\)
norman	\(\frac {-\frac {a^{5}}{b c}-\frac {b^{4} x^{5}}{c}-\frac {5 a^{4} x}{c}-\frac {5 a \,b^{3} x^{4}}{c}-\frac {10 a^{2} b^{2} x^{3}}{c}-\frac {10 a^{3} b \,x^{2}}{c}}{c^{6} \left (b x +a \right )^{6}}\)	\(82\)

[In]

int((b*x+a)^5/(b*c*x+a*c)^7,x,method=_RETURNVERBOSE)

[Out]

-1/b/c^7/(b*x+a)

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.27 \[ \int \frac {(a+b x)^5}{(a c+b c x)^7} \, dx=-\frac {1}{b^{2} c^{7} x + a b c^{7}} \]

[In]

integrate((b*x+a)^5/(b*c*x+a*c)^7,x, algorithm="fricas")

[Out]

-1/(b^2*c^7*x + a*b*c^7)

Sympy [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13 \[ \int \frac {(a+b x)^5}{(a c+b c x)^7} \, dx=- \frac {1}{a b c^{7} + b^{2} c^{7} x} \]

[In]

integrate((b*x+a)**5/(b*c*x+a*c)**7,x)

[Out]

-1/(a*b*c**7 + b**2*c**7*x)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.27 \[ \int \frac {(a+b x)^5}{(a c+b c x)^7} \, dx=-\frac {1}{b^{2} c^{7} x + a b c^{7}} \]

[In]

integrate((b*x+a)^5/(b*c*x+a*c)^7,x, algorithm="maxima")

[Out]

-1/(b^2*c^7*x + a*b*c^7)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^5}{(a c+b c x)^7} \, dx=-\frac {1}{{\left (b x + a\right )} b c^{7}} \]

[In]

integrate((b*x+a)^5/(b*c*x+a*c)^7,x, algorithm="giac")

[Out]

-1/((b*x + a)*b*c^7)

Mupad [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.27 \[ \int \frac {(a+b x)^5}{(a c+b c x)^7} \, dx=-\frac {1}{x\,b^2\,c^7+a\,b\,c^7} \]

[In]

int((a + b*x)^5/(a*c + b*c*x)^7,x)

[Out]

-1/(b^2*c^7*x + a*b*c^7)