\(\int \frac {1}{(a+b x)^7} \, dx\) [217]

Optimal result

Integrand size = 7, antiderivative size = 14 \[ \int \frac {1}{(a+b x)^7} \, dx=-\frac {1}{6 b (a+b x)^6} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {32} \[ \int \frac {1}{(a+b x)^7} \, dx=-\frac {1}{6 b (a+b x)^6} \]

[In]

Int[(a + b*x)^(-7),x]

[Out]

-1/6*1/(b*(a + b*x)^6)

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 {1}{6 b (a+b x)^6} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(a+b x)^7} \, dx=-\frac {1}{6 b (a+b x)^6} \]

[In]

Integrate[(a + b*x)^(-7),x]

[Out]

-1/6*1/(b*(a + b*x)^6)

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93

method	result	size
gosper	\(-\frac {1}{6 b \left (b x +a \right )^{6}}\)	\(13\)
default	\(-\frac {1}{6 b \left (b x +a \right )^{6}}\)	\(13\)
norman	\(-\frac {1}{6 b \left (b x +a \right )^{6}}\)	\(13\)
risch	\(-\frac {1}{6 b \left (b x +a \right )^{6}}\)	\(13\)
parallelrisch	\(-\frac {1}{6 b \left (b x +a \right )^{6}}\)	\(13\)

[In]

int(1/(b*x+a)^7,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (12) = 24\).

Time = 0.22 (sec) , antiderivative size = 68, normalized size of antiderivative = 4.86 \[ \int \frac {1}{(a+b x)^7} \, dx=-\frac {1}{6 \, {\left (b^{7} x^{6} + 6 \, a b^{6} x^{5} + 15 \, a^{2} b^{5} x^{4} + 20 \, a^{3} b^{4} x^{3} + 15 \, a^{4} b^{3} x^{2} + 6 \, a^{5} b^{2} x + a^{6} b\right )}} \]

[In]

integrate(1/(b*x+a)^7,x, algorithm="fricas")

[Out]

-1/6/(b^7*x^6 + 6*a*b^6*x^5 + 15*a^2*b^5*x^4 + 20*a^3*b^4*x^3 + 15*a^4*b^3*x^2 + 6*a^5*b^2*x + a^6*b)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (12) = 24\).

Time = 0.21 (sec) , antiderivative size = 73, normalized size of antiderivative = 5.21 \[ \int \frac {1}{(a+b x)^7} \, dx=- \frac {1}{6 a^{6} b + 36 a^{5} b^{2} x + 90 a^{4} b^{3} x^{2} + 120 a^{3} b^{4} x^{3} + 90 a^{2} b^{5} x^{4} + 36 a b^{6} x^{5} + 6 b^{7} x^{6}} \]

[In]

integrate(1/(b*x+a)**7,x)

[Out]

-1/(6*a**6*b + 36*a**5*b**2*x + 90*a**4*b**3*x**2 + 120*a**3*b**4*x**3 + 90*a**2*b**5*x**4 + 36*a*b**6*x**5 +
6*b**7*x**6)

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {1}{(a+b x)^7} \, dx=-\frac {1}{6 \, {\left (b x + a\right )}^{6} b} \]

[In]

integrate(1/(b*x+a)^7,x, algorithm="maxima")

[Out]

-1/6/((b*x + a)^6*b)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {1}{(a+b x)^7} \, dx=-\frac {1}{6 \, {\left (b x + a\right )}^{6} b} \]

[In]

integrate(1/(b*x+a)^7,x, algorithm="giac")

[Out]

-1/6/((b*x + a)^6*b)

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 70, normalized size of antiderivative = 5.00 \[ \int \frac {1}{(a+b x)^7} \, dx=-\frac {1}{6\,a^6\,b+36\,a^5\,b^2\,x+90\,a^4\,b^3\,x^2+120\,a^3\,b^4\,x^3+90\,a^2\,b^5\,x^4+36\,a\,b^6\,x^5+6\,b^7\,x^6} \]

[In]

int(1/(a + b*x)^7,x)

[Out]

-1/(6*a^6*b + 6*b^7*x^6 + 36*a^5*b^2*x + 36*a*b^6*x^5 + 90*a^4*b^3*x^2 + 120*a^3*b^4*x^3 + 90*a^2*b^5*x^4)