\(\int x^{-m} (-a-b x)^{-n} (a+b x)^n \, dx\) [827]
Optimal result
Integrand size = 25, antiderivative size = 34 \[
\int x^{-m} (-a-b x)^{-n} (a+b x)^n \, dx=\frac {x^{1-m} (-a-b x)^{-n} (a+b x)^n}{1-m}
\]
[Out]
x^(1-m)*(b*x+a)^n/(1-m)/((-b*x-a)^n)
Rubi [A] (verified)
Time = 0.00 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00,
number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {23, 30}
\[
\int x^{-m} (-a-b x)^{-n} (a+b x)^n \, dx=\frac {x^{1-m} (-a-b x)^{-n} (a+b x)^n}{1-m}
\]
[In]
Int[(a + b*x)^n/(x^m*(-a - b*x)^n),x]
[Out]
(x^(1 - m)*(a + b*x)^n)/((1 - m)*(-a - b*x)^n)
Rule 23
Int[(u_.)*((a_) + (b_.)*(v_))^(m_)*((c_) + (d_.)*(v_))^(n_), x_Symbol] :> Dist[(a + b*v)^m/(c + d*v)^m, Int[u*
(c + d*v)^(m + n), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[b*c - a*d, 0] && !(IntegerQ[m] || IntegerQ[n
] || GtQ[b/d, 0])
Rule 30
Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]
Rubi steps \begin{align*}
\text {integral}& = \left ((-a-b x)^{-n} (a+b x)^n\right ) \int x^{-m} \, dx \\ & = \frac {x^{1-m} (-a-b x)^{-n} (a+b x)^n}{1-m} \\
\end{align*}
Mathematica [A] (verified)
Time = 0.00 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00
\[
\int x^{-m} (-a-b x)^{-n} (a+b x)^n \, dx=\frac {x^{1-m} (-a-b x)^{-n} (a+b x)^n}{1-m}
\]
[In]
Integrate[(a + b*x)^n/(x^m*(-a - b*x)^n),x]
[Out]
(x^(1 - m)*(a + b*x)^n)/((1 - m)*(-a - b*x)^n)
Maple [A] (verified)
Time = 1.94 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.97
| | |
| method | result | size |
| | |
| gosper |
\(-\frac {x \left (b x +a \right )^{n} x^{-m} \left (-b x -a \right )^{-n}}{-1+m}\) |
\(33\) |
| parallelrisch |
\(-\frac {x \left (b x +a \right )^{n} x^{-m} \left (-b x -a \right )^{-n}}{-1+m}\) |
\(33\) |
| risch |
\(-\frac {x \,x^{-m} {\mathrm e}^{-i n \pi \left (\operatorname {csgn}\left (i \left (b x +a \right )\right )^{3}-\operatorname {csgn}\left (i \left (b x +a \right )\right )^{2}+1\right )}}{-1+m}\) |
\(46\) |
| | |
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[In]
int((b*x+a)^n/(x^m)/((-b*x-a)^n),x,method=_RETURNVERBOSE)
[Out]
-x/(-1+m)*(b*x+a)^n/(x^m)/((-b*x-a)^n)
Fricas [C] (verification not implemented)
Result contains complex when optimal does not.
Time = 0.24 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.53
\[
\int x^{-m} (-a-b x)^{-n} (a+b x)^n \, dx=-\frac {x e^{\left (i \, \pi n\right )}}{{\left (m - 1\right )} x^{m}}
\]
[In]
integrate((b*x+a)^n/(x^m)/((-b*x-a)^n),x, algorithm="fricas")
[Out]
-x*e^(I*pi*n)/((m - 1)*x^m)
Sympy [C] (verification not implemented)
Result contains complex when optimal does not.
Time = 37.81 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.35
\[
\int x^{-m} (-a-b x)^{-n} (a+b x)^n \, dx=\begin {cases} - \frac {x \left (a + b x\right )^{n}}{m x^{m} \left (- a - b x\right )^{n} - x^{m} \left (- a - b x\right )^{n}} & \text {for}\: m \neq 1 \\\begin {cases} e^{- i \pi n} \log {\left (-1 + \frac {b \left (\frac {a}{b} + x\right )}{a} \right )} & \text {for}\: \left |{\frac {b \left (\frac {a}{b} + x\right )}{a}}\right | > 1 \\e^{- i \pi n} \log {\left (1 - \frac {b \left (\frac {a}{b} + x\right )}{a} \right )} & \text {otherwise} \end {cases} & \text {otherwise} \end {cases}
\]
[In]
integrate((b*x+a)**n/(x**m)/((-b*x-a)**n),x)
[Out]
Piecewise((-x*(a + b*x)**n/(m*x**m*(-a - b*x)**n - x**m*(-a - b*x)**n), Ne(m, 1)), (Piecewise((exp(-I*pi*n)*lo
g(-1 + b*(a/b + x)/a), Abs(b*(a/b + x)/a) > 1), (exp(-I*pi*n)*log(1 - b*(a/b + x)/a), True)), True))
Maxima [A] (verification not implemented)
none
Time = 0.21 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.62
\[
\int x^{-m} (-a-b x)^{-n} (a+b x)^n \, dx=-\frac {x}{{\left (\left (-1\right )^{n} m - \left (-1\right )^{n}\right )} x^{m}}
\]
[In]
integrate((b*x+a)^n/(x^m)/((-b*x-a)^n),x, algorithm="maxima")
[Out]
-x/(((-1)^n*m - (-1)^n)*x^m)
Giac [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 1586 vs. \(2 (33) = 66\).
Time = 0.41 (sec) , antiderivative size = 1586, normalized size of antiderivative = 46.65
\[
\int x^{-m} (-a-b x)^{-n} (a+b x)^n \, dx=\text {Too large to display}
\]
[In]
integrate((b*x+a)^n/(x^m)/((-b*x-a)^n),x, algorithm="giac")
[Out]
-(pi*b*x*abs(b*x + a)^(-n)*abs(b*x + a)^n*sgn(b*x + a)*tan(1/4*pi*n*sgn(b*x + a) - 1/4*pi*n)^2*tan(1/4*pi*n*sg
n(b*x + a) - 3/4*pi*n) - pi*b*x*abs(b*x + a)^(-n)*abs(b*x + a)^n*sgn(b*x + a)*tan(1/4*pi*n*sgn(b*x + a) - 1/4*
pi*n)*tan(1/4*pi*n*sgn(b*x + a) - 3/4*pi*n)^2 - b*x*abs(b*x + a)^(-n)*abs(b*x + a)^n*log(abs(b*x + a))*tan(1/4
*pi*n*sgn(b*x + a) - 1/4*pi*n)^2*tan(1/4*pi*n*sgn(b*x + a) - 3/4*pi*n)^2 - pi*b*x*abs(b*x + a)^(-n)*abs(b*x +
a)^n*tan(1/4*pi*n*sgn(b*x + a) - 1/4*pi*n)^2*tan(1/4*pi*n*sgn(b*x + a) - 3/4*pi*n) + pi*a*abs(b*x + a)^(-n)*ab
s(b*x + a)^n*sgn(b*x + a)*tan(1/4*pi*n*sgn(b*x + a) - 1/4*pi*n)^2*tan(1/4*pi*n*sgn(b*x + a) - 3/4*pi*n) + pi*b
*x*abs(b*x + a)^(-n)*abs(b*x + a)^n*tan(1/4*pi*n*sgn(b*x + a) - 1/4*pi*n)*tan(1/4*pi*n*sgn(b*x + a) - 3/4*pi*n
)^2 - pi*a*abs(b*x + a)^(-n)*abs(b*x + a)^n*sgn(b*x + a)*tan(1/4*pi*n*sgn(b*x + a) - 1/4*pi*n)*tan(1/4*pi*n*sg
n(b*x + a) - 3/4*pi*n)^2 + b*x*abs(b*x + a)^(-n)*abs(b*x + a)^n*tan(1/4*pi*n*sgn(b*x + a) - 1/4*pi*n)^2*tan(1/
4*pi*n*sgn(b*x + a) - 3/4*pi*n)^2 - a*abs(b*x + a)^(-n)*abs(b*x + a)^n*log(abs(b*x + a))*tan(1/4*pi*n*sgn(b*x
+ a) - 1/4*pi*n)^2*tan(1/4*pi*n*sgn(b*x + a) - 3/4*pi*n)^2 + pi*b*x*abs(b*x + a)^(-n)*abs(b*x + a)^n*sgn(b*x +
a)*tan(1/4*pi*n*sgn(b*x + a) - 1/4*pi*n) + b*x*abs(b*x + a)^(-n)*abs(b*x + a)^n*log(abs(b*x + a))*tan(1/4*pi*
n*sgn(b*x + a) - 1/4*pi*n)^2 - pi*b*x*abs(b*x + a)^(-n)*abs(b*x + a)^n*sgn(b*x + a)*tan(1/4*pi*n*sgn(b*x + a)
- 3/4*pi*n) - 4*b*x*abs(b*x + a)^(-n)*abs(b*x + a)^n*log(abs(b*x + a))*tan(1/4*pi*n*sgn(b*x + a) - 1/4*pi*n)*t
an(1/4*pi*n*sgn(b*x + a) - 3/4*pi*n) - pi*a*abs(b*x + a)^(-n)*abs(b*x + a)^n*tan(1/4*pi*n*sgn(b*x + a) - 1/4*p
i*n)^2*tan(1/4*pi*n*sgn(b*x + a) - 3/4*pi*n) + b*x*abs(b*x + a)^(-n)*abs(b*x + a)^n*log(abs(b*x + a))*tan(1/4*
pi*n*sgn(b*x + a) - 3/4*pi*n)^2 + pi*a*abs(b*x + a)^(-n)*abs(b*x + a)^n*tan(1/4*pi*n*sgn(b*x + a) - 1/4*pi*n)*
tan(1/4*pi*n*sgn(b*x + a) - 3/4*pi*n)^2 - pi*b*x*abs(b*x + a)^(-n)*abs(b*x + a)^n*tan(1/4*pi*n*sgn(b*x + a) -
1/4*pi*n) + pi*a*abs(b*x + a)^(-n)*abs(b*x + a)^n*sgn(b*x + a)*tan(1/4*pi*n*sgn(b*x + a) - 1/4*pi*n) - b*x*abs
(b*x + a)^(-n)*abs(b*x + a)^n*tan(1/4*pi*n*sgn(b*x + a) - 1/4*pi*n)^2 + a*abs(b*x + a)^(-n)*abs(b*x + a)^n*log
(abs(b*x + a))*tan(1/4*pi*n*sgn(b*x + a) - 1/4*pi*n)^2 + pi*b*x*abs(b*x + a)^(-n)*abs(b*x + a)^n*tan(1/4*pi*n*
sgn(b*x + a) - 3/4*pi*n) - pi*a*abs(b*x + a)^(-n)*abs(b*x + a)^n*sgn(b*x + a)*tan(1/4*pi*n*sgn(b*x + a) - 3/4*
pi*n) + 4*b*x*abs(b*x + a)^(-n)*abs(b*x + a)^n*tan(1/4*pi*n*sgn(b*x + a) - 1/4*pi*n)*tan(1/4*pi*n*sgn(b*x + a)
- 3/4*pi*n) - 4*a*abs(b*x + a)^(-n)*abs(b*x + a)^n*log(abs(b*x + a))*tan(1/4*pi*n*sgn(b*x + a) - 1/4*pi*n)*ta
n(1/4*pi*n*sgn(b*x + a) - 3/4*pi*n) - b*x*abs(b*x + a)^(-n)*abs(b*x + a)^n*tan(1/4*pi*n*sgn(b*x + a) - 3/4*pi*
n)^2 + a*abs(b*x + a)^(-n)*abs(b*x + a)^n*log(abs(b*x + a))*tan(1/4*pi*n*sgn(b*x + a) - 3/4*pi*n)^2 - b*x*abs(
b*x + a)^(-n)*abs(b*x + a)^n*log(abs(b*x + a)) - pi*a*abs(b*x + a)^(-n)*abs(b*x + a)^n*tan(1/4*pi*n*sgn(b*x +
a) - 1/4*pi*n) + pi*a*abs(b*x + a)^(-n)*abs(b*x + a)^n*tan(1/4*pi*n*sgn(b*x + a) - 3/4*pi*n) + b*x*abs(b*x + a
)^(-n)*abs(b*x + a)^n - a*abs(b*x + a)^(-n)*abs(b*x + a)^n*log(abs(b*x + a)))/(b*tan(1/4*pi*n*sgn(b*x + a) - 1
/4*pi*n)^2*tan(1/4*pi*n*sgn(b*x + a) - 3/4*pi*n)^2 + b*tan(1/4*pi*n*sgn(b*x + a) - 1/4*pi*n)^2 + b*tan(1/4*pi*
n*sgn(b*x + a) - 3/4*pi*n)^2 + b)
Mupad [F(-1)]
Timed out. \[
\int x^{-m} (-a-b x)^{-n} (a+b x)^n \, dx=\int \frac {{\left (a+b\,x\right )}^n}{x^m\,{\left (-a-b\,x\right )}^n} \,d x
\]
[In]
int((a + b*x)^n/(x^m*(- a - b*x)^n),x)
[Out]
int((a + b*x)^n/(x^m*(- a - b*x)^n), x)