∫ x−m(−a − bx)−n(a + bx)n dx

3.9.2 \(\int x^{-m} (-a-b x)^{-n} (a+b x)^n \, dx\)

Optimal. Leaf size=34 \[ \frac {x^{1-m} (-a-b x)^{-n} (a+b x)^n}{1-m} \]

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Rubi [A] time = 0.00, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {23, 30} \begin {gather*} \frac {x^{1-m} (-a-b x)^{-n} (a+b x)^n}{1-m} \end {gather*}

Antiderivative was successfully veriﬁed.

[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*} \int x^{-m} (-a-b x)^{-n} (a+b x)^n \, dx &=\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*}

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Mathematica [A] time = 0.01, size = 34, normalized size = 1.00 \begin {gather*} \frac {x^{1-m} (-a-b x)^{-n} (a+b x)^n}{1-m} \end {gather*}

Antiderivative was successfully veriﬁed.

[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)

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IntegrateAlgebraic [F] time = 0.04, size = 0, normalized size = 0.00 \begin {gather*} \int x^{-m} (-a-b x)^{-n} (a+b x)^n \, dx \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x)^n/(x^m*(-a - b*x)^n),x]

[Out]

Defer[IntegrateAlgebraic][(a + b*x)^n/(x^m*(-a - b*x)^n), x]

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fricas [A] time = 1.64, size = 17, normalized size = 0.50 \begin {gather*} -\frac {x \cos \left (\pi n\right )}{{\left (m - 1\right )} x^{m}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x*cos(pi*n)/((m - 1)*x^m)

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giac [B] time = 1.99, size = 1586, normalized size = 46.65

result too large to display

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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maple [A] time = 0.00, size = 33, normalized size = 0.97 \begin {gather*} -\frac {x \,x^{-m} \left (-b x -a \right )^{-n} \left (b x +a \right )^{n}}{m -1} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^n/(x^m)/((-b*x-a)^n),x)

[Out]

-x/(m-1)*(b*x+a)^n/(x^m)/((-b*x-a)^n)

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maxima [A] time = 0.94, size = 21, normalized size = 0.62 \begin {gather*} -\frac {x}{{\left (\left (-1\right )^{n} m - \left (-1\right )^{n}\right )} x^{m}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/(((-1)^n*m - (-1)^n)*x^m)

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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\left (a+b\,x\right )}^n}{x^m\,{\left (-a-b\,x\right )}^n} \,d x \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**n/(x**m)/((-b*x-a)**n),x)

[Out]

Timed out

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