3.1861 \(\int (a+b x)^{-2+n} (c+d x)^{-n} \, dx\)

Optimal. Leaf size=39 \[ -\frac{(a+b x)^{n-1} (c+d x)^{1-n}}{(1-n) (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.0041875, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {37} \[ -\frac{(a+b x)^{n-1} (c+d x)^{1-n}}{(1-n) (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(-2 + n)/(c + d*x)^n,x]

[Out]

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

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0120888, size = 36, normalized size = 0.92 \[ \frac{(a+b x)^{n-1} (c+d x)^{1-n}}{(n-1) (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(-2 + n)/(c + d*x)^n,x]

[Out]

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

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Maple [A]  time = 0.002, size = 45, normalized size = 1.2 \begin{align*} -{\frac{ \left ( bx+a \right ) ^{-1+n} \left ( dx+c \right ) }{ \left ( adn-bcn-ad+bc \right ) \left ( dx+c \right ) ^{n}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(-2+n)/((d*x+c)^n),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{n - 2}}{{\left (d x + c\right )}^{n}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(-2+n)/((d*x+c)^n),x, algorithm="maxima")

[Out]

integrate((b*x + a)^(n - 2)/(d*x + c)^n, x)

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Fricas [A]  time = 1.82054, size = 127, normalized size = 3.26 \begin{align*} -\frac{{\left (b d x^{2} + a c +{\left (b c + a d\right )} x\right )}{\left (b x + a\right )}^{n - 2}}{{\left (b c - a d -{\left (b c - a d\right )} n\right )}{\left (d x + c\right )}^{n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(-2+n)/((d*x+c)^n),x, algorithm="fricas")

[Out]

-(b*d*x^2 + a*c + (b*c + a*d)*x)*(b*x + a)^(n - 2)/((b*c - a*d - (b*c - a*d)*n)*(d*x + c)^n)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**(-2+n)/((d*x+c)**n),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x + a\right )}^{n - 2}}{{\left (d x + c\right )}^{n}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(-2+n)/((d*x+c)^n),x, algorithm="giac")

[Out]

integrate((b*x + a)^(n - 2)/(d*x + c)^n, x)