3.1868 \(\int (a+b x)^{-2-n} (c+d x)^n \, dx\)

Optimal. Leaf size=37 \[ -\frac{(a+b x)^{-n-1} (c+d x)^{n+1}}{(n+1) (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.0059839, antiderivative size = 37, 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} \[ -\frac{(a+b x)^{-n-1} (c+d x)^{n+1}}{(n+1) (b c-a d)} \]

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 = 0.0127175, size = 38, normalized size = 1.03 \[ \frac{(a+b x)^{-n-1} (c+d x)^{n+1}}{(-n-1) (b c-a d)} \]

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 = 0.003, size = 41, normalized size = 1.1 \begin{align*}{\frac{ \left ( bx+a \right ) ^{-1-n} \left ( dx+c \right ) ^{1+n}}{adn-bcn+ad-bc}} \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)^(1+n)/(a*d*n-b*c*n+a*d-b*c)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\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 = 1.91859, size = 126, normalized size = 3.41 \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 (d x + c\right )}^{n}}{b c - a d +{\left (b c - a d\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)*(d*x + c)^n/(b*c - a*d + (b*c - a*d)*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{\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)