3.1874 \(\int (a+b x)^n (c+d x)^{-2-n} \, dx\)

Optimal. Leaf size=36 \[ \frac {(a+b x)^{n+1} (c+d x)^{-n-1}}{(n+1) (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.00, antiderivative size = 36, normalized size of antiderivative = 1.00, 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)^n*(c + d*x)^(-2 - 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)^n (c+d x)^{-2-n} \, dx &=\frac {(a+b x)^{1+n} (c+d x)^{-1-n}}{(b c-a d) (1+n)}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 36, normalized size = 1.00 \[ \frac {(a+b x)^{n+1} (c+d x)^{-n-1}}{(n+1) (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.84, size = 58, normalized size = 1.61 \[ \frac {{\left (b d x^{2} + a c + {\left (b c + a d\right )} x\right )} {\left (b x + a\right )}^{n} {\left (d x + c\right )}^{-n - 2}}{b c - a d + {\left (b c - a d\right )} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x + a\right )}^{n} {\left (d x + c\right )}^{-n - 2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 42, normalized size = 1.17 \[ -\frac {\left (b x +a \right )^{n +1} \left (d x +c \right )^{-n -1}}{a d n -b c n +a d -b c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x + a\right )}^{n} {\left (d x + c\right )}^{-n - 2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.56, size = 98, normalized size = 2.72 \[ -\frac {\frac {a\,c\,{\left (a+b\,x\right )}^n}{\left (a\,d-b\,c\right )\,\left (n+1\right )}+\frac {x\,\left (a\,d+b\,c\right )\,{\left (a+b\,x\right )}^n}{\left (a\,d-b\,c\right )\,\left (n+1\right )}+\frac {b\,d\,x^2\,{\left (a+b\,x\right )}^n}{\left (a\,d-b\,c\right )\,\left (n+1\right )}}{{\left (c+d\,x\right )}^{n+2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^n/(c + d*x)^(n + 2),x)

[Out]

-((a*c*(a + b*x)^n)/((a*d - b*c)*(n + 1)) + (x*(a*d + b*c)*(a + b*x)^n)/((a*d - b*c)*(n + 1)) + (b*d*x^2*(a +
b*x)^n)/((a*d - b*c)*(n + 1)))/(c + d*x)^(n + 2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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