3.19.61 \(\int (a+b x)^{-2+n} (c+d x)^{-n} \, dx\) [1861]

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

[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 = 39, 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} \begin {gather*} -\frac {(a+b x)^{n-1} (c+d x)^{1-n}}{(1-n) (b c-a d)} \end {gather*}

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*}

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Mathematica [A]
time = 0.04, size = 36, normalized size = 0.92 \begin {gather*} \frac {(a+b x)^{-1+n} (c+d x)^{1-n}}{(b c-a d) (-1+n)} \end {gather*}

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 = 0.19, size = 45, normalized size = 1.15

method	result	size
gosper	\(-\frac {\left (b x +a \right )^{-1+n} \left (d x +c \right ) \left (d x +c \right )^{-n}}{a d n -b c n -a d +b c}\)	\(45\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 1.30, size = 60, normalized size = 1.54 \begin {gather*} -\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 {gather*}

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)] Timed out
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)**(-2+n)/((d*x+c)**n),x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [B]
time = 0.56, size = 102, normalized size = 2.62 \begin {gather*} -{\left (a+b\,x\right )}^{n-2}\,\left (\frac {a\,c}{\left (a\,d-b\,c\right )\,\left (n-1\right )\,{\left (c+d\,x\right )}^n}+\frac {x\,\left (a\,d+b\,c\right )}{\left (a\,d-b\,c\right )\,\left (n-1\right )\,{\left (c+d\,x\right )}^n}+\frac {b\,d\,x^2}{\left (a\,d-b\,c\right )\,\left (n-1\right )\,{\left (c+d\,x\right )}^n}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^(n - 2)/(c + d*x)^n,x)

[Out]

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

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