3.19.68 \(\int (a+b x)^{-2-n} (c+d x)^n \, dx\) [1868]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.00, antiderivative size = 37, 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)^{n+1}}{(n+1) (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*}

________________________________________________________________________________________

Mathematica [A]
time = 0.04, size = 38, normalized size = 1.03 \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))

________________________________________________________________________________________

Maple [A]
time = 0.19, size = 41, normalized size = 1.11

method	result	size
gosper	\(\frac {\left (b x +a \right )^{-1-n} \left (d x +c \right )^{1+n}}{a d n -b c n +a d -b c}\)	\(41\)

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

________________________________________________________________________________________

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)

________________________________________________________________________________________

Fricas [A]
time = 0.88, size = 59, normalized size = 1.59 \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 (d x + c\right )}^{n}}{b c - a d + {\left (b c - a d\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)*(d*x + c)^n/(b*c - a*d + (b*c - a*d)*n)

________________________________________________________________________________________

Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: HeuristicGCDFailed} \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]

Exception raised: HeuristicGCDFailed >> no luck

________________________________________________________________________________________

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)

________________________________________________________________________________________

Mupad [B]
time = 0.53, size = 97, normalized size = 2.62 \begin {gather*} \frac {\frac {a\,c\,{\left (c+d\,x\right )}^n}{\left (a\,d-b\,c\right )\,\left (n+1\right )}+\frac {x\,\left (a\,d+b\,c\right )\,{\left (c+d\,x\right )}^n}{\left (a\,d-b\,c\right )\,\left (n+1\right )}+\frac {b\,d\,x^2\,{\left (c+d\,x\right )}^n}{\left (a\,d-b\,c\right )\,\left (n+1\right )}}{{\left (a+b\,x\right )}^{n+2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^n/(a + b*x)^(n + 2),x)

[Out]

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

________________________________________________________________________________________