3.1867 \(\int (a+b x)^{-1-n} (c+d x)^n \, dx\)

Optimal. Leaf size=75 \[ -\frac {(a+b x)^{-n} (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \, _2F_1\left (-n,-n;1-n;-\frac {d (a+b x)}{b c-a d}\right )}{b n} \]

[Out]

-(d*x+c)^n*hypergeom([-n, -n],[1-n],-d*(b*x+a)/(-a*d+b*c))/b/n/((b*x+a)^n)/((b*(d*x+c)/(-a*d+b*c))^n)

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Rubi [A]  time = 0.02, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {70, 69} \[ -\frac {(a+b x)^{-n} (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \, _2F_1\left (-n,-n;1-n;-\frac {d (a+b x)}{b c-a d}\right )}{b n} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(-1 - n)*(c + d*x)^n,x]

[Out]

-(((c + d*x)^n*Hypergeometric2F1[-n, -n, 1 - n, -((d*(a + b*x))/(b*c - a*d))])/(b*n*(a + b*x)^n*((b*(c + d*x))
/(b*c - a*d))^n))

Rule 69

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*Hypergeometric2F1[
-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b*(m + 1)*(b/(b*c - a*d))^n), x] /; FreeQ[{a, b, c, d, m, n}
, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-(d/(b*c - a*d)), 0]))

Rule 70

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Dist[(c + d*x)^FracPart[n]/((b/(b*c - a*d)
)^IntPart[n]*((b*(c + d*x))/(b*c - a*d))^FracPart[n]), Int[(a + b*x)^m*Simp[(b*c)/(b*c - a*d) + (b*d*x)/(b*c -
a*d), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] &&
(RationalQ[m] ||  !SimplerQ[n + 1, m + 1])

Rubi steps

\begin {align*} \int (a+b x)^{-1-n} (c+d x)^n \, dx &=\left ((c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n}\right ) \int (a+b x)^{-1-n} \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^n \, dx\\ &=-\frac {(a+b x)^{-n} (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \, _2F_1\left (-n,-n;1-n;-\frac {d (a+b x)}{b c-a d}\right )}{b n}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 74, normalized size = 0.99 \[ -\frac {(a+b x)^{-n} (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \, _2F_1\left (-n,-n;1-n;\frac {d (a+b x)}{a d-b c}\right )}{b n} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(-1 - n)*(c + d*x)^n,x]

[Out]

-(((c + d*x)^n*Hypergeometric2F1[-n, -n, 1 - n, (d*(a + b*x))/(-(b*c) + a*d)])/(b*n*(a + b*x)^n*((b*(c + d*x))
/(b*c - a*d))^n))

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fricas [F]  time = 0.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b x + a\right )}^{-n - 1} {\left (d x + c\right )}^{n}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(-1-n)*(d*x+c)^n,x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x + a\right )}^{-n - 1} {\left (d x + c\right )}^{n}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(-1-n)*(d*x+c)^n,x, algorithm="giac")

[Out]

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

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maple [F]  time = 0.09, size = 0, normalized size = 0.00 \[ \int \left (b x +a \right )^{-n -1} \left (d x +c \right )^{n}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x + a\right )}^{-n - 1} {\left (d x + c\right )}^{n}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(-1-n)*(d*x+c)^n,x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c+d\,x\right )}^n}{{\left (a+b\,x\right )}^{n+1}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^n/(a + b*x)^(n + 1),x)

[Out]

int((c + d*x)^n/(a + b*x)^(n + 1), x)

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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)**(-1-n)*(d*x+c)**n,x)

[Out]

Timed out

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