3.19.67 \(\int (a+b x)^{-1-n} (c+d x)^n \, dx\) [1867]

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 = {72, 71} \begin {gather*} -\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 {gather*}

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 71

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c
- a*d))^n))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], 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 72

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 = 0.05, size = 74, normalized size = 0.99 \begin {gather*} -\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 {gather*}

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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Maple [F]
time = 0.11, size = 0, normalized size = 0.00 \[\int \left (b x +a \right )^{-1-n} \left (d x +c \right )^{n}\, dx\]

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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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

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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