3.8.44 \(\int x^{1+n} (a+b x)^{-n} \, dx\) [744]

Optimal. Leaf size=45 \[ \frac {x^{2+n} (a+b x)^{-n} \left (1+\frac {b x}{a}\right )^n \, _2F_1\left (n,2+n;3+n;-\frac {b x}{a}\right )}{2+n} \]

[Out]

x^(2+n)*(1+b*x/a)^n*hypergeom([n, 2+n],[3+n],-b*x/a)/(2+n)/((b*x+a)^n)

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Rubi [A]
time = 0.01, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {68, 66} \begin {gather*} \frac {x^{n+2} (a+b x)^{-n} \left (\frac {b x}{a}+1\right )^n \, _2F_1\left (n,n+2;n+3;-\frac {b x}{a}\right )}{n+2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(1 + n)/(a + b*x)^n,x]

[Out]

(x^(2 + n)*(1 + (b*x)/a)^n*Hypergeometric2F1[n, 2 + n, 3 + n, -((b*x)/a)])/((2 + n)*(a + b*x)^n)

Rule 66

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^n*((b*x)^(m + 1)/(b*(m + 1)))*Hypergeometr
ic2F1[-n, m + 1, m + 2, (-d)*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] && (IntegerQ[n] || (GtQ[
c, 0] &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-d/(b*c), 0])))

Rule 68

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Dist[c^IntPart[n]*((c + d*x)^FracPart[n]/(1 + d*(
x/c))^FracPart[n]), Int[(b*x)^m*(1 + d*(x/c))^n, x], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] &&  !Int
egerQ[n] &&  !GtQ[c, 0] &&  !GtQ[-d/(b*c), 0] && ((RationalQ[m] &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0])) |
|  !RationalQ[n])

Rubi steps

\begin {align*} \int x^{1+n} (a+b x)^{-n} \, dx &=\left ((a+b x)^{-n} \left (1+\frac {b x}{a}\right )^n\right ) \int x^{1+n} \left (1+\frac {b x}{a}\right )^{-n} \, dx\\ &=\frac {x^{2+n} (a+b x)^{-n} \left (1+\frac {b x}{a}\right )^n \, _2F_1\left (n,2+n;3+n;-\frac {b x}{a}\right )}{2+n}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 45, normalized size = 1.00 \begin {gather*} \frac {x^{2+n} (a+b x)^{-n} \left (1+\frac {b x}{a}\right )^n \, _2F_1\left (n,2+n;3+n;-\frac {b x}{a}\right )}{2+n} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^(1 + n)/(a + b*x)^n,x]

[Out]

(x^(2 + n)*(1 + (b*x)/a)^n*Hypergeometric2F1[n, 2 + n, 3 + n, -((b*x)/a)])/((2 + n)*(a + b*x)^n)

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 5 in optimal.
time = 109.86, size = 36, normalized size = 0.80 \begin {gather*} \frac {a^{-n} x^{2+n} \text {hyper}\left [\left \{n,2+n\right \},\left \{3+n\right \},\frac {b x \text {exp\_polar}\left [I \text {Pi}\right ]}{a}\right ]}{2+n} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[x^(1 + n)/(a + b*x)^n,x]')

[Out]

a ^ (-n) x ^ (2 + n) hyper[{n, 2 + n}, {3 + n}, b x exp_polar[I Pi] / a] / (2 + n)

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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int x^{1+n} \left (b x +a \right )^{-n}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1+n)/((b*x+a)^n),x)

[Out]

int(x^(1+n)/((b*x+a)^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(x^(1+n)/((b*x+a)^n),x, algorithm="maxima")

[Out]

integrate(x^(n + 1)/(b*x + a)^n, x)

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Fricas [F]
time = 0.32, 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(x^(1+n)/((b*x+a)^n),x, algorithm="fricas")

[Out]

integral(x^(n + 1)/(b*x + a)^n, x)

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Sympy [C] Result contains complex when optimal does not.
time = 156.36, size = 34, normalized size = 0.76 \begin {gather*} \frac {a^{- n} x^{2} x^{n} \Gamma \left (n + 2\right ) {{}_{2}F_{1}\left (\begin {matrix} n, n + 2 \\ n + 3 \end {matrix}\middle | {\frac {b x e^{i \pi }}{a}} \right )}}{\Gamma \left (n + 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(1+n)/((b*x+a)**n),x)

[Out]

x**2*x**n*gamma(n + 2)*hyper((n, n + 2), (n + 3,), b*x*exp_polar(I*pi)/a)/(a**n*gamma(n + 3))

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Giac [F] N/A
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(x^(1+n)/((b*x+a)^n),x)

[Out]

Could not integrate

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x^{n+1}}{{\left (a+b\,x\right )}^n} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(n + 1)/(a + b*x)^n,x)

[Out]

int(x^(n + 1)/(a + b*x)^n, x)

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