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

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

[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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Rubi [A]  time = 0.0333227, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{x^{n+2} (a+b x)^{-n} \left (\frac{b x}{a}+1\right )^n \, _2F_1\left (n,n+2;n+3;-\frac{b x}{a}\right )}{n+2} \]

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)

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Rubi in Sympy [A]  time = 6.06901, size = 34, normalized size = 0.76 \[ \frac{x^{n + 2} \left (1 + \frac{b x}{a}\right )^{n} \left (a + b x\right )^{- n}{{}_{2}F_{1}\left (\begin{matrix} n, n + 2 \\ n + 3 \end{matrix}\middle |{- \frac{b x}{a}} \right )}}{n + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(1+n)/((b*x+a)**n),x)

[Out]

x**(n + 2)*(1 + b*x/a)**n*(a + b*x)**(-n)*hyper((n, n + 2), (n + 3,), -b*x/a)/(n
+ 2)

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Mathematica [A]  time = 0.0286519, size = 45, normalized size = 1. \[ \frac{x^{n+2} (a+b x)^{-n} \left (\frac{b x}{a}+1\right )^n \, _2F_1\left (n,n+2;n+3;-\frac{b x}{a}\right )}{n+2} \]

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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Maple [F]  time = 0.079, size = 0, normalized size = 0. \[ \int{\frac{{x}^{1+n}}{ \left ( bx+a \right ) ^{n}}}\, dx \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(n + 1)/(b*x + a)^n,x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{n + 1}}{{\left (b x + a\right )}^{n}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(n + 1)/(b*x + a)^n,x, algorithm="fricas")

[Out]

integral(x^(n + 1)/(b*x + a)^n, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{n + 1}}{{\left (b x + a\right )}^{n}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(n + 1)/(b*x + a)^n,x, algorithm="giac")

[Out]

integrate(x^(n + 1)/(b*x + a)^n, x)