3.736 \(\int \frac{(a+b x)^n}{x} \, dx\)

Optimal. Leaf size=35 \[ -\frac{(a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{a (n+1)} \]

[Out]

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

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Rubi [A]  time = 0.0217294, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{(a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{a (n+1)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^n/x,x]

[Out]

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

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Rubi in Sympy [A]  time = 3.17155, size = 26, normalized size = 0.74 \[ - \frac{\left (a + b x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{1 + \frac{b x}{a}} \right )}}{a \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**n/x,x)

[Out]

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

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Mathematica [A]  time = 0.0180605, size = 46, normalized size = 1.31 \[ \frac{\left (\frac{a}{b x}+1\right )^{-n} (a+b x)^n \, _2F_1\left (-n,-n;1-n;-\frac{a}{b x}\right )}{n} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^n/x,x]

[Out]

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

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Maple [F]  time = 0.033, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{n}}{x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^n/x,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x + a)^n/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x + a)^n/x, x)

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Sympy [A]  time = 4.8334, size = 83, normalized size = 2.37 \[ - \frac{b b^{n} n \left (\frac{a}{b} + x\right ) \left (\frac{a}{b} + x\right )^{n} \Phi \left (\frac{b \left (\frac{a}{b} + x\right )}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{a \Gamma \left (n + 2\right )} - \frac{b b^{n} \left (\frac{a}{b} + x\right ) \left (\frac{a}{b} + x\right )^{n} \Phi \left (\frac{b \left (\frac{a}{b} + x\right )}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{a \Gamma \left (n + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**n/x,x)

[Out]

-b*b**n*n*(a/b + x)*(a/b + x)**n*lerchphi(b*(a/b + x)/a, 1, n + 1)*gamma(n + 1)/
(a*gamma(n + 2)) - b*b**n*(a/b + x)*(a/b + x)**n*lerchphi(b*(a/b + x)/a, 1, n +
1)*gamma(n + 1)/(a*gamma(n + 2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x + a)^n/x, x)