3.737 \(\int \frac{(a+b x)^n}{x^2} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0222475, 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 \[ \frac{b (a+b x)^{n+1} \, _2F_1\left (2,n+1;n+2;\frac{b x}{a}+1\right )}{a^2 (n+1)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^n/x^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 3.41503, size = 27, normalized size = 0.77 \[ \frac{b \left (a + b x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 2, n + 1 \\ n + 2 \end{matrix}\middle |{1 + \frac{b x}{a}} \right )}}{a^{2} \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**n/x**2,x)

[Out]

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

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Mathematica [A]  time = 0.0183417, size = 53, normalized size = 1.51 \[ \frac{\left (\frac{a}{b x}+1\right )^{-n} (a+b x)^n \, _2F_1\left (1-n,-n;2-n;-\frac{a}{b x}\right )}{(n-1) x} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^n/x^2,x]

[Out]

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

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Maple [F]  time = 0.043, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{n}}{{x}^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^n/x^2,x)

[Out]

int((b*x+a)^n/x^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{n}}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^n/x^2,x, algorithm="maxima")

[Out]

integrate((b*x + a)^n/x^2, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x + a\right )}^{n}}{x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^n/x^2,x, algorithm="fricas")

[Out]

integral((b*x + a)^n/x^2, x)

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Sympy [A]  time = 6.46835, size = 354, normalized size = 10.11 \[ \frac{a b^{2} b^{n} n^{2} \left (\frac{a}{b} + x\right ) \left (\frac{a}{b} + x\right )^{n} \Phi \left (\frac{b \left (\frac{a}{b} + x\right )}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{- a^{3} \Gamma \left (n + 2\right ) + a^{2} b \left (\frac{a}{b} + x\right ) \Gamma \left (n + 2\right )} + \frac{a b^{2} b^{n} n \left (\frac{a}{b} + x\right ) \left (\frac{a}{b} + x\right )^{n} \Phi \left (\frac{b \left (\frac{a}{b} + x\right )}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{- a^{3} \Gamma \left (n + 2\right ) + a^{2} b \left (\frac{a}{b} + x\right ) \Gamma \left (n + 2\right )} - \frac{a b^{2} b^{n} n \left (\frac{a}{b} + x\right ) \left (\frac{a}{b} + x\right )^{n} \Gamma \left (n + 1\right )}{- a^{3} \Gamma \left (n + 2\right ) + a^{2} b \left (\frac{a}{b} + x\right ) \Gamma \left (n + 2\right )} - \frac{a b^{2} b^{n} \left (\frac{a}{b} + x\right ) \left (\frac{a}{b} + x\right )^{n} \Gamma \left (n + 1\right )}{- a^{3} \Gamma \left (n + 2\right ) + a^{2} b \left (\frac{a}{b} + x\right ) \Gamma \left (n + 2\right )} - \frac{b^{3} b^{n} n^{2} \left (\frac{a}{b} + x\right )^{2} \left (\frac{a}{b} + x\right )^{n} \Phi \left (\frac{b \left (\frac{a}{b} + x\right )}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{- a^{3} \Gamma \left (n + 2\right ) + a^{2} b \left (\frac{a}{b} + x\right ) \Gamma \left (n + 2\right )} - \frac{b^{3} b^{n} n \left (\frac{a}{b} + x\right )^{2} \left (\frac{a}{b} + x\right )^{n} \Phi \left (\frac{b \left (\frac{a}{b} + x\right )}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{- a^{3} \Gamma \left (n + 2\right ) + a^{2} b \left (\frac{a}{b} + x\right ) \Gamma \left (n + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**n/x**2,x)

[Out]

a*b**2*b**n*n**2*(a/b + x)*(a/b + x)**n*lerchphi(b*(a/b + x)/a, 1, n + 1)*gamma(
n + 1)/(-a**3*gamma(n + 2) + a**2*b*(a/b + x)*gamma(n + 2)) + a*b**2*b**n*n*(a/b
+ x)*(a/b + x)**n*lerchphi(b*(a/b + x)/a, 1, n + 1)*gamma(n + 1)/(-a**3*gamma(n
+ 2) + a**2*b*(a/b + x)*gamma(n + 2)) - a*b**2*b**n*n*(a/b + x)*(a/b + x)**n*ga
mma(n + 1)/(-a**3*gamma(n + 2) + a**2*b*(a/b + x)*gamma(n + 2)) - a*b**2*b**n*(a
/b + x)*(a/b + x)**n*gamma(n + 1)/(-a**3*gamma(n + 2) + a**2*b*(a/b + x)*gamma(n
+ 2)) - b**3*b**n*n**2*(a/b + x)**2*(a/b + x)**n*lerchphi(b*(a/b + x)/a, 1, n +
1)*gamma(n + 1)/(-a**3*gamma(n + 2) + a**2*b*(a/b + x)*gamma(n + 2)) - b**3*b**
n*n*(a/b + x)**2*(a/b + x)**n*lerchphi(b*(a/b + x)/a, 1, n + 1)*gamma(n + 1)/(-a
**3*gamma(n + 2) + a**2*b*(a/b + x)*gamma(n + 2))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{n}}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^n/x^2,x, algorithm="giac")

[Out]

integrate((b*x + a)^n/x^2, x)