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 = 0.0068788, 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, Rules used = {65} \[ \frac{b (a+b x)^{n+1} \, _2F_1\left (2,n+1;n+2;\frac{b x}{a}+1\right )}{a^2 (n+1)} \]

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))

Rule 65

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x)^(n + 1)*Hypergeometric2F1[-m, n +
1, n + 2, 1 + (d*x)/c])/(d*(n + 1)*(-(d/(b*c)))^m), x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[n] && (Inte
gerQ[m] || GtQ[-(d/(b*c)), 0])

Rubi steps

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

Mathematica [A]  time = 0.0057215, size = 35, normalized size = 1. \[ \frac{b (a+b x)^{n+1} \, _2F_1\left (2,n+1;n+2;\frac{b x}{a}+1\right )}{a^2 (n+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[(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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Maple [F]  time = 0.027, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bx+a \right ) ^{n}}{{x}^{2}}}\, dx \end{align*}

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. \begin{align*} \int \frac{{\left (b x + a\right )}^{n}}{x^{2}}\,{d x} \end{align*}

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. \begin{align*}{\rm integral}\left (\frac{{\left (b x + a\right )}^{n}}{x^{2}}, x\right ) \end{align*}

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 [B]  time = 1.90122, size = 354, normalized size = 10.11 \begin{align*} \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 )} \end{align*}

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

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)