\(\int \frac {(a+b x)^n}{x} \, dx\) [736]

Optimal result

Integrand size = 11, antiderivative size = 35 \[ \int \frac {(a+b x)^n}{x} \, dx=-\frac {(a+b x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,1+\frac {b x}{a}\right )}{a (1+n)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {67} \[ \int \frac {(a+b x)^n}{x} \, dx=-\frac {(a+b x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {b x}{a}+1\right )}{a (n+1)} \]

[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)))

Rule 67

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

Rubi steps \begin{align*} \text {integral}& = -\frac {(a+b x)^{1+n} \, _2F_1\left (1,1+n;2+n;1+\frac {b x}{a}\right )}{a (1+n)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^n}{x} \, dx=-\frac {(a+b x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,1+\frac {b x}{a}\right )}{a (1+n)} \]

[In]

Integrate[(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)))

Maple [F]

\[\int \frac {\left (b x +a \right )^{n}}{x}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {(a+b x)^n}{x} \, dx=\int { \frac {{\left (b x + a\right )}^{n}}{x} \,d x } \]

[In]

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

[Out]

integral((b*x + a)^n/x, x)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (26) = 52\).

Time = 2.32 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.17 \[ \int \frac {(a+b x)^n}{x} \, dx=- \frac {b^{n + 1} n \left (\frac {a}{b} + x\right )^{n + 1} \Phi \left (\frac {b \left (\frac {a}{b} + x\right )}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{a \Gamma \left (n + 2\right )} - \frac {b^{n + 1} \left (\frac {a}{b} + x\right )^{n + 1} \Phi \left (\frac {b \left (\frac {a}{b} + x\right )}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{a \Gamma \left (n + 2\right )} \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {(a+b x)^n}{x} \, dx=\int { \frac {{\left (b x + a\right )}^{n}}{x} \,d x } \]

[In]

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

[Out]

integrate((b*x + a)^n/x, x)

Giac [F]

\[ \int \frac {(a+b x)^n}{x} \, dx=\int { \frac {{\left (b x + a\right )}^{n}}{x} \,d x } \]

[In]

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

[Out]

integrate((b*x + a)^n/x, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b x)^n}{x} \, dx=\int \frac {{\left (a+b\,x\right )}^n}{x} \,d x \]

[In]

int((a + b*x)^n/x,x)

[Out]

int((a + b*x)^n/x, x)