\(\int \frac {(a+b x)^n (c+d x)}{x} \, dx\) [921]

Optimal result

Integrand size = 16, antiderivative size = 56 \[ \int \frac {(a+b x)^n (c+d x)}{x} \, dx=\frac {d (a+b x)^{1+n}}{b (1+n)}-\frac {c (a+b x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,1+\frac {b x}{a}\right )}{a (1+n)} \]

[Out]

d*(b*x+a)^(1+n)/b/(1+n)-c*(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 = 56, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {81, 67} \[ \int \frac {(a+b x)^n (c+d x)}{x} \, dx=\frac {d (a+b x)^{n+1}}{b (n+1)}-\frac {c (a+b x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {b x}{a}+1\right )}{a (n+1)} \]

[In]

Int[((a + b*x)^n*(c + d*x))/x,x]

[Out]

(d*(a + b*x)^(1 + n))/(b*(1 + n)) - (c*(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])

Rule 81

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(c + d*x)^
(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rubi steps \begin{align*} \text {integral}& = \frac {d (a+b x)^{1+n}}{b (1+n)}+c \int \frac {(a+b x)^n}{x} \, dx \\ & = \frac {d (a+b x)^{1+n}}{b (1+n)}-\frac {c (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 = 0.08 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.80 \[ \int \frac {(a+b x)^n (c+d x)}{x} \, dx=\frac {(a+b x)^{1+n} \left (a d-b c \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,1+\frac {b x}{a}\right )\right )}{a b (1+n)} \]

[In]

Integrate[((a + b*x)^n*(c + d*x))/x,x]

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral((d*x + c)*(b*x + a)^n/x, x)

Sympy [A] (verification not implemented)

Time = 2.19 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.86 \[ \int \frac {(a+b x)^n (c+d x)}{x} \, dx=d \left (\begin {cases} a^{n} x & \text {for}\: b = 0 \\\frac {\begin {cases} \frac {\left (a + b x\right )^{n + 1}}{n + 1} & \text {for}\: n \neq -1 \\\log {\left (a + b x \right )} & \text {otherwise} \end {cases}}{b} & \text {otherwise} \end {cases}\right ) - \frac {b^{n + 1} c n \left (\frac {a}{b} + x\right )^{n + 1} \Phi \left (1 + \frac {b x}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{a \Gamma \left (n + 2\right )} - \frac {b^{n + 1} c \left (\frac {a}{b} + x\right )^{n + 1} \Phi \left (1 + \frac {b x}{a}, 1, n + 1\right ) \Gamma \left (n + 1\right )}{a \Gamma \left (n + 2\right )} \]

[In]

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

[Out]

d*Piecewise((a**n*x, Eq(b, 0)), (Piecewise(((a + b*x)**(n + 1)/(n + 1), Ne(n, -1)), (log(a + b*x), True))/b, T
rue)) - b**(n + 1)*c*n*(a/b + x)**(n + 1)*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(a*gamma(n + 2)) - b**(n
+ 1)*c*(a/b + x)**(n + 1)*lerchphi(1 + b*x/a, 1, n + 1)*gamma(n + 1)/(a*gamma(n + 2))

Maxima [F]

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

[In]

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

[Out]

c*integrate((b*x + a)^n/x, x) + (b*x + a)^(n + 1)*d/(b*(n + 1))

Giac [F]

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

[In]

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

[Out]

integrate((d*x + c)*(b*x + a)^n/x, x)

Mupad [F(-1)]

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

[In]

int(((a + b*x)^n*(c + d*x))/x,x)

[Out]

int(((a + b*x)^n*(c + d*x))/x, x)