\(\int (b x)^m (c+d x)^n \, dx\) [752]

Optimal result

Integrand size = 13, antiderivative size = 52 \[ \int (b x)^m (c+d x)^n \, dx=\frac {(b x)^{1+m} (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} \operatorname {Hypergeometric2F1}\left (1+m,-n,2+m,-\frac {d x}{c}\right )}{b (1+m)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {68, 66} \[ \int (b x)^m (c+d x)^n \, dx=\frac {(b x)^{m+1} (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} \operatorname {Hypergeometric2F1}\left (m+1,-n,m+2,-\frac {d x}{c}\right )}{b (m+1)} \]

[In]

Int[(b*x)^m*(c + d*x)^n,x]

[Out]

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

Rule 66

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^n*((b*x)^(m + 1)/(b*(m + 1)))*Hypergeometr
ic2F1[-n, m + 1, m + 2, (-d)*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] && (IntegerQ[n] || (GtQ[
c, 0] &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-d/(b*c), 0])))

Rule 68

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Dist[c^IntPart[n]*((c + d*x)^FracPart[n]/(1 + d*(
x/c))^FracPart[n]), Int[(b*x)^m*(1 + d*(x/c))^n, x], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] &&  !Int
egerQ[n] &&  !GtQ[c, 0] &&  !GtQ[-d/(b*c), 0] && ((RationalQ[m] &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0])) |
|  !RationalQ[n])

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.92 \[ \int (b x)^m (c+d x)^n \, dx=\frac {x (b x)^m (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} \operatorname {Hypergeometric2F1}\left (1+m,-n,2+m,-\frac {d x}{c}\right )}{1+m} \]

[In]

Integrate[(b*x)^m*(c + d*x)^n,x]

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral((b*x)^m*(d*x + c)^n, x)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 2.95 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.71 \[ \int (b x)^m (c+d x)^n \, dx=\frac {b^{m} c^{n} x^{m + 1} \Gamma \left (m + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} - n, m + 1 \\ m + 2 \end {matrix}\middle | {\frac {d x e^{i \pi }}{c}} \right )}}{\Gamma \left (m + 2\right )} \]

[In]

integrate((b*x)**m*(d*x+c)**n,x)

[Out]

b**m*c**n*x**(m + 1)*gamma(m + 1)*hyper((-n, m + 1), (m + 2,), d*x*exp_polar(I*pi)/c)/gamma(m + 2)

Maxima [F]

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

[In]

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

[Out]

integrate((b*x)^m*(d*x + c)^n, x)

Giac [F]

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

[In]

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

[Out]

integrate((b*x)^m*(d*x + c)^n, x)

Mupad [F(-1)]

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

[In]

int((b*x)^m*(c + d*x)^n,x)

[Out]

int((b*x)^m*(c + d*x)^n, x)