3.752 \(\int (b x)^m (c+d x)^n \, dx\)

Optimal. Leaf size=52 \[ \frac{(b x)^{m+1} (c+d x)^n \left (\frac{d x}{c}+1\right )^{-n} \, _2F_1\left (m+1,-n;m+2;-\frac{d x}{c}\right )}{b (m+1)} \]

[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)

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Rubi [A]  time = 0.0128916, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {66, 64} \[ \frac{(b x)^{m+1} (c+d x)^n \left (\frac{d x}{c}+1\right )^{-n} \, _2F_1\left (m+1,-n;m+2;-\frac{d x}{c}\right )}{b (m+1)} \]

Antiderivative was successfully verified.

[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] :> 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])

Rule 64

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c^n*(b*x)^(m + 1)*Hypergeometric2F1[-n, m +
1, m + 2, -((d*x)/c)])/(b*(m + 1)), 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])))

Rubi steps

\begin{align*} \int (b x)^m (c+d x)^n \, dx &=\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]  time = 0.0120037, size = 48, normalized size = 0.92 \[ \frac{x (b x)^m (c+d x)^n \left (\frac{d x}{c}+1\right )^{-n} \, _2F_1\left (m+1,-n;m+2;-\frac{d x}{c}\right )}{m+1} \]

Antiderivative was successfully verified.

[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)

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Maple [F]  time = 0.064, size = 0, normalized size = 0. \begin{align*} \int \left ( bx \right ) ^{m} \left ( dx+c \right ) ^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x)^m*(d*x + c)^n, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (b x\right )^{m}{\left (d x + c\right )}^{n}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*x)^m*(d*x + c)^n, x)

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Sympy [C]  time = 2.15417, size = 37, normalized size = 0.71 \begin{align*} \frac{b^{m} c^{n} x x^{m} \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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x)^m*(d*x + c)^n, x)