3.750 \(\int (b x)^m (2+d x)^n \, dx\)

Optimal. Leaf size=35 \[ \frac{2^n (b x)^{m+1} \, _2F_1\left (m+1,-n;m+2;-\frac{d x}{2}\right )}{b (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.0108964, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {64} \[ \frac{2^n (b x)^{m+1} \, _2F_1\left (m+1,-n;m+2;-\frac{d x}{2}\right )}{b (m+1)} \]

Antiderivative was successfully verified.

[In]

Int[(b*x)^m*(2 + d*x)^n,x]

[Out]

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

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

Mathematica [A]  time = 0.0062759, size = 31, normalized size = 0.89 \[ \frac{2^n x (b x)^m \, _2F_1\left (m+1,-n;m+2;-\frac{d x}{2}\right )}{m+1} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x)^m*(2 + d*x)^n,x]

[Out]

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

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Maple [A]  time = 0.069, size = 32, normalized size = 0.9 \begin{align*}{\frac{{2}^{n} \left ( bx \right ) ^{m}x}{1+m}{\mbox{$_2$F$_1$}(-n,1+m;\,2+m;\,-{\frac{dx}{2}})}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x)^m*(d*x+2)^n,x)

[Out]

2^n*(b*x)^m/(1+m)*x*hypergeom([-n,1+m],[2+m],-1/2*d*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 + 2\right )}^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x)^m*(d*x+2)^n,x, algorithm="maxima")

[Out]

integrate((b*x)^m*(d*x + 2)^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 + 2\right )}^{n}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x)^m*(d*x+2)^n,x, algorithm="fricas")

[Out]

integral((b*x)^m*(d*x + 2)^n, x)

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Sympy [C]  time = 2.12556, size = 37, normalized size = 1.06 \begin{align*} \frac{2^{n} b^{m} 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 }}{2}} \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+2)**n,x)

[Out]

2**n*b**m*x*x**m*gamma(m + 1)*hyper((-n, m + 1), (m + 2,), d*x*exp_polar(I*pi)/2)/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 + 2\right )}^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x)^m*(d*x+2)^n,x, algorithm="giac")

[Out]

integrate((b*x)^m*(d*x + 2)^n, x)