3.8.50 \(\int (b x)^m (2+d x)^n \, dx\) [750]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, 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 = {66} \begin {gather*} \frac {2^n (b x)^{m+1} \, _2F_1\left (m+1,-n;m+2;-\frac {d x}{2}\right )}{b (m+1)} \end {gather*}

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, -1/2*(d*x)])/(b*(1 + m))

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])))

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 = 0.02, size = 31, normalized size = 0.89 \begin {gather*} \frac {2^n x (b x)^m \, _2F_1\left (1+m,-n;2+m;-\frac {d x}{2}\right )}{1+m} \end {gather*}

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, -1/2*(d*x)])/(1 + m)

________________________________________________________________________________________

Maple [A]
time = 0.13, size = 32, normalized size = 0.91

method	result	size
meijerg	\(\frac {2^{n} \left (b x \right )^{m} x \hypergeom \left (\left [-n , 1+m \right ], \left [2+m \right ], -\frac {d x}{2}\right )}{1+m}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)

________________________________________________________________________________________

Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

________________________________________________________________________________________

Sympy [C] Result contains complex when optimal does not.
time = 1.14, size = 37, normalized size = 1.06 \begin {gather*} \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 {gather*}

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)

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int {\left (b\,x\right )}^m\,{\left (d\,x+2\right )}^n \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x)^m*(d*x + 2)^n,x)

[Out]

int((b*x)^m*(d*x + 2)^n, x)

________________________________________________________________________________________