3.19.89 \(\int (1+2 x)^{-m} (2+3 x)^m \, dx\) [1889]

Optimal. Leaf size=47 \[ \frac {2^{-1-m} (1+2 x)^{1-m} \, _2F_1(1-m,-m;2-m;-3 (1+2 x))}{1-m} \]

[Out]

2^(-1-m)*(1+2*x)^(1-m)*hypergeom([-m, 1-m],[2-m],-3-6*x)/(1-m)

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Rubi [A]
time = 0.01, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {71} \begin {gather*} \frac {2^{-m-1} (2 x+1)^{1-m} \, _2F_1(1-m,-m;2-m;-3 (2 x+1))}{1-m} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x)^m/(1 + 2*x)^m,x]

[Out]

(2^(-1 - m)*(1 + 2*x)^(1 - m)*Hypergeometric2F1[1 - m, -m, 2 - m, -3*(1 + 2*x)])/(1 - m)

Rule 71

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c
- a*d))^n))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}
, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-d/(b*c - a*d), 0]))

Rubi steps

\begin {align*} \int (1+2 x)^{-m} (2+3 x)^m \, dx &=\frac {2^{-1-m} (1+2 x)^{1-m} \, _2F_1(1-m,-m;2-m;-3 (1+2 x))}{1-m}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 47, normalized size = 1.00 \begin {gather*} \frac {2^{-1-m} (1+2 x)^{1-m} \, _2F_1(1-m,-m;2-m;-3 (1+2 x))}{1-m} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + 3*x)^m/(1 + 2*x)^m,x]

[Out]

(2^(-1 - m)*(1 + 2*x)^(1 - m)*Hypergeometric2F1[1 - m, -m, 2 - m, -3*(1 + 2*x)])/(1 - m)

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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \left (2+3 x \right )^{m} \left (1+2 x \right )^{-m}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+3*x)^m/((1+2*x)^m),x)

[Out]

int((2+3*x)^m/((1+2*x)^m),x)

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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((2+3*x)^m/((1+2*x)^m),x, algorithm="maxima")

[Out]

integrate((3*x + 2)^m/(2*x + 1)^m, x)

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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((2+3*x)^m/((1+2*x)^m),x, algorithm="fricas")

[Out]

integral((3*x + 2)^m/(2*x + 1)^m, x)

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Sympy [C] Result contains complex when optimal does not.
time = 15.38, size = 42, normalized size = 0.89 \begin {gather*} \frac {3^{2 m} \left (x + \frac {2}{3}\right ) \left (x + \frac {2}{3}\right )^{m} e^{- i \pi m} \Gamma \left (m + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} m, m + 1 \\ m + 2 \end {matrix}\middle | {6 x + 4} \right )}}{\Gamma \left (m + 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)**m/((1+2*x)**m),x)

[Out]

3**(2*m)*(x + 2/3)*(x + 2/3)**m*exp(-I*pi*m)*gamma(m + 1)*hyper((m, m + 1), (m + 2,), 6*x + 4)/gamma(m + 2)

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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((2+3*x)^m/((1+2*x)^m),x, algorithm="giac")

[Out]

integrate((3*x + 2)^m/(2*x + 1)^m, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\left (3\,x+2\right )}^m}{{\left (2\,x+1\right )}^m} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x + 2)^m/(2*x + 1)^m,x)

[Out]

int((3*x + 2)^m/(2*x + 1)^m, x)

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