3.3195 \(\int \frac{(2+3 x)^m (3+5 x)}{1-2 x} \, dx\)

Optimal. Leaf size=54 \[ \frac{11 (3 x+2)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{2}{7} (3 x+2)\right )}{14 (m+1)}-\frac{5 (3 x+2)^{m+1}}{6 (m+1)} \]

[Out]

(-5*(2 + 3*x)^(1 + m))/(6*(1 + m)) + (11*(2 + 3*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (2*(2 + 3*x))/7]
)/(14*(1 + m))

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Rubi [A]  time = 0.0112327, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {80, 68} \[ \frac{11 (3 x+2)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{2}{7} (3 x+2)\right )}{14 (m+1)}-\frac{5 (3 x+2)^{m+1}}{6 (m+1)} \]

Antiderivative was successfully verified.

[In]

Int[((2 + 3*x)^m*(3 + 5*x))/(1 - 2*x),x]

[Out]

(-5*(2 + 3*x)^(1 + m))/(6*(1 + m)) + (11*(2 + 3*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (2*(2 + 3*x))/7]
)/(14*(1 + m))

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 68

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((b*c - a*d)^n*(a + b*x)^(m + 1)*Hype
rgeometric2F1[-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b^(n + 1)*(m + 1)), x] /; FreeQ[{a, b, c, d, m
}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] && IntegerQ[n]

Rubi steps

\begin{align*} \int \frac{(2+3 x)^m (3+5 x)}{1-2 x} \, dx &=-\frac{5 (2+3 x)^{1+m}}{6 (1+m)}+\frac{11}{2} \int \frac{(2+3 x)^m}{1-2 x} \, dx\\ &=-\frac{5 (2+3 x)^{1+m}}{6 (1+m)}+\frac{11 (2+3 x)^{1+m} \, _2F_1\left (1,1+m;2+m;\frac{2}{7} (2+3 x)\right )}{14 (1+m)}\\ \end{align*}

Mathematica [A]  time = 0.0070367, size = 39, normalized size = 0.72 \[ \frac{(3 x+2)^{m+1} \left (33 \, _2F_1\left (1,m+1;m+2;\frac{2}{7} (3 x+2)\right )-35\right )}{42 (m+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[((2 + 3*x)^m*(3 + 5*x))/(1 - 2*x),x]

[Out]

((2 + 3*x)^(1 + m)*(-35 + 33*Hypergeometric2F1[1, 1 + m, 2 + m, (2*(2 + 3*x))/7]))/(42*(1 + m))

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Maple [F]  time = 0.037, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( 3+5\,x \right ) \left ( 2+3\,x \right ) ^{m}}{1-2\,x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+3*x)^m*(3+5*x)/(1-2*x),x)

[Out]

int((2+3*x)^m*(3+5*x)/(1-2*x),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{{\left (3 \, x + 2\right )}^{m}{\left (5 \, x + 3\right )}}{2 \, x - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^m*(3+5*x)/(1-2*x),x, algorithm="maxima")

[Out]

-integrate((3*x + 2)^m*(5*x + 3)/(2*x - 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (3 \, x + 2\right )}^{m}{\left (5 \, x + 3\right )}}{2 \, x - 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^m*(3+5*x)/(1-2*x),x, algorithm="fricas")

[Out]

integral(-(3*x + 2)^m*(5*x + 3)/(2*x - 1), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{3 \left (3 x + 2\right )^{m}}{2 x - 1}\, dx - \int \frac{5 x \left (3 x + 2\right )^{m}}{2 x - 1}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)**m*(3+5*x)/(1-2*x),x)

[Out]

-Integral(3*(3*x + 2)**m/(2*x - 1), x) - Integral(5*x*(3*x + 2)**m/(2*x - 1), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{{\left (3 \, x + 2\right )}^{m}{\left (5 \, x + 3\right )}}{2 \, x - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^m*(3+5*x)/(1-2*x),x, algorithm="giac")

[Out]

integrate(-(3*x + 2)^m*(5*x + 3)/(2*x - 1), x)