3.3167 \(\int \frac{(2+3 x)^m (3+5 x)}{1-2 x} \, dx\)

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))

_______________________________________________________________________________________

Rubi [A]  time = 0.0471738, 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 \[ \frac{11 (3 x+2)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{2}{7} (3 x+2)\right )}{14 (m+1)}-\frac{5 (3 x+2)^{m+1}}{6 (m+1)} \]

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))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 6.47515, size = 42, normalized size = 0.78 \[ \frac{11 \left (3 x + 2\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{6 x}{7} + \frac{4}{7}} \right )}}{14 \left (m + 1\right )} - \frac{5 \left (3 x + 2\right )^{m + 1}}{6 \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2+3*x)**m*(3+5*x)/(1-2*x),x)

[Out]

11*(3*x + 2)**(m + 1)*hyper((1, m + 1), (m + 2,), 6*x/7 + 4/7)/(14*(m + 1)) - 5*
(3*x + 2)**(m + 1)/(6*(m + 1))

_______________________________________________________________________________________

Mathematica [A]  time = 0.112026, size = 67, normalized size = 1.24 \[ \frac{1}{12} (3 x+2)^m \left (-\frac{33 \left (\frac{6 x+4}{6 x-3}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{7}{3-6 x}\right )}{m}-\frac{10 (3 x+2)}{m+1}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((2 + 3*x)^m*((-10*(2 + 3*x))/(1 + m) - (33*Hypergeometric2F1[-m, -m, 1 - m, 7/(
3 - 6*x)])/(m*((4 + 6*x)/(-3 + 6*x))^m)))/12

_______________________________________________________________________________________

Maple [F]  time = 0.054, size = 0, normalized size = 0. \[ \int{\frac{ \left ( 2+3\,x \right ) ^{m} \left ( 3+5\,x \right ) }{1-2\,x}}\, dx \]

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)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

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)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -\frac{{\left (3 \, x + 2\right )}^{m}{\left (5 \, x + 3\right )}}{2 \, x - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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