3.3164 \(\int \frac{(1-2 x) (2+3 x)^m}{(3+5 x)^3} \, dx\)

Optimal. Leaf size=59 \[ -\frac{3 (37-33 m) (3 x+2)^{m+1} \, _2F_1(2,m+1;m+2;5 (3 x+2))}{10 (m+1)}-\frac{11 (3 x+2)^{m+1}}{10 (5 x+3)^2} \]

[Out]

(-11*(2 + 3*x)^(1 + m))/(10*(3 + 5*x)^2) - (3*(37 - 33*m)*(2 + 3*x)^(1 + m)*Hype
rgeometric2F1[2, 1 + m, 2 + m, 5*(2 + 3*x)])/(10*(1 + m))

_______________________________________________________________________________________

Rubi [A]  time = 0.0706887, antiderivative size = 59, 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 \[ -\frac{3 (37-33 m) (3 x+2)^{m+1} \, _2F_1(2,m+1;m+2;5 (3 x+2))}{10 (m+1)}-\frac{11 (3 x+2)^{m+1}}{10 (5 x+3)^2} \]

Antiderivative was successfully verified.

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

[Out]

(-11*(2 + 3*x)^(1 + m))/(10*(3 + 5*x)^2) - (3*(37 - 33*m)*(2 + 3*x)^(1 + m)*Hype
rgeometric2F1[2, 1 + m, 2 + m, 5*(2 + 3*x)])/(10*(1 + m))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 6.78721, size = 49, normalized size = 0.83 \[ - \frac{3 \left (- 33 m + 37\right ) \left (3 x + 2\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 2, m + 1 \\ m + 2 \end{matrix}\middle |{15 x + 10} \right )}}{10 \left (m + 1\right )} - \frac{11 \left (3 x + 2\right )^{m + 1}}{10 \left (5 x + 3\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*(-33*m + 37)*(3*x + 2)**(m + 1)*hyper((2, m + 1), (m + 2,), 15*x + 10)/(10*(m
+ 1)) - 11*(3*x + 2)**(m + 1)/(10*(5*x + 3)**2)

_______________________________________________________________________________________

Mathematica [A]  time = 0.08822, size = 103, normalized size = 1.75 \[ \frac{(3 x+2)^m \left (\frac{1}{15 x+9}+1\right )^{-m} \left (11 (m-1) \, _2F_1\left (2-m,-m;3-m;-\frac{1}{15 x+9}\right )-2 (m-2) (5 x+3) \, _2F_1\left (1-m,-m;2-m;-\frac{1}{15 x+9}\right )\right )}{25 (m-2) (m-1) (5 x+3)^2} \]

Antiderivative was successfully verified.

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

[Out]

((2 + 3*x)^m*(-2*(-2 + m)*(3 + 5*x)*Hypergeometric2F1[1 - m, -m, 2 - m, -(9 + 15
*x)^(-1)] + 11*(-1 + m)*Hypergeometric2F1[2 - m, -m, 3 - m, -(9 + 15*x)^(-1)]))/
(25*(-2 + m)*(-1 + m)*(3 + 5*x)^2*(1 + (9 + 15*x)^(-1))^m)

_______________________________________________________________________________________

Maple [F]  time = 0.069, size = 0, normalized size = 0. \[ \int{\frac{ \left ( 1-2\,x \right ) \left ( 2+3\,x \right ) ^{m}}{ \left ( 3+5\,x \right ) ^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (3 \, x + 2\right )}^{m}{\left (2 \, x - 1\right )}}{125 \, x^{3} + 225 \, x^{2} + 135 \, x + 27}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(3*x + 2)^m*(2*x - 1)/(125*x^3 + 225*x^2 + 135*x + 27), x)

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \left (- \frac{\left (3 x + 2\right )^{m}}{125 x^{3} + 225 x^{2} + 135 x + 27}\right )\, dx - \int \frac{2 x \left (3 x + 2\right )^{m}}{125 x^{3} + 225 x^{2} + 135 x + 27}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(-(3*x + 2)**m/(125*x**3 + 225*x**2 + 135*x + 27), x) - Integral(2*x*(3
*x + 2)**m/(125*x**3 + 225*x**2 + 135*x + 27), x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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