3.3170 \(\int \frac{(2+3 x)^m}{(1-2 x) (3+5 x)^3} \, dx\)

Optimal. Leaf size=124 \[ -\frac{5 \left (1089 m^2-957 m+8\right ) (3 x+2)^{m+1} \, _2F_1(1,m+1;m+2;5 (3 x+2))}{2662 (m+1)}+\frac{8 (3 x+2)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{2}{7} (3 x+2)\right )}{9317 (m+1)}+\frac{5 (29-33 m) (3 x+2)^{m+1}}{242 (5 x+3)}-\frac{5 (3 x+2)^{m+1}}{22 (5 x+3)^2} \]

[Out]

(-5*(2 + 3*x)^(1 + m))/(22*(3 + 5*x)^2) + (5*(29 - 33*m)*(2 + 3*x)^(1 + m))/(242
*(3 + 5*x)) + (8*(2 + 3*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (2*(2 + 3*
x))/7])/(9317*(1 + m)) - (5*(8 - 957*m + 1089*m^2)*(2 + 3*x)^(1 + m)*Hypergeomet
ric2F1[1, 1 + m, 2 + m, 5*(2 + 3*x)])/(2662*(1 + m))

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Rubi [A]  time = 0.316209, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{5 \left (1089 m^2-957 m+8\right ) (3 x+2)^{m+1} \, _2F_1(1,m+1;m+2;5 (3 x+2))}{2662 (m+1)}+\frac{8 (3 x+2)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{2}{7} (3 x+2)\right )}{9317 (m+1)}+\frac{5 (29-33 m) (3 x+2)^{m+1}}{242 (5 x+3)}-\frac{5 (3 x+2)^{m+1}}{22 (5 x+3)^2} \]

Antiderivative was successfully verified.

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

[Out]

(-5*(2 + 3*x)^(1 + m))/(22*(3 + 5*x)^2) + (5*(29 - 33*m)*(2 + 3*x)^(1 + m))/(242
*(3 + 5*x)) + (8*(2 + 3*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (2*(2 + 3*
x))/7])/(9317*(1 + m)) - (5*(8 - 957*m + 1089*m^2)*(2 + 3*x)^(1 + m)*Hypergeomet
ric2F1[1, 1 + m, 2 + m, 5*(2 + 3*x)])/(2662*(1 + m))

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Rubi in Sympy [A]  time = 43.1896, size = 104, normalized size = 0.84 \[ \frac{\left (- \frac{15 m}{22} + \frac{145}{242}\right ) \left (3 x + 2\right )^{m + 1}}{5 x + 3} - \frac{5 \left (3 x + 2\right )^{m + 1}}{22 \left (5 x + 3\right )^{2}} - \frac{5 \left (3 x + 2\right )^{m + 1} \left (1089 m^{2} - 957 m + 8\right ){{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{15 x + 10} \right )}}{2662 \left (m + 1\right )} + \frac{8 \left (3 x + 2\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{6 x}{7} + \frac{4}{7}} \right )}}{9317 \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-15*m/22 + 145/242)*(3*x + 2)**(m + 1)/(5*x + 3) - 5*(3*x + 2)**(m + 1)/(22*(5*
x + 3)**2) - 5*(3*x + 2)**(m + 1)*(1089*m**2 - 957*m + 8)*hyper((1, m + 1), (m +
2,), 15*x + 10)/(2662*(m + 1)) + 8*(3*x + 2)**(m + 1)*hyper((1, m + 1), (m + 2,
), 6*x/7 + 4/7)/(9317*(m + 1))

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Mathematica [A]  time = 0.699722, size = 220, normalized size = 1.77 \[ \frac{(3 x+2)^m \left (\frac{22 \left (\frac{3}{5}\right )^m \left (\frac{3 x+2}{5 x+3}\right )^{-m} \, _2F_1\left (1-m,-m;2-m;-\frac{1}{15 x+9}\right )}{(m-1) (5 x+3)}+\frac{121 \left (\frac{3}{5}\right )^m \left (\frac{3 x+2}{5 x+3}\right )^{-m} \, _2F_1\left (2-m,-m;3-m;-\frac{1}{15 x+9}\right )}{(m-2) (5 x+3)^2}+\frac{4 \left (\left (\frac{3}{5}\right )^m \left (\frac{3 x+2}{5 x+3}\right )^{-m} \, _2F_1\left (-m,-m;1-m;-\frac{1}{15 x+9}\right )-\left (\frac{6 x+4}{6 x-3}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{7}{3-6 x}\right )\right )}{m}\right )}{1331} \]

Antiderivative was successfully verified.

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

[Out]

((2 + 3*x)^m*((22*(3/5)^m*Hypergeometric2F1[1 - m, -m, 2 - m, -(9 + 15*x)^(-1)])
/((-1 + m)*((2 + 3*x)/(3 + 5*x))^m*(3 + 5*x)) + (121*(3/5)^m*Hypergeometric2F1[2
- m, -m, 3 - m, -(9 + 15*x)^(-1)])/((-2 + m)*((2 + 3*x)/(3 + 5*x))^m*(3 + 5*x)^
2) + (4*(-(Hypergeometric2F1[-m, -m, 1 - m, 7/(3 - 6*x)]/((4 + 6*x)/(-3 + 6*x))^
m) + ((3/5)^m*Hypergeometric2F1[-m, -m, 1 - m, -(9 + 15*x)^(-1)])/((2 + 3*x)/(3
+ 5*x))^m))/m))/1331

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Maple [F]  time = 0.095, size = 0, normalized size = 0. \[ \int{\frac{ \left ( 2+3\,x \right ) ^{m}}{ \left ( 1-2\,x \right ) \left ( 3+5\,x \right ) ^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (3 \, x + 2\right )}^{m}}{250 \, x^{4} + 325 \, x^{3} + 45 \, x^{2} - 81 \, x - 27}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(3*x + 2)^m/(250*x^4 + 325*x^3 + 45*x^2 - 81*x - 27), x)

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Sympy [A]  time = 6.59582, size = 1226, normalized size = 9.89 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

245025*45**m*m**3*(x + 2/3)**2*(x + 2/3)**m*lerchphi(1/(15*(x + 2/3)), 1, m*exp_
polar(I*pi))*gamma(-m)/(598950*15**m*(x + 2/3)**2*gamma(-m + 1) - 79860*15**m*(x
+ 2/3)*gamma(-m + 1) + 2662*15**m*gamma(-m + 1)) - 32670*45**m*m**3*(x + 2/3)*(
x + 2/3)**m*lerchphi(1/(15*(x + 2/3)), 1, m*exp_polar(I*pi))*gamma(-m)/(598950*1
5**m*(x + 2/3)**2*gamma(-m + 1) - 79860*15**m*(x + 2/3)*gamma(-m + 1) + 2662*15*
*m*gamma(-m + 1)) + 1089*45**m*m**3*(x + 2/3)**m*lerchphi(1/(15*(x + 2/3)), 1, m
*exp_polar(I*pi))*gamma(-m)/(598950*15**m*(x + 2/3)**2*gamma(-m + 1) - 79860*15*
*m*(x + 2/3)*gamma(-m + 1) + 2662*15**m*gamma(-m + 1)) - 215325*45**m*m**2*(x +
2/3)**2*(x + 2/3)**m*lerchphi(1/(15*(x + 2/3)), 1, m*exp_polar(I*pi))*gamma(-m)/
(598950*15**m*(x + 2/3)**2*gamma(-m + 1) - 79860*15**m*(x + 2/3)*gamma(-m + 1) +
2662*15**m*gamma(-m + 1)) + 245025*45**m*m**2*(x + 2/3)**2*(x + 2/3)**m*gamma(-
m)/(598950*15**m*(x + 2/3)**2*gamma(-m + 1) - 79860*15**m*(x + 2/3)*gamma(-m + 1
) + 2662*15**m*gamma(-m + 1)) + 28710*45**m*m**2*(x + 2/3)*(x + 2/3)**m*lerchphi
(1/(15*(x + 2/3)), 1, m*exp_polar(I*pi))*gamma(-m)/(598950*15**m*(x + 2/3)**2*ga
mma(-m + 1) - 79860*15**m*(x + 2/3)*gamma(-m + 1) + 2662*15**m*gamma(-m + 1)) -
16335*45**m*m**2*(x + 2/3)*(x + 2/3)**m*gamma(-m)/(598950*15**m*(x + 2/3)**2*gam
ma(-m + 1) - 79860*15**m*(x + 2/3)*gamma(-m + 1) + 2662*15**m*gamma(-m + 1)) - 9
57*45**m*m**2*(x + 2/3)**m*lerchphi(1/(15*(x + 2/3)), 1, m*exp_polar(I*pi))*gamm
a(-m)/(598950*15**m*(x + 2/3)**2*gamma(-m + 1) - 79860*15**m*(x + 2/3)*gamma(-m
+ 1) + 2662*15**m*gamma(-m + 1)) + 1800*45**m*m*(x + 2/3)**2*(x + 2/3)**m*lerchp
hi(1/(15*(x + 2/3)), 1, m*exp_polar(I*pi))*gamma(-m)/(598950*15**m*(x + 2/3)**2*
gamma(-m + 1) - 79860*15**m*(x + 2/3)*gamma(-m + 1) + 2662*15**m*gamma(-m + 1))
- 1800*45**m*m*(x + 2/3)**2*(x + 2/3)**m*lerchphi(7/(6*(x + 2/3)), 1, m*exp_pola
r(I*pi))*gamma(-m)/(598950*15**m*(x + 2/3)**2*gamma(-m + 1) - 79860*15**m*(x + 2
/3)*gamma(-m + 1) + 2662*15**m*gamma(-m + 1)) - 215325*45**m*m*(x + 2/3)**2*(x +
2/3)**m*gamma(-m)/(598950*15**m*(x + 2/3)**2*gamma(-m + 1) - 79860*15**m*(x + 2
/3)*gamma(-m + 1) + 2662*15**m*gamma(-m + 1)) - 240*45**m*m*(x + 2/3)*(x + 2/3)*
*m*lerchphi(1/(15*(x + 2/3)), 1, m*exp_polar(I*pi))*gamma(-m)/(598950*15**m*(x +
2/3)**2*gamma(-m + 1) - 79860*15**m*(x + 2/3)*gamma(-m + 1) + 2662*15**m*gamma(
-m + 1)) + 240*45**m*m*(x + 2/3)*(x + 2/3)**m*lerchphi(7/(6*(x + 2/3)), 1, m*exp
_polar(I*pi))*gamma(-m)/(598950*15**m*(x + 2/3)**2*gamma(-m + 1) - 79860*15**m*(
x + 2/3)*gamma(-m + 1) + 2662*15**m*gamma(-m + 1)) + 30690*45**m*m*(x + 2/3)*(x
+ 2/3)**m*gamma(-m)/(598950*15**m*(x + 2/3)**2*gamma(-m + 1) - 79860*15**m*(x +
2/3)*gamma(-m + 1) + 2662*15**m*gamma(-m + 1)) + 8*45**m*m*(x + 2/3)**m*lerchphi
(1/(15*(x + 2/3)), 1, m*exp_polar(I*pi))*gamma(-m)/(598950*15**m*(x + 2/3)**2*ga
mma(-m + 1) - 79860*15**m*(x + 2/3)*gamma(-m + 1) + 2662*15**m*gamma(-m + 1)) -
8*45**m*m*(x + 2/3)**m*lerchphi(7/(6*(x + 2/3)), 1, m*exp_polar(I*pi))*gamma(-m)
/(598950*15**m*(x + 2/3)**2*gamma(-m + 1) - 79860*15**m*(x + 2/3)*gamma(-m + 1)
+ 2662*15**m*gamma(-m + 1))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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