3.3169 \(\int \frac{(2+3 x)^m}{(1-2 x) (3+5 x)^2} \, dx\)

Optimal. Leaf size=94 \[ \frac{4 (3 x+2)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{2}{7} (3 x+2)\right )}{847 (m+1)}-\frac{5 (33 m+2) (3 x+2)^{m+1} \, _2F_1(1,m+1;m+2;5 (3 x+2))}{121 (m+1)}-\frac{5 (3 x+2)^{m+1}}{11 (5 x+3)} \]

[Out]

(-5*(2 + 3*x)^(1 + m))/(11*(3 + 5*x)) + (4*(2 + 3*x)^(1 + m)*Hypergeometric2F1[1
, 1 + m, 2 + m, (2*(2 + 3*x))/7])/(847*(1 + m)) - (5*(2 + 33*m)*(2 + 3*x)^(1 + m
)*Hypergeometric2F1[1, 1 + m, 2 + m, 5*(2 + 3*x)])/(121*(1 + m))

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Rubi [A]  time = 0.170146, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{4 (3 x+2)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{2}{7} (3 x+2)\right )}{847 (m+1)}-\frac{5 (33 m+2) (3 x+2)^{m+1} \, _2F_1(1,m+1;m+2;5 (3 x+2))}{121 (m+1)}-\frac{5 (3 x+2)^{m+1}}{11 (5 x+3)} \]

Antiderivative was successfully verified.

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

[Out]

(-5*(2 + 3*x)^(1 + m))/(11*(3 + 5*x)) + (4*(2 + 3*x)^(1 + m)*Hypergeometric2F1[1
, 1 + m, 2 + m, (2*(2 + 3*x))/7])/(847*(1 + m)) - (5*(2 + 33*m)*(2 + 3*x)^(1 + m
)*Hypergeometric2F1[1, 1 + m, 2 + m, 5*(2 + 3*x)])/(121*(1 + m))

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Rubi in Sympy [A]  time = 22.6106, size = 73, normalized size = 0.78 \[ - \frac{5 \left (3 x + 2\right )^{m + 1}}{11 \left (5 x + 3\right )} - \frac{5 \left (33 m + 2\right ) \left (3 x + 2\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{15 x + 10} \right )}}{121 \left (m + 1\right )} + \frac{4 \left (3 x + 2\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{6 x}{7} + \frac{4}{7}} \right )}}{847 \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*(3*x + 2)**(m + 1)/(11*(5*x + 3)) - 5*(33*m + 2)*(3*x + 2)**(m + 1)*hyper((1,
m + 1), (m + 2,), 15*x + 10)/(121*(m + 1)) + 4*(3*x + 2)**(m + 1)*hyper((1, m +
1), (m + 2,), 6*x/7 + 4/7)/(847*(m + 1))

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Mathematica [A]  time = 0.332874, size = 161, normalized size = 1.71 \[ \frac{1}{121} (3 x+2)^m \left (\frac{11 \left (\frac{3}{5}\right )^m \left (\frac{3 x+2}{5 x+3}\right )^{-m} \, _2F_1\left (1-m,-m;2-m;-\frac{1}{15 x+9}\right )}{(m-1) (5 x+3)}+\frac{2 \left (\left (\frac{3}{5}\right )^m \left (\frac{3 x+2}{5 x+3}\right )^{-m} \, _2F_1\left (-m,-m;1-m;-\frac{1}{15 x+9}\right )-\left (\frac{6 x+4}{6 x-3}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{7}{3-6 x}\right )\right )}{m}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((2 + 3*x)^m*((11*(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)) + (2*(-(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))/121

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Maple [F]  time = 0.077, size = 0, normalized size = 0. \[ \int{\frac{ \left ( 2+3\,x \right ) ^{m}}{ \left ( 1-2\,x \right ) \left ( 3+5\,x \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (3 \, x + 2\right )}^{m}}{50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(3*x + 2)^m/(50*x^3 + 35*x^2 - 12*x - 9), x)

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Sympy [A]  time = 4.50268, size = 447, normalized size = 4.76 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-30*15**(2*m)*3**m*m*(x + 2/3)*(x + 2/3)**m*lerchphi(7/(6*(x + 2/3)), 1, m*exp_p
olar(I*pi))*gamma(-m)/(1815*15**(2*m)*(x + 2/3)*gamma(-m + 1) - 121*15**(2*m)*ga
mma(-m + 1)) + 2*15**(2*m)*3**m*m*(x + 2/3)**m*lerchphi(7/(6*(x + 2/3)), 1, m*ex
p_polar(I*pi))*gamma(-m)/(1815*15**(2*m)*(x + 2/3)*gamma(-m + 1) - 121*15**(2*m)
*gamma(-m + 1)) + 495*675**m*m**2*(x + 2/3)*(x + 2/3)**m*lerchphi(1/(15*(x + 2/3
)), 1, m*exp_polar(I*pi))*gamma(-m)/(1815*15**(2*m)*(x + 2/3)*gamma(-m + 1) - 12
1*15**(2*m)*gamma(-m + 1)) - 33*675**m*m**2*(x + 2/3)**m*lerchphi(1/(15*(x + 2/3
)), 1, m*exp_polar(I*pi))*gamma(-m)/(1815*15**(2*m)*(x + 2/3)*gamma(-m + 1) - 12
1*15**(2*m)*gamma(-m + 1)) + 30*675**m*m*(x + 2/3)*(x + 2/3)**m*lerchphi(1/(15*(
x + 2/3)), 1, m*exp_polar(I*pi))*gamma(-m)/(1815*15**(2*m)*(x + 2/3)*gamma(-m +
1) - 121*15**(2*m)*gamma(-m + 1)) + 495*675**m*m*(x + 2/3)*(x + 2/3)**m*gamma(-m
)/(1815*15**(2*m)*(x + 2/3)*gamma(-m + 1) - 121*15**(2*m)*gamma(-m + 1)) - 2*675
**m*m*(x + 2/3)**m*lerchphi(1/(15*(x + 2/3)), 1, m*exp_polar(I*pi))*gamma(-m)/(1
815*15**(2*m)*(x + 2/3)*gamma(-m + 1) - 121*15**(2*m)*gamma(-m + 1))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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