3.3168 \(\int \frac{(2+3 x)^m}{(1-2 x) (3+5 x)} \, dx\)

Optimal. Leaf size=69 \[ \frac{2 (3 x+2)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{2}{7} (3 x+2)\right )}{77 (m+1)}-\frac{5 (3 x+2)^{m+1} \, _2F_1(1,m+1;m+2;5 (3 x+2))}{11 (m+1)} \]

[Out]

(2*(2 + 3*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (2*(2 + 3*x))/7])/(77*(1
+ m)) - (5*(2 + 3*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, 5*(2 + 3*x)])/(
11*(1 + m))

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Rubi [A]  time = 0.0741557, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{2 (3 x+2)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{2}{7} (3 x+2)\right )}{77 (m+1)}-\frac{5 (3 x+2)^{m+1} \, _2F_1(1,m+1;m+2;5 (3 x+2))}{11 (m+1)} \]

Antiderivative was successfully verified.

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

[Out]

(2*(2 + 3*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (2*(2 + 3*x))/7])/(77*(1
+ m)) - (5*(2 + 3*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, 5*(2 + 3*x)])/(
11*(1 + m))

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Rubi in Sympy [A]  time = 8.35858, size = 53, normalized size = 0.77 \[ \frac{2 \left (3 x + 2\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{6 x}{7} + \frac{4}{7}} \right )}}{77 \left (m + 1\right )} - \frac{5 \left (3 x + 2\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{15 x + 10} \right )}}{11 \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(3*x + 2)**(m + 1)*hyper((1, m + 1), (m + 2,), 6*x/7 + 4/7)/(77*(m + 1)) - 5*(
3*x + 2)**(m + 1)*hyper((1, m + 1), (m + 2,), 15*x + 10)/(11*(m + 1))

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Mathematica [A]  time = 0.15552, size = 96, normalized size = 1.39 \[ -\frac{(3 x+2)^m \left (\left (\frac{6 x+4}{6 x-3}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{7}{3-6 x}\right )-\left (\frac{3 x+2}{3 x+\frac{9}{5}}\right )^{-m} \, _2F_1\left (-m,-m;1-m;-\frac{1}{15 x+9}\right )\right )}{11 m} \]

Antiderivative was successfully verified.

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

[Out]

-((2 + 3*x)^m*(Hypergeometric2F1[-m, -m, 1 - m, 7/(3 - 6*x)]/((4 + 6*x)/(-3 + 6*
x))^m - Hypergeometric2F1[-m, -m, 1 - m, -(9 + 15*x)^(-1)]/((2 + 3*x)/(9/5 + 3*x
))^m))/(11*m)

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Maple [F]  time = 0.063, size = 0, normalized size = 0. \[ \int{\frac{ \left ( 2+3\,x \right ) ^{m}}{ \left ( 1-2\,x \right ) \left ( 3+5\,x \right ) }}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x + 2)^m/((5*x + 3)*(2*x - 1)),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. \[{\rm integral}\left (-\frac{{\left (3 \, x + 2\right )}^{m}}{10 \, x^{2} + x - 3}, x\right ) \]

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/(10*x^2 + x - 3), x)

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Sympy [A]  time = 2.35217, size = 112, normalized size = 1.62 \[ - \frac{3^{m} m \left (x + \frac{2}{3}\right )^{m} \Phi \left (\frac{7}{6 \left (x + \frac{2}{3}\right )}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{11 \Gamma \left (- m + 1\right )} + \frac{3^{m} m \left (x + \frac{2}{3}\right )^{m} \Phi \left (\frac{1}{15 \left (x + \frac{2}{3}\right )}, 1, - m + 1\right ) \Gamma \left (- m + 1\right )}{165 \left (x + \frac{2}{3}\right ) \Gamma \left (- m + 2\right )} - \frac{3^{m} \left (x + \frac{2}{3}\right )^{m} \Phi \left (\frac{1}{15 \left (x + \frac{2}{3}\right )}, 1, - m + 1\right ) \Gamma \left (- m + 1\right )}{165 \left (x + \frac{2}{3}\right ) \Gamma \left (- m + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3**m*m*(x + 2/3)**m*lerchphi(7/(6*(x + 2/3)), 1, m*exp_polar(I*pi))*gamma(-m)/(
11*gamma(-m + 1)) + 3**m*m*(x + 2/3)**m*lerchphi(1/(15*(x + 2/3)), 1, -m + 1)*ga
mma(-m + 1)/(165*(x + 2/3)*gamma(-m + 2)) - 3**m*(x + 2/3)**m*lerchphi(1/(15*(x
+ 2/3)), 1, -m + 1)*gamma(-m + 1)/(165*(x + 2/3)*gamma(-m + 2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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