3.3166 \(\int \frac{(2+3 x)^m (3+5 x)^2}{1-2 x} \, dx\)

Optimal. Leaf size=72 \[ \frac{121 (3 x+2)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{2}{7} (3 x+2)\right )}{28 (m+1)}-\frac{155 (3 x+2)^{m+1}}{36 (m+1)}-\frac{25 (3 x+2)^{m+2}}{18 (m+2)} \]

[Out]

(-155*(2 + 3*x)^(1 + m))/(36*(1 + m)) - (25*(2 + 3*x)^(2 + m))/(18*(2 + m)) + (1
21*(2 + 3*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (2*(2 + 3*x))/7])/(28*(1
+ m))

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Rubi [A]  time = 0.081636, antiderivative size = 72, 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 \[ \frac{121 (3 x+2)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{2}{7} (3 x+2)\right )}{28 (m+1)}-\frac{155 (3 x+2)^{m+1}}{36 (m+1)}-\frac{25 (3 x+2)^{m+2}}{18 (m+2)} \]

Antiderivative was successfully verified.

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

[Out]

(-155*(2 + 3*x)^(1 + m))/(36*(1 + m)) - (25*(2 + 3*x)^(2 + m))/(18*(2 + m)) + (1
21*(2 + 3*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, (2*(2 + 3*x))/7])/(28*(1
+ m))

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Rubi in Sympy [A]  time = 10.2567, size = 58, normalized size = 0.81 \[ - \frac{25 \left (3 x + 2\right )^{m + 2}}{18 \left (m + 2\right )} + \frac{121 \left (3 x + 2\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{6 x}{7} + \frac{4}{7}} \right )}}{28 \left (m + 1\right )} - \frac{155 \left (3 x + 2\right )^{m + 1}}{36 \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-25*(3*x + 2)**(m + 2)/(18*(m + 2)) + 121*(3*x + 2)**(m + 1)*hyper((1, m + 1), (
m + 2,), 6*x/7 + 4/7)/(28*(m + 1)) - 155*(3*x + 2)**(m + 1)/(36*(m + 1))

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Mathematica [A]  time = 0.448318, size = 106, normalized size = 1.47 \[ \frac{1}{72} (3 x+2)^m \left (-\frac{1089 \left (\frac{6 x+4}{6 x-3}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{7}{3-6 x}\right )}{m}-\frac{5 \left (6 m \left (30 x^2+71 x+34\right )+5\ 7^{m+2} (6 x+4)^{-m}+180 x^2+612 x+328\right )}{m^2+3 m+2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((2 + 3*x)^m*((-5*(328 + 612*x + 180*x^2 + (5*7^(2 + m))/(4 + 6*x)^m + 6*m*(34 +
71*x + 30*x^2)))/(2 + 3*m + m^2) - (1089*Hypergeometric2F1[-m, -m, 1 - m, 7/(3
- 6*x)])/(m*((4 + 6*x)/(-3 + 6*x))^m)))/72

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Maple [F]  time = 0.064, size = 0, normalized size = 0. \[ \int{\frac{ \left ( 2+3\,x \right ) ^{m} \left ( 3+5\,x \right ) ^{2}}{1-2\,x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(3*x + 2)^m*(5*x + 3)^2/(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. \[{\rm integral}\left (-\frac{{\left (25 \, x^{2} + 30 \, x + 9\right )}{\left (3 \, x + 2\right )}^{m}}{2 \, x - 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(25*x^2 + 30*x + 9)*(3*x + 2)^m/(2*x - 1), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{9 \left (3 x + 2\right )^{m}}{2 x - 1}\, dx - \int \frac{30 x \left (3 x + 2\right )^{m}}{2 x - 1}\, dx - \int \frac{25 x^{2} \left (3 x + 2\right )^{m}}{2 x - 1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(9*(3*x + 2)**m/(2*x - 1), x) - Integral(30*x*(3*x + 2)**m/(2*x - 1), x
) - Integral(25*x**2*(3*x + 2)**m/(2*x - 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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