3.3163 \(\int \frac{(1-2 x) (2+3 x)^m}{(3+5 x)^2} \, dx\)

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

[Out]

(-11*(2 + 3*x)^(1 + m))/(5*(3 + 5*x)) + ((2 - 33*m)*(2 + 3*x)^(1 + m)*Hypergeome
tric2F1[1, 1 + m, 2 + m, 5*(2 + 3*x)])/(5*(1 + m))

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Rubi [A]  time = 0.0667315, 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 \[ \frac{(2-33 m) (3 x+2)^{m+1} \, _2F_1(1,m+1;m+2;5 (3 x+2))}{5 (m+1)}-\frac{11 (3 x+2)^{m+1}}{5 (5 x+3)} \]

Antiderivative was successfully verified.

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

[Out]

(-11*(2 + 3*x)^(1 + m))/(5*(3 + 5*x)) + ((2 - 33*m)*(2 + 3*x)^(1 + m)*Hypergeome
tric2F1[1, 1 + m, 2 + m, 5*(2 + 3*x)])/(5*(1 + m))

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Rubi in Sympy [A]  time = 6.98372, size = 42, normalized size = 0.71 \[ \frac{\left (- 33 m + 2\right ) \left (3 x + 2\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{15 x + 10} \right )}}{5 \left (m + 1\right )} - \frac{11 \left (3 x + 2\right )^{m + 1}}{5 \left (5 x + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.073326, size = 97, normalized size = 1.64 \[ \frac{(3 x+2)^m \left (\frac{1}{15 x+9}+1\right )^{-m} \left (11 m \, _2F_1\left (1-m,-m;2-m;-\frac{1}{15 x+9}\right )-2 (m-1) (5 x+3) \, _2F_1\left (-m,-m;1-m;-\frac{1}{15 x+9}\right )\right )}{25 (m-1) m (5 x+3)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.056, size = 0, normalized size = 0. \[ \int{\frac{ \left ( 1-2\,x \right ) \left ( 2+3\,x \right ) ^{m}}{ \left ( 3+5\,x \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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