3.3162 \(\int \frac{(1-2 x) (2+3 x)^m}{3+5 x} \, dx\)

Optimal. Leaf size=52 \[ -\frac{11 (3 x+2)^{m+1} \, _2F_1(1,m+1;m+2;5 (3 x+2))}{5 (m+1)}-\frac{2 (3 x+2)^{m+1}}{15 (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.051404, antiderivative size = 52, 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 \[ -\frac{11 (3 x+2)^{m+1} \, _2F_1(1,m+1;m+2;5 (3 x+2))}{5 (m+1)}-\frac{2 (3 x+2)^{m+1}}{15 (m+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 6.5881, size = 41, normalized size = 0.79 \[ - \frac{11 \left (3 x + 2\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{15 x + 10} \right )}}{5 \left (m + 1\right )} - \frac{2 \left (3 x + 2\right )^{m + 1}}{15 \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.114688, size = 63, normalized size = 1.21 \[ \frac{1}{75} (3 x+2)^m \left (\frac{33 \left (\frac{1}{15 x+9}+1\right )^{-m} \, _2F_1\left (-m,-m;1-m;-\frac{1}{15 x+9}\right )}{m}-\frac{10 (3 x+2)}{m+1}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((2 + 3*x)^m*((-10*(2 + 3*x))/(1 + m) + (33*Hypergeometric2F1[-m, -m, 1 - m, -(9
+ 15*x)^(-1)])/(m*(1 + (9 + 15*x)^(-1))^m)))/75

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Maple [F]  time = 0.047, size = 0, normalized size = 0. \[ \int{\frac{ \left ( 1-2\,x \right ) \left ( 2+3\,x \right ) ^{m}}{3+5\,x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(-(3*x + 2)**m/(5*x + 3), x) - Integral(2*x*(3*x + 2)**m/(5*x + 3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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