3.3165 \(\int \frac{(2+3 x)^m (3+5 x)^3}{1-2 x} \, dx\)

Optimal. Leaf size=90 \[ \frac{1331 (3 x+2)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{2}{7} (3 x+2)\right )}{56 (m+1)}-\frac{5135 (3 x+2)^{m+1}}{216 (m+1)}-\frac{725 (3 x+2)^{m+2}}{108 (m+2)}-\frac{125 (3 x+2)^{m+3}}{54 (m+3)} \]

[Out]

(-5135*(2 + 3*x)^(1 + m))/(216*(1 + m)) - (725*(2 + 3*x)^(2 + m))/(108*(2 + m))
- (125*(2 + 3*x)^(3 + m))/(54*(3 + m)) + (1331*(2 + 3*x)^(1 + m)*Hypergeometric2
F1[1, 1 + m, 2 + m, (2*(2 + 3*x))/7])/(56*(1 + m))

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Rubi [A]  time = 0.0934411, antiderivative size = 90, 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 \[ \frac{1331 (3 x+2)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{2}{7} (3 x+2)\right )}{56 (m+1)}-\frac{5135 (3 x+2)^{m+1}}{216 (m+1)}-\frac{725 (3 x+2)^{m+2}}{108 (m+2)}-\frac{125 (3 x+2)^{m+3}}{54 (m+3)} \]

Antiderivative was successfully verified.

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

[Out]

(-5135*(2 + 3*x)^(1 + m))/(216*(1 + m)) - (725*(2 + 3*x)^(2 + m))/(108*(2 + m))
- (125*(2 + 3*x)^(3 + m))/(54*(3 + m)) + (1331*(2 + 3*x)^(1 + m)*Hypergeometric2
F1[1, 1 + m, 2 + m, (2*(2 + 3*x))/7])/(56*(1 + m))

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Rubi in Sympy [A]  time = 11.6814, size = 73, normalized size = 0.81 \[ - \frac{125 \left (3 x + 2\right )^{m + 3}}{54 \left (m + 3\right )} - \frac{725 \left (3 x + 2\right )^{m + 2}}{108 \left (m + 2\right )} + \frac{1331 \left (3 x + 2\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{6 x}{7} + \frac{4}{7}} \right )}}{56 \left (m + 1\right )} - \frac{5135 \left (3 x + 2\right )^{m + 1}}{216 \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-125*(3*x + 2)**(m + 3)/(54*(m + 3)) - 725*(3*x + 2)**(m + 2)/(108*(m + 2)) + 13
31*(3*x + 2)**(m + 1)*hyper((1, m + 1), (m + 2,), 6*x/7 + 4/7)/(56*(m + 1)) - 51
35*(3*x + 2)**(m + 1)/(216*(m + 1))

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Mathematica [B]  time = 0.569363, size = 240, normalized size = 2.67 \[ \frac{1}{432} (3 x+2)^m \left (-\frac{35937 \left (\frac{6 x+4}{6 x-3}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{7}{3-6 x}\right )}{m}+\frac{2475 \left (-6 m \left (6 x^2+x-2\right )-7^{m+2} (6 x+4)^{-m}-36 x^2+36 x+40\right )}{m^2+3 m+2}+\frac{250 \left (-9 m^2 (1-2 x)^2 (3 x+2) (6 x+4)^m-216 x^3 (6 x+4)^m+324 x^2 (6 x+4)^m-3 m \left (108 x^3-120 x^2-59 x+46\right ) (6 x+4)^m-162 x (6 x+4)^m-316 (6 x+4)^m+7^{m+3}\right ) (6 x+4)^{-m}}{(m+1) (m+2) (m+3)}-\frac{32670 (3 x+2)}{m+1}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((2 + 3*x)^m*((-32670*(2 + 3*x))/(1 + m) + (2475*(40 + 36*x - 36*x^2 - 7^(2 + m)
/(4 + 6*x)^m - 6*m*(-2 + x + 6*x^2)))/(2 + 3*m + m^2) + (250*(7^(3 + m) - 316*(4
+ 6*x)^m - 162*x*(4 + 6*x)^m + 324*x^2*(4 + 6*x)^m - 216*x^3*(4 + 6*x)^m - 9*m^
2*(1 - 2*x)^2*(2 + 3*x)*(4 + 6*x)^m - 3*m*(4 + 6*x)^m*(46 - 59*x - 120*x^2 + 108
*x^3)))/((1 + m)*(2 + m)*(3 + m)*(4 + 6*x)^m) - (35937*Hypergeometric2F1[-m, -m,
1 - m, 7/(3 - 6*x)])/(m*((4 + 6*x)/(-3 + 6*x))^m)))/432

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Maple [F]  time = 0.073, size = 0, normalized size = 0. \[ \int{\frac{ \left ( 2+3\,x \right ) ^{m} \left ( 3+5\,x \right ) ^{3}}{1-2\,x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{{\left (125 \, x^{3} + 225 \, x^{2} + 135 \, x + 27\right )}{\left (3 \, x + 2\right )}^{m}}{2 \, x - 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(125*x^3 + 225*x^2 + 135*x + 27)*(3*x + 2)^m/(2*x - 1), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{27 \left (3 x + 2\right )^{m}}{2 x - 1}\, dx - \int \frac{135 x \left (3 x + 2\right )^{m}}{2 x - 1}\, dx - \int \frac{225 x^{2} \left (3 x + 2\right )^{m}}{2 x - 1}\, dx - \int \frac{125 x^{3} \left (3 x + 2\right )^{m}}{2 x - 1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(27*(3*x + 2)**m/(2*x - 1), x) - Integral(135*x*(3*x + 2)**m/(2*x - 1),
x) - Integral(225*x**2*(3*x + 2)**m/(2*x - 1), x) - Integral(125*x**3*(3*x + 2)
**m/(2*x - 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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