3.978 \(\int x^m (3-2 a x)^{2+n} (6+4 a x)^n \, dx\)

Optimal. Leaf size=151 \[ \frac{2^n 9^{n+1} x^{m+1} \, _2F_1\left (\frac{m+1}{2},-n;\frac{m+3}{2};\frac{4 a^2 x^2}{9}\right )}{m+1}-\frac{a 2^{n+2} 3^{2 n+1} x^{m+2} \, _2F_1\left (\frac{m+2}{2},-n;\frac{m+4}{2};\frac{4 a^2 x^2}{9}\right )}{m+2}+\frac{a^2 2^{n+2} 9^n x^{m+3} \, _2F_1\left (\frac{m+3}{2},-n;\frac{m+5}{2};\frac{4 a^2 x^2}{9}\right )}{m+3} \]

[Out]

(2^n*9^(1 + n)*x^(1 + m)*Hypergeometric2F1[(1 + m)/2, -n, (3 + m)/2, (4*a^2*x^2)
/9])/(1 + m) - (2^(2 + n)*3^(1 + 2*n)*a*x^(2 + m)*Hypergeometric2F1[(2 + m)/2, -
n, (4 + m)/2, (4*a^2*x^2)/9])/(2 + m) + (2^(2 + n)*9^n*a^2*x^(3 + m)*Hypergeomet
ric2F1[(3 + m)/2, -n, (5 + m)/2, (4*a^2*x^2)/9])/(3 + m)

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Rubi [A]  time = 0.250406, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{2^n 9^{n+1} x^{m+1} \, _2F_1\left (\frac{m+1}{2},-n;\frac{m+3}{2};\frac{4 a^2 x^2}{9}\right )}{m+1}-\frac{a 2^{n+2} 3^{2 n+1} x^{m+2} \, _2F_1\left (\frac{m+2}{2},-n;\frac{m+4}{2};\frac{4 a^2 x^2}{9}\right )}{m+2}+\frac{a^2 2^{n+2} 9^n x^{m+3} \, _2F_1\left (\frac{m+3}{2},-n;\frac{m+5}{2};\frac{4 a^2 x^2}{9}\right )}{m+3} \]

Antiderivative was successfully verified.

[In]  Int[x^m*(3 - 2*a*x)^(2 + n)*(6 + 4*a*x)^n,x]

[Out]

(2^n*9^(1 + n)*x^(1 + m)*Hypergeometric2F1[(1 + m)/2, -n, (3 + m)/2, (4*a^2*x^2)
/9])/(1 + m) - (2^(2 + n)*3^(1 + 2*n)*a*x^(2 + m)*Hypergeometric2F1[(2 + m)/2, -
n, (4 + m)/2, (4*a^2*x^2)/9])/(2 + m) + (2^(2 + n)*9^n*a^2*x^(3 + m)*Hypergeomet
ric2F1[(3 + m)/2, -n, (5 + m)/2, (4*a^2*x^2)/9])/(3 + m)

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Rubi in Sympy [A]  time = 40.3112, size = 112, normalized size = 0.74 \[ \frac{4 \cdot 18^{n} a^{2} x^{m + 3}{{}_{2}F_{1}\left (\begin{matrix} - n, \frac{m}{2} + \frac{3}{2} \\ \frac{m}{2} + \frac{5}{2} \end{matrix}\middle |{\frac{4 a^{2} x^{2}}{9}} \right )}}{m + 3} - \frac{12 \cdot 18^{n} a x^{m + 2}{{}_{2}F_{1}\left (\begin{matrix} - n, \frac{m}{2} + 1 \\ \frac{m}{2} + 2 \end{matrix}\middle |{\frac{4 a^{2} x^{2}}{9}} \right )}}{m + 2} + \frac{9 \cdot 18^{n} x^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} - n, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{\frac{4 a^{2} x^{2}}{9}} \right )}}{m + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**m*(-2*a*x+3)**(2+n)*(4*a*x+6)**n,x)

[Out]

4*18**n*a**2*x**(m + 3)*hyper((-n, m/2 + 3/2), (m/2 + 5/2,), 4*a**2*x**2/9)/(m +
3) - 12*18**n*a*x**(m + 2)*hyper((-n, m/2 + 1), (m/2 + 2,), 4*a**2*x**2/9)/(m +
2) + 9*18**n*x**(m + 1)*hyper((-n, m/2 + 1/2), (m/2 + 3/2,), 4*a**2*x**2/9)/(m
+ 1)

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Mathematica [A]  time = 0.213474, size = 171, normalized size = 1.13 \[ \frac{x^{m+1} (3-2 a x)^n (4 a x+6)^n \left (1-\frac{4 a^2 x^2}{9}\right )^{-n} \left ((m+2) \left (4 a^2 (m+1) x^2 \, _2F_1\left (\frac{m+3}{2},-n;\frac{m+5}{2};\frac{4 a^2 x^2}{9}\right )+9 (m+3) \, _2F_1\left (\frac{m+1}{2},-n;\frac{m+3}{2};\frac{4 a^2 x^2}{9}\right )\right )-12 a \left (m^2+4 m+3\right ) x \, _2F_1\left (\frac{m}{2}+1,-n;\frac{m}{2}+2;\frac{4 a^2 x^2}{9}\right )\right )}{(m+1) (m+2) (m+3)} \]

Antiderivative was successfully verified.

[In]  Integrate[x^m*(3 - 2*a*x)^(2 + n)*(6 + 4*a*x)^n,x]

[Out]

(x^(1 + m)*(3 - 2*a*x)^n*(6 + 4*a*x)^n*(-12*a*(3 + 4*m + m^2)*x*Hypergeometric2F
1[1 + m/2, -n, 2 + m/2, (4*a^2*x^2)/9] + (2 + m)*(9*(3 + m)*Hypergeometric2F1[(1
+ m)/2, -n, (3 + m)/2, (4*a^2*x^2)/9] + 4*a^2*(1 + m)*x^2*Hypergeometric2F1[(3
+ m)/2, -n, (5 + m)/2, (4*a^2*x^2)/9])))/((1 + m)*(2 + m)*(3 + m)*(1 - (4*a^2*x^
2)/9)^n)

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Maple [F]  time = 0.24, size = 0, normalized size = 0. \[ \int{x}^{m} \left ( -2\,ax+3 \right ) ^{2+n} \left ( 4\,ax+6 \right ) ^{n}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m*(-2*a*x+3)^(2+n)*(4*a*x+6)^n,x)

[Out]

int(x^m*(-2*a*x+3)^(2+n)*(4*a*x+6)^n,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (4 \, a x + 6\right )}^{n}{\left (-2 \, a x + 3\right )}^{n + 2} x^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*a*x + 6)^n*(-2*a*x + 3)^(n + 2)*x^m,x, algorithm="maxima")

[Out]

integrate((4*a*x + 6)^n*(-2*a*x + 3)^(n + 2)*x^m, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (4 \, a x + 6\right )}^{n}{\left (-2 \, a x + 3\right )}^{n + 2} x^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*a*x + 6)^n*(-2*a*x + 3)^(n + 2)*x^m,x, algorithm="fricas")

[Out]

integral((4*a*x + 6)^n*(-2*a*x + 3)^(n + 2)*x^m, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**m*(-2*a*x+3)**(2+n)*(4*a*x+6)**n,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (4 \, a x + 6\right )}^{n}{\left (-2 \, a x + 3\right )}^{n + 2} x^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*a*x + 6)^n*(-2*a*x + 3)^(n + 2)*x^m,x, algorithm="giac")

[Out]

integrate((4*a*x + 6)^n*(-2*a*x + 3)^(n + 2)*x^m, x)