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

Optimal. Leaf size=158 \[ \frac{2^n 9^{n-1} x^{m+1} \, _2F_1\left (\frac{m+1}{2},2-n;\frac{m+3}{2};\frac{4 a^2 x^2}{9}\right )}{m+1}+\frac{a 2^{n+2} 3^{2 n-3} x^{m+2} \, _2F_1\left (\frac{m+2}{2},2-n;\frac{m+4}{2};\frac{4 a^2 x^2}{9}\right )}{m+2}+\frac{a^2 2^{n+2} 9^{n-2} x^{m+3} \, _2F_1\left (\frac{m+3}{2},2-n;\frac{m+5}{2};\frac{4 a^2 x^2}{9}\right )}{m+3} \]

[Out]

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

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Rubi [A]  time = 0.335826, antiderivative size = 158, 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 \[ \frac{2^n 9^{n-1} x^{m+1} \, _2F_1\left (\frac{m+1}{2},2-n;\frac{m+3}{2};\frac{4 a^2 x^2}{9}\right )}{m+1}+\frac{a 2^{n+2} 3^{2 n-3} x^{m+2} \, _2F_1\left (\frac{m+2}{2},2-n;\frac{m+4}{2};\frac{4 a^2 x^2}{9}\right )}{m+2}+\frac{a^2 2^{n+2} 9^{n-2} x^{m+3} \, _2F_1\left (\frac{m+3}{2},2-n;\frac{m+5}{2};\frac{4 a^2 x^2}{9}\right )}{m+3} \]

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

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Rubi in Sympy [A]  time = 27.1876, size = 117, normalized size = 0.74 \[ \frac{16 \cdot 18^{n - 2} a^{2} x^{m + 3}{{}_{2}F_{1}\left (\begin{matrix} - n + 2, \frac{m}{2} + \frac{3}{2} \\ \frac{m}{2} + \frac{5}{2} \end{matrix}\middle |{\frac{4 a^{2} x^{2}}{9}} \right )}}{m + 3} + \frac{48 \cdot 18^{n - 2} a x^{m + 2}{{}_{2}F_{1}\left (\begin{matrix} - n + 2, \frac{m}{2} + 1 \\ \frac{m}{2} + 2 \end{matrix}\middle |{\frac{4 a^{2} x^{2}}{9}} \right )}}{m + 2} + \frac{36 \cdot 18^{n - 2} x^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} - n + 2, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{\frac{4 a^{2} x^{2}}{9}} \right )}}{m + 1} \]

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]

16*18**(n - 2)*a**2*x**(m + 3)*hyper((-n + 2, m/2 + 3/2), (m/2 + 5/2,), 4*a**2*x
**2/9)/(m + 3) + 48*18**(n - 2)*a*x**(m + 2)*hyper((-n + 2, m/2 + 1), (m/2 + 2,)
, 4*a**2*x**2/9)/(m + 2) + 36*18**(n - 2)*x**(m + 1)*hyper((-n + 2, m/2 + 1/2),
(m/2 + 3/2,), 4*a**2*x**2/9)/(m + 1)

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Mathematica [C]  time = 0.44254, size = 163, normalized size = 1.03 \[ \frac{3 (m+2) x^{m+1} (3-2 a x)^{n-2} (4 a x+6)^n F_1\left (m+1;2-n,-n;m+2;\frac{2 a x}{3},-\frac{2 a x}{3}\right )}{(m+1) \left (3 (m+2) F_1\left (m+1;2-n,-n;m+2;\frac{2 a x}{3},-\frac{2 a x}{3}\right )+2 a x \left (n F_1\left (m+2;2-n,1-n;m+3;\frac{2 a x}{3},-\frac{2 a x}{3}\right )-(n-2) F_1\left (m+2;3-n,-n;m+3;\frac{2 a x}{3},-\frac{2 a x}{3}\right )\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(3*(2 + m)*x^(1 + m)*(3 - 2*a*x)^(-2 + n)*(6 + 4*a*x)^n*AppellF1[1 + m, 2 - n, -
n, 2 + m, (2*a*x)/3, (-2*a*x)/3])/((1 + m)*(3*(2 + m)*AppellF1[1 + m, 2 - n, -n,
2 + m, (2*a*x)/3, (-2*a*x)/3] + 2*a*x*(n*AppellF1[2 + m, 2 - n, 1 - n, 3 + m, (
2*a*x)/3, (-2*a*x)/3] - (-2 + n)*AppellF1[2 + m, 3 - n, -n, 3 + m, (2*a*x)/3, (-
2*a*x)/3])))

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Maple [F]  time = 0.234, 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)