3.979 \(\int x^m (3-2 a x)^{1+n} (6+4 a x)^n \, dx\)

Optimal. Leaf size=99 \[ \frac{2^n 3^{2 n+1} x^{m+1} \, _2F_1\left (\frac{m+1}{2},-n;\frac{m+3}{2};\frac{4 a^2 x^2}{9}\right )}{m+1}-\frac{a 2^{n+1} 9^n x^{m+2} \, _2F_1\left (\frac{m+2}{2},-n;\frac{m+4}{2};\frac{4 a^2 x^2}{9}\right )}{m+2} \]

[Out]

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

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Rubi [A]  time = 0.136241, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{2^n 3^{2 n+1} x^{m+1} \, _2F_1\left (\frac{m+1}{2},-n;\frac{m+3}{2};\frac{4 a^2 x^2}{9}\right )}{m+1}-\frac{a 2^{n+1} 9^n x^{m+2} \, _2F_1\left (\frac{m+2}{2},-n;\frac{m+4}{2};\frac{4 a^2 x^2}{9}\right )}{m+2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 15.5775, size = 71, normalized size = 0.72 \[ - \frac{2 \cdot 18^{n} a x^{m + 2}{{}_{2}F_{1}\left (\begin{matrix} - n, \frac{m}{2} + 1 \\ \frac{m}{2} + 2 \end{matrix}\middle |{\frac{4 a^{2} x^{2}}{9}} \right )}}{m + 2} + \frac{3 \cdot 18^{n} x^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} - n, \frac{m}{2} + \frac{1}{2} \\ \frac{m}{2} + \frac{3}{2} \end{matrix}\middle |{\frac{4 a^{2} x^{2}}{9}} \right )}}{m + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**m*(-2*a*x+3)**(1+n)*(4*a*x+6)**n,x)

[Out]

-2*18**n*a*x**(m + 2)*hyper((-n, m/2 + 1), (m/2 + 2,), 4*a**2*x**2/9)/(m + 2) +
3*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 = 0.147894, size = 117, normalized size = 1.18 \[ \frac{x^{m+1} (3-2 a x)^n (4 a x+6)^n \left (1-\frac{4 a^2 x^2}{9}\right )^{-n} \left (3 (m+2) \, _2F_1\left (\frac{m+1}{2},-n;\frac{m+3}{2};\frac{4 a^2 x^2}{9}\right )-2 a (m+1) x \, _2F_1\left (\frac{m}{2}+1,-n;\frac{m}{2}+2;\frac{4 a^2 x^2}{9}\right )\right )}{(m+1) (m+2)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.24, size = 0, normalized size = 0. \[ \int{x}^{m} \left ( -2\,ax+3 \right ) ^{1+n} \left ( 4\,ax+6 \right ) ^{n}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int(x^m*(-2*a*x+3)^(1+n)*(4*a*x+6)^n,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (4 \, a x + 6\right )}^{n}{\left (-2 \, a x + 3\right )}^{n + 1} x^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*a*x + 6)^n*(-2*a*x + 3)^(n + 1)*x^m,x, algorithm="maxima")

[Out]

integrate((4*a*x + 6)^n*(-2*a*x + 3)^(n + 1)*x^m, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (4 \, a x + 6\right )}^{n}{\left (-2 \, a x + 3\right )}^{n + 1} x^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*a*x + 6)^n*(-2*a*x + 3)^(n + 1)*x^m,x, algorithm="fricas")

[Out]

integral((4*a*x + 6)^n*(-2*a*x + 3)^(n + 1)*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)**(1+n)*(4*a*x+6)**n,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (4 \, a x + 6\right )}^{n}{\left (-2 \, a x + 3\right )}^{n + 1} x^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*a*x + 6)^n*(-2*a*x + 3)^(n + 1)*x^m,x, algorithm="giac")

[Out]

integrate((4*a*x + 6)^n*(-2*a*x + 3)^(n + 1)*x^m, x)