3.10.90 \(\int x^m (3-2 a x)^{2+n} (6+4 a x)^n \, dx\) [990]

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

[Out]

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

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Rubi [A]
time = 0.07, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {137, 126, 371} \begin {gather*} \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} \end {gather*}

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

Rule 126

Int[((f_.)*(x_))^(p_.)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[(a*c + b*d*x^2)
^m*(f*x)^p, x] /; FreeQ[{a, b, c, d, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[n, m] && GtQ[a, 0] && GtQ[c,
0]

Rule 137

Int[((f_.)*(x_))^(p_.)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand
[(a + b*x)^n*(c + d*x)^n*(f*x)^p, (a + b*x)^(m - n), x], x] /; FreeQ[{a, b, c, d, f, m, n, p}, x] && EqQ[b*c +
a*d, 0] && IGtQ[m - n, 0] && NeQ[m + n + p + 2, 0]

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin {align*} \int x^m (3-2 a x)^{2+n} (6+4 a x)^n \, dx &=\int \left (9 x^m (3-2 a x)^n (6+4 a x)^n-12 a x^{1+m} (3-2 a x)^n (6+4 a x)^n+4 a^2 x^{2+m} (3-2 a x)^n (6+4 a x)^n\right ) \, dx\\ &=9 \int x^m (3-2 a x)^n (6+4 a x)^n \, dx-(12 a) \int x^{1+m} (3-2 a x)^n (6+4 a x)^n \, dx+\left (4 a^2\right ) \int x^{2+m} (3-2 a x)^n (6+4 a x)^n \, dx\\ &=9 \int x^m \left (18-8 a^2 x^2\right )^n \, dx-(12 a) \int x^{1+m} \left (18-8 a^2 x^2\right )^n \, dx+\left (4 a^2\right ) \int x^{2+m} \left (18-8 a^2 x^2\right )^n \, dx\\ &=\frac {2^n 9^{1+n} x^{1+m} \, _2F_1\left (\frac {1+m}{2},-n;\frac {3+m}{2};\frac {4 a^2 x^2}{9}\right )}{1+m}-\frac {2^{2+n} 3^{1+2 n} a x^{2+m} \, _2F_1\left (\frac {2+m}{2},-n;\frac {4+m}{2};\frac {4 a^2 x^2}{9}\right )}{2+m}+\frac {2^{2+n} 9^n a^2 x^{3+m} \, _2F_1\left (\frac {3+m}{2},-n;\frac {5+m}{2};\frac {4 a^2 x^2}{9}\right )}{3+m}\\ \end {align*}

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Mathematica [A]
time = 0.23, size = 166, normalized size = 1.10 \begin {gather*} \frac {x^{1+m} \left (9-4 a^2 x^2\right )^n \left (\frac {1}{2}-\frac {2 a^2 x^2}{9}\right )^{-n} \left (9 \left (6+5 m+m^2\right ) \, _2F_1\left (\frac {1+m}{2},-n;\frac {3+m}{2};\frac {4 a^2 x^2}{9}\right )-4 a (1+m) x \left (3 (3+m) \, _2F_1\left (\frac {2+m}{2},-n;\frac {4+m}{2};\frac {4 a^2 x^2}{9}\right )-a (2+m) x \, _2F_1\left (\frac {3+m}{2},-n;\frac {5+m}{2};\frac {4 a^2 x^2}{9}\right )\right )\right )}{(1+m) (2+m) (3+m)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x^(1 + m)*(9 - 4*a^2*x^2)^n*(9*(6 + 5*m + m^2)*Hypergeometric2F1[(1 + m)/2, -n, (3 + m)/2, (4*a^2*x^2)/9] - 4
*a*(1 + m)*x*(3*(3 + m)*Hypergeometric2F1[(2 + m)/2, -n, (4 + m)/2, (4*a^2*x^2)/9] - a*(2 + m)*x*Hypergeometri
c2F1[(3 + m)/2, -n, (5 + m)/2, (4*a^2*x^2)/9])))/((1 + m)*(2 + m)*(3 + m)*(1/2 - (2*a^2*x^2)/9)^n)

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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int x^{m} \left (-2 a x +3\right )^{2+n} \left (4 a x +6\right )^{n}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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, 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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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, algorithm="fricas")

[Out]

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

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Sympy [C] Result contains complex when optimal does not.
time = 223.76, size = 748, normalized size = 4.95 \begin {gather*} - \frac {27 \cdot 18^{n} 4^{- \frac {m}{2}} \cdot 9^{\frac {m}{2}} a^{- m} {G_{6, 6}^{5, 3}\left (\begin {matrix} - \frac {m}{2} - \frac {n}{2}, - \frac {m}{2} - \frac {n}{2} + \frac {1}{2}, 1 & \frac {1}{2} - \frac {m}{2}, - \frac {m}{2} - n, - \frac {m}{2} - n + \frac {1}{2} \\- \frac {m}{2} - n - \frac {1}{2}, - \frac {m}{2} - n, - \frac {m}{2} - \frac {n}{2}, - \frac {m}{2} - n + \frac {1}{2}, - \frac {m}{2} - \frac {n}{2} + \frac {1}{2} & 0 \end {matrix} \middle | {\frac {9}{4 a^{2} x^{2}}} \right )} e^{i \pi n}}{8 \pi a \Gamma \left (- n\right )} - \frac {27 \cdot 18^{n} 4^{- \frac {m}{2}} \cdot 9^{\frac {m}{2}} a^{- m} {G_{6, 6}^{5, 3}\left (\begin {matrix} - \frac {m}{2} - \frac {n}{2} - 1, - \frac {m}{2} - \frac {n}{2} - \frac {1}{2}, 1 & - \frac {m}{2} - \frac {1}{2}, - \frac {m}{2} - n - 1, - \frac {m}{2} - n - \frac {1}{2} \\- \frac {m}{2} - n - \frac {3}{2}, - \frac {m}{2} - n - 1, - \frac {m}{2} - \frac {n}{2} - 1, - \frac {m}{2} - n - \frac {1}{2}, - \frac {m}{2} - \frac {n}{2} - \frac {1}{2} & 0 \end {matrix} \middle | {\frac {9}{4 a^{2} x^{2}}} \right )} e^{i \pi n}}{8 \pi a \Gamma \left (- n\right )} + \frac {27 \cdot 18^{n} 4^{- \frac {m}{2}} \cdot 9^{\frac {m}{2}} a^{- m} {G_{6, 6}^{5, 3}\left (\begin {matrix} - \frac {m}{2} - \frac {n}{2} - \frac {1}{2}, - \frac {m}{2} - \frac {n}{2}, 1 & - \frac {m}{2}, - \frac {m}{2} - n - \frac {1}{2}, - \frac {m}{2} - n \\- \frac {m}{2} - n - 1, - \frac {m}{2} - n - \frac {1}{2}, - \frac {m}{2} - \frac {n}{2} - \frac {1}{2}, - \frac {m}{2} - n, - \frac {m}{2} - \frac {n}{2} & 0 \end {matrix} \middle | {\frac {9}{4 a^{2} x^{2}}} \right )} e^{i \pi n}}{4 \pi a \Gamma \left (- n\right )} + \frac {27 \cdot 18^{n} 4^{- \frac {m}{2}} \cdot 9^{\frac {m}{2}} a^{- m} {G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {m}{2} - \frac {3}{2}, - \frac {m}{2} - 1, - \frac {m}{2} - \frac {1}{2}, - \frac {m}{2} - \frac {n}{2} - \frac {3}{2}, - \frac {m}{2} - \frac {n}{2} - 1, 1 & \\- \frac {m}{2} - \frac {n}{2} - \frac {3}{2}, - \frac {m}{2} - \frac {n}{2} - 1 & - \frac {m}{2} - \frac {3}{2}, - \frac {m}{2} - 1, - \frac {m}{2} - n - \frac {3}{2}, 0 \end {matrix} \middle | {\frac {9 e^{- 2 i \pi }}{4 a^{2} x^{2}}} \right )} e^{- i \pi m}}{8 \pi a \Gamma \left (- n\right )} + \frac {27 \cdot 18^{n} 4^{- \frac {m}{2}} \cdot 9^{\frac {m}{2}} a^{- m} {G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {m}{2} - 1, - \frac {m}{2} - \frac {1}{2}, - \frac {m}{2}, - \frac {m}{2} - \frac {n}{2} - 1, - \frac {m}{2} - \frac {n}{2} - \frac {1}{2}, 1 & \\- \frac {m}{2} - \frac {n}{2} - 1, - \frac {m}{2} - \frac {n}{2} - \frac {1}{2} & - \frac {m}{2} - 1, - \frac {m}{2} - \frac {1}{2}, - \frac {m}{2} - n - 1, 0 \end {matrix} \middle | {\frac {9 e^{- 2 i \pi }}{4 a^{2} x^{2}}} \right )} e^{- i \pi m}}{4 \pi a \Gamma \left (- n\right )} + \frac {27 \cdot 18^{n} 4^{- \frac {m}{2}} \cdot 9^{\frac {m}{2}} a^{- m} {G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {m}{2} - \frac {1}{2}, - \frac {m}{2}, \frac {1}{2} - \frac {m}{2}, - \frac {m}{2} - \frac {n}{2} - \frac {1}{2}, - \frac {m}{2} - \frac {n}{2}, 1 & \\- \frac {m}{2} - \frac {n}{2} - \frac {1}{2}, - \frac {m}{2} - \frac {n}{2} & - \frac {m}{2} - \frac {1}{2}, - \frac {m}{2}, - \frac {m}{2} - n - \frac {1}{2}, 0 \end {matrix} \middle | {\frac {9 e^{- 2 i \pi }}{4 a^{2} x^{2}}} \right )} e^{- i \pi m}}{8 \pi a \Gamma \left (- n\right )} \end {gather*}

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]

-27*18**n*9**(m/2)*meijerg(((-m/2 - n/2, -m/2 - n/2 + 1/2, 1), (1/2 - m/2, -m/2 - n, -m/2 - n + 1/2)), ((-m/2
- n - 1/2, -m/2 - n, -m/2 - n/2, -m/2 - n + 1/2, -m/2 - n/2 + 1/2), (0,)), 9/(4*a**2*x**2))*exp(I*pi*n)/(8*4**
(m/2)*pi*a*a**m*gamma(-n)) - 27*18**n*9**(m/2)*meijerg(((-m/2 - n/2 - 1, -m/2 - n/2 - 1/2, 1), (-m/2 - 1/2, -m
/2 - n - 1, -m/2 - n - 1/2)), ((-m/2 - n - 3/2, -m/2 - n - 1, -m/2 - n/2 - 1, -m/2 - n - 1/2, -m/2 - n/2 - 1/2
), (0,)), 9/(4*a**2*x**2))*exp(I*pi*n)/(8*4**(m/2)*pi*a*a**m*gamma(-n)) + 27*18**n*9**(m/2)*meijerg(((-m/2 - n
/2 - 1/2, -m/2 - n/2, 1), (-m/2, -m/2 - n - 1/2, -m/2 - n)), ((-m/2 - n - 1, -m/2 - n - 1/2, -m/2 - n/2 - 1/2,
-m/2 - n, -m/2 - n/2), (0,)), 9/(4*a**2*x**2))*exp(I*pi*n)/(4*4**(m/2)*pi*a*a**m*gamma(-n)) + 27*18**n*9**(m/
2)*meijerg(((-m/2 - 3/2, -m/2 - 1, -m/2 - 1/2, -m/2 - n/2 - 3/2, -m/2 - n/2 - 1, 1), ()), ((-m/2 - n/2 - 3/2,
-m/2 - n/2 - 1), (-m/2 - 3/2, -m/2 - 1, -m/2 - n - 3/2, 0)), 9*exp_polar(-2*I*pi)/(4*a**2*x**2))*exp(-I*pi*m)/
(8*4**(m/2)*pi*a*a**m*gamma(-n)) + 27*18**n*9**(m/2)*meijerg(((-m/2 - 1, -m/2 - 1/2, -m/2, -m/2 - n/2 - 1, -m/
2 - n/2 - 1/2, 1), ()), ((-m/2 - n/2 - 1, -m/2 - n/2 - 1/2), (-m/2 - 1, -m/2 - 1/2, -m/2 - n - 1, 0)), 9*exp_p
olar(-2*I*pi)/(4*a**2*x**2))*exp(-I*pi*m)/(4*4**(m/2)*pi*a*a**m*gamma(-n)) + 27*18**n*9**(m/2)*meijerg(((-m/2
- 1/2, -m/2, 1/2 - m/2, -m/2 - n/2 - 1/2, -m/2 - n/2, 1), ()), ((-m/2 - n/2 - 1/2, -m/2 - n/2), (-m/2 - 1/2, -
m/2, -m/2 - n - 1/2, 0)), 9*exp_polar(-2*I*pi)/(4*a**2*x**2))*exp(-I*pi*m)/(8*4**(m/2)*pi*a*a**m*gamma(-n))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^m\,{\left (3-2\,a\,x\right )}^{n+2}\,{\left (4\,a\,x+6\right )}^n \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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