\(\int x^m (3-2 a x)^{-2+n} (6+4 a x)^n \, dx\) [994]

Optimal result

Integrand size = 22, antiderivative size = 158 \[ \int x^m (3-2 a x)^{-2+n} (6+4 a x)^n \, dx=\frac {2^n 9^{-1+n} x^{1+m} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},2-n,\frac {3+m}{2},\frac {4 a^2 x^2}{9}\right )}{1+m}+\frac {2^{2+n} 3^{-3+2 n} a x^{2+m} \operatorname {Hypergeometric2F1}\left (\frac {2+m}{2},2-n,\frac {4+m}{2},\frac {4 a^2 x^2}{9}\right )}{2+m}+\frac {2^{2+n} 9^{-2+n} a^2 x^{3+m} \operatorname {Hypergeometric2F1}\left (\frac {3+m}{2},2-n,\frac {5+m}{2},\frac {4 a^2 x^2}{9}\right )}{3+m} \]

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {137, 126, 371} \[ \int x^m (3-2 a x)^{-2+n} (6+4 a x)^n \, dx=\frac {2^n 9^{n-1} x^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {m+1}{2},2-n,\frac {m+3}{2},\frac {4 a^2 x^2}{9}\right )}{m+1}+\frac {a 2^{n+2} 3^{2 n-3} x^{m+2} \operatorname {Hypergeometric2F1}\left (\frac {m+2}{2},2-n,\frac {m+4}{2},\frac {4 a^2 x^2}{9}\right )}{m+2}+\frac {a^2 2^{n+2} 9^{n-2} x^{m+3} \operatorname {Hypergeometric2F1}\left (\frac {m+3}{2},2-n,\frac {m+5}{2},\frac {4 a^2 x^2}{9}\right )}{m+3} \]

[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)

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*} \text {integral}& = \int \left (36 x^m (3-2 a x)^{-2+n} (6+4 a x)^{-2+n}+48 a x^{1+m} (3-2 a x)^{-2+n} (6+4 a x)^{-2+n}+16 a^2 x^{2+m} (3-2 a x)^{-2+n} (6+4 a x)^{-2+n}\right ) \, dx \\ & = 36 \int x^m (3-2 a x)^{-2+n} (6+4 a x)^{-2+n} \, dx+(48 a) \int x^{1+m} (3-2 a x)^{-2+n} (6+4 a x)^{-2+n} \, dx+\left (16 a^2\right ) \int x^{2+m} (3-2 a x)^{-2+n} (6+4 a x)^{-2+n} \, dx \\ & = 36 \int x^m \left (18-8 a^2 x^2\right )^{-2+n} \, dx+(48 a) \int x^{1+m} \left (18-8 a^2 x^2\right )^{-2+n} \, dx+\left (16 a^2\right ) \int x^{2+m} \left (18-8 a^2 x^2\right )^{-2+n} \, dx \\ & = \frac {2^n 9^{-1+n} x^{1+m} \, _2F_1\left (\frac {1+m}{2},2-n;\frac {3+m}{2};\frac {4 a^2 x^2}{9}\right )}{1+m}+\frac {2^{2+n} 3^{-3+2 n} a x^{2+m} \, _2F_1\left (\frac {2+m}{2},2-n;\frac {4+m}{2};\frac {4 a^2 x^2}{9}\right )}{2+m}+\frac {2^{2+n} 9^{-2+n} a^2 x^{3+m} \, _2F_1\left (\frac {3+m}{2},2-n;\frac {5+m}{2};\frac {4 a^2 x^2}{9}\right )}{3+m} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.09 \[ \int x^m (3-2 a x)^{-2+n} (6+4 a x)^n \, dx=\frac {9^{-2+n} x^{1+m} \left (36-16 a^2 x^2\right )^n \left (18-8 a^2 x^2\right )^{-n} \left (9 \left (6+5 m+m^2\right ) \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},2-n,\frac {3+m}{2},\frac {4 a^2 x^2}{9}\right )+4 a (1+m) x \left (3 (3+m) \operatorname {Hypergeometric2F1}\left (\frac {2+m}{2},2-n,\frac {4+m}{2},\frac {4 a^2 x^2}{9}\right )+a (2+m) x \operatorname {Hypergeometric2F1}\left (\frac {3+m}{2},2-n,\frac {5+m}{2},\frac {4 a^2 x^2}{9}\right )\right )\right )}{(1+m) (2+m) (3+m)} \]

[In]

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

[Out]

(9^(-2 + n)*x^(1 + m)*(36 - 16*a^2*x^2)^n*(9*(6 + 5*m + m^2)*Hypergeometric2F1[(1 + m)/2, 2 - n, (3 + m)/2, (4
*a^2*x^2)/9] + 4*a*(1 + m)*x*(3*(3 + m)*Hypergeometric2F1[(2 + m)/2, 2 - n, (4 + m)/2, (4*a^2*x^2)/9] + a*(2 +
m)*x*Hypergeometric2F1[(3 + m)/2, 2 - n, (5 + m)/2, (4*a^2*x^2)/9])))/((1 + m)*(2 + m)*(3 + m)*(18 - 8*a^2*x^
2)^n)

Maple [F]

\[\int x^{m} \left (-2 a x +3\right )^{-2+n} \left (4 a x +6\right )^{n}d x\]

[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)

Fricas [F]

\[ \int x^m (3-2 a x)^{-2+n} (6+4 a x)^n \, dx=\int { {\left (4 \, a x + 6\right )}^{n} {\left (-2 \, a x + 3\right )}^{n - 2} x^{m} \,d x } \]

[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)

Sympy [F]

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

[In]

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

[Out]

2**n*Integral(x**m*(-2*a*x + 3)**(n - 2)*(2*a*x + 3)**n, x)

Maxima [F]

\[ \int x^m (3-2 a x)^{-2+n} (6+4 a x)^n \, dx=\int { {\left (4 \, a x + 6\right )}^{n} {\left (-2 \, a x + 3\right )}^{n - 2} x^{m} \,d x } \]

[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)

Giac [F]

\[ \int x^m (3-2 a x)^{-2+n} (6+4 a x)^n \, dx=\int { {\left (4 \, a x + 6\right )}^{n} {\left (-2 \, a x + 3\right )}^{n - 2} x^{m} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int x^m (3-2 a x)^{-2+n} (6+4 a x)^n \, dx=\int x^m\,{\left (3-2\,a\,x\right )}^{n-2}\,{\left (4\,a\,x+6\right )}^n \,d x \]

[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)