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

Optimal. Leaf size=104 \[ \frac{2^n 3^{2 n-1} x^{m+1} \, _2F_1\left (\frac{m+1}{2},1-n;\frac{m+3}{2};\frac{4 a^2 x^2}{9}\right )}{m+1}+\frac{a 2^{n+1} 9^{n-1} x^{m+2} \, _2F_1\left (\frac{m+2}{2},1-n;\frac{m+4}{2};\frac{4 a^2 x^2}{9}\right )}{m+2} \]

[Out]

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

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Rubi [A]  time = 0.0544598, antiderivative size = 104, 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, Rules used = {82, 125, 364} \[ \frac{2^n 3^{2 n-1} x^{m+1} \, _2F_1\left (\frac{m+1}{2},1-n;\frac{m+3}{2};\frac{4 a^2 x^2}{9}\right )}{m+1}+\frac{a 2^{n+1} 9^{n-1} x^{m+2} \, _2F_1\left (\frac{m+2}{2},1-n;\frac{m+4}{2};\frac{4 a^2 x^2}{9}\right )}{m+2} \]

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

Rule 82

Int[((f_.)*(x_))^(p_.)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Dist[a, Int[(a + b*
x)^n*(c + d*x)^n*(f*x)^p, x], x] + Dist[b/f, Int[(a + b*x)^n*(c + d*x)^n*(f*x)^(p + 1), x], x] /; FreeQ[{a, b,
c, d, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[m - n - 1, 0] &&  !RationalQ[p] &&  !IGtQ[m, 0] && NeQ[m +
n + p + 2, 0]

Rule 125

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[m - n, 0] && GtQ[a, 0] && GtQ
[c, 0]

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), 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)^{-1+n} (6+4 a x)^n \, dx &=6 \int x^m (3-2 a x)^{-1+n} (6+4 a x)^{-1+n} \, dx+(4 a) \int x^{1+m} (3-2 a x)^{-1+n} (6+4 a x)^{-1+n} \, dx\\ &=6 \int x^m \left (18-8 a^2 x^2\right )^{-1+n} \, dx+(4 a) \int x^{1+m} \left (18-8 a^2 x^2\right )^{-1+n} \, dx\\ &=\frac{2^n 3^{-1+2 n} x^{1+m} \, _2F_1\left (\frac{1+m}{2},1-n;\frac{3+m}{2};\frac{4 a^2 x^2}{9}\right )}{1+m}+\frac{2^{1+n} 9^{-1+n} a x^{2+m} \, _2F_1\left (\frac{2+m}{2},1-n;\frac{4+m}{2};\frac{4 a^2 x^2}{9}\right )}{2+m}\\ \end{align*}

Mathematica [A]  time = 0.0483442, size = 120, normalized size = 1.15 \[ \frac{9^{n-1} x^{m+1} \left (36-16 a^2 x^2\right )^n \left (18-8 a^2 x^2\right )^{-n} \left (2 a (m+1) x \, _2F_1\left (\frac{m}{2}+1,1-n;\frac{m}{2}+2;\frac{4 a^2 x^2}{9}\right )+3 (m+2) \, _2F_1\left (\frac{m+1}{2},1-n;\frac{m+3}{2};\frac{4 a^2 x^2}{9}\right )\right )}{(m+1) (m+2)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.147, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \left ( -2\,ax+3 \right ) ^{-1+n} \left ( 4\,ax+6 \right ) ^{n}\, dx \end{align*}

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. \begin{align*} \int{\left (4 \, a x + 6\right )}^{n}{\left (-2 \, a x + 3\right )}^{n - 1} x^{m}\,{d x} \end{align*}

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, 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. \begin{align*}{\rm integral}\left ({\left (4 \, a x + 6\right )}^{n}{\left (-2 \, a x + 3\right )}^{n - 1} x^{m}, x\right ) \end{align*}

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, 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. \begin{align*} \text{Timed out} \end{align*}

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

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, algorithm="giac")

[Out]

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