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]
________________________________________________________________________________________
Rubi [A] time = 0.112638, 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, Rules used = {127, 125, 364} \[ \frac{2^n 9^{n-1} x^{m+1} \, _2F_1\left (\frac{m+1}{2},2-n;\frac{m+3}{2};\frac{4 a^2 x^2}{9}\right )}{m+1}+\frac{a 2^{n+2} 3^{2 n-3} x^{m+2} \, _2F_1\left (\frac{m+2}{2},2-n;\frac{m+4}{2};\frac{4 a^2 x^2}{9}\right )}{m+2}+\frac{a^2 2^{n+2} 9^{n-2} x^{m+3} \, _2F_1\left (\frac{m+3}{2},2-n;\frac{m+5}{2};\frac{4 a^2 x^2}{9}\right )}{m+3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 127
Rule 125
Rule 364
Rubi steps
\begin{align*} \int x^m (3-2 a x)^{-2+n} (6+4 a x)^n \, dx &=\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] time = 0.119339, size = 172, normalized size = 1.09 \[ \frac{9^{n-2} x^{m+1} \left (36-16 a^2 x^2\right )^n \left (18-8 a^2 x^2\right )^{-n} \left (9 \left (m^2+5 m+6\right ) \, _2F_1\left (\frac{m+1}{2},2-n;\frac{m+3}{2};\frac{4 a^2 x^2}{9}\right )+4 a (m+1) x \left (3 (m+3) \, _2F_1\left (\frac{m+2}{2},2-n;\frac{m+4}{2};\frac{4 a^2 x^2}{9}\right )+a (m+2) x \, _2F_1\left (\frac{m+3}{2},2-n;\frac{m+5}{2};\frac{4 a^2 x^2}{9}\right )\right )\right )}{(m+1) (m+2) (m+3)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.149, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \left ( -2\,ax+3 \right ) ^{-2+n} \left ( 4\,ax+6 \right ) ^{n}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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 - 2} x^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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 - 2} x^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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 - 2} x^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]