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 = 0.11, antiderivative size = 158, 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 = {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 125
Rule 127
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 = 0.13, 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]
________________________________________________________________________________________
fricas [F] time = 1.14, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (4 \, a x + 6\right )}^{n} {\left (-2 \, a x + 3\right )}^{n - 2} x^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (4 \, a x + 6\right )}^{n} {\left (-2 \, a x + 3\right )}^{n - 2} x^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.16, size = 0, normalized size = 0.00 \[ \int x^{m} \left (-2 a x +3\right )^{n -2} \left (4 a x +6\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (4 \, a x + 6\right )}^{n} {\left (-2 \, a x + 3\right )}^{n - 2} x^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^m\,{\left (3-2\,a\,x\right )}^{n-2}\,{\left (4\,a\,x+6\right )}^n \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________