Optimal. Leaf size=179 \[ \frac {\left (m^3+132 m^2+4358 m+32010\right ) (3 x+2)^{m-1} (2 x+1)^{1-m} \, _2F_1\left (2,1-m;2-m;\frac {23 (2 x+1)}{14 (3 x+2)}\right )}{2453889228 (1-m)}+\frac {\left (2 m^2+220 m+4359\right ) (3 x+2)^{m+1} (2 x+1)^{1-m}}{25039686 (5-4 x)^2}+\frac {(m+66) (3 x+2)^{m+1} (2 x+1)^{1-m}}{77763 (5-4 x)^3}+\frac {(3 x+2)^{m+1} (2 x+1)^{1-m}}{322 (5-4 x)^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.10, antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {129, 151, 12, 131} \[ \frac {\left (m^3+132 m^2+4358 m+32010\right ) (3 x+2)^{m-1} (2 x+1)^{1-m} \, _2F_1\left (2,1-m;2-m;\frac {23 (2 x+1)}{14 (3 x+2)}\right )}{2453889228 (1-m)}+\frac {\left (2 m^2+220 m+4359\right ) (3 x+2)^{m+1} (2 x+1)^{1-m}}{25039686 (5-4 x)^2}+\frac {(m+66) (3 x+2)^{m+1} (2 x+1)^{1-m}}{77763 (5-4 x)^3}+\frac {(3 x+2)^{m+1} (2 x+1)^{1-m}}{322 (5-4 x)^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 129
Rule 131
Rule 151
Rubi steps
\begin {align*} \int \frac {(1+2 x)^{-m} (2+3 x)^m}{(5-4 x)^5} \, dx &=\frac {(1+2 x)^{1-m} (2+3 x)^{1+m}}{322 (5-4 x)^4}-\frac {\int \frac {(-4 (51+m)-48 x) (1+2 x)^{-m} (2+3 x)^m}{(5-4 x)^4} \, dx}{1288}\\ &=\frac {(1+2 x)^{1-m} (2+3 x)^{1+m}}{322 (5-4 x)^4}+\frac {(66+m) (1+2 x)^{1-m} (2+3 x)^{1+m}}{77763 (5-4 x)^3}+\frac {\int \frac {(1+2 x)^{-m} (2+3 x)^m \left (8 \left (3369+205 m+2 m^2\right )+96 (66+m) x\right )}{(5-4 x)^3} \, dx}{1244208}\\ &=\frac {(1+2 x)^{1-m} (2+3 x)^{1+m}}{322 (5-4 x)^4}+\frac {(66+m) (1+2 x)^{1-m} (2+3 x)^{1+m}}{77763 (5-4 x)^3}+\frac {\left (4359+220 m+2 m^2\right ) (1+2 x)^{1-m} (2+3 x)^{1+m}}{25039686 (5-4 x)^2}-\frac {\int -\frac {64 \left (32010+4358 m+132 m^2+m^3\right ) (1+2 x)^{-m} (2+3 x)^m}{(5-4 x)^2} \, dx}{801269952}\\ &=\frac {(1+2 x)^{1-m} (2+3 x)^{1+m}}{322 (5-4 x)^4}+\frac {(66+m) (1+2 x)^{1-m} (2+3 x)^{1+m}}{77763 (5-4 x)^3}+\frac {\left (4359+220 m+2 m^2\right ) (1+2 x)^{1-m} (2+3 x)^{1+m}}{25039686 (5-4 x)^2}-\frac {\left (-32010-4358 m-132 m^2-m^3\right ) \int \frac {(1+2 x)^{-m} (2+3 x)^m}{(5-4 x)^2} \, dx}{12519843}\\ &=\frac {(1+2 x)^{1-m} (2+3 x)^{1+m}}{322 (5-4 x)^4}+\frac {(66+m) (1+2 x)^{1-m} (2+3 x)^{1+m}}{77763 (5-4 x)^3}+\frac {\left (4359+220 m+2 m^2\right ) (1+2 x)^{1-m} (2+3 x)^{1+m}}{25039686 (5-4 x)^2}+\frac {\left (32010+4358 m+132 m^2+m^3\right ) (1+2 x)^{1-m} (2+3 x)^{-1+m} \, _2F_1\left (2,1-m;2-m;\frac {23 (1+2 x)}{14 (2+3 x)}\right )}{2453889228 (1-m)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.14, size = 131, normalized size = 0.73 \[ \frac {(2 x+1)^{1-m} (3 x+2)^{m-1} \left (-\frac {\left (m^3+132 m^2+4358 m+32010\right ) \, _2F_1\left (2,1-m;2-m;\frac {46 x+23}{42 x+28}\right )}{m-1}+\frac {98 \left (2 m^2+220 m+4359\right ) (3 x+2)^2}{(5-4 x)^2}-\frac {31556 (m+66) (3 x+2)^2}{(4 x-5)^3}+\frac {7620774 (3 x+2)^2}{(5-4 x)^4}\right )}{2453889228} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (3 \, x + 2\right )}^{m}}{{\left (1024 \, x^{5} - 6400 \, x^{4} + 16000 \, x^{3} - 20000 \, x^{2} + 12500 \, x - 3125\right )} {\left (2 \, x + 1\right )}^{m}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {{\left (3 \, x + 2\right )}^{m}}{{\left (2 \, x + 1\right )}^{m} {\left (4 \, x - 5\right )}^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.23, size = 0, normalized size = 0.00 \[ \int \frac {\left (2 x +1\right )^{-m} \left (3 x +2\right )^{m}}{\left (-4 x +5\right )^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {{\left (3 \, x + 2\right )}^{m}}{{\left (2 \, x + 1\right )}^{m} {\left (4 \, x - 5\right )}^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ -\int \frac {{\left (3\,x+2\right )}^m}{{\left (2\,x+1\right )}^m\,{\left (4\,x-5\right )}^5} \,d x \]
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]
________________________________________________________________________________________