Optimal. Leaf size=179 \[ \frac {(1+2 x)^{1-m} (2+3 x)^{1+m}}{322 (5-4 x)^4}+\frac {(66+m) (1+2 x)^{1-m} (2+3 x)^{1+m}}{77763 (5-4 x)^3}+\frac {\left (4359+220 m+2 m^2\right ) (1+2 x)^{1-m} (2+3 x)^{1+m}}{25039686 (5-4 x)^2}+\frac {\left (32010+4358 m+132 m^2+m^3\right ) (1+2 x)^{1-m} (2+3 x)^{-1+m} \, _2F_1\left (2,1-m;2-m;\frac {23 (1+2 x)}{14 (2+3 x)}\right )}{2453889228 (1-m)} \]
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Rubi [A]
time = 0.06, 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 = {105, 156, 12,
133} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 105
Rule 133
Rule 156
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*}
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Mathematica [A]
time = 0.16, size = 131, normalized size = 0.73 \begin {gather*} \frac {(1+2 x)^{1-m} (2+3 x)^{-1+m} \left (\frac {7620774 (2+3 x)^2}{(5-4 x)^4}+\frac {98 \left (4359+220 m+2 m^2\right ) (2+3 x)^2}{(5-4 x)^2}-\frac {31556 (66+m) (2+3 x)^2}{(-5+4 x)^3}-\frac {\left (32010+4358 m+132 m^2+m^3\right ) \, _2F_1\left (2,1-m;2-m;\frac {23+46 x}{28+42 x}\right )}{-1+m}\right )}{2453889228} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (2+3 x \right )^{m} \left (1+2 x \right )^{-m}}{\left (5-4 x \right )^{5}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {{\left (3\,x+2\right )}^m}{{\left (2\,x+1\right )}^m\,{\left (4\,x-5\right )}^5} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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