Optimal. Leaf size=139 \[ -\frac {2}{3} (5-4 x)^2 (1+2 x)^{-2-m} (2+3 x)^{1+m}+\frac {7 (1+2 x)^{-2-m} (2+3 x)^{1+m} \left (3 \left (186-m+2 m^2\right )+2 \left (677+102 m-8 m^2\right ) x\right )}{3 \left (2+3 m+m^2\right )}-\frac {2^{1-m} (63-2 m) (1+2 x)^{-m} \, _2F_1(-m,-m;1-m;-3 (1+2 x))}{3 m} \]
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Rubi [A]
time = 0.07, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {102, 150, 71}
\begin {gather*} -\frac {2^{1-m} (63-2 m) (2 x+1)^{-m} \, _2F_1(-m,-m;1-m;-3 (2 x+1))}{3 m}+\frac {7 (3 x+2)^{m+1} \left (2 \left (-8 m^2+102 m+677\right ) x+3 \left (2 m^2-m+186\right )\right ) (2 x+1)^{-m-2}}{3 \left (m^2+3 m+2\right )}-\frac {2}{3} (5-4 x)^2 (3 x+2)^{m+1} (2 x+1)^{-m-2} \end {gather*}
Antiderivative was successfully verified.
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Rule 71
Rule 102
Rule 150
Rubi steps
\begin {align*} \int (5-4 x)^3 (1+2 x)^{-3-m} (2+3 x)^m \, dx &=-\frac {2}{3} (5-4 x)^2 (1+2 x)^{-2-m} (2+3 x)^{1+m}+\frac {1}{6} \int (5-4 x) (1+2 x)^{-3-m} (2+3 x)^m (-2 (7+10 m)-8 (63-2 m) x) \, dx\\ &=-\frac {2}{3} (5-4 x)^2 (1+2 x)^{-2-m} (2+3 x)^{1+m}+\frac {7 (1+2 x)^{-2-m} (2+3 x)^{1+m} \left (3 \left (186-m+2 m^2\right )+2 \left (677+102 m-8 m^2\right ) x\right )}{3 \left (2+3 m+m^2\right )}+\frac {1}{3} (4 (63-2 m)) \int (1+2 x)^{-1-m} (2+3 x)^m \, dx\\ &=-\frac {2}{3} (5-4 x)^2 (1+2 x)^{-2-m} (2+3 x)^{1+m}+\frac {7 (1+2 x)^{-2-m} (2+3 x)^{1+m} \left (3 \left (186-m+2 m^2\right )+2 \left (677+102 m-8 m^2\right ) x\right )}{3 \left (2+3 m+m^2\right )}-\frac {2^{1-m} (63-2 m) (1+2 x)^{-m} \, _2F_1(-m,-m;1-m;-3 (1+2 x))}{3 m}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 123, normalized size = 0.88 \begin {gather*} \frac {2^{-m} (1+2 x)^{-2-m} \left (-2^m m (2+3 x)^{1+m} \left (-3806-9638 x+64 x^2+8 (m+2 m x)^2+3 m \left (57-556 x+32 x^2\right )\right )+2 \left (-126-185 m-57 m^2+2 m^3\right ) (1+2 x)^2 \, _2F_1(-m,-m;1-m;-3-6 x)\right )}{3 m (1+m) (2+m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \left (5-4 x \right )^{3} \left (1+2 x \right )^{-3-m} \left (2+3 x \right )^{m}\, dx\]
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 (4\,x-5\right )}^3}{{\left (2\,x+1\right )}^{m+3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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