Optimal. Leaf size=132 \[ -\frac {1}{3} (5-4 x)^2 (1+2 x)^{-1-m} (2+3 x)^{1+m}-\frac {(1+2 x)^{-1-m} (2+3 x)^{1+m} \left (2768-315 m+4 m^2-8 (43-m) (1+m) x\right )}{9 (1+m)}+\frac {2^{-m} \left (1323-128 m+2 m^2\right ) (1+2 x)^{-m} \, _2F_1(-m,-m;1-m;-3 (1+2 x))}{9 m} \]
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Rubi [A]
time = 0.09, antiderivative size = 132, 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, 148, 71}
\begin {gather*} \frac {2^{-m} \left (2 m^2-128 m+1323\right ) (2 x+1)^{-m} \, _2F_1(-m,-m;1-m;-3 (2 x+1))}{9 m}-\frac {(3 x+2)^{m+1} \left (4 m^2-8 (43-m) (m+1) x-315 m+2768\right ) (2 x+1)^{-m-1}}{9 (m+1)}-\frac {1}{3} (5-4 x)^2 (3 x+2)^{m+1} (2 x+1)^{-m-1} \end {gather*}
Antiderivative was successfully verified.
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Rule 71
Rule 102
Rule 148
Rubi steps
\begin {align*} \int (5-4 x)^3 (1+2 x)^{-2-m} (2+3 x)^m \, dx &=-\frac {1}{3} (5-4 x)^2 (1+2 x)^{-1-m} (2+3 x)^{1+m}+\frac {1}{12} \int (5-4 x) (1+2 x)^{-2-m} (2+3 x)^m (4 (54-5 m)-16 (43-m) x) \, dx\\ &=-\frac {1}{3} (5-4 x)^2 (1+2 x)^{-1-m} (2+3 x)^{1+m}-\frac {(1+2 x)^{-1-m} (2+3 x)^{1+m} \left (2768-315 m+4 m^2-8 (43-m) (1+m) x\right )}{9 (1+m)}-\frac {1}{9} \left (2 \left (1323-128 m+2 m^2\right )\right ) \int (1+2 x)^{-1-m} (2+3 x)^m \, dx\\ &=-\frac {1}{3} (5-4 x)^2 (1+2 x)^{-1-m} (2+3 x)^{1+m}-\frac {(1+2 x)^{-1-m} (2+3 x)^{1+m} \left (2768-315 m+4 m^2-8 (43-m) (1+m) x\right )}{9 (1+m)}+\frac {2^{-m} \left (1323-128 m+2 m^2\right ) (1+2 x)^{-m} \, _2F_1(-m,-m;1-m;-3 (1+2 x))}{9 m}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 114, normalized size = 0.86 \begin {gather*} \frac {2^{-m} (1+2 x)^{-1-m} \left (-2^m m (2+3 x)^{1+m} \left (2843-464 x+48 x^2+m^2 (4+8 x)+24 m \left (-10-19 x+2 x^2\right )\right )+\left (1323+1195 m-126 m^2+2 m^3\right ) (1+2 x) \, _2F_1(-m,-m;1-m;-3-6 x)\right )}{9 m (1+m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \left (5-4 x \right )^{3} \left (1+2 x \right )^{-2-m} \left (2+3 x \right )^{m}\, dx\]
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+2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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