Optimal. Leaf size=188 \[ -\frac {1}{45} (88-m) (5-4 x)^2 (1+2 x)^{1-m} (2+3 x)^{1+m}-\frac {2}{15} (5-4 x)^3 (1+2 x)^{1-m} (2+3 x)^{1+m}-\frac {(1+2 x)^{1-m} (2+3 x)^{1+m} \left (386850-25441 m+426 m^2-2 m^3-24 \left (4359-154 m+m^2\right ) x\right )}{1215}+\frac {2^{-1-m} \left (3528363-639760 m+29050 m^2-440 m^3+2 m^4\right ) (1+2 x)^{1-m} \, _2F_1(1-m,-m;2-m;-3 (1+2 x))}{1215 (1-m)} \]
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Rubi [A]
time = 0.15, antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {102, 158, 152,
71} \begin {gather*} \frac {2^{-m-1} \left (2 m^4-440 m^3+29050 m^2-639760 m+3528363\right ) (2 x+1)^{1-m} \, _2F_1(1-m,-m;2-m;-3 (2 x+1))}{1215 (1-m)}-\frac {(3 x+2)^{m+1} \left (-2 m^3-24 \left (m^2-154 m+4359\right ) x+426 m^2-25441 m+386850\right ) (2 x+1)^{1-m}}{1215}-\frac {2}{15} (5-4 x)^3 (3 x+2)^{m+1} (2 x+1)^{1-m}-\frac {1}{45} (88-m) (5-4 x)^2 (3 x+2)^{m+1} (2 x+1)^{1-m} \end {gather*}
Antiderivative was successfully verified.
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Rule 71
Rule 102
Rule 152
Rule 158
Rubi steps
\begin {align*} \int (5-4 x)^4 (1+2 x)^{-m} (2+3 x)^m \, dx &=-\frac {2}{15} (5-4 x)^3 (1+2 x)^{1-m} (2+3 x)^{1+m}+\frac {1}{30} \int (5-4 x)^2 (1+2 x)^{-m} (2+3 x)^m (2 (397-10 m)-16 (88-m) x) \, dx\\ &=-\frac {1}{45} (88-m) (5-4 x)^2 (1+2 x)^{1-m} (2+3 x)^{1+m}-\frac {2}{15} (5-4 x)^3 (1+2 x)^{1-m} (2+3 x)^{1+m}+\frac {1}{720} \int (5-4 x) (1+2 x)^{-m} (2+3 x)^m \left (16 \left (7627-609 m+5 m^2\right )-64 \left (4359-154 m+m^2\right ) x\right ) \, dx\\ &=-\frac {1}{45} (88-m) (5-4 x)^2 (1+2 x)^{1-m} (2+3 x)^{1+m}-\frac {2}{15} (5-4 x)^3 (1+2 x)^{1-m} (2+3 x)^{1+m}-\frac {(1+2 x)^{1-m} (2+3 x)^{1+m} \left (386850-25441 m+426 m^2-2 m^3-24 \left (4359-154 m+m^2\right ) x\right )}{1215}+\frac {\left (3528363-639760 m+29050 m^2-440 m^3+2 m^4\right ) \int (1+2 x)^{-m} (2+3 x)^m \, dx}{1215}\\ &=-\frac {1}{45} (88-m) (5-4 x)^2 (1+2 x)^{1-m} (2+3 x)^{1+m}-\frac {2}{15} (5-4 x)^3 (1+2 x)^{1-m} (2+3 x)^{1+m}-\frac {(1+2 x)^{1-m} (2+3 x)^{1+m} \left (386850-25441 m+426 m^2-2 m^3-24 \left (4359-154 m+m^2\right ) x\right )}{1215}+\frac {2^{-1-m} \left (3528363-639760 m+29050 m^2-440 m^3+2 m^4\right ) (1+2 x)^{1-m} \, _2F_1(1-m,-m;2-m;-3 (1+2 x))}{1215 (1-m)}\\ \end {align*}
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Mathematica [A]
time = 0.52, size = 159, normalized size = 0.85 \begin {gather*} \frac {2^{-1-m} (1+2 x)^{1-m} \left (2^{2+m} (-1+m) (2+3 x)^{1+m} \left (2 m^2 (-19+6 x)+3 m \left (1331-796 x+72 x^2\right )+2 \left (-54629+62064 x-19224 x^2+2592 x^3\right )\right )-4 \left (-123992+9065 m-175 m^2+m^3\right ) \, _2F_1(-1-m,1-m;2-m;-3-6 x)+23 \left (-164189+17822 m-490 m^2+4 m^3\right ) \, _2F_1(1-m,-m;2-m;-3-6 x)\right )}{1215 (-1+m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \left (5-4 x \right )^{4} \left (2+3 x \right )^{m} \left (1+2 x \right )^{-m}\, dx\]
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 )}^4}{{\left (2\,x+1\right )}^m} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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