Optimal. Leaf size=55 \[ -\frac {7 (3 x+2)^{m+1}}{27 (m+1)}+\frac {37 (3 x+2)^{m+2}}{27 (m+2)}-\frac {10 (3 x+2)^{m+3}}{27 (m+3)} \]
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Rubi [A] time = 0.01, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {77} \[ -\frac {7 (3 x+2)^{m+1}}{27 (m+1)}+\frac {37 (3 x+2)^{m+2}}{27 (m+2)}-\frac {10 (3 x+2)^{m+3}}{27 (m+3)} \]
Antiderivative was successfully verified.
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Rule 77
Rubi steps
\begin {align*} \int (1-2 x) (2+3 x)^m (3+5 x) \, dx &=\int \left (-\frac {7}{9} (2+3 x)^m+\frac {37}{9} (2+3 x)^{1+m}-\frac {10}{9} (2+3 x)^{2+m}\right ) \, dx\\ &=-\frac {7 (2+3 x)^{1+m}}{27 (1+m)}+\frac {37 (2+3 x)^{2+m}}{27 (2+m)}-\frac {10 (2+3 x)^{3+m}}{27 (3+m)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 47, normalized size = 0.85 \[ \frac {1}{27} (3 x+2)^{m+1} \left (-\frac {10 (3 x+2)^2}{m+3}+\frac {37 (3 x+2)}{m+2}-\frac {7}{m+1}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 75, normalized size = 1.36 \[ -\frac {{\left (270 \, {\left (m^{2} + 3 \, m + 2\right )} x^{3} + 9 \, {\left (23 \, m^{2} + 32 \, m + 9\right )} x^{2} - 54 \, m^{2} - 3 \, {\left (21 \, m^{2} + 197 \, m + 162\right )} x - 282 \, m - 200\right )} {\left (3 \, x + 2\right )}^{m}}{27 \, {\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.83, size = 163, normalized size = 2.96 \[ -\frac {270 \, m^{2} {\left (3 \, x + 2\right )}^{m} x^{3} + 207 \, m^{2} {\left (3 \, x + 2\right )}^{m} x^{2} + 810 \, m {\left (3 \, x + 2\right )}^{m} x^{3} - 63 \, m^{2} {\left (3 \, x + 2\right )}^{m} x + 288 \, m {\left (3 \, x + 2\right )}^{m} x^{2} + 540 \, {\left (3 \, x + 2\right )}^{m} x^{3} - 54 \, m^{2} {\left (3 \, x + 2\right )}^{m} - 591 \, m {\left (3 \, x + 2\right )}^{m} x + 81 \, {\left (3 \, x + 2\right )}^{m} x^{2} - 282 \, m {\left (3 \, x + 2\right )}^{m} - 486 \, {\left (3 \, x + 2\right )}^{m} x - 200 \, {\left (3 \, x + 2\right )}^{m}}{27 \, {\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 69, normalized size = 1.25 \[ -\frac {\left (90 m^{2} x^{2}+9 m^{2} x +270 m \,x^{2}-27 m^{2}-84 m x +180 x^{2}-141 m -93 x -100\right ) \left (3 x +2\right )^{m +1}}{27 \left (m^{3}+6 m^{2}+11 m +6\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.45, size = 102, normalized size = 1.85 \[ -\frac {10 \, {\left (27 \, {\left (m^{2} + 3 \, m + 2\right )} x^{3} + 18 \, {\left (m^{2} + m\right )} x^{2} - 24 \, m x + 16\right )} {\left (3 \, x + 2\right )}^{m}}{27 \, {\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )}} - \frac {{\left (9 \, {\left (m + 1\right )} x^{2} + 6 \, m x - 4\right )} {\left (3 \, x + 2\right )}^{m}}{9 \, {\left (m^{2} + 3 \, m + 2\right )}} + \frac {{\left (3 \, x + 2\right )}^{m + 1}}{m + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 130, normalized size = 2.36 \[ {\left (3\,x+2\right )}^m\,\left (\frac {54\,m^2+282\,m+200}{27\,m^3+162\,m^2+297\,m+162}+\frac {x\,\left (63\,m^2+591\,m+486\right )}{27\,m^3+162\,m^2+297\,m+162}-\frac {x^2\,\left (207\,m^2+288\,m+81\right )}{27\,m^3+162\,m^2+297\,m+162}-\frac {x^3\,\left (270\,m^2+810\,m+540\right )}{27\,m^3+162\,m^2+297\,m+162}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.05, size = 488, normalized size = 8.87 \[ \begin {cases} - \frac {180 x^{2} \log {\left (x + \frac {2}{3} \right )}}{486 x^{2} + 648 x + 216} - \frac {240 x \log {\left (x + \frac {2}{3} \right )}}{486 x^{2} + 648 x + 216} - \frac {222 x}{486 x^{2} + 648 x + 216} - \frac {80 \log {\left (x + \frac {2}{3} \right )}}{486 x^{2} + 648 x + 216} - \frac {141}{486 x^{2} + 648 x + 216} & \text {for}\: m = -3 \\- \frac {90 x^{2}}{81 x + 54} + \frac {111 x \log {\left (x + \frac {2}{3} \right )}}{81 x + 54} + \frac {74 \log {\left (x + \frac {2}{3} \right )}}{81 x + 54} + \frac {47}{81 x + 54} & \text {for}\: m = -2 \\- \frac {5 x^{2}}{3} + \frac {17 x}{9} - \frac {7 \log {\left (x + \frac {2}{3} \right )}}{27} & \text {for}\: m = -1 \\- \frac {270 m^{2} x^{3} \left (3 x + 2\right )^{m}}{27 m^{3} + 162 m^{2} + 297 m + 162} - \frac {207 m^{2} x^{2} \left (3 x + 2\right )^{m}}{27 m^{3} + 162 m^{2} + 297 m + 162} + \frac {63 m^{2} x \left (3 x + 2\right )^{m}}{27 m^{3} + 162 m^{2} + 297 m + 162} + \frac {54 m^{2} \left (3 x + 2\right )^{m}}{27 m^{3} + 162 m^{2} + 297 m + 162} - \frac {810 m x^{3} \left (3 x + 2\right )^{m}}{27 m^{3} + 162 m^{2} + 297 m + 162} - \frac {288 m x^{2} \left (3 x + 2\right )^{m}}{27 m^{3} + 162 m^{2} + 297 m + 162} + \frac {591 m x \left (3 x + 2\right )^{m}}{27 m^{3} + 162 m^{2} + 297 m + 162} + \frac {282 m \left (3 x + 2\right )^{m}}{27 m^{3} + 162 m^{2} + 297 m + 162} - \frac {540 x^{3} \left (3 x + 2\right )^{m}}{27 m^{3} + 162 m^{2} + 297 m + 162} - \frac {81 x^{2} \left (3 x + 2\right )^{m}}{27 m^{3} + 162 m^{2} + 297 m + 162} + \frac {486 x \left (3 x + 2\right )^{m}}{27 m^{3} + 162 m^{2} + 297 m + 162} + \frac {200 \left (3 x + 2\right )^{m}}{27 m^{3} + 162 m^{2} + 297 m + 162} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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