Optimal. Leaf size=97 \[ \frac {a^5 x^{1+m}}{1+m}+\frac {5 a^4 b x^{3+m}}{3+m}+\frac {10 a^3 b^2 x^{5+m}}{5+m}+\frac {10 a^2 b^3 x^{7+m}}{7+m}+\frac {5 a b^4 x^{9+m}}{9+m}+\frac {b^5 x^{11+m}}{11+m} \]
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Rubi [A]
time = 0.03, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {276}
\begin {gather*} \frac {a^5 x^{m+1}}{m+1}+\frac {5 a^4 b x^{m+3}}{m+3}+\frac {10 a^3 b^2 x^{m+5}}{m+5}+\frac {10 a^2 b^3 x^{m+7}}{m+7}+\frac {5 a b^4 x^{m+9}}{m+9}+\frac {b^5 x^{m+11}}{m+11} \end {gather*}
Antiderivative was successfully verified.
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Rule 276
Rubi steps
\begin {align*} \int x^m \left (a+b x^2\right )^5 \, dx &=\int \left (a^5 x^m+5 a^4 b x^{2+m}+10 a^3 b^2 x^{4+m}+10 a^2 b^3 x^{6+m}+5 a b^4 x^{8+m}+b^5 x^{10+m}\right ) \, dx\\ &=\frac {a^5 x^{1+m}}{1+m}+\frac {5 a^4 b x^{3+m}}{3+m}+\frac {10 a^3 b^2 x^{5+m}}{5+m}+\frac {10 a^2 b^3 x^{7+m}}{7+m}+\frac {5 a b^4 x^{9+m}}{9+m}+\frac {b^5 x^{11+m}}{11+m}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 88, normalized size = 0.91 \begin {gather*} x^{1+m} \left (\frac {a^5}{1+m}+\frac {5 a^4 b x^2}{3+m}+\frac {10 a^3 b^2 x^4}{5+m}+\frac {10 a^2 b^3 x^6}{7+m}+\frac {5 a b^4 x^8}{9+m}+\frac {b^5 x^{10}}{11+m}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(430\) vs.
\(2(97)=194\).
time = 0.03, size = 431, normalized size = 4.44
method | result | size |
risch | \(\frac {x \left (b^{5} m^{5} x^{10}+25 b^{5} m^{4} x^{10}+5 a \,b^{4} m^{5} x^{8}+230 b^{5} m^{3} x^{10}+135 a \,b^{4} m^{4} x^{8}+950 b^{5} m^{2} x^{10}+10 a^{2} b^{3} m^{5} x^{6}+1310 a \,b^{4} m^{3} x^{8}+1689 m \,x^{10} b^{5}+290 a^{2} b^{3} m^{4} x^{6}+5610 a \,b^{4} m^{2} x^{8}+945 b^{5} x^{10}+10 a^{3} b^{2} m^{5} x^{4}+3020 a^{2} b^{3} m^{3} x^{6}+10205 m \,x^{8} a \,b^{4}+310 a^{3} b^{2} m^{4} x^{4}+13660 a^{2} b^{3} m^{2} x^{6}+5775 a \,b^{4} x^{8}+5 a^{4} b \,m^{5} x^{2}+3500 a^{3} b^{2} m^{3} x^{4}+25770 m \,x^{6} a^{2} b^{3}+165 a^{4} b \,m^{4} x^{2}+17300 a^{3} b^{2} m^{2} x^{4}+14850 a^{2} b^{3} x^{6}+a^{5} m^{5}+2030 a^{4} b \,m^{3} x^{2}+34890 m \,x^{4} a^{3} b^{2}+35 a^{5} m^{4}+11310 a^{4} b \,m^{2} x^{2}+20790 a^{3} b^{2} x^{4}+470 a^{5} m^{3}+26765 m \,x^{2} a^{4} b +3010 a^{5} m^{2}+17325 a^{4} b \,x^{2}+9129 m \,a^{5}+10395 a^{5}\right ) x^{m}}{\left (11+m \right ) \left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\) | \(431\) |
gosper | \(\frac {x^{1+m} \left (b^{5} m^{5} x^{10}+25 b^{5} m^{4} x^{10}+5 a \,b^{4} m^{5} x^{8}+230 b^{5} m^{3} x^{10}+135 a \,b^{4} m^{4} x^{8}+950 b^{5} m^{2} x^{10}+10 a^{2} b^{3} m^{5} x^{6}+1310 a \,b^{4} m^{3} x^{8}+1689 m \,x^{10} b^{5}+290 a^{2} b^{3} m^{4} x^{6}+5610 a \,b^{4} m^{2} x^{8}+945 b^{5} x^{10}+10 a^{3} b^{2} m^{5} x^{4}+3020 a^{2} b^{3} m^{3} x^{6}+10205 m \,x^{8} a \,b^{4}+310 a^{3} b^{2} m^{4} x^{4}+13660 a^{2} b^{3} m^{2} x^{6}+5775 a \,b^{4} x^{8}+5 a^{4} b \,m^{5} x^{2}+3500 a^{3} b^{2} m^{3} x^{4}+25770 m \,x^{6} a^{2} b^{3}+165 a^{4} b \,m^{4} x^{2}+17300 a^{3} b^{2} m^{2} x^{4}+14850 a^{2} b^{3} x^{6}+a^{5} m^{5}+2030 a^{4} b \,m^{3} x^{2}+34890 m \,x^{4} a^{3} b^{2}+35 a^{5} m^{4}+11310 a^{4} b \,m^{2} x^{2}+20790 a^{3} b^{2} x^{4}+470 a^{5} m^{3}+26765 m \,x^{2} a^{4} b +3010 a^{5} m^{2}+17325 a^{4} b \,x^{2}+9129 m \,a^{5}+10395 a^{5}\right )}{\left (11+m \right ) \left (9+m \right ) \left (7+m \right ) \left (5+m \right ) \left (3+m \right ) \left (1+m \right )}\) | \(432\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 97, normalized size = 1.00 \begin {gather*} \frac {b^{5} x^{m + 11}}{m + 11} + \frac {5 \, a b^{4} x^{m + 9}}{m + 9} + \frac {10 \, a^{2} b^{3} x^{m + 7}}{m + 7} + \frac {10 \, a^{3} b^{2} x^{m + 5}}{m + 5} + \frac {5 \, a^{4} b x^{m + 3}}{m + 3} + \frac {a^{5} x^{m + 1}}{m + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 367 vs.
\(2 (97) = 194\).
time = 1.21, size = 367, normalized size = 3.78 \begin {gather*} \frac {{\left ({\left (b^{5} m^{5} + 25 \, b^{5} m^{4} + 230 \, b^{5} m^{3} + 950 \, b^{5} m^{2} + 1689 \, b^{5} m + 945 \, b^{5}\right )} x^{11} + 5 \, {\left (a b^{4} m^{5} + 27 \, a b^{4} m^{4} + 262 \, a b^{4} m^{3} + 1122 \, a b^{4} m^{2} + 2041 \, a b^{4} m + 1155 \, a b^{4}\right )} x^{9} + 10 \, {\left (a^{2} b^{3} m^{5} + 29 \, a^{2} b^{3} m^{4} + 302 \, a^{2} b^{3} m^{3} + 1366 \, a^{2} b^{3} m^{2} + 2577 \, a^{2} b^{3} m + 1485 \, a^{2} b^{3}\right )} x^{7} + 10 \, {\left (a^{3} b^{2} m^{5} + 31 \, a^{3} b^{2} m^{4} + 350 \, a^{3} b^{2} m^{3} + 1730 \, a^{3} b^{2} m^{2} + 3489 \, a^{3} b^{2} m + 2079 \, a^{3} b^{2}\right )} x^{5} + 5 \, {\left (a^{4} b m^{5} + 33 \, a^{4} b m^{4} + 406 \, a^{4} b m^{3} + 2262 \, a^{4} b m^{2} + 5353 \, a^{4} b m + 3465 \, a^{4} b\right )} x^{3} + {\left (a^{5} m^{5} + 35 \, a^{5} m^{4} + 470 \, a^{5} m^{3} + 3010 \, a^{5} m^{2} + 9129 \, a^{5} m + 10395 \, a^{5}\right )} x\right )} x^{m}}{m^{6} + 36 \, m^{5} + 505 \, m^{4} + 3480 \, m^{3} + 12139 \, m^{2} + 19524 \, m + 10395} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1999 vs.
\(2 (87) = 174\).
time = 0.67, size = 1999, normalized size = 20.61 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 540 vs.
\(2 (97) = 194\).
time = 0.91, size = 540, normalized size = 5.57 \begin {gather*} \frac {b^{5} m^{5} x^{11} x^{m} + 25 \, b^{5} m^{4} x^{11} x^{m} + 5 \, a b^{4} m^{5} x^{9} x^{m} + 230 \, b^{5} m^{3} x^{11} x^{m} + 135 \, a b^{4} m^{4} x^{9} x^{m} + 950 \, b^{5} m^{2} x^{11} x^{m} + 10 \, a^{2} b^{3} m^{5} x^{7} x^{m} + 1310 \, a b^{4} m^{3} x^{9} x^{m} + 1689 \, b^{5} m x^{11} x^{m} + 290 \, a^{2} b^{3} m^{4} x^{7} x^{m} + 5610 \, a b^{4} m^{2} x^{9} x^{m} + 945 \, b^{5} x^{11} x^{m} + 10 \, a^{3} b^{2} m^{5} x^{5} x^{m} + 3020 \, a^{2} b^{3} m^{3} x^{7} x^{m} + 10205 \, a b^{4} m x^{9} x^{m} + 310 \, a^{3} b^{2} m^{4} x^{5} x^{m} + 13660 \, a^{2} b^{3} m^{2} x^{7} x^{m} + 5775 \, a b^{4} x^{9} x^{m} + 5 \, a^{4} b m^{5} x^{3} x^{m} + 3500 \, a^{3} b^{2} m^{3} x^{5} x^{m} + 25770 \, a^{2} b^{3} m x^{7} x^{m} + 165 \, a^{4} b m^{4} x^{3} x^{m} + 17300 \, a^{3} b^{2} m^{2} x^{5} x^{m} + 14850 \, a^{2} b^{3} x^{7} x^{m} + a^{5} m^{5} x x^{m} + 2030 \, a^{4} b m^{3} x^{3} x^{m} + 34890 \, a^{3} b^{2} m x^{5} x^{m} + 35 \, a^{5} m^{4} x x^{m} + 11310 \, a^{4} b m^{2} x^{3} x^{m} + 20790 \, a^{3} b^{2} x^{5} x^{m} + 470 \, a^{5} m^{3} x x^{m} + 26765 \, a^{4} b m x^{3} x^{m} + 3010 \, a^{5} m^{2} x x^{m} + 17325 \, a^{4} b x^{3} x^{m} + 9129 \, a^{5} m x x^{m} + 10395 \, a^{5} x x^{m}}{m^{6} + 36 \, m^{5} + 505 \, m^{4} + 3480 \, m^{3} + 12139 \, m^{2} + 19524 \, m + 10395} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.11, size = 389, normalized size = 4.01 \begin {gather*} \frac {a^5\,x\,x^m\,\left (m^5+35\,m^4+470\,m^3+3010\,m^2+9129\,m+10395\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395}+\frac {b^5\,x^m\,x^{11}\,\left (m^5+25\,m^4+230\,m^3+950\,m^2+1689\,m+945\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395}+\frac {5\,a\,b^4\,x^m\,x^9\,\left (m^5+27\,m^4+262\,m^3+1122\,m^2+2041\,m+1155\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395}+\frac {5\,a^4\,b\,x^m\,x^3\,\left (m^5+33\,m^4+406\,m^3+2262\,m^2+5353\,m+3465\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395}+\frac {10\,a^2\,b^3\,x^m\,x^7\,\left (m^5+29\,m^4+302\,m^3+1366\,m^2+2577\,m+1485\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395}+\frac {10\,a^3\,b^2\,x^m\,x^5\,\left (m^5+31\,m^4+350\,m^3+1730\,m^2+3489\,m+2079\right )}{m^6+36\,m^5+505\,m^4+3480\,m^3+12139\,m^2+19524\,m+10395} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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