Integrand size = 15, antiderivative size = 75 \[ \int \left (a+b \sqrt {x}\right )^5 x^4 \, dx=\frac {a^5 x^5}{5}+\frac {10}{11} a^4 b x^{11/2}+\frac {5}{3} a^3 b^2 x^6+\frac {20}{13} a^2 b^3 x^{13/2}+\frac {5}{7} a b^4 x^7+\frac {2}{15} b^5 x^{15/2} \]
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Time = 0.03 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 45} \[ \int \left (a+b \sqrt {x}\right )^5 x^4 \, dx=\frac {a^5 x^5}{5}+\frac {10}{11} a^4 b x^{11/2}+\frac {5}{3} a^3 b^2 x^6+\frac {20}{13} a^2 b^3 x^{13/2}+\frac {5}{7} a b^4 x^7+\frac {2}{15} b^5 x^{15/2} \]
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Rule 45
Rule 272
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int x^9 (a+b x)^5 \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (a^5 x^9+5 a^4 b x^{10}+10 a^3 b^2 x^{11}+10 a^2 b^3 x^{12}+5 a b^4 x^{13}+b^5 x^{14}\right ) \, dx,x,\sqrt {x}\right ) \\ & = \frac {a^5 x^5}{5}+\frac {10}{11} a^4 b x^{11/2}+\frac {5}{3} a^3 b^2 x^6+\frac {20}{13} a^2 b^3 x^{13/2}+\frac {5}{7} a b^4 x^7+\frac {2}{15} b^5 x^{15/2} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.89 \[ \int \left (a+b \sqrt {x}\right )^5 x^4 \, dx=\frac {3003 a^5 x^5+13650 a^4 b x^{11/2}+25025 a^3 b^2 x^6+23100 a^2 b^3 x^{13/2}+10725 a b^4 x^7+2002 b^5 x^{15/2}}{15015} \]
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Time = 5.64 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.77
method | result | size |
derivativedivides | \(\frac {a^{5} x^{5}}{5}+\frac {10 a^{4} b \,x^{\frac {11}{2}}}{11}+\frac {5 a^{3} b^{2} x^{6}}{3}+\frac {20 a^{2} b^{3} x^{\frac {13}{2}}}{13}+\frac {5 a \,b^{4} x^{7}}{7}+\frac {2 b^{5} x^{\frac {15}{2}}}{15}\) | \(58\) |
default | \(\frac {a^{5} x^{5}}{5}+\frac {10 a^{4} b \,x^{\frac {11}{2}}}{11}+\frac {5 a^{3} b^{2} x^{6}}{3}+\frac {20 a^{2} b^{3} x^{\frac {13}{2}}}{13}+\frac {5 a \,b^{4} x^{7}}{7}+\frac {2 b^{5} x^{\frac {15}{2}}}{15}\) | \(58\) |
trager | \(\frac {a \left (75 b^{4} x^{6}+175 a^{2} b^{2} x^{5}+75 b^{4} x^{5}+21 a^{4} x^{4}+175 a^{2} x^{4} b^{2}+75 b^{4} x^{4}+21 a^{4} x^{3}+175 a^{2} b^{2} x^{3}+75 b^{4} x^{3}+21 a^{4} x^{2}+175 a^{2} b^{2} x^{2}+75 b^{4} x^{2}+21 a^{4} x +175 a^{2} b^{2} x +75 b^{4} x +21 a^{4}+175 a^{2} b^{2}+75 b^{4}\right ) \left (-1+x \right )}{105}+\frac {2 b \,x^{\frac {11}{2}} \left (143 b^{4} x^{2}+1650 a^{2} b^{2} x +975 a^{4}\right )}{2145}\) | \(185\) |
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Time = 0.25 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.84 \[ \int \left (a+b \sqrt {x}\right )^5 x^4 \, dx=\frac {5}{7} \, a b^{4} x^{7} + \frac {5}{3} \, a^{3} b^{2} x^{6} + \frac {1}{5} \, a^{5} x^{5} + \frac {2}{2145} \, {\left (143 \, b^{5} x^{7} + 1650 \, a^{2} b^{3} x^{6} + 975 \, a^{4} b x^{5}\right )} \sqrt {x} \]
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Time = 0.50 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.97 \[ \int \left (a+b \sqrt {x}\right )^5 x^4 \, dx=\frac {a^{5} x^{5}}{5} + \frac {10 a^{4} b x^{\frac {11}{2}}}{11} + \frac {5 a^{3} b^{2} x^{6}}{3} + \frac {20 a^{2} b^{3} x^{\frac {13}{2}}}{13} + \frac {5 a b^{4} x^{7}}{7} + \frac {2 b^{5} x^{\frac {15}{2}}}{15} \]
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Leaf count of result is larger than twice the leaf count of optimal. 166 vs. \(2 (57) = 114\).
Time = 0.19 (sec) , antiderivative size = 166, normalized size of antiderivative = 2.21 \[ \int \left (a+b \sqrt {x}\right )^5 x^4 \, dx=\frac {2 \, {\left (b \sqrt {x} + a\right )}^{15}}{15 \, b^{10}} - \frac {9 \, {\left (b \sqrt {x} + a\right )}^{14} a}{7 \, b^{10}} + \frac {72 \, {\left (b \sqrt {x} + a\right )}^{13} a^{2}}{13 \, b^{10}} - \frac {14 \, {\left (b \sqrt {x} + a\right )}^{12} a^{3}}{b^{10}} + \frac {252 \, {\left (b \sqrt {x} + a\right )}^{11} a^{4}}{11 \, b^{10}} - \frac {126 \, {\left (b \sqrt {x} + a\right )}^{10} a^{5}}{5 \, b^{10}} + \frac {56 \, {\left (b \sqrt {x} + a\right )}^{9} a^{6}}{3 \, b^{10}} - \frac {9 \, {\left (b \sqrt {x} + a\right )}^{8} a^{7}}{b^{10}} + \frac {18 \, {\left (b \sqrt {x} + a\right )}^{7} a^{8}}{7 \, b^{10}} - \frac {{\left (b \sqrt {x} + a\right )}^{6} a^{9}}{3 \, b^{10}} \]
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Time = 0.31 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.76 \[ \int \left (a+b \sqrt {x}\right )^5 x^4 \, dx=\frac {2}{15} \, b^{5} x^{\frac {15}{2}} + \frac {5}{7} \, a b^{4} x^{7} + \frac {20}{13} \, a^{2} b^{3} x^{\frac {13}{2}} + \frac {5}{3} \, a^{3} b^{2} x^{6} + \frac {10}{11} \, a^{4} b x^{\frac {11}{2}} + \frac {1}{5} \, a^{5} x^{5} \]
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Time = 0.03 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.76 \[ \int \left (a+b \sqrt {x}\right )^5 x^4 \, dx=\frac {a^5\,x^5}{5}+\frac {2\,b^5\,x^{15/2}}{15}+\frac {5\,a\,b^4\,x^7}{7}+\frac {10\,a^4\,b\,x^{11/2}}{11}+\frac {5\,a^3\,b^2\,x^6}{3}+\frac {20\,a^2\,b^3\,x^{13/2}}{13} \]
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