Integrand size = 15, antiderivative size = 73 \[ \int \left (a+b \sqrt {x}\right )^5 x^3 \, dx=\frac {a^5 x^4}{4}+\frac {10}{9} a^4 b x^{9/2}+2 a^3 b^2 x^5+\frac {20}{11} a^2 b^3 x^{11/2}+\frac {5}{6} a b^4 x^6+\frac {2}{13} b^5 x^{13/2} \]
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Time = 0.04 (sec) , antiderivative size = 73, 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} \[ \int \left (a+b \sqrt {x}\right )^5 x^3 \, dx=\frac {a^5 x^4}{4}+\frac {10}{9} a^4 b x^{9/2}+2 a^3 b^2 x^5+\frac {20}{11} a^2 b^3 x^{11/2}+\frac {5}{6} a b^4 x^6+\frac {2}{13} b^5 x^{13/2} \]
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Rule 45
Rule 272
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int x^7 (a+b x)^5 \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (a^5 x^7+5 a^4 b x^8+10 a^3 b^2 x^9+10 a^2 b^3 x^{10}+5 a b^4 x^{11}+b^5 x^{12}\right ) \, dx,x,\sqrt {x}\right ) \\ & = \frac {a^5 x^4}{4}+\frac {10}{9} a^4 b x^{9/2}+2 a^3 b^2 x^5+\frac {20}{11} a^2 b^3 x^{11/2}+\frac {5}{6} a b^4 x^6+\frac {2}{13} b^5 x^{13/2} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.92 \[ \int \left (a+b \sqrt {x}\right )^5 x^3 \, dx=\frac {1287 a^5 x^4+5720 a^4 b x^{9/2}+10296 a^3 b^2 x^5+9360 a^2 b^3 x^{11/2}+4290 a b^4 x^6+792 b^5 x^{13/2}}{5148} \]
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Time = 5.62 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.79
method | result | size |
derivativedivides | \(\frac {a^{5} x^{4}}{4}+\frac {10 a^{4} b \,x^{\frac {9}{2}}}{9}+2 a^{3} b^{2} x^{5}+\frac {20 a^{2} b^{3} x^{\frac {11}{2}}}{11}+\frac {5 a \,b^{4} x^{6}}{6}+\frac {2 b^{5} x^{\frac {13}{2}}}{13}\) | \(58\) |
default | \(\frac {a^{5} x^{4}}{4}+\frac {10 a^{4} b \,x^{\frac {9}{2}}}{9}+2 a^{3} b^{2} x^{5}+\frac {20 a^{2} b^{3} x^{\frac {11}{2}}}{11}+\frac {5 a \,b^{4} x^{6}}{6}+\frac {2 b^{5} x^{\frac {13}{2}}}{13}\) | \(58\) |
trager | \(\frac {a \left (10 b^{4} x^{5}+24 a^{2} x^{4} b^{2}+10 b^{4} x^{4}+3 a^{4} x^{3}+24 a^{2} b^{2} x^{3}+10 b^{4} x^{3}+3 a^{4} x^{2}+24 a^{2} b^{2} x^{2}+10 b^{4} x^{2}+3 a^{4} x +24 a^{2} b^{2} x +10 b^{4} x +3 a^{4}+24 a^{2} b^{2}+10 b^{4}\right ) \left (-1+x \right )}{12}+\frac {2 b \,x^{\frac {9}{2}} \left (99 b^{4} x^{2}+1170 a^{2} b^{2} x +715 a^{4}\right )}{1287}\) | \(158\) |
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Time = 0.26 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.86 \[ \int \left (a+b \sqrt {x}\right )^5 x^3 \, dx=\frac {5}{6} \, a b^{4} x^{6} + 2 \, a^{3} b^{2} x^{5} + \frac {1}{4} \, a^{5} x^{4} + \frac {2}{1287} \, {\left (99 \, b^{5} x^{6} + 1170 \, a^{2} b^{3} x^{5} + 715 \, a^{4} b x^{4}\right )} \sqrt {x} \]
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Time = 0.45 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.97 \[ \int \left (a+b \sqrt {x}\right )^5 x^3 \, dx=\frac {a^{5} x^{4}}{4} + \frac {10 a^{4} b x^{\frac {9}{2}}}{9} + 2 a^{3} b^{2} x^{5} + \frac {20 a^{2} b^{3} x^{\frac {11}{2}}}{11} + \frac {5 a b^{4} x^{6}}{6} + \frac {2 b^{5} x^{\frac {13}{2}}}{13} \]
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Leaf count of result is larger than twice the leaf count of optimal. 132 vs. \(2 (57) = 114\).
Time = 0.19 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.81 \[ \int \left (a+b \sqrt {x}\right )^5 x^3 \, dx=\frac {2 \, {\left (b \sqrt {x} + a\right )}^{13}}{13 \, b^{8}} - \frac {7 \, {\left (b \sqrt {x} + a\right )}^{12} a}{6 \, b^{8}} + \frac {42 \, {\left (b \sqrt {x} + a\right )}^{11} a^{2}}{11 \, b^{8}} - \frac {7 \, {\left (b \sqrt {x} + a\right )}^{10} a^{3}}{b^{8}} + \frac {70 \, {\left (b \sqrt {x} + a\right )}^{9} a^{4}}{9 \, b^{8}} - \frac {21 \, {\left (b \sqrt {x} + a\right )}^{8} a^{5}}{4 \, b^{8}} + \frac {2 \, {\left (b \sqrt {x} + a\right )}^{7} a^{6}}{b^{8}} - \frac {{\left (b \sqrt {x} + a\right )}^{6} a^{7}}{3 \, b^{8}} \]
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Time = 0.26 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.78 \[ \int \left (a+b \sqrt {x}\right )^5 x^3 \, dx=\frac {2}{13} \, b^{5} x^{\frac {13}{2}} + \frac {5}{6} \, a b^{4} x^{6} + \frac {20}{11} \, a^{2} b^{3} x^{\frac {11}{2}} + 2 \, a^{3} b^{2} x^{5} + \frac {10}{9} \, a^{4} b x^{\frac {9}{2}} + \frac {1}{4} \, a^{5} x^{4} \]
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Time = 0.03 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.78 \[ \int \left (a+b \sqrt {x}\right )^5 x^3 \, dx=\frac {a^5\,x^4}{4}+\frac {2\,b^5\,x^{13/2}}{13}+\frac {5\,a\,b^4\,x^6}{6}+\frac {10\,a^4\,b\,x^{9/2}}{9}+2\,a^3\,b^2\,x^5+\frac {20\,a^2\,b^3\,x^{11/2}}{11} \]
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