Integrand size = 13, antiderivative size = 19 \[ \int \left (a+b \sqrt {x}\right ) x^3 \, dx=\frac {a x^4}{4}+\frac {2}{9} b x^{9/2} \]
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Time = 0.00 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {14} \[ \int \left (a+b \sqrt {x}\right ) x^3 \, dx=\frac {a x^4}{4}+\frac {2}{9} b x^{9/2} \]
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Rule 14
Rubi steps \begin{align*} \text {integral}& = \int \left (a x^3+b x^{7/2}\right ) \, dx \\ & = \frac {a x^4}{4}+\frac {2}{9} b x^{9/2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \left (a+b \sqrt {x}\right ) x^3 \, dx=\frac {1}{36} \left (9 a+8 b \sqrt {x}\right ) x^4 \]
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Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.74
method | result | size |
derivativedivides | \(\frac {a \,x^{4}}{4}+\frac {2 b \,x^{\frac {9}{2}}}{9}\) | \(14\) |
default | \(\frac {a \,x^{4}}{4}+\frac {2 b \,x^{\frac {9}{2}}}{9}\) | \(14\) |
trager | \(\frac {a \left (x^{3}+x^{2}+x +1\right ) \left (-1+x \right )}{4}+\frac {2 b \,x^{\frac {9}{2}}}{9}\) | \(23\) |
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Time = 0.27 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68 \[ \int \left (a+b \sqrt {x}\right ) x^3 \, dx=\frac {2}{9} \, b x^{\frac {9}{2}} + \frac {1}{4} \, a x^{4} \]
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Time = 0.32 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \left (a+b \sqrt {x}\right ) x^3 \, dx=\frac {a x^{4}}{4} + \frac {2 b x^{\frac {9}{2}}}{9} \]
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Leaf count of result is larger than twice the leaf count of optimal. 132 vs. \(2 (13) = 26\).
Time = 0.20 (sec) , antiderivative size = 132, normalized size of antiderivative = 6.95 \[ \int \left (a+b \sqrt {x}\right ) x^3 \, dx=\frac {2 \, {\left (b \sqrt {x} + a\right )}^{9}}{9 \, b^{8}} - \frac {7 \, {\left (b \sqrt {x} + a\right )}^{8} a}{4 \, b^{8}} + \frac {6 \, {\left (b \sqrt {x} + a\right )}^{7} a^{2}}{b^{8}} - \frac {35 \, {\left (b \sqrt {x} + a\right )}^{6} a^{3}}{3 \, b^{8}} + \frac {14 \, {\left (b \sqrt {x} + a\right )}^{5} a^{4}}{b^{8}} - \frac {21 \, {\left (b \sqrt {x} + a\right )}^{4} a^{5}}{2 \, b^{8}} + \frac {14 \, {\left (b \sqrt {x} + a\right )}^{3} a^{6}}{3 \, b^{8}} - \frac {{\left (b \sqrt {x} + a\right )}^{2} a^{7}}{b^{8}} \]
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Time = 0.27 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68 \[ \int \left (a+b \sqrt {x}\right ) x^3 \, dx=\frac {2}{9} \, b x^{\frac {9}{2}} + \frac {1}{4} \, a x^{4} \]
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Time = 0.03 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68 \[ \int \left (a+b \sqrt {x}\right ) x^3 \, dx=\frac {a\,x^4}{4}+\frac {2\,b\,x^{9/2}}{9} \]
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