Integrand size = 15, antiderivative size = 38 \[ \int x^5 \sqrt {b+a x^3} \, dx=\frac {2 \sqrt {b+a x^3} \left (-2 b^2+a b x^3+3 a^2 x^6\right )}{45 a^2} \]
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Time = 0.02 (sec) , antiderivative size = 38, 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} \[ \int x^5 \sqrt {b+a x^3} \, dx=\frac {2 \left (a x^3+b\right )^{5/2}}{15 a^2}-\frac {2 b \left (a x^3+b\right )^{3/2}}{9 a^2} \]
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Rule 45
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int x \sqrt {b+a x} \, dx,x,x^3\right ) \\ & = \frac {1}{3} \text {Subst}\left (\int \left (-\frac {b \sqrt {b+a x}}{a}+\frac {(b+a x)^{3/2}}{a}\right ) \, dx,x,x^3\right ) \\ & = -\frac {2 b \left (b+a x^3\right )^{3/2}}{9 a^2}+\frac {2 \left (b+a x^3\right )^{5/2}}{15 a^2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00 \[ \int x^5 \sqrt {b+a x^3} \, dx=\frac {2 \sqrt {b+a x^3} \left (-2 b^2+a b x^3+3 a^2 x^6\right )}{45 a^2} \]
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Time = 0.87 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.66
method | result | size |
gosper | \(\frac {2 \left (a \,x^{3}+b \right )^{\frac {3}{2}} \left (3 a \,x^{3}-2 b \right )}{45 a^{2}}\) | \(25\) |
pseudoelliptic | \(\frac {2 \left (a \,x^{3}+b \right )^{\frac {3}{2}} \left (3 a \,x^{3}-2 b \right )}{45 a^{2}}\) | \(25\) |
trager | \(\frac {2 \sqrt {a \,x^{3}+b}\, \left (3 a^{2} x^{6}+a b \,x^{3}-2 b^{2}\right )}{45 a^{2}}\) | \(35\) |
risch | \(\frac {2 \sqrt {a \,x^{3}+b}\, \left (3 a^{2} x^{6}+a b \,x^{3}-2 b^{2}\right )}{45 a^{2}}\) | \(35\) |
default | \(\frac {2 x^{6} \sqrt {a \,x^{3}+b}}{15}+\frac {2 b \,x^{3} \sqrt {a \,x^{3}+b}}{45 a}-\frac {4 b^{2} \sqrt {a \,x^{3}+b}}{45 a^{2}}\) | \(51\) |
elliptic | \(\frac {2 x^{6} \sqrt {a \,x^{3}+b}}{15}+\frac {2 b \,x^{3} \sqrt {a \,x^{3}+b}}{45 a}-\frac {4 b^{2} \sqrt {a \,x^{3}+b}}{45 a^{2}}\) | \(51\) |
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Time = 0.25 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.89 \[ \int x^5 \sqrt {b+a x^3} \, dx=\frac {2 \, {\left (3 \, a^{2} x^{6} + a b x^{3} - 2 \, b^{2}\right )} \sqrt {a x^{3} + b}}{45 \, a^{2}} \]
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Time = 0.19 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.74 \[ \int x^5 \sqrt {b+a x^3} \, dx=\begin {cases} \frac {2 x^{6} \sqrt {a x^{3} + b}}{15} + \frac {2 b x^{3} \sqrt {a x^{3} + b}}{45 a} - \frac {4 b^{2} \sqrt {a x^{3} + b}}{45 a^{2}} & \text {for}\: a \neq 0 \\\frac {\sqrt {b} x^{6}}{6} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.79 \[ \int x^5 \sqrt {b+a x^3} \, dx=\frac {2 \, {\left (a x^{3} + b\right )}^{\frac {5}{2}}}{15 \, a^{2}} - \frac {2 \, {\left (a x^{3} + b\right )}^{\frac {3}{2}} b}{9 \, a^{2}} \]
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Time = 0.28 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.76 \[ \int x^5 \sqrt {b+a x^3} \, dx=\frac {2 \, {\left (3 \, {\left (a x^{3} + b\right )}^{\frac {5}{2}} - 5 \, {\left (a x^{3} + b\right )}^{\frac {3}{2}} b\right )}}{45 \, a^{2}} \]
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Time = 5.34 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.76 \[ \int x^5 \sqrt {b+a x^3} \, dx=-\frac {10\,b\,{\left (a\,x^3+b\right )}^{3/2}-6\,{\left (a\,x^3+b\right )}^{5/2}}{45\,a^2} \]
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