Integrand size = 15, antiderivative size = 27 \[ \int \frac {x^5}{\sqrt {b+a x^3}} \, dx=\frac {2 \left (-2 b+a x^3\right ) \sqrt {b+a x^3}}{9 a^2} \]
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Time = 0.02 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.41, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 45} \[ \int \frac {x^5}{\sqrt {b+a x^3}} \, dx=\frac {2 \left (a x^3+b\right )^{3/2}}{9 a^2}-\frac {2 b \sqrt {a x^3+b}}{3 a^2} \]
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Rule 45
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {x}{\sqrt {b+a x}} \, dx,x,x^3\right ) \\ & = \frac {1}{3} \text {Subst}\left (\int \left (-\frac {b}{a \sqrt {b+a x}}+\frac {\sqrt {b+a x}}{a}\right ) \, dx,x,x^3\right ) \\ & = -\frac {2 b \sqrt {b+a x^3}}{3 a^2}+\frac {2 \left (b+a x^3\right )^{3/2}}{9 a^2} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {x^5}{\sqrt {b+a x^3}} \, dx=\frac {2 \left (-2 b+a x^3\right ) \sqrt {b+a x^3}}{9 a^2} \]
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Time = 0.79 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89
method | result | size |
gosper | \(\frac {2 \left (a \,x^{3}-2 b \right ) \sqrt {a \,x^{3}+b}}{9 a^{2}}\) | \(24\) |
trager | \(\frac {2 \left (a \,x^{3}-2 b \right ) \sqrt {a \,x^{3}+b}}{9 a^{2}}\) | \(24\) |
risch | \(\frac {2 \left (a \,x^{3}-2 b \right ) \sqrt {a \,x^{3}+b}}{9 a^{2}}\) | \(24\) |
pseudoelliptic | \(\frac {2 \left (a \,x^{3}-2 b \right ) \sqrt {a \,x^{3}+b}}{9 a^{2}}\) | \(24\) |
default | \(\frac {2 x^{3} \sqrt {a \,x^{3}+b}}{9 a}-\frac {4 b \sqrt {a \,x^{3}+b}}{9 a^{2}}\) | \(34\) |
elliptic | \(\frac {2 x^{3} \sqrt {a \,x^{3}+b}}{9 a}-\frac {4 b \sqrt {a \,x^{3}+b}}{9 a^{2}}\) | \(34\) |
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none
Time = 0.26 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int \frac {x^5}{\sqrt {b+a x^3}} \, dx=\frac {2 \, \sqrt {a x^{3} + b} {\left (a x^{3} - 2 \, b\right )}}{9 \, a^{2}} \]
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Time = 0.19 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.70 \[ \int \frac {x^5}{\sqrt {b+a x^3}} \, dx=\begin {cases} \frac {2 x^{3} \sqrt {a x^{3} + b}}{9 a} - \frac {4 b \sqrt {a x^{3} + b}}{9 a^{2}} & \text {for}\: a \neq 0 \\\frac {x^{6}}{6 \sqrt {b}} & \text {otherwise} \end {cases} \]
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none
Time = 0.20 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.11 \[ \int \frac {x^5}{\sqrt {b+a x^3}} \, dx=\frac {2 \, {\left (a x^{3} + b\right )}^{\frac {3}{2}}}{9 \, a^{2}} - \frac {2 \, \sqrt {a x^{3} + b} b}{3 \, a^{2}} \]
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none
Time = 0.27 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.11 \[ \int \frac {x^5}{\sqrt {b+a x^3}} \, dx=\frac {2 \, {\left (a x^{3} + b\right )}^{\frac {3}{2}}}{9 \, a^{2}} - \frac {2 \, \sqrt {a x^{3} + b} b}{3 \, a^{2}} \]
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Time = 5.38 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {x^5}{\sqrt {b+a x^3}} \, dx=-\frac {2\,\sqrt {a\,x^3+b}\,\left (2\,b-a\,x^3\right )}{9\,a^2} \]
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