Integrand size = 17, antiderivative size = 29 \[ \int \frac {x^5}{\sqrt {-b+a x^3}} \, dx=\frac {2 \sqrt {-b+a x^3} \left (2 b+a x^3\right )}{9 a^2} \]
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Time = 0.02 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.45, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, 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 = 29, normalized size of antiderivative = 1.00 \[ \int \frac {x^5}{\sqrt {-b+a x^3}} \, dx=\frac {2 \sqrt {-b+a x^3} \left (2 b+a x^3\right )}{9 a^2} \]
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Time = 0.79 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90
method | result | size |
gosper | \(\frac {2 \sqrt {a \,x^{3}-b}\, \left (a \,x^{3}+2 b \right )}{9 a^{2}}\) | \(26\) |
trager | \(\frac {2 \sqrt {a \,x^{3}-b}\, \left (a \,x^{3}+2 b \right )}{9 a^{2}}\) | \(26\) |
risch | \(\frac {2 \sqrt {a \,x^{3}-b}\, \left (a \,x^{3}+2 b \right )}{9 a^{2}}\) | \(26\) |
pseudoelliptic | \(\frac {2 \sqrt {a \,x^{3}-b}\, \left (a \,x^{3}+2 b \right )}{9 a^{2}}\) | \(26\) |
default | \(\frac {2 x^{3} \sqrt {a \,x^{3}-b}}{9 a}+\frac {4 b \sqrt {a \,x^{3}-b}}{9 a^{2}}\) | \(38\) |
elliptic | \(\frac {2 x^{3} \sqrt {a \,x^{3}-b}}{9 a}+\frac {4 b \sqrt {a \,x^{3}-b}}{9 a^{2}}\) | \(38\) |
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none
Time = 0.24 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.86 \[ \int \frac {x^5}{\sqrt {-b+a x^3}} \, dx=\frac {2 \, {\left (a x^{3} + 2 \, b\right )} \sqrt {a x^{3} - b}}{9 \, a^{2}} \]
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Time = 0.18 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.66 \[ \int \frac {x^5}{\sqrt {-b+a x^3}} \, dx=\begin {cases} \frac {2 x^{3} \sqrt {a x^{3} - b}}{9 a} + \frac {4 b \sqrt {a x^{3} - b}}{9 a^{2}} & \text {for}\: a \neq 0 \\\frac {x^{6}}{6 \sqrt {- b}} & \text {otherwise} \end {cases} \]
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none
Time = 0.19 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.17 \[ \int \frac {x^5}{\sqrt {-b+a x^3}} \, dx=\frac {2 \, {\left (a x^{3} - b\right )}^{\frac {3}{2}}}{9 \, a^{2}} + \frac {2 \, \sqrt {a x^{3} - b} b}{3 \, a^{2}} \]
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none
Time = 0.27 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.17 \[ \int \frac {x^5}{\sqrt {-b+a x^3}} \, dx=\frac {2 \, {\left (a x^{3} - b\right )}^{\frac {3}{2}}}{9 \, a^{2}} + \frac {2 \, \sqrt {a x^{3} - b} b}{3 \, a^{2}} \]
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Time = 5.37 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.86 \[ \int \frac {x^5}{\sqrt {-b+a x^3}} \, dx=\frac {2\,\sqrt {a\,x^3-b}\,\left (a\,x^3+2\,b\right )}{9\,a^2} \]
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