Integrand size = 15, antiderivative size = 38 \[ \int \frac {x^5}{\sqrt {a+b x^3}} \, dx=-\frac {2 a \sqrt {a+b x^3}}{3 b^2}+\frac {2 \left (a+b x^3\right )^{3/2}}{9 b^2} \]
[Out]
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} \[ \int \frac {x^5}{\sqrt {a+b x^3}} \, dx=\frac {2 \left (a+b x^3\right )^{3/2}}{9 b^2}-\frac {2 a \sqrt {a+b x^3}}{3 b^2} \]
[In]
[Out]
Rule 45
Rule 272
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {x}{\sqrt {a+b x}} \, dx,x,x^3\right ) \\ & = \frac {1}{3} \text {Subst}\left (\int \left (-\frac {a}{b \sqrt {a+b x}}+\frac {\sqrt {a+b x}}{b}\right ) \, dx,x,x^3\right ) \\ & = -\frac {2 a \sqrt {a+b x^3}}{3 b^2}+\frac {2 \left (a+b x^3\right )^{3/2}}{9 b^2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.71 \[ \int \frac {x^5}{\sqrt {a+b x^3}} \, dx=\frac {2 \left (-2 a+b x^3\right ) \sqrt {a+b x^3}}{9 b^2} \]
[In]
[Out]
Time = 3.82 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.66
method | result | size |
gosper | \(-\frac {2 \sqrt {b \,x^{3}+a}\, \left (-b \,x^{3}+2 a \right )}{9 b^{2}}\) | \(25\) |
trager | \(-\frac {2 \sqrt {b \,x^{3}+a}\, \left (-b \,x^{3}+2 a \right )}{9 b^{2}}\) | \(25\) |
risch | \(-\frac {2 \sqrt {b \,x^{3}+a}\, \left (-b \,x^{3}+2 a \right )}{9 b^{2}}\) | \(25\) |
pseudoelliptic | \(-\frac {2 \sqrt {b \,x^{3}+a}\, \left (-b \,x^{3}+2 a \right )}{9 b^{2}}\) | \(25\) |
default | \(\frac {2 x^{3} \sqrt {b \,x^{3}+a}}{9 b}-\frac {4 a \sqrt {b \,x^{3}+a}}{9 b^{2}}\) | \(34\) |
elliptic | \(\frac {2 x^{3} \sqrt {b \,x^{3}+a}}{9 b}-\frac {4 a \sqrt {b \,x^{3}+a}}{9 b^{2}}\) | \(34\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.61 \[ \int \frac {x^5}{\sqrt {a+b x^3}} \, dx=\frac {2 \, \sqrt {b x^{3} + a} {\left (b x^{3} - 2 \, a\right )}}{9 \, b^{2}} \]
[In]
[Out]
Time = 0.19 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.21 \[ \int \frac {x^5}{\sqrt {a+b x^3}} \, dx=\begin {cases} - \frac {4 a \sqrt {a + b x^{3}}}{9 b^{2}} + \frac {2 x^{3} \sqrt {a + b x^{3}}}{9 b} & \text {for}\: b \neq 0 \\\frac {x^{6}}{6 \sqrt {a}} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.79 \[ \int \frac {x^5}{\sqrt {a+b x^3}} \, dx=\frac {2 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}}}{9 \, b^{2}} - \frac {2 \, \sqrt {b x^{3} + a} a}{3 \, b^{2}} \]
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.79 \[ \int \frac {x^5}{\sqrt {a+b x^3}} \, dx=\frac {2 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}}}{9 \, b^{2}} - \frac {2 \, \sqrt {b x^{3} + a} a}{3 \, b^{2}} \]
[In]
[Out]
Time = 5.79 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.63 \[ \int \frac {x^5}{\sqrt {a+b x^3}} \, dx=-\frac {2\,\sqrt {b\,x^3+a}\,\left (2\,a-b\,x^3\right )}{9\,b^2} \]
[In]
[Out]