Integrand size = 26, antiderivative size = 20 \[ \int \frac {\sqrt {-b+a x^3} \left (2 b+a x^3\right )}{x^4} \, dx=\frac {2 \left (-b+a x^3\right )^{3/2}}{3 x^3} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {457, 75} \[ \int \frac {\sqrt {-b+a x^3} \left (2 b+a x^3\right )}{x^4} \, dx=\frac {2 \left (a x^3-b\right )^{3/2}}{3 x^3} \]
[In]
[Out]
Rule 75
Rule 457
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {\sqrt {-b+a x} (2 b+a x)}{x^2} \, dx,x,x^3\right ) \\ & = \frac {2 \left (-b+a x^3\right )^{3/2}}{3 x^3} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {-b+a x^3} \left (2 b+a x^3\right )}{x^4} \, dx=\frac {2 \left (-b+a x^3\right )^{3/2}}{3 x^3} \]
[In]
[Out]
Time = 0.80 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85
method | result | size |
gosper | \(\frac {2 \left (a \,x^{3}-b \right )^{\frac {3}{2}}}{3 x^{3}}\) | \(17\) |
trager | \(\frac {2 \left (a \,x^{3}-b \right )^{\frac {3}{2}}}{3 x^{3}}\) | \(17\) |
risch | \(\frac {2 \left (a \,x^{3}-b \right )^{\frac {3}{2}}}{3 x^{3}}\) | \(17\) |
pseudoelliptic | \(\frac {2 \left (a \,x^{3}-b \right )^{\frac {3}{2}}}{3 x^{3}}\) | \(17\) |
elliptic | \(-\frac {2 b \sqrt {a \,x^{3}-b}}{3 x^{3}}+\frac {2 a \sqrt {a \,x^{3}-b}}{3}\) | \(33\) |
default | \(a \left (\frac {2 \sqrt {a \,x^{3}-b}}{3}+\frac {2 b \,\operatorname {arctanh}\left (\frac {\sqrt {a \,x^{3}-b}}{\sqrt {-b}}\right )}{3 \sqrt {-b}}\right )+2 b \left (-\frac {\sqrt {a \,x^{3}-b}}{3 x^{3}}-\frac {a \,\operatorname {arctanh}\left (\frac {\sqrt {a \,x^{3}-b}}{\sqrt {-b}}\right )}{3 \sqrt {-b}}\right )\) | \(90\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int \frac {\sqrt {-b+a x^3} \left (2 b+a x^3\right )}{x^4} \, dx=\frac {2 \, {\left (a x^{3} - b\right )}^{\frac {3}{2}}}{3 \, x^{3}} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 10.43 (sec) , antiderivative size = 309, normalized size of antiderivative = 15.45 \[ \int \frac {\sqrt {-b+a x^3} \left (2 b+a x^3\right )}{x^4} \, dx=a \left (\begin {cases} - \frac {2 i \sqrt {a} x^{\frac {3}{2}}}{3 \sqrt {-1 + \frac {b}{a x^{3}}}} - \frac {2 i \sqrt {b} \operatorname {acosh}{\left (\frac {\sqrt {b}}{\sqrt {a} x^{\frac {3}{2}}} \right )}}{3} + \frac {2 i b}{3 \sqrt {a} x^{\frac {3}{2}} \sqrt {-1 + \frac {b}{a x^{3}}}} & \text {for}\: \left |{\frac {b}{a x^{3}}}\right | > 1 \\\frac {2 \sqrt {a} x^{\frac {3}{2}}}{3 \sqrt {1 - \frac {b}{a x^{3}}}} + \frac {2 \sqrt {b} \operatorname {asin}{\left (\frac {\sqrt {b}}{\sqrt {a} x^{\frac {3}{2}}} \right )}}{3} - \frac {2 b}{3 \sqrt {a} x^{\frac {3}{2}} \sqrt {1 - \frac {b}{a x^{3}}}} & \text {otherwise} \end {cases}\right ) + 2 b \left (\begin {cases} \frac {i \sqrt {a}}{3 x^{\frac {3}{2}} \sqrt {-1 + \frac {b}{a x^{3}}}} + \frac {i a \operatorname {acosh}{\left (\frac {\sqrt {b}}{\sqrt {a} x^{\frac {3}{2}}} \right )}}{3 \sqrt {b}} - \frac {i b}{3 \sqrt {a} x^{\frac {9}{2}} \sqrt {-1 + \frac {b}{a x^{3}}}} & \text {for}\: \left |{\frac {b}{a x^{3}}}\right | > 1 \\- \frac {\sqrt {a} \sqrt {1 - \frac {b}{a x^{3}}}}{3 x^{\frac {3}{2}}} - \frac {a \operatorname {asin}{\left (\frac {\sqrt {b}}{\sqrt {a} x^{\frac {3}{2}}} \right )}}{3 \sqrt {b}} & \text {otherwise} \end {cases}\right ) \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (16) = 32\).
Time = 0.29 (sec) , antiderivative size = 79, normalized size of antiderivative = 3.95 \[ \int \frac {\sqrt {-b+a x^3} \left (2 b+a x^3\right )}{x^4} \, dx=-\frac {2}{3} \, {\left (\sqrt {b} \arctan \left (\frac {\sqrt {a x^{3} - b}}{\sqrt {b}}\right ) - \sqrt {a x^{3} - b}\right )} a + \frac {2}{3} \, {\left (\frac {a \arctan \left (\frac {\sqrt {a x^{3} - b}}{\sqrt {b}}\right )}{\sqrt {b}} - \frac {\sqrt {a x^{3} - b}}{x^{3}}\right )} b \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (16) = 32\).
Time = 0.27 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.95 \[ \int \frac {\sqrt {-b+a x^3} \left (2 b+a x^3\right )}{x^4} \, dx=\frac {2 \, {\left (\sqrt {a x^{3} - b} a^{2} - \frac {\sqrt {a x^{3} - b} a b}{x^{3}}\right )}}{3 \, a} \]
[In]
[Out]
Time = 5.19 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80 \[ \int \frac {\sqrt {-b+a x^3} \left (2 b+a x^3\right )}{x^4} \, dx=\frac {2\,{\left (a\,x^3-b\right )}^{3/2}}{3\,x^3} \]
[In]
[Out]