Integrand size = 21, antiderivative size = 55 \[ \int \frac {\sqrt {a+b \sqrt {c x^3}}}{x} \, dx=\frac {4}{3} \sqrt {a+b \sqrt {c x^3}}-\frac {4}{3} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \sqrt {c x^3}}}{\sqrt {a}}\right ) \]
[Out]
Time = 0.02 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {376, 272, 52, 65, 214} \[ \int \frac {\sqrt {a+b \sqrt {c x^3}}}{x} \, dx=\frac {4}{3} \sqrt {a+b \sqrt {c x^3}}-\frac {4}{3} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \sqrt {c x^3}}}{\sqrt {a}}\right ) \]
[In]
[Out]
Rule 52
Rule 65
Rule 214
Rule 272
Rule 376
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {\sqrt {a+b \sqrt {c} x^{3/2}}}{x} \, dx,\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right ) \\ & = \text {Subst}\left (\frac {2}{3} \text {Subst}\left (\int \frac {\sqrt {a+b \sqrt {c} x}}{x} \, dx,x,x^{3/2}\right ),\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right ) \\ & = \frac {4}{3} \sqrt {a+b \sqrt {c x^3}}+\text {Subst}\left (\frac {1}{3} (2 a) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b \sqrt {c} x}} \, dx,x,x^{3/2}\right ),\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right ) \\ & = \frac {4}{3} \sqrt {a+b \sqrt {c x^3}}+\text {Subst}\left (\frac {(4 a) \text {Subst}\left (\int \frac {1}{-\frac {a}{b \sqrt {c}}+\frac {x^2}{b \sqrt {c}}} \, dx,x,\sqrt {a+b \sqrt {c} x^{3/2}}\right )}{3 b \sqrt {c}},\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right ) \\ & = \frac {4}{3} \sqrt {a+b \sqrt {c x^3}}-\frac {4}{3} \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c x^3}}}{\sqrt {a}}\right ) \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {a+b \sqrt {c x^3}}}{x} \, dx=\frac {4}{3} \sqrt {a+b \sqrt {c x^3}}-\frac {4}{3} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \sqrt {c x^3}}}{\sqrt {a}}\right ) \]
[In]
[Out]
Time = 5.11 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.73
method | result | size |
default | \(-\frac {4 \,\operatorname {arctanh}\left (\frac {\sqrt {a +b \sqrt {c \,x^{3}}}}{\sqrt {a}}\right ) \sqrt {a}}{3}+\frac {4 \sqrt {a +b \sqrt {c \,x^{3}}}}{3}\) | \(40\) |
[In]
[Out]
Timed out. \[ \int \frac {\sqrt {a+b \sqrt {c x^3}}}{x} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {\sqrt {a+b \sqrt {c x^3}}}{x} \, dx=\int \frac {\sqrt {a + b \sqrt {c x^{3}}}}{x}\, dx \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.11 \[ \int \frac {\sqrt {a+b \sqrt {c x^3}}}{x} \, dx=\frac {2}{3} \, \sqrt {a} \log \left (\frac {\sqrt {\sqrt {c x^{3}} b + a} - \sqrt {a}}{\sqrt {\sqrt {c x^{3}} b + a} + \sqrt {a}}\right ) + \frac {4}{3} \, \sqrt {\sqrt {c x^{3}} b + a} \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.22 \[ \int \frac {\sqrt {a+b \sqrt {c x^3}}}{x} \, dx=\frac {4 \, {\left (\frac {a \arctan \left (\frac {\sqrt {\sqrt {c x} b c^{2} x + a c^{2}}}{\sqrt {-a} c}\right )}{\sqrt {-a}} + \frac {\sqrt {\sqrt {c x} b c^{2} x + a c^{2}}}{c}\right )} {\left | c \right |}}{3 \, c} \]
[In]
[Out]
Timed out. \[ \int \frac {\sqrt {a+b \sqrt {c x^3}}}{x} \, dx=\int \frac {\sqrt {a+b\,\sqrt {c\,x^3}}}{x} \,d x \]
[In]
[Out]