Integrand size = 17, antiderivative size = 275 \[ \int \frac {\left (a x+b x^3\right )^{3/2}}{x} \, dx=\frac {8 a^2 x \left (a+b x^2\right )}{15 \sqrt {b} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {a x+b x^3}}+\frac {4}{15} a x \sqrt {a x+b x^3}+\frac {2}{9} \left (a x+b x^3\right )^{3/2}-\frac {8 a^{9/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 b^{3/4} \sqrt {a x+b x^3}}+\frac {4 a^{9/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{15 b^{3/4} \sqrt {a x+b x^3}} \]
[Out]
Time = 0.15 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {2046, 2029, 2057, 335, 311, 226, 1210} \[ \int \frac {\left (a x+b x^3\right )^{3/2}}{x} \, dx=\frac {4 a^{9/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{15 b^{3/4} \sqrt {a x+b x^3}}-\frac {8 a^{9/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 b^{3/4} \sqrt {a x+b x^3}}+\frac {8 a^2 x \left (a+b x^2\right )}{15 \sqrt {b} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {a x+b x^3}}+\frac {4}{15} a x \sqrt {a x+b x^3}+\frac {2}{9} \left (a x+b x^3\right )^{3/2} \]
[In]
[Out]
Rule 226
Rule 311
Rule 335
Rule 1210
Rule 2029
Rule 2046
Rule 2057
Rubi steps \begin{align*} \text {integral}& = \frac {2}{9} \left (a x+b x^3\right )^{3/2}+\frac {1}{3} (2 a) \int \sqrt {a x+b x^3} \, dx \\ & = \frac {4}{15} a x \sqrt {a x+b x^3}+\frac {2}{9} \left (a x+b x^3\right )^{3/2}+\frac {1}{15} \left (4 a^2\right ) \int \frac {x}{\sqrt {a x+b x^3}} \, dx \\ & = \frac {4}{15} a x \sqrt {a x+b x^3}+\frac {2}{9} \left (a x+b x^3\right )^{3/2}+\frac {\left (4 a^2 \sqrt {x} \sqrt {a+b x^2}\right ) \int \frac {\sqrt {x}}{\sqrt {a+b x^2}} \, dx}{15 \sqrt {a x+b x^3}} \\ & = \frac {4}{15} a x \sqrt {a x+b x^3}+\frac {2}{9} \left (a x+b x^3\right )^{3/2}+\frac {\left (8 a^2 \sqrt {x} \sqrt {a+b x^2}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {a+b x^4}} \, dx,x,\sqrt {x}\right )}{15 \sqrt {a x+b x^3}} \\ & = \frac {4}{15} a x \sqrt {a x+b x^3}+\frac {2}{9} \left (a x+b x^3\right )^{3/2}+\frac {\left (8 a^{5/2} \sqrt {x} \sqrt {a+b x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^4}} \, dx,x,\sqrt {x}\right )}{15 \sqrt {b} \sqrt {a x+b x^3}}-\frac {\left (8 a^{5/2} \sqrt {x} \sqrt {a+b x^2}\right ) \text {Subst}\left (\int \frac {1-\frac {\sqrt {b} x^2}{\sqrt {a}}}{\sqrt {a+b x^4}} \, dx,x,\sqrt {x}\right )}{15 \sqrt {b} \sqrt {a x+b x^3}} \\ & = \frac {8 a^2 x \left (a+b x^2\right )}{15 \sqrt {b} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {a x+b x^3}}+\frac {4}{15} a x \sqrt {a x+b x^3}+\frac {2}{9} \left (a x+b x^3\right )^{3/2}-\frac {8 a^{9/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 b^{3/4} \sqrt {a x+b x^3}}+\frac {4 a^{9/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{15 b^{3/4} \sqrt {a x+b x^3}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.02 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.19 \[ \int \frac {\left (a x+b x^3\right )^{3/2}}{x} \, dx=\frac {2 a x \sqrt {x \left (a+b x^2\right )} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {3}{4},\frac {7}{4},-\frac {b x^2}{a}\right )}{3 \sqrt {1+\frac {b x^2}{a}}} \]
[In]
[Out]
Time = 2.11 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.71
method | result | size |
default | \(\frac {2 b \,x^{3} \sqrt {b \,x^{3}+a x}}{9}+\frac {22 a x \sqrt {b \,x^{3}+a x}}{45}+\frac {4 a^{2} \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{15 b \sqrt {b \,x^{3}+a x}}\) | \(195\) |
elliptic | \(\frac {2 b \,x^{3} \sqrt {b \,x^{3}+a x}}{9}+\frac {22 a x \sqrt {b \,x^{3}+a x}}{45}+\frac {4 a^{2} \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{15 b \sqrt {b \,x^{3}+a x}}\) | \(195\) |
risch | \(\frac {2 x^{2} \left (5 b \,x^{2}+11 a \right ) \left (b \,x^{2}+a \right )}{45 \sqrt {x \left (b \,x^{2}+a \right )}}+\frac {4 a^{2} \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{15 b \sqrt {b \,x^{3}+a x}}\) | \(196\) |
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.21 \[ \int \frac {\left (a x+b x^3\right )^{3/2}}{x} \, dx=-\frac {2 \, {\left (12 \, a^{2} \sqrt {b} {\rm weierstrassZeta}\left (-\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right )\right ) - {\left (5 \, b^{2} x^{3} + 11 \, a b x\right )} \sqrt {b x^{3} + a x}\right )}}{45 \, b} \]
[In]
[Out]
\[ \int \frac {\left (a x+b x^3\right )^{3/2}}{x} \, dx=\int \frac {\left (x \left (a + b x^{2}\right )\right )^{\frac {3}{2}}}{x}\, dx \]
[In]
[Out]
\[ \int \frac {\left (a x+b x^3\right )^{3/2}}{x} \, dx=\int { \frac {{\left (b x^{3} + a x\right )}^{\frac {3}{2}}}{x} \,d x } \]
[In]
[Out]
\[ \int \frac {\left (a x+b x^3\right )^{3/2}}{x} \, dx=\int { \frac {{\left (b x^{3} + a x\right )}^{\frac {3}{2}}}{x} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\left (a x+b x^3\right )^{3/2}}{x} \, dx=\int \frac {{\left (b\,x^3+a\,x\right )}^{3/2}}{x} \,d x \]
[In]
[Out]