Integrand size = 17, antiderivative size = 306 \[ \int \frac {1}{x^2 \left (a x+b x^3\right )^{3/2}} \, dx=\frac {1}{a x^2 \sqrt {a x+b x^3}}-\frac {21 b^{3/2} x \left (a+b x^2\right )}{5 a^3 \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {a x+b x^3}}-\frac {7 \sqrt {a x+b x^3}}{5 a^2 x^3}+\frac {21 b \sqrt {a x+b x^3}}{5 a^3 x}+\frac {21 b^{5/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 a^{11/4} \sqrt {a x+b x^3}}-\frac {21 b^{5/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{10 a^{11/4} \sqrt {a x+b x^3}} \]
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Time = 0.21 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {2048, 2050, 2057, 335, 311, 226, 1210} \[ \int \frac {1}{x^2 \left (a x+b x^3\right )^{3/2}} \, dx=-\frac {21 b^{5/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{10 a^{11/4} \sqrt {a x+b x^3}}+\frac {21 b^{5/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 a^{11/4} \sqrt {a x+b x^3}}-\frac {21 b^{3/2} x \left (a+b x^2\right )}{5 a^3 \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {a x+b x^3}}+\frac {21 b \sqrt {a x+b x^3}}{5 a^3 x}-\frac {7 \sqrt {a x+b x^3}}{5 a^2 x^3}+\frac {1}{a x^2 \sqrt {a x+b x^3}} \]
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Rule 226
Rule 311
Rule 335
Rule 1210
Rule 2048
Rule 2050
Rule 2057
Rubi steps \begin{align*} \text {integral}& = \frac {1}{a x^2 \sqrt {a x+b x^3}}+\frac {7 \int \frac {1}{x^3 \sqrt {a x+b x^3}} \, dx}{2 a} \\ & = \frac {1}{a x^2 \sqrt {a x+b x^3}}-\frac {7 \sqrt {a x+b x^3}}{5 a^2 x^3}-\frac {(21 b) \int \frac {1}{x \sqrt {a x+b x^3}} \, dx}{10 a^2} \\ & = \frac {1}{a x^2 \sqrt {a x+b x^3}}-\frac {7 \sqrt {a x+b x^3}}{5 a^2 x^3}+\frac {21 b \sqrt {a x+b x^3}}{5 a^3 x}-\frac {\left (21 b^2\right ) \int \frac {x}{\sqrt {a x+b x^3}} \, dx}{10 a^3} \\ & = \frac {1}{a x^2 \sqrt {a x+b x^3}}-\frac {7 \sqrt {a x+b x^3}}{5 a^2 x^3}+\frac {21 b \sqrt {a x+b x^3}}{5 a^3 x}-\frac {\left (21 b^2 \sqrt {x} \sqrt {a+b x^2}\right ) \int \frac {\sqrt {x}}{\sqrt {a+b x^2}} \, dx}{10 a^3 \sqrt {a x+b x^3}} \\ & = \frac {1}{a x^2 \sqrt {a x+b x^3}}-\frac {7 \sqrt {a x+b x^3}}{5 a^2 x^3}+\frac {21 b \sqrt {a x+b x^3}}{5 a^3 x}-\frac {\left (21 b^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 )}{5 a^3 \sqrt {a x+b x^3}} \\ & = \frac {1}{a x^2 \sqrt {a x+b x^3}}-\frac {7 \sqrt {a x+b x^3}}{5 a^2 x^3}+\frac {21 b \sqrt {a x+b x^3}}{5 a^3 x}-\frac {\left (21 b^{3/2} \sqrt {x} \sqrt {a+b x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^4}} \, dx,x,\sqrt {x}\right )}{5 a^{5/2} \sqrt {a x+b x^3}}+\frac {\left (21 b^{3/2} \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 )}{5 a^{5/2} \sqrt {a x+b x^3}} \\ & = \frac {1}{a x^2 \sqrt {a x+b x^3}}-\frac {21 b^{3/2} x \left (a+b x^2\right )}{5 a^3 \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {a x+b x^3}}-\frac {7 \sqrt {a x+b x^3}}{5 a^2 x^3}+\frac {21 b \sqrt {a x+b x^3}}{5 a^3 x}+\frac {21 b^{5/4} \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 )}{5 a^{11/4} \sqrt {a x+b x^3}}-\frac {21 b^{5/4} \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 )}{10 a^{11/4} \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 = 56, normalized size of antiderivative = 0.18 \[ \int \frac {1}{x^2 \left (a x+b x^3\right )^{3/2}} \, dx=-\frac {2 \sqrt {1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},\frac {3}{2},-\frac {1}{4},-\frac {b x^2}{a}\right )}{5 a x^2 \sqrt {x \left (a+b x^2\right )}} \]
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Time = 2.63 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.75
method | result | size |
default | \(-\frac {2 \sqrt {b \,x^{3}+a x}}{5 a^{2} x^{3}}+\frac {16 \left (b \,x^{2}+a \right ) b}{5 a^{3} \sqrt {x \left (b \,x^{2}+a \right )}}+\frac {b^{2} x^{2}}{a^{3} \sqrt {\left (x^{2}+\frac {a}{b}\right ) b x}}-\frac {21 b \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{10 a^{3} \sqrt {b \,x^{3}+a x}}\) | \(228\) |
elliptic | \(-\frac {2 \sqrt {b \,x^{3}+a x}}{5 a^{2} x^{3}}+\frac {16 \left (b \,x^{2}+a \right ) b}{5 a^{3} \sqrt {x \left (b \,x^{2}+a \right )}}+\frac {b^{2} x^{2}}{a^{3} \sqrt {\left (x^{2}+\frac {a}{b}\right ) b x}}-\frac {21 b \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{10 a^{3} \sqrt {b \,x^{3}+a x}}\) | \(228\) |
risch | \(-\frac {2 \left (b \,x^{2}+a \right ) \left (-8 b \,x^{2}+a \right )}{5 a^{3} x^{2} \sqrt {x \left (b \,x^{2}+a \right )}}-\frac {b^{2} \left (\frac {8 \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{b \sqrt {b \,x^{3}+a x}}-5 a \left (\frac {x^{2}}{a \sqrt {\left (x^{2}+\frac {a}{b}\right ) b x}}-\frac {\sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}+\frac {\sqrt {-a b}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{b}\right )}{2 a b \sqrt {b \,x^{3}+a x}}\right )\right )}{5 a^{3}}\) | \(389\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.29 \[ \int \frac {1}{x^2 \left (a x+b x^3\right )^{3/2}} \, dx=\frac {21 \, {\left (b^{2} x^{5} + a b x^{3}\right )} \sqrt {b} {\rm weierstrassZeta}\left (-\frac {4 \, a}{b}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right )\right ) + {\left (21 \, b^{2} x^{4} + 14 \, a b x^{2} - 2 \, a^{2}\right )} \sqrt {b x^{3} + a x}}{5 \, {\left (a^{3} b x^{5} + a^{4} x^{3}\right )}} \]
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\[ \int \frac {1}{x^2 \left (a x+b x^3\right )^{3/2}} \, dx=\int \frac {1}{x^{2} \left (x \left (a + b x^{2}\right )\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {1}{x^2 \left (a x+b x^3\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b x^{3} + a x\right )}^{\frac {3}{2}} x^{2}} \,d x } \]
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\[ \int \frac {1}{x^2 \left (a x+b x^3\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b x^{3} + a x\right )}^{\frac {3}{2}} x^{2}} \,d x } \]
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Timed out. \[ \int \frac {1}{x^2 \left (a x+b x^3\right )^{3/2}} \, dx=\int \frac {1}{x^2\,{\left (b\,x^3+a\,x\right )}^{3/2}} \,d x \]
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