Integrand size = 17, antiderivative size = 140 \[ \int \frac {x^4}{\sqrt {a x+b x^3}} \, dx=-\frac {10 a \sqrt {a x+b x^3}}{21 b^2}+\frac {2 x^2 \sqrt {a x+b x^3}}{7 b}+\frac {5 a^{7/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{21 b^{9/4} \sqrt {a x+b x^3}} \]
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Time = 0.10 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {2049, 2036, 335, 226} \[ \int \frac {x^4}{\sqrt {a x+b x^3}} \, dx=\frac {5 a^{7/4} \sqrt {x} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{21 b^{9/4} \sqrt {a x+b x^3}}-\frac {10 a \sqrt {a x+b x^3}}{21 b^2}+\frac {2 x^2 \sqrt {a x+b x^3}}{7 b} \]
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Rule 226
Rule 335
Rule 2036
Rule 2049
Rubi steps \begin{align*} \text {integral}& = \frac {2 x^2 \sqrt {a x+b x^3}}{7 b}-\frac {(5 a) \int \frac {x^2}{\sqrt {a x+b x^3}} \, dx}{7 b} \\ & = -\frac {10 a \sqrt {a x+b x^3}}{21 b^2}+\frac {2 x^2 \sqrt {a x+b x^3}}{7 b}+\frac {\left (5 a^2\right ) \int \frac {1}{\sqrt {a x+b x^3}} \, dx}{21 b^2} \\ & = -\frac {10 a \sqrt {a x+b x^3}}{21 b^2}+\frac {2 x^2 \sqrt {a x+b x^3}}{7 b}+\frac {\left (5 a^2 \sqrt {x} \sqrt {a+b x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x^2}} \, dx}{21 b^2 \sqrt {a x+b x^3}} \\ & = -\frac {10 a \sqrt {a x+b x^3}}{21 b^2}+\frac {2 x^2 \sqrt {a x+b x^3}}{7 b}+\frac {\left (10 a^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 )}{21 b^2 \sqrt {a x+b x^3}} \\ & = -\frac {10 a \sqrt {a x+b x^3}}{21 b^2}+\frac {2 x^2 \sqrt {a x+b x^3}}{7 b}+\frac {5 a^{7/4} \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 )}{21 b^{9/4} \sqrt {a x+b x^3}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.05 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.57 \[ \int \frac {x^4}{\sqrt {a x+b x^3}} \, dx=\frac {2 x \left (-5 a^2-2 a b x^2+3 b^2 x^4+5 a^2 \sqrt {1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {b x^2}{a}\right )\right )}{21 b^2 \sqrt {x \left (a+b x^2\right )}} \]
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Time = 2.16 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.05
method | result | size |
risch | \(-\frac {2 \left (-3 b \,x^{2}+5 a \right ) x \left (b \,x^{2}+a \right )}{21 b^{2} \sqrt {x \left (b \,x^{2}+a \right )}}+\frac {5 a^{2} \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{21 b^{3} \sqrt {b \,x^{3}+a x}}\) | \(147\) |
default | \(\frac {2 x^{2} \sqrt {b \,x^{3}+a x}}{7 b}-\frac {10 a \sqrt {b \,x^{3}+a x}}{21 b^{2}}+\frac {5 a^{2} \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{21 b^{3} \sqrt {b \,x^{3}+a x}}\) | \(149\) |
elliptic | \(\frac {2 x^{2} \sqrt {b \,x^{3}+a x}}{7 b}-\frac {10 a \sqrt {b \,x^{3}+a x}}{21 b^{2}}+\frac {5 a^{2} \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {x b}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{21 b^{3} \sqrt {b \,x^{3}+a x}}\) | \(149\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.15 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.34 \[ \int \frac {x^4}{\sqrt {a x+b x^3}} \, dx=\frac {2 \, {\left (5 \, a^{2} \sqrt {b} {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right ) + {\left (3 \, b^{2} x^{2} - 5 \, a b\right )} \sqrt {b x^{3} + a x}\right )}}{21 \, b^{3}} \]
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\[ \int \frac {x^4}{\sqrt {a x+b x^3}} \, dx=\int \frac {x^{4}}{\sqrt {x \left (a + b x^{2}\right )}}\, dx \]
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\[ \int \frac {x^4}{\sqrt {a x+b x^3}} \, dx=\int { \frac {x^{4}}{\sqrt {b x^{3} + a x}} \,d x } \]
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\[ \int \frac {x^4}{\sqrt {a x+b x^3}} \, dx=\int { \frac {x^{4}}{\sqrt {b x^{3} + a x}} \,d x } \]
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Timed out. \[ \int \frac {x^4}{\sqrt {a x+b x^3}} \, dx=\int \frac {x^4}{\sqrt {b\,x^3+a\,x}} \,d x \]
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