Integrand size = 36, antiderivative size = 154 \[ \int \frac {\left (-b+a x^2\right ) \sqrt {-b x+a x^3}}{x^2 \left (b+a x^2\right )} \, dx=\frac {2 \sqrt {-b x+a x^3}}{x}+\sqrt [4]{a} \sqrt [4]{b} \arctan \left (\frac {2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {-b x+a x^3}}{-b-2 \sqrt {a} \sqrt {b} x+a x^2}\right )-\sqrt [4]{a} \sqrt [4]{b} \text {arctanh}\left (\frac {-\frac {b^{3/4}}{2 \sqrt [4]{a}}+\sqrt [4]{a} \sqrt [4]{b} x+\frac {a^{3/4} x^2}{2 \sqrt [4]{b}}}{\sqrt {-b x+a x^3}}\right ) \]
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Time = 0.43 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.27, number of steps used = 15, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.306, Rules used = {2081, 477, 485, 12, 504, 1225, 230, 227, 1713, 214, 211} \[ \int \frac {\left (-b+a x^2\right ) \sqrt {-b x+a x^3}}{x^2 \left (b+a x^2\right )} \, dx=\frac {\sqrt {2} \sqrt [4]{-a} \sqrt [4]{b} \sqrt {a x^3-b x} \arctan \left (\frac {\sqrt {2} \sqrt [4]{-a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a x^2-b}}\right )}{\sqrt {x} \sqrt {a x^2-b}}-\frac {\sqrt {2} \sqrt [4]{-a} \sqrt [4]{b} \sqrt {a x^3-b x} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{-a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a x^2-b}}\right )}{\sqrt {x} \sqrt {a x^2-b}}+\frac {2 \sqrt {a x^3-b x}}{x} \]
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Rule 12
Rule 211
Rule 214
Rule 227
Rule 230
Rule 477
Rule 485
Rule 504
Rule 1225
Rule 1713
Rule 2081
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {-b x+a x^3} \int \frac {\left (-b+a x^2\right )^{3/2}}{x^{3/2} \left (b+a x^2\right )} \, dx}{\sqrt {x} \sqrt {-b+a x^2}} \\ & = \frac {\left (2 \sqrt {-b x+a x^3}\right ) \text {Subst}\left (\int \frac {\left (-b+a x^4\right )^{3/2}}{x^2 \left (b+a x^4\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-b+a x^2}} \\ & = \frac {2 \sqrt {-b x+a x^3}}{x}+\frac {\left (2 \sqrt {-b x+a x^3}\right ) \text {Subst}\left (\int -\frac {4 a b^2 x^2}{\sqrt {-b+a x^4} \left (b+a x^4\right )} \, dx,x,\sqrt {x}\right )}{b \sqrt {x} \sqrt {-b+a x^2}} \\ & = \frac {2 \sqrt {-b x+a x^3}}{x}-\frac {\left (8 a b \sqrt {-b x+a x^3}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {-b+a x^4} \left (b+a x^4\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-b+a x^2}} \\ & = \frac {2 \sqrt {-b x+a x^3}}{x}-\frac {\left (4 a b \sqrt {-b x+a x^3}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {b}-\sqrt {-a} x^2\right ) \sqrt {-b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-a} \sqrt {x} \sqrt {-b+a x^2}}+\frac {\left (4 a b \sqrt {-b x+a x^3}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {b}+\sqrt {-a} x^2\right ) \sqrt {-b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-a} \sqrt {x} \sqrt {-b+a x^2}} \\ & = \frac {2 \sqrt {-b x+a x^3}}{x}+\frac {\left (2 a \sqrt {b} \sqrt {-b x+a x^3}\right ) \text {Subst}\left (\int \frac {\sqrt {b}-\sqrt {-a} x^2}{\left (\sqrt {b}+\sqrt {-a} x^2\right ) \sqrt {-b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-a} \sqrt {x} \sqrt {-b+a x^2}}-\frac {\left (2 a \sqrt {b} \sqrt {-b x+a x^3}\right ) \text {Subst}\left (\int \frac {\sqrt {b}+\sqrt {-a} x^2}{\left (\sqrt {b}-\sqrt {-a} x^2\right ) \sqrt {-b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-a} \sqrt {x} \sqrt {-b+a x^2}} \\ & = \frac {2 \sqrt {-b x+a x^3}}{x}+\frac {\left (2 a b \sqrt {-b x+a x^3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b}-2 \sqrt {-a} b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {-b+a x^2}}\right )}{\sqrt {-a} \sqrt {x} \sqrt {-b+a x^2}}-\frac {\left (2 a b \sqrt {-b x+a x^3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b}+2 \sqrt {-a} b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {-b+a x^2}}\right )}{\sqrt {-a} \sqrt {x} \sqrt {-b+a x^2}} \\ & = \frac {2 \sqrt {-b x+a x^3}}{x}+\frac {\sqrt {2} \sqrt [4]{-a} \sqrt [4]{b} \sqrt {-b x+a x^3} \arctan \left (\frac {\sqrt {2} \sqrt [4]{-a} \sqrt [4]{b} \sqrt {x}}{\sqrt {-b+a x^2}}\right )}{\sqrt {x} \sqrt {-b+a x^2}}-\frac {\sqrt {2} \sqrt [4]{-a} \sqrt [4]{b} \sqrt {-b x+a x^3} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{-a} \sqrt [4]{b} \sqrt {x}}{\sqrt {-b+a x^2}}\right )}{\sqrt {x} \sqrt {-b+a x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.44 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.02 \[ \int \frac {\left (-b+a x^2\right ) \sqrt {-b x+a x^3}}{x^2 \left (b+a x^2\right )} \, dx=\frac {(1+i) \sqrt {-b x+a x^3} \left ((1-i) \sqrt {-b+a x^2}+\sqrt [4]{a} \sqrt [4]{b} \sqrt {x} \arctan \left (\frac {(1+i) \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {-b+a x^2}}\right )+i \sqrt [4]{a} \sqrt [4]{b} \sqrt {x} \arctan \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {-b+a x^2}}{\sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )\right )}{x \sqrt {-b+a x^2}} \]
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Time = 1.53 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.17
method | result | size |
risch | \(\frac {2 a \,x^{2}-2 b}{\sqrt {x \left (a \,x^{2}-b \right )}}-\frac {\left (a b \right )^{\frac {1}{4}} \left (\ln \left (\frac {a \,x^{2}+2 x \sqrt {a b}+2 \left (a b \right )^{\frac {1}{4}} \sqrt {x \left (a \,x^{2}-b \right )}-b}{a \,x^{2}+2 x \sqrt {a b}-2 \left (a b \right )^{\frac {1}{4}} \sqrt {x \left (a \,x^{2}-b \right )}-b}\right )+2 \arctan \left (\frac {x \left (a b \right )^{\frac {1}{4}}+\sqrt {x \left (a \,x^{2}-b \right )}}{\left (a b \right )^{\frac {1}{4}} x}\right )-2 \arctan \left (\frac {x \left (a b \right )^{\frac {1}{4}}-\sqrt {x \left (a \,x^{2}-b \right )}}{\left (a b \right )^{\frac {1}{4}} x}\right )\right )}{2}\) | \(180\) |
default | \(\frac {-\ln \left (\frac {a \,x^{2}+2 x \sqrt {a b}+2 \left (a b \right )^{\frac {1}{4}} \sqrt {x \left (a \,x^{2}-b \right )}-b}{a \,x^{2}+2 x \sqrt {a b}-2 \left (a b \right )^{\frac {1}{4}} \sqrt {x \left (a \,x^{2}-b \right )}-b}\right ) \left (a b \right )^{\frac {1}{4}} x -2 \arctan \left (\frac {x \left (a b \right )^{\frac {1}{4}}+\sqrt {x \left (a \,x^{2}-b \right )}}{\left (a b \right )^{\frac {1}{4}} x}\right ) \left (a b \right )^{\frac {1}{4}} x +2 \arctan \left (\frac {x \left (a b \right )^{\frac {1}{4}}-\sqrt {x \left (a \,x^{2}-b \right )}}{\left (a b \right )^{\frac {1}{4}} x}\right ) \left (a b \right )^{\frac {1}{4}} x +4 \sqrt {x \left (a \,x^{2}-b \right )}}{2 x}\) | \(188\) |
pseudoelliptic | \(\frac {-\ln \left (\frac {a \,x^{2}+2 x \sqrt {a b}+2 \left (a b \right )^{\frac {1}{4}} \sqrt {x \left (a \,x^{2}-b \right )}-b}{a \,x^{2}+2 x \sqrt {a b}-2 \left (a b \right )^{\frac {1}{4}} \sqrt {x \left (a \,x^{2}-b \right )}-b}\right ) \left (a b \right )^{\frac {1}{4}} x -2 \arctan \left (\frac {x \left (a b \right )^{\frac {1}{4}}+\sqrt {x \left (a \,x^{2}-b \right )}}{\left (a b \right )^{\frac {1}{4}} x}\right ) \left (a b \right )^{\frac {1}{4}} x +2 \arctan \left (\frac {x \left (a b \right )^{\frac {1}{4}}-\sqrt {x \left (a \,x^{2}-b \right )}}{\left (a b \right )^{\frac {1}{4}} x}\right ) \left (a b \right )^{\frac {1}{4}} x +4 \sqrt {x \left (a \,x^{2}-b \right )}}{2 x}\) | \(188\) |
elliptic | \(\frac {2 a \,x^{2}-2 b}{\sqrt {x \left (a \,x^{2}-b \right )}}-\frac {2 b \sqrt {a b}\, \sqrt {\frac {x a}{\sqrt {a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {a b}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, -\frac {\sqrt {a b}}{a \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a \,x^{3}-b x}\, \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}-\frac {2 b \sqrt {a b}\, \sqrt {\frac {x a}{\sqrt {a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {a b}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, -\frac {\sqrt {a b}}{a \left (-\frac {\sqrt {a b}}{a}+\frac {\sqrt {-a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a \,x^{3}-b x}\, \left (-\frac {\sqrt {a b}}{a}+\frac {\sqrt {-a b}}{a}\right )}\) | \(310\) |
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Result contains complex when optimal does not.
Time = 41.62 (sec) , antiderivative size = 1362, normalized size of antiderivative = 8.84 \[ \int \frac {\left (-b+a x^2\right ) \sqrt {-b x+a x^3}}{x^2 \left (b+a x^2\right )} \, dx=\text {Too large to display} \]
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\[ \int \frac {\left (-b+a x^2\right ) \sqrt {-b x+a x^3}}{x^2 \left (b+a x^2\right )} \, dx=\int \frac {\sqrt {x \left (a x^{2} - b\right )} \left (a x^{2} - b\right )}{x^{2} \left (a x^{2} + b\right )}\, dx \]
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\[ \int \frac {\left (-b+a x^2\right ) \sqrt {-b x+a x^3}}{x^2 \left (b+a x^2\right )} \, dx=\int { \frac {\sqrt {a x^{3} - b x} {\left (a x^{2} - b\right )}}{{\left (a x^{2} + b\right )} x^{2}} \,d x } \]
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\[ \int \frac {\left (-b+a x^2\right ) \sqrt {-b x+a x^3}}{x^2 \left (b+a x^2\right )} \, dx=\int { \frac {\sqrt {a x^{3} - b x} {\left (a x^{2} - b\right )}}{{\left (a x^{2} + b\right )} x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\left (-b+a x^2\right ) \sqrt {-b x+a x^3}}{x^2 \left (b+a x^2\right )} \, dx=\text {Hanged} \]
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