Optimal. Leaf size=73 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {a x^3+b x}}{a x^2+b}\right )}{\sqrt [4]{2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {a x^3+b x}}{a x^2+b}\right )}{\sqrt [4]{2}} \]
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Rubi [C] time = 15.30, antiderivative size = 2513, normalized size of antiderivative = 34.42, number of steps used = 23, number of rules used = 9, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {1594, 2056, 6715, 6728, 406, 220, 409, 1217, 1707}
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Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 220
Rule 406
Rule 409
Rule 1217
Rule 1594
Rule 1707
Rule 2056
Rule 6715
Rule 6728
Rubi steps
\begin {align*} \int \frac {\left (-b+a x^2\right ) \sqrt {b x+a x^3}}{b^2 x+2 (-1+a b) x^3+a^2 x^5} \, dx &=\int \frac {\left (-b+a x^2\right ) \sqrt {b x+a x^3}}{x \left (b^2+2 (-1+a b) x^2+a^2 x^4\right )} \, dx\\ &=\frac {\sqrt {b x+a x^3} \int \frac {\left (-b+a x^2\right ) \sqrt {b+a x^2}}{\sqrt {x} \left (b^2+2 (-1+a b) x^2+a^2 x^4\right )} \, dx}{\sqrt {x} \sqrt {b+a x^2}}\\ &=\frac {\left (2 \sqrt {b x+a x^3}\right ) \operatorname {Subst}\left (\int \frac {\left (-b+a x^4\right ) \sqrt {b+a x^4}}{b^2+2 (-1+a b) x^4+a^2 x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {b+a x^2}}\\ &=\frac {\left (2 \sqrt {b x+a x^3}\right ) \operatorname {Subst}\left (\int \left (\frac {\left (a+a \sqrt {1-2 a b}\right ) \sqrt {b+a x^4}}{-2 \sqrt {1-2 a b}+2 (-1+a b)+2 a^2 x^4}+\frac {\left (a-a \sqrt {1-2 a b}\right ) \sqrt {b+a x^4}}{2 \sqrt {1-2 a b}+2 (-1+a b)+2 a^2 x^4}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {b+a x^2}}\\ &=\frac {\left (2 a \left (1-\sqrt {1-2 a b}\right ) \sqrt {b x+a x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {b+a x^4}}{2 \sqrt {1-2 a b}+2 (-1+a b)+2 a^2 x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {b+a x^2}}+\frac {\left (2 a \left (1+\sqrt {1-2 a b}\right ) \sqrt {b x+a x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {b+a x^4}}{-2 \sqrt {1-2 a b}+2 (-1+a b)+2 a^2 x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {b+a x^2}}\\ &=\frac {\left (\left (1-\sqrt {1-2 a b}\right ) \sqrt {b x+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {b+a x^2}}+\frac {\left (2 \left (1-\sqrt {1-2 a b}\right )^2 \sqrt {b x+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b+a x^4} \left (2 \sqrt {1-2 a b}+2 (-1+a b)+2 a^2 x^4\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {b+a x^2}}+\frac {\left (\left (1+\sqrt {1-2 a b}\right ) \sqrt {b x+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b+a x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {b+a x^2}}+\frac {\left (2 \left (1+\sqrt {1-2 a b}\right )^2 \sqrt {b x+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b+a x^4} \left (-2 \sqrt {1-2 a b}+2 (-1+a b)+2 a^2 x^4\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {b+a x^2}}\\ &=\frac {\left (1-\sqrt {1-2 a b}\right ) \left (\sqrt {b}+\sqrt {a} x\right ) \sqrt {\frac {b+a x^2}{\left (\sqrt {b}+\sqrt {a} x\right )^2}} \sqrt {b x+a x^3} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {x} \left (b+a x^2\right )}+\frac {\left (1+\sqrt {1-2 a b}\right ) \left (\sqrt {b}+\sqrt {a} x\right ) \sqrt {\frac {b+a x^2}{\left (\sqrt {b}+\sqrt {a} x\right )^2}} \sqrt {b x+a x^3} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {x} \left (b+a x^2\right )}-\frac {\left (\left (1-\sqrt {1-2 a b}\right )^2 \sqrt {b x+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-\frac {a x^2}{\sqrt {1-a b-\sqrt {1-2 a b}}}\right ) \sqrt {b+a x^4}} \, dx,x,\sqrt {x}\right )}{2 \left (1-a b-\sqrt {1-2 a b}\right ) \sqrt {x} \sqrt {b+a x^2}}-\frac {\left (\left (1-\sqrt {1-2 a b}\right )^2 \sqrt {b x+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+\frac {a x^2}{\sqrt {1-a b-\sqrt {1-2 a b}}}\right ) \sqrt {b+a x^4}} \, dx,x,\sqrt {x}\right )}{2 \left (1-a b-\sqrt {1-2 a b}\right ) \sqrt {x} \sqrt {b+a x^2}}-\frac {\left (\left (1+\sqrt {1-2 a b}\right )^2 \sqrt {b x+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-\frac {a x^2}{\sqrt {1-a b+\sqrt {1-2 a b}}}\right ) \sqrt {b+a x^4}} \, dx,x,\sqrt {x}\right )}{2 \left (1-a b+\sqrt {1-2 a b}\right ) \sqrt {x} \sqrt {b+a x^2}}-\frac {\left (\left (1+\sqrt {1-2 a b}\right )^2 \sqrt {b x+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+\frac {a x^2}{\sqrt {1-a b+\sqrt {1-2 a b}}}\right ) \sqrt {b+a x^4}} \, dx,x,\sqrt {x}\right )}{2 \left (1-a b+\sqrt {1-2 a b}\right ) \sqrt {x} \sqrt {b+a x^2}}\\ &=\frac {\left (1-\sqrt {1-2 a b}\right ) \left (\sqrt {b}+\sqrt {a} x\right ) \sqrt {\frac {b+a x^2}{\left (\sqrt {b}+\sqrt {a} x\right )^2}} \sqrt {b x+a x^3} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {x} \left (b+a x^2\right )}+\frac {\left (1+\sqrt {1-2 a b}\right ) \left (\sqrt {b}+\sqrt {a} x\right ) \sqrt {\frac {b+a x^2}{\left (\sqrt {b}+\sqrt {a} x\right )^2}} \sqrt {b x+a x^3} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {x} \left (b+a x^2\right )}-\frac {\left (\left (1-\sqrt {1-2 a b}\right )^2 \left (1-\frac {\sqrt {a} \sqrt {b}}{\sqrt {1-a b-\sqrt {1-2 a b}}}\right ) \sqrt {b x+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b+a x^4}} \, dx,x,\sqrt {x}\right )}{2 \left (1-a b-\sqrt {1-2 a b}\right ) \left (1-\frac {a b}{1-a b-\sqrt {1-2 a b}}\right ) \sqrt {x} \sqrt {b+a x^2}}-\frac {\left (\left (1-\sqrt {1-2 a b}\right )^2 \left (1+\frac {\sqrt {a} \sqrt {b}}{\sqrt {1-a b-\sqrt {1-2 a b}}}\right ) \sqrt {b x+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b+a x^4}} \, dx,x,\sqrt {x}\right )}{2 \left (1-a b-\sqrt {1-2 a b}\right ) \left (1-\frac {a b}{1-a b-\sqrt {1-2 a b}}\right ) \sqrt {x} \sqrt {b+a x^2}}+\frac {\left (\sqrt {a} \sqrt {b} \left (1-\sqrt {1-2 a b}\right )^2 \left (\sqrt {a} \sqrt {b}-\sqrt {1-a b-\sqrt {1-2 a b}}\right ) \sqrt {b x+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {\sqrt {a} x^2}{\sqrt {b}}}{\left (1-\frac {a x^2}{\sqrt {1-a b-\sqrt {1-2 a b}}}\right ) \sqrt {b+a x^4}} \, dx,x,\sqrt {x}\right )}{2 \left (1-2 a b-\sqrt {1-2 a b}\right ) \left (1-a b-\sqrt {1-2 a b}\right ) \sqrt {x} \sqrt {b+a x^2}}+\frac {\left (\sqrt {a} \sqrt {b} \left (1-\sqrt {1-2 a b}\right )^2 \left (\sqrt {a} \sqrt {b}+\sqrt {1-a b-\sqrt {1-2 a b}}\right ) \sqrt {b x+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {\sqrt {a} x^2}{\sqrt {b}}}{\left (1+\frac {a x^2}{\sqrt {1-a b-\sqrt {1-2 a b}}}\right ) \sqrt {b+a x^4}} \, dx,x,\sqrt {x}\right )}{2 \left (1-2 a b-\sqrt {1-2 a b}\right ) \left (1-a b-\sqrt {1-2 a b}\right ) \sqrt {x} \sqrt {b+a x^2}}-\frac {\left (\left (1+\sqrt {1-2 a b}\right )^2 \left (1-\frac {\sqrt {a} \sqrt {b}}{\sqrt {1-a b+\sqrt {1-2 a b}}}\right ) \sqrt {b x+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b+a x^4}} \, dx,x,\sqrt {x}\right )}{2 \left (1-a b+\sqrt {1-2 a b}\right ) \left (1-\frac {a b}{1-a b+\sqrt {1-2 a b}}\right ) \sqrt {x} \sqrt {b+a x^2}}-\frac {\left (\left (1+\sqrt {1-2 a b}\right )^2 \left (1+\frac {\sqrt {a} \sqrt {b}}{\sqrt {1-a b+\sqrt {1-2 a b}}}\right ) \sqrt {b x+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b+a x^4}} \, dx,x,\sqrt {x}\right )}{2 \left (1-a b+\sqrt {1-2 a b}\right ) \left (1-\frac {a b}{1-a b+\sqrt {1-2 a b}}\right ) \sqrt {x} \sqrt {b+a x^2}}+\frac {\left (\sqrt {a} \sqrt {b} \left (1+\sqrt {1-2 a b}\right )^2 \left (\sqrt {a} \sqrt {b}-\sqrt {1-a b+\sqrt {1-2 a b}}\right ) \sqrt {b x+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {\sqrt {a} x^2}{\sqrt {b}}}{\left (1-\frac {a x^2}{\sqrt {1-a b+\sqrt {1-2 a b}}}\right ) \sqrt {b+a x^4}} \, dx,x,\sqrt {x}\right )}{2 \left (1-2 a b+\sqrt {1-2 a b}\right ) \left (1-a b+\sqrt {1-2 a b}\right ) \sqrt {x} \sqrt {b+a x^2}}+\frac {\left (\sqrt {a} \sqrt {b} \left (1+\sqrt {1-2 a b}\right )^2 \left (\sqrt {a} \sqrt {b}+\sqrt {1-a b+\sqrt {1-2 a b}}\right ) \sqrt {b x+a x^3}\right ) \operatorname {Subst}\left (\int \frac {1+\frac {\sqrt {a} x^2}{\sqrt {b}}}{\left (1+\frac {a x^2}{\sqrt {1-a b+\sqrt {1-2 a b}}}\right ) \sqrt {b+a x^4}} \, dx,x,\sqrt {x}\right )}{2 \left (1-2 a b+\sqrt {1-2 a b}\right ) \left (1-a b+\sqrt {1-2 a b}\right ) \sqrt {x} \sqrt {b+a x^2}}\\ &=\frac {\left (1-\sqrt {1-2 a b}\right )^{3/2} \left (\sqrt {a} \sqrt {b}-\sqrt {1-a b-\sqrt {1-2 a b}}\right ) \left (\sqrt {a} \sqrt {b}+\sqrt {1-a b-\sqrt {1-2 a b}}\right ) \sqrt {b x+a x^3} \tan ^{-1}\left (\frac {\sqrt {1-\sqrt {1-2 a b}} \sqrt {x}}{\sqrt [4]{1-a b-\sqrt {1-2 a b}} \sqrt {b+a x^2}}\right )}{4 \left (1-2 a b-\sqrt {1-2 a b}\right ) \left (1-a b-\sqrt {1-2 a b}\right )^{3/4} \sqrt {x} \sqrt {b+a x^2}}+\frac {\left (-1+\sqrt {1-2 a b}\right )^{3/2} \left (\sqrt {a} \sqrt {b}-\sqrt {1-a b-\sqrt {1-2 a b}}\right ) \left (\sqrt {a} \sqrt {b}+\sqrt {1-a b-\sqrt {1-2 a b}}\right ) \sqrt {b x+a x^3} \tan ^{-1}\left (\frac {\sqrt {-1+\sqrt {1-2 a b}} \sqrt {x}}{\sqrt [4]{1-a b-\sqrt {1-2 a b}} \sqrt {b+a x^2}}\right )}{4 \left (1-2 a b-\sqrt {1-2 a b}\right ) \left (1-a b-\sqrt {1-2 a b}\right )^{3/4} \sqrt {x} \sqrt {b+a x^2}}-\frac {\left (2 a^2 b^2+2 \left (1+\sqrt {1-2 a b}\right )-a b \left (5+3 \sqrt {1-2 a b}\right )\right ) \sqrt {b x+a x^3} \tan ^{-1}\left (\frac {\sqrt {-1-\sqrt {1-2 a b}} \sqrt {x}}{\sqrt [4]{1-a b+\sqrt {1-2 a b}} \sqrt {b+a x^2}}\right )}{2 \sqrt {-1-\sqrt {1-2 a b}} \left (1-2 a b+\sqrt {1-2 a b}\right ) \left (1-a b+\sqrt {1-2 a b}\right )^{3/4} \sqrt {x} \sqrt {b+a x^2}}+\frac {\left (1+\sqrt {1-2 a b}\right )^{3/2} \left (\sqrt {a} \sqrt {b}-\sqrt {1-a b+\sqrt {1-2 a b}}\right ) \left (\sqrt {a} \sqrt {b}+\sqrt {1-a b+\sqrt {1-2 a b}}\right ) \sqrt {b x+a x^3} \tan ^{-1}\left (\frac {\sqrt {1+\sqrt {1-2 a b}} \sqrt {x}}{\sqrt [4]{1-a b+\sqrt {1-2 a b}} \sqrt {b+a x^2}}\right )}{4 \left (1-2 a b+\sqrt {1-2 a b}\right ) \left (1-a b+\sqrt {1-2 a b}\right )^{3/4} \sqrt {x} \sqrt {b+a x^2}}+\frac {\left (1-\sqrt {1-2 a b}\right ) \left (\sqrt {b}+\sqrt {a} x\right ) \sqrt {\frac {b+a x^2}{\left (\sqrt {b}+\sqrt {a} x\right )^2}} \sqrt {b x+a x^3} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {x} \left (b+a x^2\right )}+\frac {\left (1+\sqrt {1-2 a b}\right ) \left (\sqrt {b}+\sqrt {a} x\right ) \sqrt {\frac {b+a x^2}{\left (\sqrt {b}+\sqrt {a} x\right )^2}} \sqrt {b x+a x^3} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {x} \left (b+a x^2\right )}+\frac {\left (1-\sqrt {1-2 a b}\right )^2 \left (\sqrt {a} \sqrt {b}-\sqrt {1-a b-\sqrt {1-2 a b}}\right ) \left (\sqrt {b}+\sqrt {a} x\right ) \sqrt {\frac {b+a x^2}{\left (\sqrt {b}+\sqrt {a} x\right )^2}} \sqrt {b x+a x^3} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} \left (1-2 a b-\sqrt {1-2 a b}\right ) \sqrt {1-a b-\sqrt {1-2 a b}} \sqrt {x} \left (b+a x^2\right )}-\frac {\left (1-\sqrt {1-2 a b}\right )^2 \left (\sqrt {a} \sqrt {b}+\sqrt {1-a b-\sqrt {1-2 a b}}\right ) \left (\sqrt {b}+\sqrt {a} x\right ) \sqrt {\frac {b+a x^2}{\left (\sqrt {b}+\sqrt {a} x\right )^2}} \sqrt {b x+a x^3} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} \left (1-2 a b-\sqrt {1-2 a b}\right ) \sqrt {1-a b-\sqrt {1-2 a b}} \sqrt {x} \left (b+a x^2\right )}+\frac {\sqrt {1-a b+\sqrt {1-2 a b}} \left (\sqrt {a} \sqrt {b}-\sqrt {1-a b+\sqrt {1-2 a b}}\right ) \left (\sqrt {b}+\sqrt {a} x\right ) \sqrt {\frac {b+a x^2}{\left (\sqrt {b}+\sqrt {a} x\right )^2}} \sqrt {b x+a x^3} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} \left (1-2 a b+\sqrt {1-2 a b}\right ) \sqrt {x} \left (b+a x^2\right )}-\frac {\sqrt {1-a b+\sqrt {1-2 a b}} \left (\sqrt {a} \sqrt {b}+\sqrt {1-a b+\sqrt {1-2 a b}}\right ) \left (\sqrt {b}+\sqrt {a} x\right ) \sqrt {\frac {b+a x^2}{\left (\sqrt {b}+\sqrt {a} x\right )^2}} \sqrt {b x+a x^3} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} \left (1-2 a b+\sqrt {1-2 a b}\right ) \sqrt {x} \left (b+a x^2\right )}+\frac {\left (1-\sqrt {1-2 a b}+2 \sqrt {a} \sqrt {b} \sqrt {1-a b-\sqrt {1-2 a b}}\right ) \left (\sqrt {b}+\sqrt {a} x\right ) \sqrt {\frac {b+a x^2}{\left (\sqrt {b}+\sqrt {a} x\right )^2}} \sqrt {b x+a x^3} \Pi \left (-\frac {\left (\sqrt {a} \sqrt {b}-\sqrt {1-a b-\sqrt {1-2 a b}}\right )^2}{4 \sqrt {a} \sqrt {b} \sqrt {1-a b-\sqrt {1-2 a b}}};2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} \left (1-2 a b-\sqrt {1-2 a b}\right ) \sqrt {x} \left (b+a x^2\right )}+\frac {\left (1-\sqrt {1-2 a b}-2 \sqrt {a} \sqrt {b} \sqrt {1-a b-\sqrt {1-2 a b}}\right ) \left (\sqrt {b}+\sqrt {a} x\right ) \sqrt {\frac {b+a x^2}{\left (\sqrt {b}+\sqrt {a} x\right )^2}} \sqrt {b x+a x^3} \Pi \left (\frac {\left (\sqrt {a} \sqrt {b}+\sqrt {1-a b-\sqrt {1-2 a b}}\right )^2}{4 \sqrt {a} \sqrt {b} \sqrt {1-a b-\sqrt {1-2 a b}}};2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} \left (1-2 a b-\sqrt {1-2 a b}\right ) \sqrt {x} \left (b+a x^2\right )}+\frac {\left (1+\sqrt {1-2 a b}+2 \sqrt {a} \sqrt {b} \sqrt {1-a b+\sqrt {1-2 a b}}\right ) \left (\sqrt {b}+\sqrt {a} x\right ) \sqrt {\frac {b+a x^2}{\left (\sqrt {b}+\sqrt {a} x\right )^2}} \sqrt {b x+a x^3} \Pi \left (-\frac {\left (\sqrt {a} \sqrt {b}-\sqrt {1-a b+\sqrt {1-2 a b}}\right )^2}{4 \sqrt {a} \sqrt {b} \sqrt {1-a b+\sqrt {1-2 a b}}};2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} \left (1-2 a b+\sqrt {1-2 a b}\right ) \sqrt {x} \left (b+a x^2\right )}+\frac {\left (1+\sqrt {1-2 a b}-2 \sqrt {a} \sqrt {b} \sqrt {1-a b+\sqrt {1-2 a b}}\right ) \left (\sqrt {b}+\sqrt {a} x\right ) \sqrt {\frac {b+a x^2}{\left (\sqrt {b}+\sqrt {a} x\right )^2}} \sqrt {b x+a x^3} \Pi \left (\frac {\left (\sqrt {a} \sqrt {b}+\sqrt {1-a b+\sqrt {1-2 a b}}\right )^2}{4 \sqrt {a} \sqrt {b} \sqrt {1-a b+\sqrt {1-2 a b}}};2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{b} \left (1-2 a b+\sqrt {1-2 a b}\right ) \sqrt {x} \left (b+a x^2\right )}\\ \end {align*}
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Mathematica [C] time = 1.42, size = 372, normalized size = 5.10 \begin {gather*} -\frac {i x^{3/2} \sqrt {\frac {b}{a x^2}+1} \left (-\Pi \left (-\frac {i \sqrt {a}}{\sqrt {b} \sqrt {\frac {-a b+\sqrt {1-2 a b}+1}{b^2}}};\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}}{\sqrt {x}}\right )\right |-1\right )-\Pi \left (\frac {i \sqrt {a}}{\sqrt {b} \sqrt {\frac {-a b+\sqrt {1-2 a b}+1}{b^2}}};\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}}{\sqrt {x}}\right )\right |-1\right )-\Pi \left (-\frac {i \sqrt {a}}{\sqrt {b} \sqrt {-\frac {a b+\sqrt {1-2 a b}-1}{b^2}}};\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}}{\sqrt {x}}\right )\right |-1\right )-\Pi \left (\frac {i \sqrt {a}}{\sqrt {b} \sqrt {-\frac {a b+\sqrt {1-2 a b}-1}{b^2}}};\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}}{\sqrt {x}}\right )\right |-1\right )+2 F\left (\left .i \sinh ^{-1}\left (\frac {\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}}{\sqrt {x}}\right )\right |-1\right )\right )}{\sqrt {\frac {i \sqrt {b}}{\sqrt {a}}} \sqrt {x \left (a x^2+b\right )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.48, size = 73, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {b x+a x^3}}{b+a x^2}\right )}{\sqrt [4]{2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {b x+a x^3}}{b+a x^2}\right )}{\sqrt [4]{2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.73, size = 221, normalized size = 3.03 \begin {gather*} -\frac {1}{2} \cdot 2^{\frac {3}{4}} \arctan \left (\frac {2^{\frac {1}{4}} \sqrt {a x^{3} + b x}}{a x^{2} + b}\right ) - \frac {1}{8} \cdot 2^{\frac {3}{4}} \log \left (\frac {a^{2} x^{4} + 2 \, {\left (a b + 1\right )} x^{2} + b^{2} + 2 \, \sqrt {2} {\left (a x^{3} + b x\right )} + 2 \, \sqrt {a x^{3} + b x} {\left (2^{\frac {3}{4}} x + 2^{\frac {1}{4}} {\left (a x^{2} + b\right )}\right )}}{a^{2} x^{4} + 2 \, {\left (a b - 1\right )} x^{2} + b^{2}}\right ) + \frac {1}{8} \cdot 2^{\frac {3}{4}} \log \left (\frac {a^{2} x^{4} + 2 \, {\left (a b + 1\right )} x^{2} + b^{2} + 2 \, \sqrt {2} {\left (a x^{3} + b x\right )} - 2 \, \sqrt {a x^{3} + b x} {\left (2^{\frac {3}{4}} x + 2^{\frac {1}{4}} {\left (a x^{2} + b\right )}\right )}}{a^{2} x^{4} + 2 \, {\left (a b - 1\right )} x^{2} + b^{2}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a x^{3} + b x} {\left (a x^{2} - b\right )}}{a^{2} x^{5} + 2 \, {\left (a b - 1\right )} x^{3} + b^{2} x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.17, size = 385, normalized size = 5.27
method | result | size |
elliptic | \(\frac {\sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a \,x^{3}+b x}}-\frac {\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a^{2} \textit {\_Z}^{4}+\left (2 a b -2\right ) \textit {\_Z}^{2}+b^{2}\right )}{\sum }\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{2} a b -\underline {\hspace {1.25 ex}}\alpha ^{2}+b^{2}\right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {\left (x -\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \left (a \left (a^{2} \underline {\hspace {1.25 ex}}\alpha ^{3}+a b \underline {\hspace {1.25 ex}}\alpha -2 \underline {\hspace {1.25 ex}}\alpha \right )-a^{2} \sqrt {-a b}\, \underline {\hspace {1.25 ex}}\alpha ^{2}-\sqrt {-a b}\, a b +2 \sqrt {-a b}\right ) \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}\, \underline {\hspace {1.25 ex}}\alpha ^{3} a^{2}+\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2} b +\sqrt {-a b}\, \underline {\hspace {1.25 ex}}\alpha a b +a \,b^{2}-2 \sqrt {-a b}\, \underline {\hspace {1.25 ex}}\alpha -2 b}{2 b}, \frac {\sqrt {2}}{2}\right )}{\underline {\hspace {1.25 ex}}\alpha \left (\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}+a b -1\right ) \sqrt {x \left (a \,x^{2}+b \right )}}\right )}{4 a b}\) | \(385\) |
default | \(\frac {\frac {2 \sqrt {a \,x^{3}+b x}}{3}+\frac {5 b \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{3 a \sqrt {a \,x^{3}+b x}}-\frac {\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a^{2} \textit {\_Z}^{4}+\left (2 a b -2\right ) \textit {\_Z}^{2}+b^{2}\right )}{\sum }\frac {\left (\underline {\hspace {1.25 ex}}\alpha ^{2} a b -\underline {\hspace {1.25 ex}}\alpha ^{2}+b^{2}\right ) \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {\left (x -\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \left (a \left (a^{2} \underline {\hspace {1.25 ex}}\alpha ^{3}+a b \underline {\hspace {1.25 ex}}\alpha -2 \underline {\hspace {1.25 ex}}\alpha \right )-a^{2} \sqrt {-a b}\, \underline {\hspace {1.25 ex}}\alpha ^{2}-\sqrt {-a b}\, a b +2 \sqrt {-a b}\right ) \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, -\frac {\sqrt {-a b}\, \underline {\hspace {1.25 ex}}\alpha ^{3} a^{2}+\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2} b +\sqrt {-a b}\, \underline {\hspace {1.25 ex}}\alpha a b +a \,b^{2}-2 \sqrt {-a b}\, \underline {\hspace {1.25 ex}}\alpha -2 b}{2 b}, \frac {\sqrt {2}}{2}\right )}{\underline {\hspace {1.25 ex}}\alpha \left (\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}+a b -1\right ) \sqrt {x \left (a \,x^{2}+b \right )}}\right )}{4 a}}{b}-\frac {\frac {2 \sqrt {a \,x^{3}+b x}}{3}+\frac {2 b \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {x a}{\sqrt {-a b}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{3 a \sqrt {a \,x^{3}+b x}}}{b}\) | \(530\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a x^{3} + b x} {\left (a x^{2} - b\right )}}{a^{2} x^{5} + 2 \, {\left (a b - 1\right )} x^{3} + b^{2} x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.36, size = 119, normalized size = 1.63 \begin {gather*} \frac {2^{3/4}\,\ln \left (\frac {2^{3/4}\,b+2\,2^{1/4}\,x-4\,\sqrt {x\,\left (a\,x^2+b\right )}+2^{3/4}\,a\,x^2}{4\,a\,x^2-4\,\sqrt {2}\,x+4\,b}\right )}{4}+\frac {2^{3/4}\,\ln \left (\frac {2^{3/4}\,b\,1{}\mathrm {i}-2^{1/4}\,x\,2{}\mathrm {i}-4\,\sqrt {x\,\left (a\,x^2+b\right )}+2^{3/4}\,a\,x^2\,1{}\mathrm {i}}{a\,x^2+\sqrt {2}\,x+b}\right )\,1{}\mathrm {i}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x \left (a x^{2} + b\right )} \left (a x^{2} - b\right )}{x \left (a^{2} x^{4} + 2 a b x^{2} + b^{2} - 2 x^{2}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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