Optimal. Leaf size=125 \[ \frac {x \left (a-b x^2\right )}{4 a^2 \sqrt {a+b x^2} \sqrt {a^2-b^2 x^4}}+\frac {3 \sqrt {a+b x^2} \sqrt {a-b x^2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {b} x}{\sqrt {a-b x^2}}\right )}{4 \sqrt {2} a^2 \sqrt {b} \sqrt {a^2-b^2 x^4}} \]
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Rubi [A] time = 0.05, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1152, 382, 377, 205} \[ \frac {x \left (a-b x^2\right )}{4 a^2 \sqrt {a+b x^2} \sqrt {a^2-b^2 x^4}}+\frac {3 \sqrt {a+b x^2} \sqrt {a-b x^2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {b} x}{\sqrt {a-b x^2}}\right )}{4 \sqrt {2} a^2 \sqrt {b} \sqrt {a^2-b^2 x^4}} \]
Antiderivative was successfully verified.
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Rule 205
Rule 377
Rule 382
Rule 1152
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b x^2\right )^{3/2} \sqrt {a^2-b^2 x^4}} \, dx &=\frac {\left (\sqrt {a-b x^2} \sqrt {a+b x^2}\right ) \int \frac {1}{\sqrt {a-b x^2} \left (a+b x^2\right )^2} \, dx}{\sqrt {a^2-b^2 x^4}}\\ &=\frac {x \left (a-b x^2\right )}{4 a^2 \sqrt {a+b x^2} \sqrt {a^2-b^2 x^4}}+\frac {\left (3 \sqrt {a-b x^2} \sqrt {a+b x^2}\right ) \int \frac {1}{\sqrt {a-b x^2} \left (a+b x^2\right )} \, dx}{4 a \sqrt {a^2-b^2 x^4}}\\ &=\frac {x \left (a-b x^2\right )}{4 a^2 \sqrt {a+b x^2} \sqrt {a^2-b^2 x^4}}+\frac {\left (3 \sqrt {a-b x^2} \sqrt {a+b x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{a+2 a b x^2} \, dx,x,\frac {x}{\sqrt {a-b x^2}}\right )}{4 a \sqrt {a^2-b^2 x^4}}\\ &=\frac {x \left (a-b x^2\right )}{4 a^2 \sqrt {a+b x^2} \sqrt {a^2-b^2 x^4}}+\frac {3 \sqrt {a-b x^2} \sqrt {a+b x^2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {b} x}{\sqrt {a-b x^2}}\right )}{4 \sqrt {2} a^2 \sqrt {b} \sqrt {a^2-b^2 x^4}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 111, normalized size = 0.89 \[ \frac {\sqrt {a^2-b^2 x^4} \left (2 \sqrt {b} x \sqrt {a-b x^2}+3 \sqrt {2} \left (a+b x^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {b} x}{\sqrt {a-b x^2}}\right )\right )}{8 a^2 \sqrt {b} \sqrt {a-b x^2} \left (a+b x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 297, normalized size = 2.38 \[ \left [\frac {4 \, \sqrt {-b^{2} x^{4} + a^{2}} \sqrt {b x^{2} + a} b x - 3 \, \sqrt {2} {\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \sqrt {-b} \log \left (-\frac {3 \, b^{2} x^{4} + 2 \, a b x^{2} - 2 \, \sqrt {2} \sqrt {-b^{2} x^{4} + a^{2}} \sqrt {b x^{2} + a} \sqrt {-b} x - a^{2}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right )}{16 \, {\left (a^{2} b^{3} x^{4} + 2 \, a^{3} b^{2} x^{2} + a^{4} b\right )}}, \frac {2 \, \sqrt {-b^{2} x^{4} + a^{2}} \sqrt {b x^{2} + a} b x - 3 \, \sqrt {2} {\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \sqrt {b} \arctan \left (\frac {\sqrt {2} \sqrt {-b^{2} x^{4} + a^{2}} \sqrt {b x^{2} + a} \sqrt {b}}{2 \, {\left (b^{2} x^{3} + a b x\right )}}\right )}{8 \, {\left (a^{2} b^{3} x^{4} + 2 \, a^{3} b^{2} x^{2} + a^{4} b\right )}}\right ] \]
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 \[ \int \frac {1}{\sqrt {-b^{2} x^{4} + a^{2}} {\left (b x^{2} + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 488, normalized size = 3.90 \[ -\frac {\sqrt {-b^{2} x^{4}+a^{2}}\, \left (3 \sqrt {2}\, \sqrt {a}\, b^{\frac {3}{2}} x^{2} \ln \left (\frac {2 \left (a -\sqrt {-a b}\, x +\sqrt {2}\, \sqrt {-b \,x^{2}+a}\, \sqrt {a}\right ) b}{b x -\sqrt {-a b}}\right )-3 \sqrt {2}\, \sqrt {a}\, b^{\frac {3}{2}} x^{2} \ln \left (\frac {2 \left (a +\sqrt {-a b}\, x +\sqrt {2}\, \sqrt {-b \,x^{2}+a}\, \sqrt {a}\right ) b}{b x +\sqrt {-a b}}\right )-4 \sqrt {-a b}\, b \,x^{2} \arctan \left (\frac {\sqrt {b}\, x}{\sqrt {\frac {\left (-b x +\sqrt {a b}\right ) \left (b x +\sqrt {a b}\right )}{b}}}\right )+4 \sqrt {-a b}\, b \,x^{2} \arctan \left (\frac {\sqrt {b}\, x}{\sqrt {-b \,x^{2}+a}}\right )+3 \sqrt {2}\, a^{\frac {3}{2}} \sqrt {b}\, \ln \left (\frac {2 \left (a -\sqrt {-a b}\, x +\sqrt {2}\, \sqrt {-b \,x^{2}+a}\, \sqrt {a}\right ) b}{b x -\sqrt {-a b}}\right )-3 \sqrt {2}\, a^{\frac {3}{2}} \sqrt {b}\, \ln \left (\frac {2 \left (a +\sqrt {-a b}\, x +\sqrt {2}\, \sqrt {-b \,x^{2}+a}\, \sqrt {a}\right ) b}{b x +\sqrt {-a b}}\right )-4 \sqrt {-a b}\, a \arctan \left (\frac {\sqrt {b}\, x}{\sqrt {\frac {\left (-b x +\sqrt {a b}\right ) \left (b x +\sqrt {a b}\right )}{b}}}\right )+4 \sqrt {-a b}\, a \arctan \left (\frac {\sqrt {b}\, x}{\sqrt {-b \,x^{2}+a}}\right )-4 \sqrt {-a b}\, \sqrt {-b \,x^{2}+a}\, \sqrt {b}\, x \right ) b^{\frac {5}{2}}}{4 \sqrt {b \,x^{2}+a}\, \sqrt {-b \,x^{2}+a}\, \left (\sqrt {-a b}+\sqrt {a b}\right )^{2} \left (\sqrt {-a b}-\sqrt {a b}\right )^{2} \sqrt {-a b}\, \left (b x +\sqrt {-a b}\right ) \left (b x -\sqrt {-a b}\right )} \]
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 \[ \int \frac {1}{\sqrt {-b^{2} x^{4} + a^{2}} {\left (b x^{2} + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\sqrt {a^2-b^2\,x^4}\,{\left (b\,x^2+a\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{\sqrt {- \left (- a + b x^{2}\right ) \left (a + b x^{2}\right )} \left (a + b x^{2}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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