Optimal. Leaf size=109 \[ \frac {3 a \sqrt {a-b x^2} \sqrt {a+b x^2} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b} \sqrt {a^2-b^2 x^4}}-\frac {x \sqrt {a-b x^2} \left (a+b x^2\right )}{2 \sqrt {a^2-b^2 x^4}} \]
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Rubi [A] time = 0.04, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {1152, 388, 217, 206} \begin {gather*} \frac {3 a \sqrt {a-b x^2} \sqrt {a+b x^2} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b} \sqrt {a^2-b^2 x^4}}-\frac {x \sqrt {a-b x^2} \left (a+b x^2\right )}{2 \sqrt {a^2-b^2 x^4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 388
Rule 1152
Rubi steps
\begin {align*} \int \frac {\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 {a-b x^2}{\sqrt {a+b x^2}} \, dx}{\sqrt {a^2-b^2 x^4}}\\ &=-\frac {x \sqrt {a-b x^2} \left (a+b x^2\right )}{2 \sqrt {a^2-b^2 x^4}}+\frac {\left (3 a \sqrt {a-b x^2} \sqrt {a+b x^2}\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{2 \sqrt {a^2-b^2 x^4}}\\ &=-\frac {x \sqrt {a-b x^2} \left (a+b x^2\right )}{2 \sqrt {a^2-b^2 x^4}}+\frac {\left (3 a \sqrt {a-b x^2} \sqrt {a+b x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {a^2-b^2 x^4}}\\ &=-\frac {x \sqrt {a-b x^2} \left (a+b x^2\right )}{2 \sqrt {a^2-b^2 x^4}}+\frac {3 a \sqrt {a-b x^2} \sqrt {a+b x^2} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b} \sqrt {a^2-b^2 x^4}}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 110, normalized size = 1.01 \begin {gather*} \frac {1}{2} \left (-\frac {x \sqrt {a^2-b^2 x^4}}{\sqrt {a-b x^2}}+\frac {3 a \log \left (\sqrt {b} \sqrt {a-b x^2} \sqrt {a^2-b^2 x^4}+a b x-b^2 x^3\right )}{\sqrt {b}}-\frac {3 a \log \left (b x^2-a\right )}{\sqrt {b}}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 2.58, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a-b x^2\right )^{3/2}}{\sqrt {a^2-b^2 x^4}} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.04, size = 236, normalized size = 2.17 \begin {gather*} \left [\frac {2 \, \sqrt {-b^{2} x^{4} + a^{2}} \sqrt {-b x^{2} + a} b x + 3 \, {\left (a b x^{2} - a^{2}\right )} \sqrt {b} \log \left (\frac {2 \, b^{2} x^{4} - a b x^{2} - 2 \, \sqrt {-b^{2} x^{4} + a^{2}} \sqrt {-b x^{2} + a} \sqrt {b} x - a^{2}}{b x^{2} - a}\right )}{4 \, {\left (b^{2} x^{2} - a b\right )}}, \frac {\sqrt {-b^{2} x^{4} + a^{2}} \sqrt {-b x^{2} + a} b x + 3 \, {\left (a b x^{2} - a^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b^{2} x^{4} + a^{2}} \sqrt {-b x^{2} + a} \sqrt {-b}}{b^{2} x^{3} - a b x}\right )}{2 \, {\left (b^{2} x^{2} - a b\right )}}\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 {{\left (-b x^{2} + a\right )}^{\frac {3}{2}}}{\sqrt {-b^{2} x^{4} + a^{2}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 85, normalized size = 0.78 \begin {gather*} -\frac {\sqrt {-b \,x^{2}+a}\, \sqrt {-b^{2} x^{4}+a^{2}}\, \left (3 a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )-\sqrt {b \,x^{2}+a}\, \sqrt {b}\, x \right )}{2 \left (b \,x^{2}-a \right ) \sqrt {b \,x^{2}+a}\, \sqrt {b}} \end {gather*}
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 {{\left (-b x^{2} + a\right )}^{\frac {3}{2}}}{\sqrt {-b^{2} x^{4} + a^{2}}}\,{d x} \end {gather*}
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 \begin {gather*} \int \frac {{\left (a-b\,x^2\right )}^{3/2}}{\sqrt {a^2-b^2\,x^4}} \,d x \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 {\left (a - b x^{2}\right )^{\frac {3}{2}}}{\sqrt {- \left (- a + b x^{2}\right ) \left (a + b x^{2}\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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