Optimal. Leaf size=110 \[ -\frac {x \left (a-b x^2\right ) \sqrt {a+b x^2}}{2 \sqrt {a^2-b^2 x^4}}+\frac {3 a \sqrt {a-b x^2} \sqrt {a+b x^2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a-b x^2}}\right )}{2 \sqrt {b} \sqrt {a^2-b^2 x^4}} \]
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Rubi [A]
time = 0.02, antiderivative size = 110, 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 = {1166, 396, 223,
209} \begin {gather*} \frac {3 a \sqrt {a-b x^2} \sqrt {a+b x^2} \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a-b x^2}}\right )}{2 \sqrt {b} \sqrt {a^2-b^2 x^4}}-\frac {x \left (a-b x^2\right ) \sqrt {a+b x^2}}{2 \sqrt {a^2-b^2 x^4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 223
Rule 396
Rule 1166
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 \left (a-b x^2\right ) \sqrt {a+b x^2}}{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 \left (a-b x^2\right ) \sqrt {a+b x^2}}{2 \sqrt {a^2-b^2 x^4}}+\frac {\left (3 a \sqrt {a-b x^2} \sqrt {a+b x^2}\right ) \text {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 \left (a-b x^2\right ) \sqrt {a+b x^2}}{2 \sqrt {a^2-b^2 x^4}}+\frac {3 a \sqrt {a-b x^2} \sqrt {a+b x^2} \tan ^{-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 [C] Result contains complex when optimal does not.
time = 2.26, size = 86, normalized size = 0.78 \begin {gather*} -\frac {x \sqrt {a^2-b^2 x^4}}{2 \sqrt {a+b x^2}}+\frac {3 i a \log \left (-2 i \sqrt {b} x+\frac {2 \sqrt {a^2-b^2 x^4}}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 75, normalized size = 0.68
method | result | size |
default | \(\frac {\sqrt {-b^{2} x^{4}+a^{2}}\, \left (-x \sqrt {-b \,x^{2}+a}\, \sqrt {b}+3 \arctan \left (\frac {x \sqrt {b}}{\sqrt {-b \,x^{2}+a}}\right ) a \right )}{2 \sqrt {b \,x^{2}+a}\, \sqrt {-b \,x^{2}+a}\, \sqrt {b}}\) | \(75\) |
risch | \(-\frac {x \sqrt {-b \,x^{2}+a}\, \sqrt {\frac {-b^{2} x^{4}+a^{2}}{b \,x^{2}+a}}\, \sqrt {b \,x^{2}+a}}{2 \sqrt {-b^{2} x^{4}+a^{2}}}+\frac {3 a \arctan \left (\frac {x \sqrt {b}}{\sqrt {-b \,x^{2}+a}}\right ) \sqrt {\frac {-b^{2} x^{4}+a^{2}}{b \,x^{2}+a}}\, \sqrt {b \,x^{2}+a}}{2 \sqrt {b}\, \sqrt {-b^{2} x^{4}+a^{2}}}\) | \(131\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 223, normalized size = 2.03 \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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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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 (b\,x^2+a\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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