Optimal. Leaf size=70 \[ -\frac {b \tan ^{-1}\left (\frac {b-2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2-c x^4}}\right )}{4 c^{3/2}}-\frac {\sqrt {a+b x^2-c x^4}}{2 c} \]
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Rubi [A] time = 0.06, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {1114, 640, 621, 204} \[ -\frac {b \tan ^{-1}\left (\frac {b-2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2-c x^4}}\right )}{4 c^{3/2}}-\frac {\sqrt {a+b x^2-c x^4}}{2 c} \]
Antiderivative was successfully verified.
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Rule 204
Rule 621
Rule 640
Rule 1114
Rubi steps
\begin {align*} \int \frac {x^3}{\sqrt {a+b x^2-c x^4}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x}{\sqrt {a+b x-c x^2}} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {a+b x^2-c x^4}}{2 c}+\frac {b \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x-c x^2}} \, dx,x,x^2\right )}{4 c}\\ &=-\frac {\sqrt {a+b x^2-c x^4}}{2 c}+\frac {b \operatorname {Subst}\left (\int \frac {1}{-4 c-x^2} \, dx,x,\frac {b-2 c x^2}{\sqrt {a+b x^2-c x^4}}\right )}{2 c}\\ &=-\frac {\sqrt {a+b x^2-c x^4}}{2 c}-\frac {b \tan ^{-1}\left (\frac {b-2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2-c x^4}}\right )}{4 c^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 70, normalized size = 1.00 \[ -\frac {b \tan ^{-1}\left (\frac {b-2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2-c x^4}}\right )}{4 c^{3/2}}-\frac {\sqrt {a+b x^2-c x^4}}{2 c} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.83, size = 169, normalized size = 2.41 \[ \left [-\frac {b \sqrt {-c} \log \left (8 \, c^{2} x^{4} - 8 \, b c x^{2} + b^{2} - 4 \, \sqrt {-c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} - b\right )} \sqrt {-c} - 4 \, a c\right ) + 4 \, \sqrt {-c x^{4} + b x^{2} + a} c}{8 \, c^{2}}, -\frac {b \sqrt {c} \arctan \left (\frac {\sqrt {-c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} - b\right )} \sqrt {c}}{2 \, {\left (c^{2} x^{4} - b c x^{2} - a c\right )}}\right ) + 2 \, \sqrt {-c x^{4} + b x^{2} + a} c}{4 \, c^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 70, normalized size = 1.00 \[ -\frac {b \log \left ({\left | 2 \, {\left (\sqrt {-c} x^{2} - \sqrt {-c x^{4} + b x^{2} + a}\right )} \sqrt {-c} + b \right |}\right )}{4 \, \sqrt {-c} c} - \frac {\sqrt {-c x^{4} + b x^{2} + a}}{2 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 58, normalized size = 0.83 \[ \frac {b \arctan \left (\frac {\left (x^{2}-\frac {b}{2 c}\right ) \sqrt {c}}{\sqrt {-c \,x^{4}+b \,x^{2}+a}}\right )}{4 c^{\frac {3}{2}}}-\frac {\sqrt {-c \,x^{4}+b \,x^{2}+a}}{2 c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.47, size = 50, normalized size = 0.71 \[ -\frac {b \arcsin \left (-\frac {2 \, c x^{2} - b}{\sqrt {b^{2} + 4 \, a c}}\right )}{4 \, c^{\frac {3}{2}}} - \frac {\sqrt {-c x^{4} + b x^{2} + a}}{2 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.59, size = 62, normalized size = 0.89 \[ -\frac {\sqrt {-c\,x^4+b\,x^2+a}}{2\,c}-\frac {b\,\ln \left (\frac {\frac {b}{2}-c\,x^2}{\sqrt {-c}}+\sqrt {-c\,x^4+b\,x^2+a}\right )}{4\,{\left (-c\right )}^{3/2}} \]
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 \[ \int \frac {x^{3}}{\sqrt {a + b x^{2} - c x^{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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