Optimal. Leaf size=68 \[ \frac {\sqrt {a+b x^2+c x^4}}{2 c}-\frac {b \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{4 c^{3/2}} \]
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Rubi [A] time = 0.05, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1114, 640, 621, 206} \begin {gather*} \frac {\sqrt {a+b x^2+c x^4}}{2 c}-\frac {b \tanh ^{-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 {gather*}
Antiderivative was successfully verified.
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Rule 206
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 \tanh ^{-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 = 0.01, size = 68, normalized size = 1.00 \begin {gather*} \frac {\sqrt {a+b x^2+c x^4}}{2 c}-\frac {b \tanh ^{-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 {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.17, size = 70, normalized size = 1.03 \begin {gather*} \frac {b \log \left (-2 c^{3/2} \sqrt {a+b x^2+c x^4}+b c+2 c^2 x^2\right )}{4 c^{3/2}}+\frac {\sqrt {a+b x^2+c x^4}}{2 c} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 161, normalized size = 2.37 \begin {gather*} \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 ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 61, normalized size = 0.90 \begin {gather*} \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 \, c^{\frac {3}{2}}} + \frac {\sqrt {c x^{4} + b x^{2} + a}}{2 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 56, normalized size = 0.82 \begin {gather*} -\frac {b \ln \left (\frac {c \,x^{2}+\frac {b}{2}}{\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.43, size = 55, normalized size = 0.81 \begin {gather*} \frac {\sqrt {c\,x^4+b\,x^2+a}}{2\,c}-\frac {b\,\ln \left (\sqrt {c\,x^4+b\,x^2+a}+\frac {c\,x^2+\frac {b}{2}}{\sqrt {c}}\right )}{4\,c^{3/2}} \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 {x^{3}}{\sqrt {a + b x^{2} + c x^{4}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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