Optimal. Leaf size=83 \[ \frac {\left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{8 c}-\frac {\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{16 c^{3/2}} \]
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Rubi [A] time = 0.05, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1107, 612, 621, 206} \[ \frac {\left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{8 c}-\frac {\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{16 c^{3/2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 621
Rule 1107
Rubi steps
\begin {align*} \int x \sqrt {a+b x^2+c x^4} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \sqrt {a+b x+c x^2} \, dx,x,x^2\right )\\ &=\frac {\left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{8 c}-\frac {\left (b^2-4 a c\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{16 c}\\ &=\frac {\left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{8 c}-\frac {\left (b^2-4 a c\right ) \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 )}{8 c}\\ &=\frac {\left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{8 c}-\frac {\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{16 c^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 83, normalized size = 1.00 \[ \frac {\left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{8 c}-\frac {\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{16 c^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 197, normalized size = 2.37 \[ \left [-\frac {{\left (b^{2} - 4 \, a c\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{4} - 8 \, b c x^{2} - b^{2} - 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {c} - 4 \, a c\right ) - 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c^{2} x^{2} + b c\right )}}{32 \, c^{2}}, \frac {{\left (b^{2} - 4 \, a c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{4} + b c x^{2} + a c\right )}}\right ) + 2 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c^{2} x^{2} + b c\right )}}{16 \, c^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 76, normalized size = 0.92 \[ \frac {1}{8} \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, x^{2} + \frac {b}{c}\right )} + \frac {{\left (b^{2} - 4 \, a c\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} \sqrt {c} - b \right |}\right )}{16 \, c^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 101, normalized size = 1.22 \[ \frac {a \ln \left (\frac {c \,x^{2}+\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{4 \sqrt {c}}-\frac {b^{2} \ln \left (\frac {c \,x^{2}+\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{16 c^{\frac {3}{2}}}+\frac {\left (2 c \,x^{2}+b \right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{8 c} \]
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 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.62, size = 72, normalized size = 0.87 \[ \frac {\left (\frac {b}{4\,c}+\frac {x^2}{2}\right )\,\sqrt {c\,x^4+b\,x^2+a}}{2}+\frac {\ln \left (\sqrt {c\,x^4+b\,x^2+a}+\frac {c\,x^2+\frac {b}{2}}{\sqrt {c}}\right )\,\left (a\,c-\frac {b^2}{4}\right )}{4\,c^{3/2}} \]
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 \[ \int x \sqrt {a + b x^{2} + c x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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