Optimal. Leaf size=104 \[ -\frac {3 b \sqrt {a+b x^2+c x^4}}{8 c^2}+\frac {x^2 \sqrt {a+b x^2+c x^4}}{4 c}+\frac {\left (3 b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{16 c^{5/2}} \]
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Rubi [A]
time = 0.06, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1128, 756, 654,
635, 212} \begin {gather*} \frac {\left (3 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^{5/2}}-\frac {3 b \sqrt {a+b x^2+c x^4}}{8 c^2}+\frac {x^2 \sqrt {a+b x^2+c x^4}}{4 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 635
Rule 654
Rule 756
Rule 1128
Rubi steps
\begin {align*} \int \frac {x^5}{\sqrt {a+b x^2+c x^4}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^2}{\sqrt {a+b x+c x^2}} \, dx,x,x^2\right )\\ &=\frac {x^2 \sqrt {a+b x^2+c x^4}}{4 c}+\frac {\text {Subst}\left (\int \frac {-a-\frac {3 b x}{2}}{\sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{4 c}\\ &=-\frac {3 b \sqrt {a+b x^2+c x^4}}{8 c^2}+\frac {x^2 \sqrt {a+b x^2+c x^4}}{4 c}+\frac {\left (3 b^2-4 a c\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{16 c^2}\\ &=-\frac {3 b \sqrt {a+b x^2+c x^4}}{8 c^2}+\frac {x^2 \sqrt {a+b x^2+c x^4}}{4 c}+\frac {\left (3 b^2-4 a c\right ) \text {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^2}\\ &=-\frac {3 b \sqrt {a+b x^2+c x^4}}{8 c^2}+\frac {x^2 \sqrt {a+b x^2+c x^4}}{4 c}+\frac {\left (3 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^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.18, size = 91, normalized size = 0.88 \begin {gather*} \frac {\left (-3 b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{8 c^2}+\frac {\left (-3 b^2+4 a c\right ) \log \left (b c^2+2 c^3 x^2-2 c^{5/2} \sqrt {a+b x^2+c x^4}\right )}{16 c^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 116, normalized size = 1.12
method | result | size |
risch | \(-\frac {\left (-2 c \,x^{2}+3 b \right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}{8 c^{2}}-\frac {a \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{4 c^{\frac {3}{2}}}+\frac {3 b^{2} \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{16 c^{\frac {5}{2}}}\) | \(103\) |
default | \(\frac {x^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{4 c}-\frac {3 b \sqrt {c \,x^{4}+b \,x^{2}+a}}{8 c^{2}}+\frac {3 b^{2} \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{16 c^{\frac {5}{2}}}-\frac {a \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{4 c^{\frac {3}{2}}}\) | \(116\) |
elliptic | \(\frac {x^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}}{4 c}-\frac {3 b \sqrt {c \,x^{4}+b \,x^{2}+a}}{8 c^{2}}+\frac {3 b^{2} \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{16 c^{\frac {5}{2}}}-\frac {a \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{4 c^{\frac {3}{2}}}\) | \(116\) |
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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Fricas [A]
time = 0.37, size = 203, normalized size = 1.95 \begin {gather*} \left [-\frac {{\left (3 \, 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} - 3 \, b c\right )}}{32 \, c^{3}}, -\frac {{\left (3 \, 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} - 3 \, b c\right )}}{16 \, c^{3}}\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 {x^{5}}{\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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Giac [A]
time = 4.13, size = 82, normalized size = 0.79 \begin {gather*} \frac {1}{8} \, \sqrt {c x^{4} + b x^{2} + a} {\left (\frac {2 \, x^{2}}{c} - \frac {3 \, b}{c^{2}}\right )} - \frac {{\left (3 \, 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 {5}{2}}} \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 {x^5}{\sqrt {c\,x^4+b\,x^2+a}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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