Integrand size = 20, antiderivative size = 115 \[ \int \frac {x^5}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\frac {x^2 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {b \sqrt {a+b x^2+c x^4}}{c \left (b^2-4 a c\right )}+\frac {\text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{2 c^{3/2}} \]
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Time = 0.06 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1128, 752, 654, 635, 212} \[ \int \frac {x^5}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\frac {\text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{2 c^{3/2}}+\frac {x^2 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {b \sqrt {a+b x^2+c x^4}}{c \left (b^2-4 a c\right )} \]
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Rule 212
Rule 635
Rule 654
Rule 752
Rule 1128
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {x^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx,x,x^2\right ) \\ & = \frac {x^2 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {\text {Subst}\left (\int \frac {2 a+b x}{\sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{b^2-4 a c} \\ & = \frac {x^2 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {b \sqrt {a+b x^2+c x^4}}{c \left (b^2-4 a c\right )}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{2 c} \\ & = \frac {x^2 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {b \sqrt {a+b x^2+c x^4}}{c \left (b^2-4 a c\right )}+\frac {\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 )}{c} \\ & = \frac {x^2 \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {b \sqrt {a+b x^2+c x^4}}{c \left (b^2-4 a c\right )}+\frac {\tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{2 c^{3/2}} \\ \end{align*}
Time = 0.37 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.82 \[ \int \frac {x^5}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\frac {b^2 x^2+a \left (b-2 c x^2\right )}{c \left (-b^2+4 a c\right ) \sqrt {a+b x^2+c x^4}}+\frac {\text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{2 c^{3/2}} \]
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Time = 0.12 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.92
method | result | size |
pseudoelliptic | \(-\frac {x^{2}}{2 c \sqrt {c \,x^{4}+b \,x^{2}+a}}+\frac {b \left (b \,x^{2}+2 a \right )}{8 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (a c -\frac {b^{2}}{4}\right ) c}+\frac {-\ln \left (2\right )+\ln \left (\frac {2 c \,x^{2}+2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {c}+b}{\sqrt {c}}\right )}{2 c^{\frac {3}{2}}}\) | \(106\) |
default | \(-\frac {x^{2}}{2 c \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {b \left (2 c \,x^{2}+b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}\right )}{4 c}+\frac {\ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{2 c^{\frac {3}{2}}}\) | \(124\) |
elliptic | \(-\frac {x^{2}}{2 c \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {b \left (2 c \,x^{2}+b \right )}{c \left (4 a c -b^{2}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}}\right )}{4 c}+\frac {\ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right )}{2 c^{\frac {3}{2}}}\) | \(124\) |
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Time = 0.31 (sec) , antiderivative size = 387, normalized size of antiderivative = 3.37 \[ \int \frac {x^5}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\left [\frac {{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{4} + a b^{2} - 4 \, a^{2} c + {\left (b^{3} - 4 \, a b c\right )} x^{2}\right )} \sqrt {c} \log \left (-8 \, c^{2} x^{4} - 8 \, b c x^{2} - b^{2} - 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {c} - 4 \, a c\right ) - 4 \, \sqrt {c x^{4} + b x^{2} + a} {\left (a b c + {\left (b^{2} c - 2 \, a c^{2}\right )} x^{2}\right )}}{4 \, {\left (a b^{2} c^{2} - 4 \, a^{2} c^{3} + {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{4} + {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x^{2}\right )}}, -\frac {{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{4} + a b^{2} - 4 \, a^{2} c + {\left (b^{3} - 4 \, a b c\right )} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2} + a} {\left (2 \, c x^{2} + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{4} + b c x^{2} + a c\right )}}\right ) + 2 \, \sqrt {c x^{4} + b x^{2} + a} {\left (a b c + {\left (b^{2} c - 2 \, a c^{2}\right )} x^{2}\right )}}{2 \, {\left (a b^{2} c^{2} - 4 \, a^{2} c^{3} + {\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{4} + {\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x^{2}\right )}}\right ] \]
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\[ \int \frac {x^5}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int \frac {x^{5}}{\left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}}\, dx \]
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Exception generated. \[ \int \frac {x^5}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.31 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.86 \[ \int \frac {x^5}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=-\frac {\frac {{\left (b^{2} - 2 \, a c\right )} x^{2}}{b^{2} c - 4 \, a c^{2}} + \frac {a b}{b^{2} c - 4 \, a c^{2}}}{\sqrt {c x^{4} + b x^{2} + a}} - \frac {\log \left ({\left | 2 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} \sqrt {c} + b \right |}\right )}{2 \, c^{\frac {3}{2}}} \]
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Time = 14.12 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.73 \[ \int \frac {x^5}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\frac {\ln \left (\sqrt {c\,x^4+b\,x^2+a}+\frac {c\,x^2+\frac {b}{2}}{\sqrt {c}}\right )}{2\,c^{3/2}}+\frac {\frac {a\,b}{2}-x^2\,\left (a\,c-\frac {b^2}{2}\right )}{2\,c\,\left (a\,c-\frac {b^2}{4}\right )\,\sqrt {c\,x^4+b\,x^2+a}} \]
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