Integrand size = 24, antiderivative size = 103 \[ \int \frac {\sqrt {x}}{\left (a x+b x^3+c x^5\right )^{3/2}} \, dx=\frac {\sqrt {x} \left (b^2-2 a c+b c x^2\right )}{a \left (b^2-4 a c\right ) \sqrt {a x+b x^3+c x^5}}-\frac {\text {arctanh}\left (\frac {\sqrt {x} \left (2 a+b x^2\right )}{2 \sqrt {a} \sqrt {a x+b x^3+c x^5}}\right )}{2 a^{3/2}} \]
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Time = 0.04 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1936, 1927, 212} \[ \int \frac {\sqrt {x}}{\left (a x+b x^3+c x^5\right )^{3/2}} \, dx=\frac {\sqrt {x} \left (-2 a c+b^2+b c x^2\right )}{a \left (b^2-4 a c\right ) \sqrt {a x+b x^3+c x^5}}-\frac {\text {arctanh}\left (\frac {\sqrt {x} \left (2 a+b x^2\right )}{2 \sqrt {a} \sqrt {a x+b x^3+c x^5}}\right )}{2 a^{3/2}} \]
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Rule 212
Rule 1927
Rule 1936
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {x} \left (b^2-2 a c+b c x^2\right )}{a \left (b^2-4 a c\right ) \sqrt {a x+b x^3+c x^5}}+\frac {\int \frac {1}{\sqrt {x} \sqrt {a x+b x^3+c x^5}} \, dx}{a} \\ & = \frac {\sqrt {x} \left (b^2-2 a c+b c x^2\right )}{a \left (b^2-4 a c\right ) \sqrt {a x+b x^3+c x^5}}-\frac {\text {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {\sqrt {x} \left (2 a+b x^2\right )}{\sqrt {a x+b x^3+c x^5}}\right )}{a} \\ & = \frac {\sqrt {x} \left (b^2-2 a c+b c x^2\right )}{a \left (b^2-4 a c\right ) \sqrt {a x+b x^3+c x^5}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {x} \left (2 a+b x^2\right )}{2 \sqrt {a} \sqrt {a x+b x^3+c x^5}}\right )}{2 a^{3/2}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.19 \[ \int \frac {\sqrt {x}}{\left (a x+b x^3+c x^5\right )^{3/2}} \, dx=-\frac {\sqrt {x} \left (\sqrt {a} \left (b^2-2 a c+b c x^2\right )+\left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4} \text {arctanh}\left (\frac {\sqrt {c} x^2-\sqrt {a+b x^2+c x^4}}{\sqrt {a}}\right )\right )}{a^{3/2} \left (-b^2+4 a c\right ) \sqrt {x \left (a+b x^2+c x^4\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(178\) vs. \(2(87)=174\).
Time = 0.06 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.74
method | result | size |
default | \(-\frac {\sqrt {x \left (c \,x^{4}+b \,x^{2}+a \right )}\, \left (2 b c \,x^{2} \sqrt {a}+4 \ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right ) a c \sqrt {c \,x^{4}+b \,x^{2}+a}-\ln \left (\frac {2 a +b \,x^{2}+2 \sqrt {a}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}{x^{2}}\right ) b^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}-4 a^{\frac {3}{2}} c +2 b^{2} \sqrt {a}\right )}{2 a^{\frac {3}{2}} \sqrt {x}\, \left (c \,x^{4}+b \,x^{2}+a \right ) \left (4 a c -b^{2}\right )}\) | \(179\) |
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Leaf count of result is larger than twice the leaf count of optimal. 208 vs. \(2 (87) = 174\).
Time = 0.31 (sec) , antiderivative size = 424, normalized size of antiderivative = 4.12 \[ \int \frac {\sqrt {x}}{\left (a x+b x^3+c x^5\right )^{3/2}} \, dx=\left [\frac {{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{5} + {\left (b^{3} - 4 \, a b c\right )} x^{3} + {\left (a b^{2} - 4 \, a^{2} c\right )} x\right )} \sqrt {a} \log \left (-\frac {{\left (b^{2} + 4 \, a c\right )} x^{5} + 8 \, a b x^{3} + 8 \, a^{2} x - 4 \, \sqrt {c x^{5} + b x^{3} + a x} {\left (b x^{2} + 2 \, a\right )} \sqrt {a} \sqrt {x}}{x^{5}}\right ) + 4 \, \sqrt {c x^{5} + b x^{3} + a x} {\left (a b c x^{2} + a b^{2} - 2 \, a^{2} c\right )} \sqrt {x}}{4 \, {\left ({\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )} x^{5} + {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} x^{3} + {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} x\right )}}, \frac {{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{5} + {\left (b^{3} - 4 \, a b c\right )} x^{3} + {\left (a b^{2} - 4 \, a^{2} c\right )} x\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {c x^{5} + b x^{3} + a x} {\left (b x^{2} + 2 \, a\right )} \sqrt {-a} \sqrt {x}}{2 \, {\left (a c x^{5} + a b x^{3} + a^{2} x\right )}}\right ) + 2 \, \sqrt {c x^{5} + b x^{3} + a x} {\left (a b c x^{2} + a b^{2} - 2 \, a^{2} c\right )} \sqrt {x}}{2 \, {\left ({\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )} x^{5} + {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} x^{3} + {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} x\right )}}\right ] \]
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\[ \int \frac {\sqrt {x}}{\left (a x+b x^3+c x^5\right )^{3/2}} \, dx=\int \frac {\sqrt {x}}{\left (x \left (a + b x^{2} + c x^{4}\right )\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {\sqrt {x}}{\left (a x+b x^3+c x^5\right )^{3/2}} \, dx=\int { \frac {\sqrt {x}}{{\left (c x^{5} + b x^{3} + a x\right )}^{\frac {3}{2}}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 193 vs. \(2 (87) = 174\).
Time = 0.29 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.87 \[ \int \frac {\sqrt {x}}{\left (a x+b x^3+c x^5\right )^{3/2}} \, dx=\frac {\frac {a b c x^{2}}{a^{2} b^{2} - 4 \, a^{3} c} + \frac {a b^{2} - 2 \, a^{2} c}{a^{2} b^{2} - 4 \, a^{3} c}}{\sqrt {c x^{4} + b x^{2} + a}} - \frac {a b^{2} \arctan \left (\frac {\sqrt {a}}{\sqrt {-a}}\right ) - 4 \, a^{2} c \arctan \left (\frac {\sqrt {a}}{\sqrt {-a}}\right ) + \sqrt {-a} \sqrt {a} b^{2} - 2 \, \sqrt {-a} a^{\frac {3}{2}} c}{\sqrt {-a} a^{2} b^{2} - 4 \, \sqrt {-a} a^{3} c} + \frac {\arctan \left (-\frac {\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a} \]
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Timed out. \[ \int \frac {\sqrt {x}}{\left (a x+b x^3+c x^5\right )^{3/2}} \, dx=\int \frac {\sqrt {x}}{{\left (c\,x^5+b\,x^3+a\,x\right )}^{3/2}} \,d x \]
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