Integrand size = 24, antiderivative size = 129 \[ \int \sqrt {x} \sqrt {a x+b x^3+c x^5} \, dx=\frac {\left (b+2 c x^2\right ) \sqrt {a x+b x^3+c x^5}}{8 c \sqrt {x}}-\frac {\left (b^2-4 a c\right ) \sqrt {x} \sqrt {a+b x^2+c x^4} \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{16 c^{3/2} \sqrt {a x+b x^3+c x^5}} \]
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Time = 0.06 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1932, 1928, 1121, 635, 212} \[ \int \sqrt {x} \sqrt {a x+b x^3+c x^5} \, dx=\frac {\left (b+2 c x^2\right ) \sqrt {a x+b x^3+c x^5}}{8 c \sqrt {x}}-\frac {\sqrt {x} \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4} \text {arctanh}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{16 c^{3/2} \sqrt {a x+b x^3+c x^5}} \]
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Rule 212
Rule 635
Rule 1121
Rule 1928
Rule 1932
Rubi steps \begin{align*} \text {integral}& = \frac {\left (b+2 c x^2\right ) \sqrt {a x+b x^3+c x^5}}{8 c \sqrt {x}}-\frac {\left (b^2-4 a c\right ) \int \frac {x^{3/2}}{\sqrt {a x+b x^3+c x^5}} \, dx}{8 c} \\ & = \frac {\left (b+2 c x^2\right ) \sqrt {a x+b x^3+c x^5}}{8 c \sqrt {x}}-\frac {\left (\left (b^2-4 a c\right ) \sqrt {x} \sqrt {a+b x^2+c x^4}\right ) \int \frac {x}{\sqrt {a+b x^2+c x^4}} \, dx}{8 c \sqrt {a x+b x^3+c x^5}} \\ & = \frac {\left (b+2 c x^2\right ) \sqrt {a x+b x^3+c x^5}}{8 c \sqrt {x}}-\frac {\left (\left (b^2-4 a c\right ) \sqrt {x} \sqrt {a+b x^2+c x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{16 c \sqrt {a x+b x^3+c x^5}} \\ & = \frac {\left (b+2 c x^2\right ) \sqrt {a x+b x^3+c x^5}}{8 c \sqrt {x}}-\frac {\left (\left (b^2-4 a c\right ) \sqrt {x} \sqrt {a+b x^2+c x^4}\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 \sqrt {a x+b x^3+c x^5}} \\ & = \frac {\left (b+2 c x^2\right ) \sqrt {a x+b x^3+c x^5}}{8 c \sqrt {x}}-\frac {\left (b^2-4 a c\right ) \sqrt {x} \sqrt {a+b x^2+c x^4} \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} \sqrt {a x+b x^3+c x^5}} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.84 \[ \int \sqrt {x} \sqrt {a x+b x^3+c x^5} \, dx=\frac {\sqrt {x \left (a+b x^2+c x^4\right )} \left (\sqrt {c} \left (b+2 c x^2\right )+\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {\sqrt {c} x^2}{\sqrt {a}-\sqrt {a+b x^2+c x^4}}\right )}{\sqrt {a+b x^2+c x^4}}\right )}{8 c^{3/2} \sqrt {x}} \]
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Time = 0.07 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.95
method | result | size |
risch | \(\frac {\left (2 c \,x^{2}+b \right ) \left (c \,x^{4}+b \,x^{2}+a \right ) \sqrt {x}}{8 c \sqrt {x \left (c \,x^{4}+b \,x^{2}+a \right )}}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c \,x^{2}}{\sqrt {c}}+\sqrt {c \,x^{4}+b \,x^{2}+a}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {x}}{16 c^{\frac {3}{2}} \sqrt {x \left (c \,x^{4}+b \,x^{2}+a \right )}}\) | \(123\) |
default | \(\frac {\sqrt {x \left (c \,x^{4}+b \,x^{2}+a \right )}\, \left (4 c^{\frac {3}{2}} x^{2} \sqrt {c \,x^{4}+b \,x^{2}+a}+4 \ln \left (\frac {2 c \,x^{2}+2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {c}+b}{2 \sqrt {c}}\right ) a c -\ln \left (\frac {2 c \,x^{2}+2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {c}+b}{2 \sqrt {c}}\right ) b^{2}+2 b \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {c}\right )}{16 c^{\frac {3}{2}} \sqrt {x}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}\) | \(157\) |
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Time = 0.29 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.80 \[ \int \sqrt {x} \sqrt {a x+b x^3+c x^5} \, dx=\left [-\frac {{\left (b^{2} - 4 \, a c\right )} \sqrt {c} x \log \left (-\frac {8 \, c^{2} x^{5} + 8 \, b c x^{3} + 4 \, \sqrt {c x^{5} + b x^{3} + a x} {\left (2 \, c x^{2} + b\right )} \sqrt {c} \sqrt {x} + {\left (b^{2} + 4 \, a c\right )} x}{x}\right ) - 4 \, \sqrt {c x^{5} + b x^{3} + a x} {\left (2 \, c^{2} x^{2} + b c\right )} \sqrt {x}}{32 \, c^{2} x}, \frac {{\left (b^{2} - 4 \, a c\right )} \sqrt {-c} x \arctan \left (\frac {\sqrt {c x^{5} + b x^{3} + a x} {\left (2 \, c x^{2} + b\right )} \sqrt {-c} \sqrt {x}}{2 \, {\left (c^{2} x^{5} + b c x^{3} + a c x\right )}}\right ) + 2 \, \sqrt {c x^{5} + b x^{3} + a x} {\left (2 \, c^{2} x^{2} + b c\right )} \sqrt {x}}{16 \, c^{2} x}\right ] \]
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\[ \int \sqrt {x} \sqrt {a x+b x^3+c x^5} \, dx=\int \sqrt {x} \sqrt {x \left (a + b x^{2} + c x^{4}\right )}\, dx \]
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\[ \int \sqrt {x} \sqrt {a x+b x^3+c x^5} \, dx=\int { \sqrt {c x^{5} + b x^{3} + a x} \sqrt {x} \,d x } \]
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Time = 0.32 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.94 \[ \int \sqrt {x} \sqrt {a x+b x^3+c x^5} \, dx=\frac {1}{8} \, \sqrt {c x^{4} + b x^{2} + a} {\left (2 \, x^{2} + \frac {b}{c}\right )} + \frac {{\left (b^{2} - 4 \, a c\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}\right )} \sqrt {c} + b \right |}\right )}{16 \, c^{\frac {3}{2}}} - \frac {b^{2} \log \left ({\left | b - 2 \, \sqrt {a} \sqrt {c} \right |}\right ) - 4 \, a c \log \left ({\left | b - 2 \, \sqrt {a} \sqrt {c} \right |}\right ) + 2 \, \sqrt {a} b \sqrt {c}}{16 \, c^{\frac {3}{2}}} \]
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Timed out. \[ \int \sqrt {x} \sqrt {a x+b x^3+c x^5} \, dx=\int \sqrt {x}\,\sqrt {c\,x^5+b\,x^3+a\,x} \,d x \]
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