Optimal. Leaf size=129 \[ \frac {\left (b+2 c x^2\right ) \sqrt {a x+b x^3+c x^5}}{8 c \sqrt {x}}-\frac {\sqrt {x} \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4} \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{16 c^{3/2} \sqrt {a x+b x^3+c x^5}} \]
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Rubi [A] time = 0.09, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1918, 1914, 1107, 621, 206} \begin {gather*} \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} \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 {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 1107
Rule 1914
Rule 1918
Rubi steps
\begin {align*} \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 ) \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 ) \operatorname {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 ) \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 \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*}
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Mathematica [A] time = 0.08, size = 126, normalized size = 0.98 \begin {gather*} \frac {\sqrt {x \left (a+b x^2+c x^4\right )} \left (\frac {\left (b+2 c x^2\right ) \sqrt {a+b x^2+c x^4}}{4 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 )}{8 c^{3/2}}\right )}{2 \sqrt {x} \sqrt {a+b x^2+c x^4}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.46, size = 126, normalized size = 0.98 \begin {gather*} \frac {\log \left (\sqrt {x}\right ) \left (4 a c-b^2\right )}{16 c^{3/2}}+\frac {\left (b^2-4 a c\right ) \log \left (-2 c^{3/2} \sqrt {a x+b x^3+c x^5}+b c \sqrt {x}+2 c^2 x^{5/2}\right )}{16 c^{3/2}}+\frac {\left (b+2 c x^2\right ) \sqrt {a x+b x^3+c x^5}}{8 c \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.04, size = 232, normalized size = 1.80 \begin {gather*} \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 ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.97, size = 127, normalized size = 0.98 \begin {gather*} \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}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 157, normalized size = 1.22 \begin {gather*} \frac {\sqrt {\left (c \,x^{4}+b \,x^{2}+a \right ) x}\, \left (4 \sqrt {c \,x^{4}+b \,x^{2}+a}\, c^{\frac {3}{2}} x^{2}+4 a c \ln \left (\frac {2 c \,x^{2}+b +2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {c}}{2 \sqrt {c}}\right )-b^{2} \ln \left (\frac {2 c \,x^{2}+b +2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {c}}{2 \sqrt {c}}\right )+2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, b \sqrt {c}\right )}{16 \sqrt {c \,x^{4}+b \,x^{2}+a}\, c^{\frac {3}{2}} \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {c x^{5} + b x^{3} + a x} \sqrt {x}\,{d x} \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 \sqrt {x}\,\sqrt {c\,x^5+b\,x^3+a\,x} \,d x \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 \sqrt {x} \sqrt {x \left (a + b x^{2} + c x^{4}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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