Optimal. Leaf size=194 \[ \frac {\sqrt {a x+b x^3+c x^5}}{2 \sqrt {x}}-\frac {\sqrt {a} \sqrt {x} \sqrt {a+b x^2+c x^4} \tanh ^{-1}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{2 \sqrt {a x+b x^3+c x^5}}+\frac {b \sqrt {x} \sqrt {a+b x^2+c x^4} \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )}{4 \sqrt {c} \sqrt {a x+b x^3+c x^5}} \]
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Rubi [A] time = 0.21, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {1921, 1953, 1251, 843, 621, 206, 724} \begin {gather*} \frac {\sqrt {a x+b x^3+c x^5}}{2 \sqrt {x}}-\frac {\sqrt {a} \sqrt {x} \sqrt {a+b x^2+c x^4} \tanh ^{-1}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{2 \sqrt {a x+b x^3+c x^5}}+\frac {b \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 )}{4 \sqrt {c} \sqrt {a x+b x^3+c x^5}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 724
Rule 843
Rule 1251
Rule 1921
Rule 1953
Rubi steps
\begin {align*} \int \frac {\sqrt {a x+b x^3+c x^5}}{x^{3/2}} \, dx &=\frac {\sqrt {a x+b x^3+c x^5}}{2 \sqrt {x}}+\frac {1}{2} \int \frac {2 a+b x^2}{\sqrt {x} \sqrt {a x+b x^3+c x^5}} \, dx\\ &=\frac {\sqrt {a x+b x^3+c x^5}}{2 \sqrt {x}}+\frac {\left (\sqrt {x} \sqrt {a+b x^2+c x^4}\right ) \int \frac {2 a+b x^2}{x \sqrt {a+b x^2+c x^4}} \, dx}{2 \sqrt {a x+b x^3+c x^5}}\\ &=\frac {\sqrt {a x+b x^3+c x^5}}{2 \sqrt {x}}+\frac {\left (\sqrt {x} \sqrt {a+b x^2+c x^4}\right ) \operatorname {Subst}\left (\int \frac {2 a+b x}{x \sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{4 \sqrt {a x+b x^3+c x^5}}\\ &=\frac {\sqrt {a x+b x^3+c x^5}}{2 \sqrt {x}}+\frac {\left (a \sqrt {x} \sqrt {a+b x^2+c x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x+c x^2}} \, dx,x,x^2\right )}{2 \sqrt {a x+b x^3+c x^5}}+\frac {\left (b \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 )}{4 \sqrt {a x+b x^3+c x^5}}\\ &=\frac {\sqrt {a x+b x^3+c x^5}}{2 \sqrt {x}}-\frac {\left (a \sqrt {x} \sqrt {a+b x^2+c x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+b x^2}{\sqrt {a+b x^2+c x^4}}\right )}{\sqrt {a x+b x^3+c x^5}}+\frac {\left (b \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 )}{2 \sqrt {a x+b x^3+c x^5}}\\ &=\frac {\sqrt {a x+b x^3+c x^5}}{2 \sqrt {x}}-\frac {\sqrt {a} \sqrt {x} \sqrt {a+b x^2+c x^4} \tanh ^{-1}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{2 \sqrt {a x+b x^3+c x^5}}+\frac {b \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 )}{4 \sqrt {c} \sqrt {a x+b x^3+c x^5}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 155, normalized size = 0.80 \begin {gather*} \frac {\sqrt {x} \sqrt {a+b x^2+c x^4} \left (2 \sqrt {c} \sqrt {a+b x^2+c x^4}-2 \sqrt {a} \sqrt {c} \tanh ^{-1}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )+b \tanh ^{-1}\left (\frac {b+2 c x^2}{2 \sqrt {c} \sqrt {a+b x^2+c x^4}}\right )\right )}{4 \sqrt {c} \sqrt {x \left (a+b x^2+c x^4\right )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.60, size = 148, normalized size = 0.76 \begin {gather*} \frac {\sqrt {a x+b x^3+c x^5}}{2 \sqrt {x}}-\frac {b \log \left (-2 \sqrt {c} \sqrt {a x+b x^3+c x^5}+b \sqrt {x}+2 c x^{5/2}\right )}{4 \sqrt {c}}+\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {c} x^{5/2}-\sqrt {a x+b x^3+c x^5}}\right )+\frac {b \log \left (\sqrt {x}\right )}{4 \sqrt {c}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.40, size = 666, normalized size = 3.43 \begin {gather*} \left [\frac {b \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 ) + 2 \, \sqrt {a} c x \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} c \sqrt {x}}{8 \, c x}, -\frac {b \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 ) - \sqrt {a} c x \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 ) - 2 \, \sqrt {c x^{5} + b x^{3} + a x} c \sqrt {x}}{4 \, c x}, \frac {4 \, \sqrt {-a} c x \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 ) + b \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} c \sqrt {x}}{8 \, c x}, \frac {2 \, \sqrt {-a} c x \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 ) - b \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} c \sqrt {x}}{4 \, c x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 136, normalized size = 0.70 \begin {gather*} -\frac {\sqrt {\left (c \,x^{4}+b \,x^{2}+a \right ) x}\, \left (2 \sqrt {a}\, \sqrt {c}\, \ln \left (\frac {b \,x^{2}+2 a +2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {a}}{x^{2}}\right )-b \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}\, \sqrt {c}\right )}{4 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {c}\, \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 \frac {\sqrt {c x^{5} + b x^{3} + a x}}{x^{\frac {3}{2}}}\,{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 \frac {\sqrt {c\,x^5+b\,x^3+a\,x}}{x^{3/2}} \,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 \frac {\sqrt {x \left (a + b x^{2} + c x^{4}\right )}}{x^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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