Optimal. Leaf size=78 \[ \frac {A \sqrt {b x^2+c x^4}}{x}-A \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )+\frac {B \left (b x^2+c x^4\right )^{3/2}}{3 c x^3} \]
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Rubi [A] time = 0.15, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2039, 2021, 2008, 206} \begin {gather*} \frac {A \sqrt {b x^2+c x^4}}{x}-A \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )+\frac {B \left (b x^2+c x^4\right )^{3/2}}{3 c x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 2008
Rule 2021
Rule 2039
Rubi steps
\begin {align*} \int \frac {\left (A+B x^2\right ) \sqrt {b x^2+c x^4}}{x^2} \, dx &=\frac {B \left (b x^2+c x^4\right )^{3/2}}{3 c x^3}+A \int \frac {\sqrt {b x^2+c x^4}}{x^2} \, dx\\ &=\frac {A \sqrt {b x^2+c x^4}}{x}+\frac {B \left (b x^2+c x^4\right )^{3/2}}{3 c x^3}+(A b) \int \frac {1}{\sqrt {b x^2+c x^4}} \, dx\\ &=\frac {A \sqrt {b x^2+c x^4}}{x}+\frac {B \left (b x^2+c x^4\right )^{3/2}}{3 c x^3}-(A b) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {b x^2+c x^4}}\right )\\ &=\frac {A \sqrt {b x^2+c x^4}}{x}+\frac {B \left (b x^2+c x^4\right )^{3/2}}{3 c x^3}-A \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )\\ \end {align*}
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Mathematica [A] time = 0.09, size = 84, normalized size = 1.08 \begin {gather*} \frac {x \left (\left (b+c x^2\right ) \left (3 A c+b B+B c x^2\right )-3 A \sqrt {b} c \sqrt {b+c x^2} \tanh ^{-1}\left (\frac {\sqrt {b+c x^2}}{\sqrt {b}}\right )\right )}{3 c \sqrt {x^2 \left (b+c x^2\right )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.14, size = 71, normalized size = 0.91 \begin {gather*} \frac {\sqrt {b x^2+c x^4} \left (3 A c+b B+B c x^2\right )}{3 c x}-A \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 159, normalized size = 2.04 \begin {gather*} \left [\frac {3 \, A \sqrt {b} c x \log \left (-\frac {c x^{3} + 2 \, b x - 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {b}}{x^{3}}\right ) + 2 \, \sqrt {c x^{4} + b x^{2}} {\left (B c x^{2} + B b + 3 \, A c\right )}}{6 \, c x}, \frac {3 \, A \sqrt {-b} c x \arctan \left (\frac {\sqrt {c x^{4} + b x^{2}} \sqrt {-b}}{c x^{3} + b x}\right ) + \sqrt {c x^{4} + b x^{2}} {\left (B c x^{2} + B b + 3 \, A c\right )}}{3 \, c x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 116, normalized size = 1.49 \begin {gather*} \frac {A b \arctan \left (\frac {\sqrt {c x^{2} + b}}{\sqrt {-b}}\right ) \mathrm {sgn}\relax (x)}{\sqrt {-b}} - \frac {{\left (3 \, A b c \arctan \left (\frac {\sqrt {b}}{\sqrt {-b}}\right ) + B \sqrt {-b} b^{\frac {3}{2}} + 3 \, A \sqrt {-b} \sqrt {b} c\right )} \mathrm {sgn}\relax (x)}{3 \, \sqrt {-b} c} + \frac {{\left (c x^{2} + b\right )}^{\frac {3}{2}} B c^{2} \mathrm {sgn}\relax (x) + 3 \, \sqrt {c x^{2} + b} A c^{3} \mathrm {sgn}\relax (x)}{3 \, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 85, normalized size = 1.09 \begin {gather*} -\frac {\sqrt {c \,x^{4}+b \,x^{2}}\, \left (3 A \sqrt {b}\, c \ln \left (\frac {2 b +2 \sqrt {c \,x^{2}+b}\, \sqrt {b}}{x}\right )-3 \sqrt {c \,x^{2}+b}\, A c -\left (c \,x^{2}+b \right )^{\frac {3}{2}} B \right )}{3 \sqrt {c \,x^{2}+b}\, c 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*} A \int \frac {\sqrt {c x^{2} + b}}{x}\,{d x} + \frac {{\left (c x^{2} + b\right )}^{\frac {3}{2}} B}{3 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.50, size = 99, normalized size = 1.27 \begin {gather*} \frac {A\,\sqrt {c\,x^4+b\,x^2}}{x}+\frac {B\,\left (c\,x^2+b\right )\,\sqrt {c\,x^4+b\,x^2}}{3\,c\,x}+\frac {A\,\sqrt {b}\,\mathrm {asin}\left (\frac {\sqrt {b}\,1{}\mathrm {i}}{\sqrt {c}\,x}\right )\,\sqrt {c\,x^4+b\,x^2}\,1{}\mathrm {i}}{\sqrt {c}\,x^2\,\sqrt {\frac {b}{c\,x^2}+1}} \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^{2} \left (b + c x^{2}\right )} \left (A + B x^{2}\right )}{x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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