Optimal. Leaf size=100 \[ -\frac {b (b B-4 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{8 c^{3/2}}-\frac {\sqrt {b x^2+c x^4} (b B-4 A c)}{8 c}+\frac {B \left (b x^2+c x^4\right )^{3/2}}{4 c x^2} \]
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Rubi [A] time = 0.20, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {2034, 794, 664, 620, 206} \[ -\frac {b (b B-4 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{8 c^{3/2}}-\frac {\sqrt {b x^2+c x^4} (b B-4 A c)}{8 c}+\frac {B \left (b x^2+c x^4\right )^{3/2}}{4 c x^2} \]
Antiderivative was successfully verified.
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Rule 206
Rule 620
Rule 664
Rule 794
Rule 2034
Rubi steps
\begin {align*} \int \frac {\left (A+B x^2\right ) \sqrt {b x^2+c x^4}}{x} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(A+B x) \sqrt {b x+c x^2}}{x} \, dx,x,x^2\right )\\ &=\frac {B \left (b x^2+c x^4\right )^{3/2}}{4 c x^2}+\frac {\left (b B-A c+\frac {3}{2} (-b B+2 A c)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {b x+c x^2}}{x} \, dx,x,x^2\right )}{4 c}\\ &=-\frac {(b B-4 A c) \sqrt {b x^2+c x^4}}{8 c}+\frac {B \left (b x^2+c x^4\right )^{3/2}}{4 c x^2}-\frac {(b (b B-4 A c)) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b x+c x^2}} \, dx,x,x^2\right )}{16 c}\\ &=-\frac {(b B-4 A c) \sqrt {b x^2+c x^4}}{8 c}+\frac {B \left (b x^2+c x^4\right )^{3/2}}{4 c x^2}-\frac {(b (b B-4 A c)) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x^2}{\sqrt {b x^2+c x^4}}\right )}{8 c}\\ &=-\frac {(b B-4 A c) \sqrt {b x^2+c x^4}}{8 c}+\frac {B \left (b x^2+c x^4\right )^{3/2}}{4 c x^2}-\frac {b (b B-4 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{8 c^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 91, normalized size = 0.91 \[ \frac {\sqrt {x^2 \left (b+c x^2\right )} \left (\sqrt {c} \left (4 A c+b B+2 B c x^2\right )-\frac {\sqrt {b} (b B-4 A c) \sinh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{x \sqrt {\frac {c x^2}{b}+1}}\right )}{8 c^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.06, size = 172, normalized size = 1.72 \[ \left [-\frac {{\left (B b^{2} - 4 \, A b c\right )} \sqrt {c} \log \left (-2 \, c x^{2} - b - 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {c}\right ) - 2 \, {\left (2 \, B c^{2} x^{2} + B b c + 4 \, A c^{2}\right )} \sqrt {c x^{4} + b x^{2}}}{16 \, c^{2}}, \frac {{\left (B b^{2} - 4 \, A b c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2}} \sqrt {-c}}{c x^{2} + b}\right ) + {\left (2 \, B c^{2} x^{2} + B b c + 4 \, A c^{2}\right )} \sqrt {c x^{4} + b x^{2}}}{8 \, c^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 103, normalized size = 1.03 \[ \frac {1}{8} \, {\left (2 \, B x^{2} \mathrm {sgn}\relax (x) + \frac {B b c \mathrm {sgn}\relax (x) + 4 \, A c^{2} \mathrm {sgn}\relax (x)}{c^{2}}\right )} \sqrt {c x^{2} + b} x + \frac {{\left (B b^{2} \mathrm {sgn}\relax (x) - 4 \, A b c \mathrm {sgn}\relax (x)\right )} \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2} + b} \right |}\right )}{8 \, c^{\frac {3}{2}}} - \frac {{\left (B b^{2} \log \left ({\left | b \right |}\right ) - 4 \, A b c \log \left ({\left | b \right |}\right )\right )} \mathrm {sgn}\relax (x)}{16 \, c^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 124, normalized size = 1.24 \[ \frac {\sqrt {c \,x^{4}+b \,x^{2}}\, \left (4 A b c \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+b}\right )-B \,b^{2} \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+b}\right )+4 \sqrt {c \,x^{2}+b}\, A \,c^{\frac {3}{2}} x -\sqrt {c \,x^{2}+b}\, B b \sqrt {c}\, x +2 \left (c \,x^{2}+b \right )^{\frac {3}{2}} B \sqrt {c}\, x \right )}{8 \sqrt {c \,x^{2}+b}\, c^{\frac {3}{2}} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.45, size = 128, normalized size = 1.28 \[ \frac {1}{4} \, {\left (\frac {b \log \left (2 \, c x^{2} + b + 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {c}\right )}{\sqrt {c}} + 2 \, \sqrt {c x^{4} + b x^{2}}\right )} A + \frac {1}{16} \, {\left (4 \, \sqrt {c x^{4} + b x^{2}} x^{2} - \frac {b^{2} \log \left (2 \, c x^{2} + b + 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {c}\right )}{c^{\frac {3}{2}}} + \frac {2 \, \sqrt {c x^{4} + b x^{2}} b}{c}\right )} B \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.61, size = 117, normalized size = 1.17 \[ \frac {A\,\sqrt {c\,x^4+b\,x^2}}{2}+\frac {B\,\left (\frac {b}{4\,c}+\frac {x^2}{2}\right )\,\sqrt {c\,x^4+b\,x^2}}{2}+\frac {A\,b\,\ln \left (\frac {c\,x^2+\frac {b}{2}}{\sqrt {c}}+\sqrt {c\,x^4+b\,x^2}\right )}{4\,\sqrt {c}}-\frac {B\,b^2\,\ln \left (\frac {c\,x^2+\frac {b}{2}}{\sqrt {c}}+\sqrt {c\,x^4+b\,x^2}\right )}{16\,c^{3/2}} \]
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 \[ \int \frac {\sqrt {x^{2} \left (b + c x^{2}\right )} \left (A + B x^{2}\right )}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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