Optimal. Leaf size=67 \[ \frac {B \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{c^{3/2}}-\frac {x^2 (b B-A c)}{b c \sqrt {b x^2+c x^4}} \]
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Rubi [A] time = 0.18, antiderivative size = 67, 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 = {2034, 777, 620, 206} \[ \frac {B \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{c^{3/2}}-\frac {x^2 (b B-A c)}{b c \sqrt {b x^2+c x^4}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 620
Rule 777
Rule 2034
Rubi steps
\begin {align*} \int \frac {x^3 \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^{3/2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x (A+B x)}{\left (b x+c x^2\right )^{3/2}} \, dx,x,x^2\right )\\ &=-\frac {(b B-A c) x^2}{b c \sqrt {b x^2+c x^4}}+\frac {B \operatorname {Subst}\left (\int \frac {1}{\sqrt {b x+c x^2}} \, dx,x,x^2\right )}{2 c}\\ &=-\frac {(b B-A c) x^2}{b c \sqrt {b x^2+c x^4}}+\frac {B \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x^2}{\sqrt {b x^2+c x^4}}\right )}{c}\\ &=-\frac {(b B-A c) x^2}{b c \sqrt {b x^2+c x^4}}+\frac {B \tanh ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {b x^2+c x^4}}\right )}{c^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 75, normalized size = 1.12 \[ \frac {x \left (\sqrt {c} x (A c-b B)+b^{3/2} B \sqrt {\frac {c x^2}{b}+1} \sinh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )\right )}{b c^{3/2} \sqrt {x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.92, size = 188, normalized size = 2.81 \[ \left [\frac {{\left (B b c x^{2} + B b^{2}\right )} \sqrt {c} \log \left (-2 \, c x^{2} - b - 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {c}\right ) - 2 \, \sqrt {c x^{4} + b x^{2}} {\left (B b c - A c^{2}\right )}}{2 \, {\left (b c^{3} x^{2} + b^{2} c^{2}\right )}}, -\frac {{\left (B b c x^{2} + B b^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2}} \sqrt {-c}}{c x^{2} + b}\right ) + \sqrt {c x^{4} + b x^{2}} {\left (B b c - A c^{2}\right )}}{b c^{3} x^{2} + b^{2} c^{2}}\right ] \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 75, normalized size = 1.12 \[ \frac {\left (c \,x^{2}+b \right ) \left (A \,c^{\frac {5}{2}} x -B b \,c^{\frac {3}{2}} x +\sqrt {c \,x^{2}+b}\, B b c \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+b}\right )\right ) x^{3}}{\left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}} b \,c^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.48, size = 79, normalized size = 1.18 \[ -\frac {1}{2} \, B {\left (\frac {2 \, x^{2}}{\sqrt {c x^{4} + b x^{2}} c} - \frac {\log \left (2 \, c x^{2} + b + 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {c}\right )}{c^{\frac {3}{2}}}\right )} + \frac {A x^{2}}{\sqrt {c x^{4} + b x^{2}} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.63, size = 78, normalized size = 1.16 \[ \frac {B\,\ln \left (\frac {c\,x^2+\frac {b}{2}}{\sqrt {c}}+\sqrt {c\,x^4+b\,x^2}\right )}{2\,c^{3/2}}+\frac {A\,x^2}{b\,\sqrt {c\,x^4+b\,x^2}}-\frac {B\,x^2}{c\,\sqrt {c\,x^4+b\,x^2}} \]
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 \[ \int \frac {x^{3} \left (A + B x^{2}\right )}{\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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