Optimal. Leaf size=133 \[ \frac {\sqrt {b x^2+c x^4} (3 A c+2 b B)}{2 x}-\frac {1}{2} \sqrt {b} (3 A c+2 b B) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )+\frac {\left (b x^2+c x^4\right )^{3/2} (3 A c+2 b B)}{6 b x^3}-\frac {A \left (b x^2+c x^4\right )^{5/2}}{2 b x^7} \]
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Rubi [A] time = 0.22, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2038, 2021, 2008, 206} \[ \frac {\left (b x^2+c x^4\right )^{3/2} (3 A c+2 b B)}{6 b x^3}+\frac {\sqrt {b x^2+c x^4} (3 A c+2 b B)}{2 x}-\frac {1}{2} \sqrt {b} (3 A c+2 b B) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )-\frac {A \left (b x^2+c x^4\right )^{5/2}}{2 b x^7} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2008
Rule 2021
Rule 2038
Rubi steps
\begin {align*} \int \frac {\left (A+B x^2\right ) \left (b x^2+c x^4\right )^{3/2}}{x^6} \, dx &=-\frac {A \left (b x^2+c x^4\right )^{5/2}}{2 b x^7}-\frac {(-2 b B-3 A c) \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^4} \, dx}{2 b}\\ &=\frac {(2 b B+3 A c) \left (b x^2+c x^4\right )^{3/2}}{6 b x^3}-\frac {A \left (b x^2+c x^4\right )^{5/2}}{2 b x^7}-\frac {1}{2} (-2 b B-3 A c) \int \frac {\sqrt {b x^2+c x^4}}{x^2} \, dx\\ &=\frac {(2 b B+3 A c) \sqrt {b x^2+c x^4}}{2 x}+\frac {(2 b B+3 A c) \left (b x^2+c x^4\right )^{3/2}}{6 b x^3}-\frac {A \left (b x^2+c x^4\right )^{5/2}}{2 b x^7}+\frac {1}{2} (b (2 b B+3 A c)) \int \frac {1}{\sqrt {b x^2+c x^4}} \, dx\\ &=\frac {(2 b B+3 A c) \sqrt {b x^2+c x^4}}{2 x}+\frac {(2 b B+3 A c) \left (b x^2+c x^4\right )^{3/2}}{6 b x^3}-\frac {A \left (b x^2+c x^4\right )^{5/2}}{2 b x^7}-\frac {1}{2} (b (2 b B+3 A c)) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {b x^2+c x^4}}\right )\\ &=\frac {(2 b B+3 A c) \sqrt {b x^2+c x^4}}{2 x}+\frac {(2 b B+3 A c) \left (b x^2+c x^4\right )^{3/2}}{6 b x^3}-\frac {A \left (b x^2+c x^4\right )^{5/2}}{2 b x^7}-\frac {1}{2} \sqrt {b} (2 b B+3 A c) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )\\ \end {align*}
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Mathematica [A] time = 0.07, size = 109, normalized size = 0.82 \[ \frac {\sqrt {x^2 \left (b+c x^2\right )} \left (\sqrt {b+c x^2} \left (-3 A b+6 A c x^2+8 b B x^2+2 B c x^4\right )-3 \sqrt {b} x^2 (3 A c+2 b B) \tanh ^{-1}\left (\frac {\sqrt {b+c x^2}}{\sqrt {b}}\right )\right )}{6 x^3 \sqrt {b+c x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.13, size = 195, normalized size = 1.47 \[ \left [\frac {3 \, {\left (2 \, B b + 3 \, A c\right )} \sqrt {b} x^{3} \log \left (-\frac {c x^{3} + 2 \, b x - 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {b}}{x^{3}}\right ) + 2 \, {\left (2 \, B c x^{4} + 2 \, {\left (4 \, B b + 3 \, A c\right )} x^{2} - 3 \, A b\right )} \sqrt {c x^{4} + b x^{2}}}{12 \, x^{3}}, \frac {3 \, {\left (2 \, B b + 3 \, A c\right )} \sqrt {-b} x^{3} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2}} \sqrt {-b}}{c x^{3} + b x}\right ) + {\left (2 \, B c x^{4} + 2 \, {\left (4 \, B b + 3 \, A c\right )} x^{2} - 3 \, A b\right )} \sqrt {c x^{4} + b x^{2}}}{6 \, x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 115, normalized size = 0.86 \[ \frac {2 \, {\left (c x^{2} + b\right )}^{\frac {3}{2}} B c \mathrm {sgn}\relax (x) + 6 \, \sqrt {c x^{2} + b} B b c \mathrm {sgn}\relax (x) + 6 \, \sqrt {c x^{2} + b} A c^{2} \mathrm {sgn}\relax (x) - \frac {3 \, \sqrt {c x^{2} + b} A b c \mathrm {sgn}\relax (x)}{x^{2}} + \frac {3 \, {\left (2 \, B b^{2} c \mathrm {sgn}\relax (x) + 3 \, A b c^{2} \mathrm {sgn}\relax (x)\right )} \arctan \left (\frac {\sqrt {c x^{2} + b}}{\sqrt {-b}}\right )}{\sqrt {-b}}}{6 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 172, normalized size = 1.29 \[ -\frac {\left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}} \left (9 A \,b^{\frac {3}{2}} c \,x^{2} \ln \left (\frac {2 b +2 \sqrt {c \,x^{2}+b}\, \sqrt {b}}{x}\right )+6 B \,b^{\frac {5}{2}} x^{2} \ln \left (\frac {2 b +2 \sqrt {c \,x^{2}+b}\, \sqrt {b}}{x}\right )-9 \sqrt {c \,x^{2}+b}\, A b c \,x^{2}-6 \sqrt {c \,x^{2}+b}\, B \,b^{2} x^{2}-3 \left (c \,x^{2}+b \right )^{\frac {3}{2}} A c \,x^{2}-2 \left (c \,x^{2}+b \right )^{\frac {3}{2}} B b \,x^{2}+3 \left (c \,x^{2}+b \right )^{\frac {5}{2}} A \right )}{6 \left (c \,x^{2}+b \right )^{\frac {3}{2}} b \,x^{5}} \]
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 \[ \int \frac {{\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} {\left (B x^{2} + A\right )}}{x^{6}}\,{d x} \]
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 \[ \int \frac {\left (B\,x^2+A\right )\,{\left (c\,x^4+b\,x^2\right )}^{3/2}}{x^6} \,d x \]
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 \[ \int \frac {\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac {3}{2}} \left (A + B x^{2}\right )}{x^{6}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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