Optimal. Leaf size=140 \[ -\frac {c^2 (6 b B-A c) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )}{16 b^{3/2}}-\frac {\left (b x^2+c x^4\right )^{3/2} (6 b B-A c)}{24 b x^7}-\frac {c \sqrt {b x^2+c x^4} (6 b B-A c)}{16 b x^3}-\frac {A \left (b x^2+c x^4\right )^{5/2}}{6 b x^{11}} \]
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Rubi [A] time = 0.22, antiderivative size = 140, 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, 2020, 2008, 206} \[ -\frac {c^2 (6 b B-A c) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )}{16 b^{3/2}}-\frac {\left (b x^2+c x^4\right )^{3/2} (6 b B-A c)}{24 b x^7}-\frac {c \sqrt {b x^2+c x^4} (6 b B-A c)}{16 b x^3}-\frac {A \left (b x^2+c x^4\right )^{5/2}}{6 b x^{11}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2008
Rule 2020
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^{10}} \, dx &=-\frac {A \left (b x^2+c x^4\right )^{5/2}}{6 b x^{11}}-\frac {(-6 b B+A c) \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^8} \, dx}{6 b}\\ &=-\frac {(6 b B-A c) \left (b x^2+c x^4\right )^{3/2}}{24 b x^7}-\frac {A \left (b x^2+c x^4\right )^{5/2}}{6 b x^{11}}+\frac {(c (6 b B-A c)) \int \frac {\sqrt {b x^2+c x^4}}{x^4} \, dx}{8 b}\\ &=-\frac {c (6 b B-A c) \sqrt {b x^2+c x^4}}{16 b x^3}-\frac {(6 b B-A c) \left (b x^2+c x^4\right )^{3/2}}{24 b x^7}-\frac {A \left (b x^2+c x^4\right )^{5/2}}{6 b x^{11}}+\frac {\left (c^2 (6 b B-A c)\right ) \int \frac {1}{\sqrt {b x^2+c x^4}} \, dx}{16 b}\\ &=-\frac {c (6 b B-A c) \sqrt {b x^2+c x^4}}{16 b x^3}-\frac {(6 b B-A c) \left (b x^2+c x^4\right )^{3/2}}{24 b x^7}-\frac {A \left (b x^2+c x^4\right )^{5/2}}{6 b x^{11}}-\frac {\left (c^2 (6 b B-A c)\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {b x^2+c x^4}}\right )}{16 b}\\ &=-\frac {c (6 b B-A c) \sqrt {b x^2+c x^4}}{16 b x^3}-\frac {(6 b B-A c) \left (b x^2+c x^4\right )^{3/2}}{24 b x^7}-\frac {A \left (b x^2+c x^4\right )^{5/2}}{6 b x^{11}}-\frac {c^2 (6 b B-A c) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )}{16 b^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 121, normalized size = 0.86 \[ -\frac {\left (b+c x^2\right ) \left (A \left (8 b^2+14 b c x^2+3 c^2 x^4\right )+6 b B x^2 \left (2 b+5 c x^2\right )\right )+3 c^2 x^6 \sqrt {\frac {c x^2}{b}+1} (6 b B-A c) \tanh ^{-1}\left (\sqrt {\frac {c x^2}{b}+1}\right )}{48 b x^5 \sqrt {x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 250, normalized size = 1.79 \[ \left [-\frac {3 \, {\left (6 \, B b c^{2} - A c^{3}\right )} \sqrt {b} x^{7} \log \left (-\frac {c x^{3} + 2 \, b x + 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {b}}{x^{3}}\right ) + 2 \, {\left (3 \, {\left (10 \, B b^{2} c + A b c^{2}\right )} x^{4} + 8 \, A b^{3} + 2 \, {\left (6 \, B b^{3} + 7 \, A b^{2} c\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}}}{96 \, b^{2} x^{7}}, \frac {3 \, {\left (6 \, B b c^{2} - A c^{3}\right )} \sqrt {-b} x^{7} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2}} \sqrt {-b}}{c x^{3} + b x}\right ) - {\left (3 \, {\left (10 \, B b^{2} c + A b c^{2}\right )} x^{4} + 8 \, A b^{3} + 2 \, {\left (6 \, B b^{3} + 7 \, A b^{2} c\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}}}{48 \, b^{2} x^{7}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 175, normalized size = 1.25 \[ \frac {\frac {3 \, {\left (6 \, B b c^{3} \mathrm {sgn}\relax (x) - A c^{4} \mathrm {sgn}\relax (x)\right )} \arctan \left (\frac {\sqrt {c x^{2} + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b} - \frac {30 \, {\left (c x^{2} + b\right )}^{\frac {5}{2}} B b c^{3} \mathrm {sgn}\relax (x) - 48 \, {\left (c x^{2} + b\right )}^{\frac {3}{2}} B b^{2} c^{3} \mathrm {sgn}\relax (x) + 18 \, \sqrt {c x^{2} + b} B b^{3} c^{3} \mathrm {sgn}\relax (x) + 3 \, {\left (c x^{2} + b\right )}^{\frac {5}{2}} A c^{4} \mathrm {sgn}\relax (x) + 8 \, {\left (c x^{2} + b\right )}^{\frac {3}{2}} A b c^{4} \mathrm {sgn}\relax (x) - 3 \, \sqrt {c x^{2} + b} A b^{2} c^{4} \mathrm {sgn}\relax (x)}{b c^{3} x^{6}}}{48 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 259, normalized size = 1.85 \[ \frac {\left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}} \left (3 A \,b^{\frac {3}{2}} c^{3} x^{6} \ln \left (\frac {2 b +2 \sqrt {c \,x^{2}+b}\, \sqrt {b}}{x}\right )-18 B \,b^{\frac {5}{2}} c^{2} x^{6} \ln \left (\frac {2 b +2 \sqrt {c \,x^{2}+b}\, \sqrt {b}}{x}\right )-3 \sqrt {c \,x^{2}+b}\, A b \,c^{3} x^{6}+18 \sqrt {c \,x^{2}+b}\, B \,b^{2} c^{2} x^{6}-\left (c \,x^{2}+b \right )^{\frac {3}{2}} A \,c^{3} x^{6}+6 \left (c \,x^{2}+b \right )^{\frac {3}{2}} B b \,c^{2} x^{6}+\left (c \,x^{2}+b \right )^{\frac {5}{2}} A \,c^{2} x^{4}-6 \left (c \,x^{2}+b \right )^{\frac {5}{2}} B b c \,x^{4}+2 \left (c \,x^{2}+b \right )^{\frac {5}{2}} A b c \,x^{2}-12 \left (c \,x^{2}+b \right )^{\frac {5}{2}} B \,b^{2} x^{2}-8 \left (c \,x^{2}+b \right )^{\frac {5}{2}} A \,b^{2}\right )}{48 \left (c \,x^{2}+b \right )^{\frac {3}{2}} b^{3} x^{9}} \]
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 \[ \int \frac {{\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} {\left (B x^{2} + A\right )}}{x^{10}}\,{d x} \]
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 \[ \int \frac {\left (B\,x^2+A\right )\,{\left (c\,x^4+b\,x^2\right )}^{3/2}}{x^{10}} \,d x \]
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 \[ \int \frac {\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac {3}{2}} \left (A + B x^{2}\right )}{x^{10}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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