Optimal. Leaf size=135 \[ \frac {3 c \sqrt {b x^2+c x^4} (A c+4 b B)}{8 b x}-\frac {3 c (A c+4 b B) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )}{8 \sqrt {b}}-\frac {\left (b x^2+c x^4\right )^{3/2} (A c+4 b B)}{8 b x^5}-\frac {A \left (b x^2+c x^4\right )^{5/2}}{4 b x^9} \]
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Rubi [A] time = 0.22, antiderivative size = 135, 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 = {2038, 2020, 2021, 2008, 206} \begin {gather*} -\frac {\left (b x^2+c x^4\right )^{3/2} (A c+4 b B)}{8 b x^5}+\frac {3 c \sqrt {b x^2+c x^4} (A c+4 b B)}{8 b x}-\frac {3 c (A c+4 b B) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )}{8 \sqrt {b}}-\frac {A \left (b x^2+c x^4\right )^{5/2}}{4 b x^9} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 2008
Rule 2020
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^8} \, dx &=-\frac {A \left (b x^2+c x^4\right )^{5/2}}{4 b x^9}-\frac {(-4 b B-A c) \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^6} \, dx}{4 b}\\ &=-\frac {(4 b B+A c) \left (b x^2+c x^4\right )^{3/2}}{8 b x^5}-\frac {A \left (b x^2+c x^4\right )^{5/2}}{4 b x^9}+\frac {(3 c (4 b B+A c)) \int \frac {\sqrt {b x^2+c x^4}}{x^2} \, dx}{8 b}\\ &=\frac {3 c (4 b B+A c) \sqrt {b x^2+c x^4}}{8 b x}-\frac {(4 b B+A c) \left (b x^2+c x^4\right )^{3/2}}{8 b x^5}-\frac {A \left (b x^2+c x^4\right )^{5/2}}{4 b x^9}+\frac {1}{8} (3 c (4 b B+A c)) \int \frac {1}{\sqrt {b x^2+c x^4}} \, dx\\ &=\frac {3 c (4 b B+A c) \sqrt {b x^2+c x^4}}{8 b x}-\frac {(4 b B+A c) \left (b x^2+c x^4\right )^{3/2}}{8 b x^5}-\frac {A \left (b x^2+c x^4\right )^{5/2}}{4 b x^9}-\frac {1}{8} (3 c (4 b B+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 {3 c (4 b B+A c) \sqrt {b x^2+c x^4}}{8 b x}-\frac {(4 b B+A c) \left (b x^2+c x^4\right )^{3/2}}{8 b x^5}-\frac {A \left (b x^2+c x^4\right )^{5/2}}{4 b x^9}-\frac {3 c (4 b B+A c) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )}{8 \sqrt {b}}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 63, normalized size = 0.47 \begin {gather*} \frac {\left (x^2 \left (b+c x^2\right )\right )^{5/2} \left (c x^4 (A c+4 b B) \, _2F_1\left (2,\frac {5}{2};\frac {7}{2};\frac {c x^2}{b}+1\right )-5 A b^2\right )}{20 b^3 x^9} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.88, size = 92, normalized size = 0.68 \begin {gather*} \frac {\sqrt {b x^2+c x^4} \left (-2 A b-5 A c x^2-4 b B x^2+8 B c x^4\right )}{8 x^5}-\frac {3 \left (A c^2+4 b B c\right ) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )}{8 \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 217, normalized size = 1.61 \begin {gather*} \left [\frac {3 \, {\left (4 \, B b c + A c^{2}\right )} \sqrt {b} x^{5} \log \left (-\frac {c x^{3} + 2 \, b x - 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {b}}{x^{3}}\right ) + 2 \, {\left (8 \, B b c x^{4} - 2 \, A b^{2} - {\left (4 \, B b^{2} + 5 \, A b c\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}}}{16 \, b x^{5}}, \frac {3 \, {\left (4 \, B b c + A c^{2}\right )} \sqrt {-b} x^{5} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2}} \sqrt {-b}}{c x^{3} + b x}\right ) + {\left (8 \, B b c x^{4} - 2 \, A b^{2} - {\left (4 \, B b^{2} + 5 \, A b c\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}}}{8 \, b x^{5}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 145, normalized size = 1.07 \begin {gather*} \frac {8 \, \sqrt {c x^{2} + b} B c^{2} \mathrm {sgn}\relax (x) + \frac {3 \, {\left (4 \, B b c^{2} \mathrm {sgn}\relax (x) + A c^{3} \mathrm {sgn}\relax (x)\right )} \arctan \left (\frac {\sqrt {c x^{2} + b}}{\sqrt {-b}}\right )}{\sqrt {-b}} - \frac {4 \, {\left (c x^{2} + b\right )}^{\frac {3}{2}} B b c^{2} \mathrm {sgn}\relax (x) - 4 \, \sqrt {c x^{2} + b} B b^{2} c^{2} \mathrm {sgn}\relax (x) + 5 \, {\left (c x^{2} + b\right )}^{\frac {3}{2}} A c^{3} \mathrm {sgn}\relax (x) - 3 \, \sqrt {c x^{2} + b} A b c^{3} \mathrm {sgn}\relax (x)}{c^{2} x^{4}}}{8 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 213, normalized size = 1.58 \begin {gather*} -\frac {\left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}} \left (3 A \,b^{\frac {3}{2}} c^{2} x^{4} \ln \left (\frac {2 b +2 \sqrt {c \,x^{2}+b}\, \sqrt {b}}{x}\right )+12 B \,b^{\frac {5}{2}} c \,x^{4} \ln \left (\frac {2 b +2 \sqrt {c \,x^{2}+b}\, \sqrt {b}}{x}\right )-3 \sqrt {c \,x^{2}+b}\, A b \,c^{2} x^{4}-12 \sqrt {c \,x^{2}+b}\, B \,b^{2} c \,x^{4}-\left (c \,x^{2}+b \right )^{\frac {3}{2}} A \,c^{2} x^{4}-4 \left (c \,x^{2}+b \right )^{\frac {3}{2}} B b c \,x^{4}+\left (c \,x^{2}+b \right )^{\frac {5}{2}} A c \,x^{2}+4 \left (c \,x^{2}+b \right )^{\frac {5}{2}} B b \,x^{2}+2 \left (c \,x^{2}+b \right )^{\frac {5}{2}} A b \right )}{8 \left (c \,x^{2}+b \right )^{\frac {3}{2}} b^{2} x^{7}} \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*} \int \frac {{\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} {\left (B x^{2} + A\right )}}{x^{8}}\,{d x} \end {gather*}
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 \begin {gather*} \int \frac {\left (B\,x^2+A\right )\,{\left (c\,x^4+b\,x^2\right )}^{3/2}}{x^8} \,d x \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 {\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac {3}{2}} \left (A + B x^{2}\right )}{x^{8}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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