Optimal. Leaf size=251 \[ \frac {c^5 (12 b B-7 A c) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )}{1024 b^{9/2}}-\frac {c^4 \sqrt {b x^2+c x^4} (12 b B-7 A c)}{1024 b^4 x^3}+\frac {c^3 \sqrt {b x^2+c x^4} (12 b B-7 A c)}{1536 b^3 x^5}-\frac {c^2 \sqrt {b x^2+c x^4} (12 b B-7 A c)}{1920 b^2 x^7}-\frac {\left (b x^2+c x^4\right )^{3/2} (12 b B-7 A c)}{120 b x^{13}}-\frac {c \sqrt {b x^2+c x^4} (12 b B-7 A c)}{320 b x^9}-\frac {A \left (b x^2+c x^4\right )^{5/2}}{12 b x^{17}} \]
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Rubi [A] time = 0.39, antiderivative size = 251, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {2038, 2020, 2025, 2008, 206} \[ -\frac {c^4 \sqrt {b x^2+c x^4} (12 b B-7 A c)}{1024 b^4 x^3}+\frac {c^3 \sqrt {b x^2+c x^4} (12 b B-7 A c)}{1536 b^3 x^5}-\frac {c^2 \sqrt {b x^2+c x^4} (12 b B-7 A c)}{1920 b^2 x^7}+\frac {c^5 (12 b B-7 A c) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )}{1024 b^{9/2}}-\frac {c \sqrt {b x^2+c x^4} (12 b B-7 A c)}{320 b x^9}-\frac {\left (b x^2+c x^4\right )^{3/2} (12 b B-7 A c)}{120 b x^{13}}-\frac {A \left (b x^2+c x^4\right )^{5/2}}{12 b x^{17}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2008
Rule 2020
Rule 2025
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^{16}} \, dx &=-\frac {A \left (b x^2+c x^4\right )^{5/2}}{12 b x^{17}}-\frac {(-12 b B+7 A c) \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^{14}} \, dx}{12 b}\\ &=-\frac {(12 b B-7 A c) \left (b x^2+c x^4\right )^{3/2}}{120 b x^{13}}-\frac {A \left (b x^2+c x^4\right )^{5/2}}{12 b x^{17}}+\frac {(c (12 b B-7 A c)) \int \frac {\sqrt {b x^2+c x^4}}{x^{10}} \, dx}{40 b}\\ &=-\frac {c (12 b B-7 A c) \sqrt {b x^2+c x^4}}{320 b x^9}-\frac {(12 b B-7 A c) \left (b x^2+c x^4\right )^{3/2}}{120 b x^{13}}-\frac {A \left (b x^2+c x^4\right )^{5/2}}{12 b x^{17}}+\frac {\left (c^2 (12 b B-7 A c)\right ) \int \frac {1}{x^6 \sqrt {b x^2+c x^4}} \, dx}{320 b}\\ &=-\frac {c (12 b B-7 A c) \sqrt {b x^2+c x^4}}{320 b x^9}-\frac {c^2 (12 b B-7 A c) \sqrt {b x^2+c x^4}}{1920 b^2 x^7}-\frac {(12 b B-7 A c) \left (b x^2+c x^4\right )^{3/2}}{120 b x^{13}}-\frac {A \left (b x^2+c x^4\right )^{5/2}}{12 b x^{17}}-\frac {\left (c^3 (12 b B-7 A c)\right ) \int \frac {1}{x^4 \sqrt {b x^2+c x^4}} \, dx}{384 b^2}\\ &=-\frac {c (12 b B-7 A c) \sqrt {b x^2+c x^4}}{320 b x^9}-\frac {c^2 (12 b B-7 A c) \sqrt {b x^2+c x^4}}{1920 b^2 x^7}+\frac {c^3 (12 b B-7 A c) \sqrt {b x^2+c x^4}}{1536 b^3 x^5}-\frac {(12 b B-7 A c) \left (b x^2+c x^4\right )^{3/2}}{120 b x^{13}}-\frac {A \left (b x^2+c x^4\right )^{5/2}}{12 b x^{17}}+\frac {\left (c^4 (12 b B-7 A c)\right ) \int \frac {1}{x^2 \sqrt {b x^2+c x^4}} \, dx}{512 b^3}\\ &=-\frac {c (12 b B-7 A c) \sqrt {b x^2+c x^4}}{320 b x^9}-\frac {c^2 (12 b B-7 A c) \sqrt {b x^2+c x^4}}{1920 b^2 x^7}+\frac {c^3 (12 b B-7 A c) \sqrt {b x^2+c x^4}}{1536 b^3 x^5}-\frac {c^4 (12 b B-7 A c) \sqrt {b x^2+c x^4}}{1024 b^4 x^3}-\frac {(12 b B-7 A c) \left (b x^2+c x^4\right )^{3/2}}{120 b x^{13}}-\frac {A \left (b x^2+c x^4\right )^{5/2}}{12 b x^{17}}-\frac {\left (c^5 (12 b B-7 A c)\right ) \int \frac {1}{\sqrt {b x^2+c x^4}} \, dx}{1024 b^4}\\ &=-\frac {c (12 b B-7 A c) \sqrt {b x^2+c x^4}}{320 b x^9}-\frac {c^2 (12 b B-7 A c) \sqrt {b x^2+c x^4}}{1920 b^2 x^7}+\frac {c^3 (12 b B-7 A c) \sqrt {b x^2+c x^4}}{1536 b^3 x^5}-\frac {c^4 (12 b B-7 A c) \sqrt {b x^2+c x^4}}{1024 b^4 x^3}-\frac {(12 b B-7 A c) \left (b x^2+c x^4\right )^{3/2}}{120 b x^{13}}-\frac {A \left (b x^2+c x^4\right )^{5/2}}{12 b x^{17}}+\frac {\left (c^5 (12 b B-7 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 )}{1024 b^4}\\ &=-\frac {c (12 b B-7 A c) \sqrt {b x^2+c x^4}}{320 b x^9}-\frac {c^2 (12 b B-7 A c) \sqrt {b x^2+c x^4}}{1920 b^2 x^7}+\frac {c^3 (12 b B-7 A c) \sqrt {b x^2+c x^4}}{1536 b^3 x^5}-\frac {c^4 (12 b B-7 A c) \sqrt {b x^2+c x^4}}{1024 b^4 x^3}-\frac {(12 b B-7 A c) \left (b x^2+c x^4\right )^{3/2}}{120 b x^{13}}-\frac {A \left (b x^2+c x^4\right )^{5/2}}{12 b x^{17}}+\frac {c^5 (12 b B-7 A c) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )}{1024 b^{9/2}}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 66, normalized size = 0.26 \[ \frac {\left (x^2 \left (b+c x^2\right )\right )^{5/2} \left (c^5 x^{12} (12 b B-7 A c) \, _2F_1\left (\frac {5}{2},6;\frac {7}{2};\frac {c x^2}{b}+1\right )-5 A b^6\right )}{60 b^7 x^{17}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.33, size = 393, normalized size = 1.57 \[ \left [-\frac {15 \, {\left (12 \, B b c^{5} - 7 \, A c^{6}\right )} \sqrt {b} x^{13} \log \left (-\frac {c x^{3} + 2 \, b x - 2 \, \sqrt {c x^{4} + b x^{2}} \sqrt {b}}{x^{3}}\right ) + 2 \, {\left (15 \, {\left (12 \, B b^{2} c^{4} - 7 \, A b c^{5}\right )} x^{10} - 10 \, {\left (12 \, B b^{3} c^{3} - 7 \, A b^{2} c^{4}\right )} x^{8} + 1280 \, A b^{6} + 8 \, {\left (12 \, B b^{4} c^{2} - 7 \, A b^{3} c^{3}\right )} x^{6} + 48 \, {\left (44 \, B b^{5} c + A b^{4} c^{2}\right )} x^{4} + 128 \, {\left (12 \, B b^{6} + 13 \, A b^{5} c\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}}}{30720 \, b^{5} x^{13}}, -\frac {15 \, {\left (12 \, B b c^{5} - 7 \, A c^{6}\right )} \sqrt {-b} x^{13} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2}} \sqrt {-b}}{c x^{3} + b x}\right ) + {\left (15 \, {\left (12 \, B b^{2} c^{4} - 7 \, A b c^{5}\right )} x^{10} - 10 \, {\left (12 \, B b^{3} c^{3} - 7 \, A b^{2} c^{4}\right )} x^{8} + 1280 \, A b^{6} + 8 \, {\left (12 \, B b^{4} c^{2} - 7 \, A b^{3} c^{3}\right )} x^{6} + 48 \, {\left (44 \, B b^{5} c + A b^{4} c^{2}\right )} x^{4} + 128 \, {\left (12 \, B b^{6} + 13 \, A b^{5} c\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}}}{15360 \, b^{5} x^{13}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 294, normalized size = 1.17 \[ -\frac {\frac {15 \, {\left (12 \, B b c^{6} \mathrm {sgn}\relax (x) - 7 \, A c^{7} \mathrm {sgn}\relax (x)\right )} \arctan \left (\frac {\sqrt {c x^{2} + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b^{4}} + \frac {180 \, {\left (c x^{2} + b\right )}^{\frac {11}{2}} B b c^{6} \mathrm {sgn}\relax (x) - 1020 \, {\left (c x^{2} + b\right )}^{\frac {9}{2}} B b^{2} c^{6} \mathrm {sgn}\relax (x) + 2376 \, {\left (c x^{2} + b\right )}^{\frac {7}{2}} B b^{3} c^{6} \mathrm {sgn}\relax (x) - 696 \, {\left (c x^{2} + b\right )}^{\frac {5}{2}} B b^{4} c^{6} \mathrm {sgn}\relax (x) - 1020 \, {\left (c x^{2} + b\right )}^{\frac {3}{2}} B b^{5} c^{6} \mathrm {sgn}\relax (x) + 180 \, \sqrt {c x^{2} + b} B b^{6} c^{6} \mathrm {sgn}\relax (x) - 105 \, {\left (c x^{2} + b\right )}^{\frac {11}{2}} A c^{7} \mathrm {sgn}\relax (x) + 595 \, {\left (c x^{2} + b\right )}^{\frac {9}{2}} A b c^{7} \mathrm {sgn}\relax (x) - 1386 \, {\left (c x^{2} + b\right )}^{\frac {7}{2}} A b^{2} c^{7} \mathrm {sgn}\relax (x) + 1686 \, {\left (c x^{2} + b\right )}^{\frac {5}{2}} A b^{3} c^{7} \mathrm {sgn}\relax (x) + 595 \, {\left (c x^{2} + b\right )}^{\frac {3}{2}} A b^{4} c^{7} \mathrm {sgn}\relax (x) - 105 \, \sqrt {c x^{2} + b} A b^{5} c^{7} \mathrm {sgn}\relax (x)}{b^{4} c^{6} x^{12}}}{15360 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 386, normalized size = 1.54 \[ -\frac {\left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}} \left (105 A \,b^{\frac {3}{2}} c^{6} x^{12} \ln \left (\frac {2 b +2 \sqrt {c \,x^{2}+b}\, \sqrt {b}}{x}\right )-180 B \,b^{\frac {5}{2}} c^{5} x^{12} \ln \left (\frac {2 b +2 \sqrt {c \,x^{2}+b}\, \sqrt {b}}{x}\right )-105 \sqrt {c \,x^{2}+b}\, A b \,c^{6} x^{12}+180 \sqrt {c \,x^{2}+b}\, B \,b^{2} c^{5} x^{12}-35 \left (c \,x^{2}+b \right )^{\frac {3}{2}} A \,c^{6} x^{12}+60 \left (c \,x^{2}+b \right )^{\frac {3}{2}} B b \,c^{5} x^{12}+35 \left (c \,x^{2}+b \right )^{\frac {5}{2}} A \,c^{5} x^{10}-60 \left (c \,x^{2}+b \right )^{\frac {5}{2}} B b \,c^{4} x^{10}+70 \left (c \,x^{2}+b \right )^{\frac {5}{2}} A b \,c^{4} x^{8}-120 \left (c \,x^{2}+b \right )^{\frac {5}{2}} B \,b^{2} c^{3} x^{8}-280 \left (c \,x^{2}+b \right )^{\frac {5}{2}} A \,b^{2} c^{3} x^{6}+480 \left (c \,x^{2}+b \right )^{\frac {5}{2}} B \,b^{3} c^{2} x^{6}+560 \left (c \,x^{2}+b \right )^{\frac {5}{2}} A \,b^{3} c^{2} x^{4}-960 \left (c \,x^{2}+b \right )^{\frac {5}{2}} B \,b^{4} c \,x^{4}-896 \left (c \,x^{2}+b \right )^{\frac {5}{2}} A \,b^{4} c \,x^{2}+1536 \left (c \,x^{2}+b \right )^{\frac {5}{2}} B \,b^{5} x^{2}+1280 \left (c \,x^{2}+b \right )^{\frac {5}{2}} A \,b^{5}\right )}{15360 \left (c \,x^{2}+b \right )^{\frac {3}{2}} b^{6} x^{15}} \]
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 \[ \int \frac {{\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}} {\left (B x^{2} + A\right )}}{x^{16}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (B\,x^2+A\right )\,{\left (c\,x^4+b\,x^2\right )}^{3/2}}{x^{16}} \,d x \]
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 \[ \int \frac {\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac {3}{2}} \left (A + B x^{2}\right )}{x^{16}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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