Optimal. Leaf size=214 \[ -\frac {3 c^4 (2 b B-A c) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )}{256 b^{7/2}}+\frac {3 c^3 \sqrt {b x^2+c x^4} (2 b B-A c)}{256 b^3 x^3}-\frac {c^2 \sqrt {b x^2+c x^4} (2 b B-A c)}{128 b^2 x^5}-\frac {\left (b x^2+c x^4\right )^{3/2} (2 b B-A c)}{16 b x^{11}}-\frac {c \sqrt {b x^2+c x^4} (2 b B-A c)}{32 b x^7}-\frac {A \left (b x^2+c x^4\right )^{5/2}}{10 b x^{15}} \]
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Rubi [A] time = 0.34, antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps used = 7, 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} \begin {gather*} \frac {3 c^3 \sqrt {b x^2+c x^4} (2 b B-A c)}{256 b^3 x^3}-\frac {c^2 \sqrt {b x^2+c x^4} (2 b B-A c)}{128 b^2 x^5}-\frac {3 c^4 (2 b B-A c) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )}{256 b^{7/2}}-\frac {c \sqrt {b x^2+c x^4} (2 b B-A c)}{32 b x^7}-\frac {\left (b x^2+c x^4\right )^{3/2} (2 b B-A c)}{16 b x^{11}}-\frac {A \left (b x^2+c x^4\right )^{5/2}}{10 b x^{15}} \end {gather*}
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^{14}} \, dx &=-\frac {A \left (b x^2+c x^4\right )^{5/2}}{10 b x^{15}}-\frac {(-10 b B+5 A c) \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^{12}} \, dx}{10 b}\\ &=-\frac {(2 b B-A c) \left (b x^2+c x^4\right )^{3/2}}{16 b x^{11}}-\frac {A \left (b x^2+c x^4\right )^{5/2}}{10 b x^{15}}+\frac {(3 c (2 b B-A c)) \int \frac {\sqrt {b x^2+c x^4}}{x^8} \, dx}{16 b}\\ &=-\frac {c (2 b B-A c) \sqrt {b x^2+c x^4}}{32 b x^7}-\frac {(2 b B-A c) \left (b x^2+c x^4\right )^{3/2}}{16 b x^{11}}-\frac {A \left (b x^2+c x^4\right )^{5/2}}{10 b x^{15}}+\frac {\left (c^2 (2 b B-A c)\right ) \int \frac {1}{x^4 \sqrt {b x^2+c x^4}} \, dx}{32 b}\\ &=-\frac {c (2 b B-A c) \sqrt {b x^2+c x^4}}{32 b x^7}-\frac {c^2 (2 b B-A c) \sqrt {b x^2+c x^4}}{128 b^2 x^5}-\frac {(2 b B-A c) \left (b x^2+c x^4\right )^{3/2}}{16 b x^{11}}-\frac {A \left (b x^2+c x^4\right )^{5/2}}{10 b x^{15}}-\frac {\left (3 c^3 (2 b B-A c)\right ) \int \frac {1}{x^2 \sqrt {b x^2+c x^4}} \, dx}{128 b^2}\\ &=-\frac {c (2 b B-A c) \sqrt {b x^2+c x^4}}{32 b x^7}-\frac {c^2 (2 b B-A c) \sqrt {b x^2+c x^4}}{128 b^2 x^5}+\frac {3 c^3 (2 b B-A c) \sqrt {b x^2+c x^4}}{256 b^3 x^3}-\frac {(2 b B-A c) \left (b x^2+c x^4\right )^{3/2}}{16 b x^{11}}-\frac {A \left (b x^2+c x^4\right )^{5/2}}{10 b x^{15}}+\frac {\left (3 c^4 (2 b B-A c)\right ) \int \frac {1}{\sqrt {b x^2+c x^4}} \, dx}{256 b^3}\\ &=-\frac {c (2 b B-A c) \sqrt {b x^2+c x^4}}{32 b x^7}-\frac {c^2 (2 b B-A c) \sqrt {b x^2+c x^4}}{128 b^2 x^5}+\frac {3 c^3 (2 b B-A c) \sqrt {b x^2+c x^4}}{256 b^3 x^3}-\frac {(2 b B-A c) \left (b x^2+c x^4\right )^{3/2}}{16 b x^{11}}-\frac {A \left (b x^2+c x^4\right )^{5/2}}{10 b x^{15}}-\frac {\left (3 c^4 (2 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 )}{256 b^3}\\ &=-\frac {c (2 b B-A c) \sqrt {b x^2+c x^4}}{32 b x^7}-\frac {c^2 (2 b B-A c) \sqrt {b x^2+c x^4}}{128 b^2 x^5}+\frac {3 c^3 (2 b B-A c) \sqrt {b x^2+c x^4}}{256 b^3 x^3}-\frac {(2 b B-A c) \left (b x^2+c x^4\right )^{3/2}}{16 b x^{11}}-\frac {A \left (b x^2+c x^4\right )^{5/2}}{10 b x^{15}}-\frac {3 c^4 (2 b B-A c) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )}{256 b^{7/2}}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 65, normalized size = 0.30 \begin {gather*} -\frac {\left (x^2 \left (b+c x^2\right )\right )^{5/2} \left (A b^5+c^4 x^{10} (2 b B-A c) \, _2F_1\left (\frac {5}{2},5;\frac {7}{2};\frac {c x^2}{b}+1\right )\right )}{10 b^6 x^{15}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.50, size = 161, normalized size = 0.75 \begin {gather*} \frac {\sqrt {b x^2+c x^4} \left (-128 A b^4-176 A b^3 c x^2-8 A b^2 c^2 x^4+10 A b c^3 x^6-15 A c^4 x^8-160 b^4 B x^2-240 b^3 B c x^4-20 b^2 B c^2 x^6+30 b B c^3 x^8\right )}{1280 b^3 x^{11}}-\frac {3 \left (2 b B c^4-A c^5\right ) \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )}{256 b^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 345, normalized size = 1.61 \begin {gather*} \left [-\frac {15 \, {\left (2 \, B b c^{4} - A c^{5}\right )} \sqrt {b} x^{11} \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 (2 \, B b^{2} c^{3} - A b c^{4}\right )} x^{8} - 10 \, {\left (2 \, B b^{3} c^{2} - A b^{2} c^{3}\right )} x^{6} - 128 \, A b^{5} - 8 \, {\left (30 \, B b^{4} c + A b^{3} c^{2}\right )} x^{4} - 16 \, {\left (10 \, B b^{5} + 11 \, A b^{4} c\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}}}{2560 \, b^{4} x^{11}}, \frac {15 \, {\left (2 \, B b c^{4} - A c^{5}\right )} \sqrt {-b} x^{11} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2}} \sqrt {-b}}{c x^{3} + b x}\right ) + {\left (15 \, {\left (2 \, B b^{2} c^{3} - A b c^{4}\right )} x^{8} - 10 \, {\left (2 \, B b^{3} c^{2} - A b^{2} c^{3}\right )} x^{6} - 128 \, A b^{5} - 8 \, {\left (30 \, B b^{4} c + A b^{3} c^{2}\right )} x^{4} - 16 \, {\left (10 \, B b^{5} + 11 \, A b^{4} c\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}}}{1280 \, b^{4} x^{11}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 234, normalized size = 1.09 \begin {gather*} \frac {\frac {15 \, {\left (2 \, B b c^{5} \mathrm {sgn}\relax (x) - A c^{6} \mathrm {sgn}\relax (x)\right )} \arctan \left (\frac {\sqrt {c x^{2} + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b^{3}} + \frac {30 \, {\left (c x^{2} + b\right )}^{\frac {9}{2}} B b c^{5} \mathrm {sgn}\relax (x) - 140 \, {\left (c x^{2} + b\right )}^{\frac {7}{2}} B b^{2} c^{5} \mathrm {sgn}\relax (x) + 140 \, {\left (c x^{2} + b\right )}^{\frac {3}{2}} B b^{4} c^{5} \mathrm {sgn}\relax (x) - 30 \, \sqrt {c x^{2} + b} B b^{5} c^{5} \mathrm {sgn}\relax (x) - 15 \, {\left (c x^{2} + b\right )}^{\frac {9}{2}} A c^{6} \mathrm {sgn}\relax (x) + 70 \, {\left (c x^{2} + b\right )}^{\frac {7}{2}} A b c^{6} \mathrm {sgn}\relax (x) - 128 \, {\left (c x^{2} + b\right )}^{\frac {5}{2}} A b^{2} c^{6} \mathrm {sgn}\relax (x) - 70 \, {\left (c x^{2} + b\right )}^{\frac {3}{2}} A b^{3} c^{6} \mathrm {sgn}\relax (x) + 15 \, \sqrt {c x^{2} + b} A b^{4} c^{6} \mathrm {sgn}\relax (x)}{b^{3} c^{5} x^{10}}}{1280 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 344, normalized size = 1.61 \begin {gather*} \frac {\left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}} \left (15 A \,b^{\frac {3}{2}} c^{5} x^{10} \ln \left (\frac {2 b +2 \sqrt {c \,x^{2}+b}\, \sqrt {b}}{x}\right )-30 B \,b^{\frac {5}{2}} c^{4} x^{10} \ln \left (\frac {2 b +2 \sqrt {c \,x^{2}+b}\, \sqrt {b}}{x}\right )-15 \sqrt {c \,x^{2}+b}\, A b \,c^{5} x^{10}+30 \sqrt {c \,x^{2}+b}\, B \,b^{2} c^{4} x^{10}-5 \left (c \,x^{2}+b \right )^{\frac {3}{2}} A \,c^{5} x^{10}+10 \left (c \,x^{2}+b \right )^{\frac {3}{2}} B b \,c^{4} x^{10}+5 \left (c \,x^{2}+b \right )^{\frac {5}{2}} A \,c^{4} x^{8}-10 \left (c \,x^{2}+b \right )^{\frac {5}{2}} B b \,c^{3} x^{8}+10 \left (c \,x^{2}+b \right )^{\frac {5}{2}} A b \,c^{3} x^{6}-20 \left (c \,x^{2}+b \right )^{\frac {5}{2}} B \,b^{2} c^{2} x^{6}-40 \left (c \,x^{2}+b \right )^{\frac {5}{2}} A \,b^{2} c^{2} x^{4}+80 \left (c \,x^{2}+b \right )^{\frac {5}{2}} B \,b^{3} c \,x^{4}+80 \left (c \,x^{2}+b \right )^{\frac {5}{2}} A \,b^{3} c \,x^{2}-160 \left (c \,x^{2}+b \right )^{\frac {5}{2}} B \,b^{4} x^{2}-128 \left (c \,x^{2}+b \right )^{\frac {5}{2}} A \,b^{4}\right )}{1280 \left (c \,x^{2}+b \right )^{\frac {3}{2}} b^{5} x^{13}} \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^{14}}\,{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 = -0.00 \begin {gather*} \int \frac {\left (B\,x^2+A\right )\,{\left (c\,x^4+b\,x^2\right )}^{3/2}}{x^{14}} \,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^{14}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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