Optimal. Leaf size=165 \[ \frac {3 c^5 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )}{256 b^{7/2}}-\frac {3 c^4 \sqrt {b x^2+c x^4}}{256 b^3 x^3}+\frac {c^3 \sqrt {b x^2+c x^4}}{128 b^2 x^5}-\frac {c^2 \sqrt {b x^2+c x^4}}{160 b x^7}-\frac {\left (b x^2+c x^4\right )^{3/2}}{10 x^{13}}-\frac {3 c \sqrt {b x^2+c x^4}}{80 x^9} \]
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Rubi [A] time = 0.25, antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {2020, 2025, 2008, 206} \begin {gather*} -\frac {3 c^4 \sqrt {b x^2+c x^4}}{256 b^3 x^3}+\frac {c^3 \sqrt {b x^2+c x^4}}{128 b^2 x^5}+\frac {3 c^5 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )}{256 b^{7/2}}-\frac {c^2 \sqrt {b x^2+c x^4}}{160 b x^7}-\frac {3 c \sqrt {b x^2+c x^4}}{80 x^9}-\frac {\left (b x^2+c x^4\right )^{3/2}}{10 x^{13}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 2008
Rule 2020
Rule 2025
Rubi steps
\begin {align*} \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^{14}} \, dx &=-\frac {\left (b x^2+c x^4\right )^{3/2}}{10 x^{13}}+\frac {1}{10} (3 c) \int \frac {\sqrt {b x^2+c x^4}}{x^{10}} \, dx\\ &=-\frac {3 c \sqrt {b x^2+c x^4}}{80 x^9}-\frac {\left (b x^2+c x^4\right )^{3/2}}{10 x^{13}}+\frac {1}{80} \left (3 c^2\right ) \int \frac {1}{x^6 \sqrt {b x^2+c x^4}} \, dx\\ &=-\frac {3 c \sqrt {b x^2+c x^4}}{80 x^9}-\frac {c^2 \sqrt {b x^2+c x^4}}{160 b x^7}-\frac {\left (b x^2+c x^4\right )^{3/2}}{10 x^{13}}-\frac {c^3 \int \frac {1}{x^4 \sqrt {b x^2+c x^4}} \, dx}{32 b}\\ &=-\frac {3 c \sqrt {b x^2+c x^4}}{80 x^9}-\frac {c^2 \sqrt {b x^2+c x^4}}{160 b x^7}+\frac {c^3 \sqrt {b x^2+c x^4}}{128 b^2 x^5}-\frac {\left (b x^2+c x^4\right )^{3/2}}{10 x^{13}}+\frac {\left (3 c^4\right ) \int \frac {1}{x^2 \sqrt {b x^2+c x^4}} \, dx}{128 b^2}\\ &=-\frac {3 c \sqrt {b x^2+c x^4}}{80 x^9}-\frac {c^2 \sqrt {b x^2+c x^4}}{160 b x^7}+\frac {c^3 \sqrt {b x^2+c x^4}}{128 b^2 x^5}-\frac {3 c^4 \sqrt {b x^2+c x^4}}{256 b^3 x^3}-\frac {\left (b x^2+c x^4\right )^{3/2}}{10 x^{13}}-\frac {\left (3 c^5\right ) \int \frac {1}{\sqrt {b x^2+c x^4}} \, dx}{256 b^3}\\ &=-\frac {3 c \sqrt {b x^2+c x^4}}{80 x^9}-\frac {c^2 \sqrt {b x^2+c x^4}}{160 b x^7}+\frac {c^3 \sqrt {b x^2+c x^4}}{128 b^2 x^5}-\frac {3 c^4 \sqrt {b x^2+c x^4}}{256 b^3 x^3}-\frac {\left (b x^2+c x^4\right )^{3/2}}{10 x^{13}}+\frac {\left (3 c^5\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 {3 c \sqrt {b x^2+c x^4}}{80 x^9}-\frac {c^2 \sqrt {b x^2+c x^4}}{160 b x^7}+\frac {c^3 \sqrt {b x^2+c x^4}}{128 b^2 x^5}-\frac {3 c^4 \sqrt {b x^2+c x^4}}{256 b^3 x^3}-\frac {\left (b x^2+c x^4\right )^{3/2}}{10 x^{13}}+\frac {3 c^5 \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 = 0.02, size = 46, normalized size = 0.28 \begin {gather*} \frac {c^5 \left (x^2 \left (b+c x^2\right )\right )^{5/2} \, _2F_1\left (\frac {5}{2},6;\frac {7}{2};\frac {c x^2}{b}+1\right )}{5 b^6 x^5} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.80, size = 104, normalized size = 0.63 \begin {gather*} \frac {3 c^5 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2+c x^4}}\right )}{256 b^{7/2}}+\frac {\sqrt {b x^2+c x^4} \left (-128 b^4-176 b^3 c x^2-8 b^2 c^2 x^4+10 b c^3 x^6-15 c^4 x^8\right )}{1280 b^3 x^{11}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.17, size = 229, normalized size = 1.39 \begin {gather*} \left [\frac {15 \, \sqrt {b} c^{5} 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 \, b c^{4} x^{8} - 10 \, b^{2} c^{3} x^{6} + 8 \, b^{3} c^{2} x^{4} + 176 \, b^{4} c x^{2} + 128 \, b^{5}\right )} \sqrt {c x^{4} + b x^{2}}}{2560 \, b^{4} x^{11}}, -\frac {15 \, \sqrt {-b} c^{5} x^{11} \arctan \left (\frac {\sqrt {c x^{4} + b x^{2}} \sqrt {-b}}{c x^{3} + b x}\right ) + {\left (15 \, b c^{4} x^{8} - 10 \, b^{2} c^{3} x^{6} + 8 \, b^{3} c^{2} x^{4} + 176 \, b^{4} c x^{2} + 128 \, b^{5}\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 = 0.38, size = 138, normalized size = 0.84 \begin {gather*} -\frac {\frac {15 \, c^{6} \arctan \left (\frac {\sqrt {c x^{2} + b}}{\sqrt {-b}}\right ) \mathrm {sgn}\relax (x)}{\sqrt {-b} b^{3}} + \frac {15 \, {\left (c x^{2} + b\right )}^{\frac {9}{2}} c^{6} \mathrm {sgn}\relax (x) - 70 \, {\left (c x^{2} + b\right )}^{\frac {7}{2}} b c^{6} \mathrm {sgn}\relax (x) + 128 \, {\left (c x^{2} + b\right )}^{\frac {5}{2}} b^{2} c^{6} \mathrm {sgn}\relax (x) + 70 \, {\left (c x^{2} + b\right )}^{\frac {3}{2}} b^{3} c^{6} \mathrm {sgn}\relax (x) - 15 \, \sqrt {c x^{2} + b} 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 = 0.03, size = 186, normalized size = 1.13 \begin {gather*} \frac {\left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}} \left (15 b^{\frac {3}{2}} c^{5} x^{10} \ln \left (\frac {2 b +2 \sqrt {c \,x^{2}+b}\, \sqrt {b}}{x}\right )-15 \sqrt {c \,x^{2}+b}\, b \,c^{5} x^{10}-5 \left (c \,x^{2}+b \right )^{\frac {3}{2}} c^{5} x^{10}+5 \left (c \,x^{2}+b \right )^{\frac {5}{2}} c^{4} x^{8}+10 \left (c \,x^{2}+b \right )^{\frac {5}{2}} b \,c^{3} x^{6}-40 \left (c \,x^{2}+b \right )^{\frac {5}{2}} b^{2} c^{2} x^{4}+80 \left (c \,x^{2}+b \right )^{\frac {5}{2}} b^{3} c \,x^{2}-128 \left (c \,x^{2}+b \right )^{\frac {5}{2}} 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}}}{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 = -0.01 \begin {gather*} \int \frac {{\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}}}{x^{14}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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