Optimal. Leaf size=52 \[ \frac {2 c \left (b x^2+c x^4\right )^{5/2}}{35 b^2 x^{10}}-\frac {\left (b x^2+c x^4\right )^{5/2}}{7 b x^{12}} \]
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Rubi [A] time = 0.09, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2016, 2014} \begin {gather*} \frac {2 c \left (b x^2+c x^4\right )^{5/2}}{35 b^2 x^{10}}-\frac {\left (b x^2+c x^4\right )^{5/2}}{7 b x^{12}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2014
Rule 2016
Rubi steps
\begin {align*} \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^{11}} \, dx &=-\frac {\left (b x^2+c x^4\right )^{5/2}}{7 b x^{12}}-\frac {(2 c) \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^9} \, dx}{7 b}\\ &=-\frac {\left (b x^2+c x^4\right )^{5/2}}{7 b x^{12}}+\frac {2 c \left (b x^2+c x^4\right )^{5/2}}{35 b^2 x^{10}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 35, normalized size = 0.67 \begin {gather*} \frac {\left (x^2 \left (b+c x^2\right )\right )^{5/2} \left (2 c x^2-5 b\right )}{35 b^2 x^{12}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.23, size = 57, normalized size = 1.10 \begin {gather*} \frac {\sqrt {b x^2+c x^4} \left (-5 b^3-8 b^2 c x^2-b c^2 x^4+2 c^3 x^6\right )}{35 b^2 x^8} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 53, normalized size = 1.02 \begin {gather*} \frac {{\left (2 \, c^{3} x^{6} - b c^{2} x^{4} - 8 \, b^{2} c x^{2} - 5 \, b^{3}\right )} \sqrt {c x^{4} + b x^{2}}}{35 \, b^{2} x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.25, size = 178, normalized size = 3.42 \begin {gather*} \frac {4 \, {\left (35 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{10} c^{\frac {7}{2}} \mathrm {sgn}\relax (x) + 35 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{8} b c^{\frac {7}{2}} \mathrm {sgn}\relax (x) + 70 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{6} b^{2} c^{\frac {7}{2}} \mathrm {sgn}\relax (x) + 14 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{4} b^{3} c^{\frac {7}{2}} \mathrm {sgn}\relax (x) + 7 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} b^{4} c^{\frac {7}{2}} \mathrm {sgn}\relax (x) - b^{5} c^{\frac {7}{2}} \mathrm {sgn}\relax (x)\right )}}{35 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} - b\right )}^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 39, normalized size = 0.75 \begin {gather*} -\frac {\left (c \,x^{2}+b \right ) \left (-2 c \,x^{2}+5 b \right ) \left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}}}{35 b^{2} x^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.46, size = 105, normalized size = 2.02 \begin {gather*} \frac {2 \, \sqrt {c x^{4} + b x^{2}} c^{3}}{35 \, b^{2} x^{2}} - \frac {\sqrt {c x^{4} + b x^{2}} c^{2}}{35 \, b x^{4}} + \frac {3 \, \sqrt {c x^{4} + b x^{2}} c}{140 \, x^{6}} + \frac {3 \, \sqrt {c x^{4} + b x^{2}} b}{28 \, x^{8}} - \frac {{\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}}}{4 \, x^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.58, size = 87, normalized size = 1.67 \begin {gather*} \frac {2\,c^3\,\sqrt {c\,x^4+b\,x^2}}{35\,b^2\,x^2}-\frac {8\,c\,\sqrt {c\,x^4+b\,x^2}}{35\,x^6}-\frac {c^2\,\sqrt {c\,x^4+b\,x^2}}{35\,b\,x^4}-\frac {b\,\sqrt {c\,x^4+b\,x^2}}{7\,x^8} \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^{11}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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