Optimal. Leaf size=80 \[ -\frac {8 c^2 \left (b x^2+c x^4\right )^{5/2}}{315 b^3 x^{10}}+\frac {4 c \left (b x^2+c x^4\right )^{5/2}}{63 b^2 x^{12}}-\frac {\left (b x^2+c x^4\right )^{5/2}}{9 b x^{14}} \]
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Rubi [A] time = 0.14, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 3, 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 {8 c^2 \left (b x^2+c x^4\right )^{5/2}}{315 b^3 x^{10}}+\frac {4 c \left (b x^2+c x^4\right )^{5/2}}{63 b^2 x^{12}}-\frac {\left (b x^2+c x^4\right )^{5/2}}{9 b x^{14}} \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^{13}} \, dx &=-\frac {\left (b x^2+c x^4\right )^{5/2}}{9 b x^{14}}-\frac {(4 c) \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^{11}} \, dx}{9 b}\\ &=-\frac {\left (b x^2+c x^4\right )^{5/2}}{9 b x^{14}}+\frac {4 c \left (b x^2+c x^4\right )^{5/2}}{63 b^2 x^{12}}+\frac {\left (8 c^2\right ) \int \frac {\left (b x^2+c x^4\right )^{3/2}}{x^9} \, dx}{63 b^2}\\ &=-\frac {\left (b x^2+c x^4\right )^{5/2}}{9 b x^{14}}+\frac {4 c \left (b x^2+c x^4\right )^{5/2}}{63 b^2 x^{12}}-\frac {8 c^2 \left (b x^2+c x^4\right )^{5/2}}{315 b^3 x^{10}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 46, normalized size = 0.58 \begin {gather*} -\frac {\left (x^2 \left (b+c x^2\right )\right )^{5/2} \left (35 b^2-20 b c x^2+8 c^2 x^4\right )}{315 b^3 x^{14}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.25, size = 68, normalized size = 0.85 \begin {gather*} \frac {\sqrt {b x^2+c x^4} \left (-35 b^4-50 b^3 c x^2-3 b^2 c^2 x^4+4 b c^3 x^6-8 c^4 x^8\right )}{315 b^3 x^{10}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.12, size = 64, normalized size = 0.80 \begin {gather*} -\frac {{\left (8 \, c^{4} x^{8} - 4 \, b c^{3} x^{6} + 3 \, b^{2} c^{2} x^{4} + 50 \, b^{3} c x^{2} + 35 \, b^{4}\right )} \sqrt {c x^{4} + b x^{2}}}{315 \, b^{3} x^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.27, size = 206, normalized size = 2.58 \begin {gather*} \frac {16 \, {\left (210 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{12} c^{\frac {9}{2}} \mathrm {sgn}\relax (x) + 315 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{10} b c^{\frac {9}{2}} \mathrm {sgn}\relax (x) + 441 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{8} b^{2} c^{\frac {9}{2}} \mathrm {sgn}\relax (x) + 126 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{6} b^{3} c^{\frac {9}{2}} \mathrm {sgn}\relax (x) + 36 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{4} b^{4} c^{\frac {9}{2}} \mathrm {sgn}\relax (x) - 9 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} b^{5} c^{\frac {9}{2}} \mathrm {sgn}\relax (x) + b^{6} c^{\frac {9}{2}} \mathrm {sgn}\relax (x)\right )}}{315 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} - b\right )}^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 50, normalized size = 0.62 \begin {gather*} -\frac {\left (c \,x^{2}+b \right ) \left (8 c^{2} x^{4}-20 b c \,x^{2}+35 b^{2}\right ) \left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}}}{315 b^{3} x^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.52, size = 129, normalized size = 1.61 \begin {gather*} -\frac {8 \, \sqrt {c x^{4} + b x^{2}} c^{4}}{315 \, b^{3} x^{2}} + \frac {4 \, \sqrt {c x^{4} + b x^{2}} c^{3}}{315 \, b^{2} x^{4}} - \frac {\sqrt {c x^{4} + b x^{2}} c^{2}}{105 \, b x^{6}} + \frac {\sqrt {c x^{4} + b x^{2}} c}{126 \, x^{8}} + \frac {\sqrt {c x^{4} + b x^{2}} b}{18 \, x^{10}} - \frac {{\left (c x^{4} + b x^{2}\right )}^{\frac {3}{2}}}{6 \, x^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.72, size = 111, normalized size = 1.39 \begin {gather*} \frac {4\,c^3\,\sqrt {c\,x^4+b\,x^2}}{315\,b^2\,x^4}-\frac {10\,c\,\sqrt {c\,x^4+b\,x^2}}{63\,x^8}-\frac {c^2\,\sqrt {c\,x^4+b\,x^2}}{105\,b\,x^6}-\frac {b\,\sqrt {c\,x^4+b\,x^2}}{9\,x^{10}}-\frac {8\,c^4\,\sqrt {c\,x^4+b\,x^2}}{315\,b^3\,x^2} \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^{13}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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