Optimal. Leaf size=108 \[ -\frac {\left (b x^2+c x^4\right )^{3/2}}{9 b x^{12}}+\frac {2 c \left (b x^2+c x^4\right )^{3/2}}{21 b^2 x^{10}}-\frac {8 c^2 \left (b x^2+c x^4\right )^{3/2}}{105 b^3 x^8}+\frac {16 c^3 \left (b x^2+c x^4\right )^{3/2}}{315 b^4 x^6} \]
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Rubi [A]
time = 0.10, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2041, 2039}
\begin {gather*} \frac {16 c^3 \left (b x^2+c x^4\right )^{3/2}}{315 b^4 x^6}-\frac {8 c^2 \left (b x^2+c x^4\right )^{3/2}}{105 b^3 x^8}+\frac {2 c \left (b x^2+c x^4\right )^{3/2}}{21 b^2 x^{10}}-\frac {\left (b x^2+c x^4\right )^{3/2}}{9 b x^{12}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2039
Rule 2041
Rubi steps
\begin {align*} \int \frac {\sqrt {b x^2+c x^4}}{x^{11}} \, dx &=-\frac {\left (b x^2+c x^4\right )^{3/2}}{9 b x^{12}}-\frac {(2 c) \int \frac {\sqrt {b x^2+c x^4}}{x^9} \, dx}{3 b}\\ &=-\frac {\left (b x^2+c x^4\right )^{3/2}}{9 b x^{12}}+\frac {2 c \left (b x^2+c x^4\right )^{3/2}}{21 b^2 x^{10}}+\frac {\left (8 c^2\right ) \int \frac {\sqrt {b x^2+c x^4}}{x^7} \, dx}{21 b^2}\\ &=-\frac {\left (b x^2+c x^4\right )^{3/2}}{9 b x^{12}}+\frac {2 c \left (b x^2+c x^4\right )^{3/2}}{21 b^2 x^{10}}-\frac {8 c^2 \left (b x^2+c x^4\right )^{3/2}}{105 b^3 x^8}-\frac {\left (16 c^3\right ) \int \frac {\sqrt {b x^2+c x^4}}{x^5} \, dx}{105 b^3}\\ &=-\frac {\left (b x^2+c x^4\right )^{3/2}}{9 b x^{12}}+\frac {2 c \left (b x^2+c x^4\right )^{3/2}}{21 b^2 x^{10}}-\frac {8 c^2 \left (b x^2+c x^4\right )^{3/2}}{105 b^3 x^8}+\frac {16 c^3 \left (b x^2+c x^4\right )^{3/2}}{315 b^4 x^6}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 68, normalized size = 0.63 \begin {gather*} \frac {\sqrt {x^2 \left (b+c x^2\right )} \left (-35 b^4-5 b^3 c x^2+6 b^2 c^2 x^4-8 b c^3 x^6+16 c^4 x^8\right )}{315 b^4 x^{10}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 61, normalized size = 0.56
method | result | size |
gosper | \(-\frac {\left (c \,x^{2}+b \right ) \left (-16 c^{3} x^{6}+24 b \,c^{2} x^{4}-30 b^{2} c \,x^{2}+35 b^{3}\right ) \sqrt {c \,x^{4}+b \,x^{2}}}{315 x^{10} b^{4}}\) | \(61\) |
default | \(-\frac {\left (c \,x^{2}+b \right ) \left (-16 c^{3} x^{6}+24 b \,c^{2} x^{4}-30 b^{2} c \,x^{2}+35 b^{3}\right ) \sqrt {c \,x^{4}+b \,x^{2}}}{315 x^{10} b^{4}}\) | \(61\) |
trager | \(-\frac {\left (-16 c^{4} x^{8}+8 b \,c^{3} x^{6}-6 b^{2} c^{2} x^{4}+5 b^{3} c \,x^{2}+35 b^{4}\right ) \sqrt {c \,x^{4}+b \,x^{2}}}{315 x^{10} b^{4}}\) | \(65\) |
risch | \(-\frac {\sqrt {x^{2} \left (c \,x^{2}+b \right )}\, \left (-16 c^{4} x^{8}+8 b \,c^{3} x^{6}-6 b^{2} c^{2} x^{4}+5 b^{3} c \,x^{2}+35 b^{4}\right )}{315 x^{10} b^{4}}\) | \(65\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 113, normalized size = 1.05 \begin {gather*} \frac {16 \, \sqrt {c x^{4} + b x^{2}} c^{4}}{315 \, b^{4} x^{2}} - \frac {8 \, \sqrt {c x^{4} + b x^{2}} c^{3}}{315 \, b^{3} x^{4}} + \frac {2 \, \sqrt {c x^{4} + b x^{2}} c^{2}}{105 \, b^{2} x^{6}} - \frac {\sqrt {c x^{4} + b x^{2}} c}{63 \, b x^{8}} - \frac {\sqrt {c x^{4} + b x^{2}}}{9 \, x^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 64, normalized size = 0.59 \begin {gather*} \frac {{\left (16 \, c^{4} x^{8} - 8 \, b c^{3} x^{6} + 6 \, b^{2} c^{2} x^{4} - 5 \, b^{3} c x^{2} - 35 \, b^{4}\right )} \sqrt {c x^{4} + b x^{2}}}{315 \, b^{4} x^{10}} \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 {\sqrt {x^{2} \left (b + c x^{2}\right )}}{x^{11}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.32, size = 178, normalized size = 1.65 \begin {gather*} \frac {32 \, {\left (315 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{10} c^{\frac {9}{2}} \mathrm {sgn}\left (x\right ) + 189 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{8} b c^{\frac {9}{2}} \mathrm {sgn}\left (x\right ) + 84 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{6} b^{2} c^{\frac {9}{2}} \mathrm {sgn}\left (x\right ) - 36 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{4} b^{3} c^{\frac {9}{2}} \mathrm {sgn}\left (x\right ) + 9 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} b^{4} c^{\frac {9}{2}} \mathrm {sgn}\left (x\right ) - b^{5} c^{\frac {9}{2}} \mathrm {sgn}\left (x\right )\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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Mupad [B]
time = 4.50, size = 113, normalized size = 1.05 \begin {gather*} \frac {2\,c^2\,\sqrt {c\,x^4+b\,x^2}}{105\,b^2\,x^6}-\frac {c\,\sqrt {c\,x^4+b\,x^2}}{63\,b\,x^8}-\frac {\sqrt {c\,x^4+b\,x^2}}{9\,x^{10}}-\frac {8\,c^3\,\sqrt {c\,x^4+b\,x^2}}{315\,b^3\,x^4}+\frac {16\,c^4\,\sqrt {c\,x^4+b\,x^2}}{315\,b^4\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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