Optimal. Leaf size=150 \[ -\frac {16 d^3 x^3 \left (c+\frac {d}{x^2}\right )^{3/2} (11 b c-8 a d)}{3465 c^5}+\frac {8 d^2 x^5 \left (c+\frac {d}{x^2}\right )^{3/2} (11 b c-8 a d)}{1155 c^4}-\frac {2 d x^7 \left (c+\frac {d}{x^2}\right )^{3/2} (11 b c-8 a d)}{231 c^3}+\frac {x^9 \left (c+\frac {d}{x^2}\right )^{3/2} (11 b c-8 a d)}{99 c^2}+\frac {a x^{11} \left (c+\frac {d}{x^2}\right )^{3/2}}{11 c} \]
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Rubi [A] time = 0.07, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {453, 271, 264} \[ \frac {8 d^2 x^5 \left (c+\frac {d}{x^2}\right )^{3/2} (11 b c-8 a d)}{1155 c^4}-\frac {16 d^3 x^3 \left (c+\frac {d}{x^2}\right )^{3/2} (11 b c-8 a d)}{3465 c^5}+\frac {x^9 \left (c+\frac {d}{x^2}\right )^{3/2} (11 b c-8 a d)}{99 c^2}-\frac {2 d x^7 \left (c+\frac {d}{x^2}\right )^{3/2} (11 b c-8 a d)}{231 c^3}+\frac {a x^{11} \left (c+\frac {d}{x^2}\right )^{3/2}}{11 c} \]
Antiderivative was successfully verified.
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Rule 264
Rule 271
Rule 453
Rubi steps
\begin {align*} \int \left (a+\frac {b}{x^2}\right ) \sqrt {c+\frac {d}{x^2}} x^{10} \, dx &=\frac {a \left (c+\frac {d}{x^2}\right )^{3/2} x^{11}}{11 c}+\frac {(11 b c-8 a d) \int \sqrt {c+\frac {d}{x^2}} x^8 \, dx}{11 c}\\ &=\frac {(11 b c-8 a d) \left (c+\frac {d}{x^2}\right )^{3/2} x^9}{99 c^2}+\frac {a \left (c+\frac {d}{x^2}\right )^{3/2} x^{11}}{11 c}-\frac {(2 d (11 b c-8 a d)) \int \sqrt {c+\frac {d}{x^2}} x^6 \, dx}{33 c^2}\\ &=-\frac {2 d (11 b c-8 a d) \left (c+\frac {d}{x^2}\right )^{3/2} x^7}{231 c^3}+\frac {(11 b c-8 a d) \left (c+\frac {d}{x^2}\right )^{3/2} x^9}{99 c^2}+\frac {a \left (c+\frac {d}{x^2}\right )^{3/2} x^{11}}{11 c}+\frac {\left (8 d^2 (11 b c-8 a d)\right ) \int \sqrt {c+\frac {d}{x^2}} x^4 \, dx}{231 c^3}\\ &=\frac {8 d^2 (11 b c-8 a d) \left (c+\frac {d}{x^2}\right )^{3/2} x^5}{1155 c^4}-\frac {2 d (11 b c-8 a d) \left (c+\frac {d}{x^2}\right )^{3/2} x^7}{231 c^3}+\frac {(11 b c-8 a d) \left (c+\frac {d}{x^2}\right )^{3/2} x^9}{99 c^2}+\frac {a \left (c+\frac {d}{x^2}\right )^{3/2} x^{11}}{11 c}-\frac {\left (16 d^3 (11 b c-8 a d)\right ) \int \sqrt {c+\frac {d}{x^2}} x^2 \, dx}{1155 c^4}\\ &=-\frac {16 d^3 (11 b c-8 a d) \left (c+\frac {d}{x^2}\right )^{3/2} x^3}{3465 c^5}+\frac {8 d^2 (11 b c-8 a d) \left (c+\frac {d}{x^2}\right )^{3/2} x^5}{1155 c^4}-\frac {2 d (11 b c-8 a d) \left (c+\frac {d}{x^2}\right )^{3/2} x^7}{231 c^3}+\frac {(11 b c-8 a d) \left (c+\frac {d}{x^2}\right )^{3/2} x^9}{99 c^2}+\frac {a \left (c+\frac {d}{x^2}\right )^{3/2} x^{11}}{11 c}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 108, normalized size = 0.72 \[ \frac {x \sqrt {c+\frac {d}{x^2}} \left (c x^2+d\right ) \left (a \left (315 c^4 x^8-280 c^3 d x^6+240 c^2 d^2 x^4-192 c d^3 x^2+128 d^4\right )+11 b c \left (35 c^3 x^6-30 c^2 d x^4+24 c d^2 x^2-16 d^3\right )\right )}{3465 c^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 131, normalized size = 0.87 \[ \frac {{\left (315 \, a c^{5} x^{11} + 35 \, {\left (11 \, b c^{5} + a c^{4} d\right )} x^{9} + 5 \, {\left (11 \, b c^{4} d - 8 \, a c^{3} d^{2}\right )} x^{7} - 6 \, {\left (11 \, b c^{3} d^{2} - 8 \, a c^{2} d^{3}\right )} x^{5} + 8 \, {\left (11 \, b c^{2} d^{3} - 8 \, a c d^{4}\right )} x^{3} - 16 \, {\left (11 \, b c d^{4} - 8 \, a d^{5}\right )} x\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{3465 \, c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 175, normalized size = 1.17 \[ \frac {16 \, {\left (11 \, b c d^{\frac {9}{2}} - 8 \, a d^{\frac {11}{2}}\right )} \mathrm {sgn}\relax (x)}{3465 \, c^{5}} + \frac {315 \, {\left (c x^{2} + d\right )}^{\frac {11}{2}} a \mathrm {sgn}\relax (x) + 385 \, {\left (c x^{2} + d\right )}^{\frac {9}{2}} b c \mathrm {sgn}\relax (x) - 1540 \, {\left (c x^{2} + d\right )}^{\frac {9}{2}} a d \mathrm {sgn}\relax (x) - 1485 \, {\left (c x^{2} + d\right )}^{\frac {7}{2}} b c d \mathrm {sgn}\relax (x) + 2970 \, {\left (c x^{2} + d\right )}^{\frac {7}{2}} a d^{2} \mathrm {sgn}\relax (x) + 2079 \, {\left (c x^{2} + d\right )}^{\frac {5}{2}} b c d^{2} \mathrm {sgn}\relax (x) - 2772 \, {\left (c x^{2} + d\right )}^{\frac {5}{2}} a d^{3} \mathrm {sgn}\relax (x) - 1155 \, {\left (c x^{2} + d\right )}^{\frac {3}{2}} b c d^{3} \mathrm {sgn}\relax (x) + 1155 \, {\left (c x^{2} + d\right )}^{\frac {3}{2}} a d^{4} \mathrm {sgn}\relax (x)}{3465 \, c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 113, normalized size = 0.75 \[ \frac {\sqrt {\frac {c \,x^{2}+d}{x^{2}}}\, \left (315 a \,x^{8} c^{4}-280 a \,c^{3} d \,x^{6}+385 b \,c^{4} x^{6}+240 a \,c^{2} d^{2} x^{4}-330 b \,c^{3} d \,x^{4}-192 a c \,d^{3} x^{2}+264 b \,c^{2} d^{2} x^{2}+128 a \,d^{4}-176 b c \,d^{3}\right ) \left (c \,x^{2}+d \right ) x}{3465 c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.64, size = 158, normalized size = 1.05 \[ \frac {{\left (35 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {9}{2}} x^{9} - 135 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {7}{2}} d x^{7} + 189 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {5}{2}} d^{2} x^{5} - 105 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} d^{3} x^{3}\right )} b}{315 \, c^{4}} + \frac {{\left (315 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {11}{2}} x^{11} - 1540 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {9}{2}} d x^{9} + 2970 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {7}{2}} d^{2} x^{7} - 2772 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {5}{2}} d^{3} x^{5} + 1155 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} d^{4} x^{3}\right )} a}{3465 \, c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.59, size = 117, normalized size = 0.78 \[ \sqrt {c+\frac {d}{x^2}}\,\left (\frac {a\,x^{11}}{11}+\frac {x\,\left (128\,a\,d^5-176\,b\,c\,d^4\right )}{3465\,c^5}+\frac {x^9\,\left (385\,b\,c^5+35\,a\,d\,c^4\right )}{3465\,c^5}-\frac {d\,x^7\,\left (8\,a\,d-11\,b\,c\right )}{693\,c^2}+\frac {2\,d^2\,x^5\,\left (8\,a\,d-11\,b\,c\right )}{1155\,c^3}-\frac {8\,d^3\,x^3\,\left (8\,a\,d-11\,b\,c\right )}{3465\,c^4}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 6.44, size = 1386, normalized size = 9.24 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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