Optimal. Leaf size=84 \[ -\frac {2 d x^3 \left (c+\frac {d}{x^2}\right )^{3/2} (7 b c-4 a d)}{105 c^3}+\frac {x^5 \left (c+\frac {d}{x^2}\right )^{3/2} (7 b c-4 a d)}{35 c^2}+\frac {a x^7 \left (c+\frac {d}{x^2}\right )^{3/2}}{7 c} \]
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Rubi [A] time = 0.04, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {453, 271, 264} \[ \frac {x^5 \left (c+\frac {d}{x^2}\right )^{3/2} (7 b c-4 a d)}{35 c^2}-\frac {2 d x^3 \left (c+\frac {d}{x^2}\right )^{3/2} (7 b c-4 a d)}{105 c^3}+\frac {a x^7 \left (c+\frac {d}{x^2}\right )^{3/2}}{7 c} \]
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^6 \, dx &=\frac {a \left (c+\frac {d}{x^2}\right )^{3/2} x^7}{7 c}+\frac {(7 b c-4 a d) \int \sqrt {c+\frac {d}{x^2}} x^4 \, dx}{7 c}\\ &=\frac {(7 b c-4 a d) \left (c+\frac {d}{x^2}\right )^{3/2} x^5}{35 c^2}+\frac {a \left (c+\frac {d}{x^2}\right )^{3/2} x^7}{7 c}-\frac {(2 d (7 b c-4 a d)) \int \sqrt {c+\frac {d}{x^2}} x^2 \, dx}{35 c^2}\\ &=-\frac {2 d (7 b c-4 a d) \left (c+\frac {d}{x^2}\right )^{3/2} x^3}{105 c^3}+\frac {(7 b c-4 a d) \left (c+\frac {d}{x^2}\right )^{3/2} x^5}{35 c^2}+\frac {a \left (c+\frac {d}{x^2}\right )^{3/2} x^7}{7 c}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 64, normalized size = 0.76 \[ \frac {x \sqrt {c+\frac {d}{x^2}} \left (c x^2+d\right ) \left (a \left (15 c^2 x^4-12 c d x^2+8 d^2\right )+7 b c \left (3 c x^2-2 d\right )\right )}{105 c^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 82, normalized size = 0.98 \[ \frac {{\left (15 \, a c^{3} x^{7} + 3 \, {\left (7 \, b c^{3} + a c^{2} d\right )} x^{5} + {\left (7 \, b c^{2} d - 4 \, a c d^{2}\right )} x^{3} - 2 \, {\left (7 \, b c d^{2} - 4 \, a d^{3}\right )} x\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{105 \, c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 105, normalized size = 1.25 \[ \frac {2 \, {\left (7 \, b c d^{\frac {5}{2}} - 4 \, a d^{\frac {7}{2}}\right )} \mathrm {sgn}\relax (x)}{105 \, c^{3}} + \frac {15 \, {\left (c x^{2} + d\right )}^{\frac {7}{2}} a \mathrm {sgn}\relax (x) + 21 \, {\left (c x^{2} + d\right )}^{\frac {5}{2}} b c \mathrm {sgn}\relax (x) - 42 \, {\left (c x^{2} + d\right )}^{\frac {5}{2}} a d \mathrm {sgn}\relax (x) - 35 \, {\left (c x^{2} + d\right )}^{\frac {3}{2}} b c d \mathrm {sgn}\relax (x) + 35 \, {\left (c x^{2} + d\right )}^{\frac {3}{2}} a d^{2} \mathrm {sgn}\relax (x)}{105 \, c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 65, normalized size = 0.77 \[ \frac {\sqrt {\frac {c \,x^{2}+d}{x^{2}}}\, \left (15 a \,x^{4} c^{2}-12 a c d \,x^{2}+21 b \,c^{2} x^{2}+8 a \,d^{2}-14 b c d \right ) \left (c \,x^{2}+d \right ) x}{105 c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 90, normalized size = 1.07 \[ \frac {{\left (3 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {5}{2}} x^{5} - 5 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} d x^{3}\right )} b}{15 \, c^{2}} + \frac {{\left (15 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {7}{2}} x^{7} - 42 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {5}{2}} d x^{5} + 35 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} d^{2} x^{3}\right )} a}{105 \, c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.49, size = 77, normalized size = 0.92 \[ \sqrt {c+\frac {d}{x^2}}\,\left (\frac {a\,x^7}{7}+\frac {x\,\left (8\,a\,d^3-14\,b\,c\,d^2\right )}{105\,c^3}+\frac {x^5\,\left (21\,b\,c^3+3\,a\,d\,c^2\right )}{105\,c^3}-\frac {d\,x^3\,\left (4\,a\,d-7\,b\,c\right )}{105\,c^2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 3.99, size = 422, normalized size = 5.02 \[ \frac {15 a c^{5} d^{\frac {9}{2}} x^{10} \sqrt {\frac {c x^{2}}{d} + 1}}{105 c^{5} d^{4} x^{4} + 210 c^{4} d^{5} x^{2} + 105 c^{3} d^{6}} + \frac {33 a c^{4} d^{\frac {11}{2}} x^{8} \sqrt {\frac {c x^{2}}{d} + 1}}{105 c^{5} d^{4} x^{4} + 210 c^{4} d^{5} x^{2} + 105 c^{3} d^{6}} + \frac {17 a c^{3} d^{\frac {13}{2}} x^{6} \sqrt {\frac {c x^{2}}{d} + 1}}{105 c^{5} d^{4} x^{4} + 210 c^{4} d^{5} x^{2} + 105 c^{3} d^{6}} + \frac {3 a c^{2} d^{\frac {15}{2}} x^{4} \sqrt {\frac {c x^{2}}{d} + 1}}{105 c^{5} d^{4} x^{4} + 210 c^{4} d^{5} x^{2} + 105 c^{3} d^{6}} + \frac {12 a c d^{\frac {17}{2}} x^{2} \sqrt {\frac {c x^{2}}{d} + 1}}{105 c^{5} d^{4} x^{4} + 210 c^{4} d^{5} x^{2} + 105 c^{3} d^{6}} + \frac {8 a d^{\frac {19}{2}} \sqrt {\frac {c x^{2}}{d} + 1}}{105 c^{5} d^{4} x^{4} + 210 c^{4} d^{5} x^{2} + 105 c^{3} d^{6}} + \frac {b \sqrt {d} x^{4} \sqrt {\frac {c x^{2}}{d} + 1}}{5} + \frac {b d^{\frac {3}{2}} x^{2} \sqrt {\frac {c x^{2}}{d} + 1}}{15 c} - \frac {2 b d^{\frac {5}{2}} \sqrt {\frac {c x^{2}}{d} + 1}}{15 c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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