Optimal. Leaf size=53 \[ \frac {x^5 \left (c+\frac {d}{x^2}\right )^{5/2} (7 b c-2 a d)}{35 c^2}+\frac {a x^7 \left (c+\frac {d}{x^2}\right )^{5/2}}{7 c} \]
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Rubi [A] time = 0.03, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {453, 264} \begin {gather*} \frac {x^5 \left (c+\frac {d}{x^2}\right )^{5/2} (7 b c-2 a d)}{35 c^2}+\frac {a x^7 \left (c+\frac {d}{x^2}\right )^{5/2}}{7 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 264
Rule 453
Rubi steps
\begin {align*} \int \left (a+\frac {b}{x^2}\right ) \left (c+\frac {d}{x^2}\right )^{3/2} x^6 \, dx &=\frac {a \left (c+\frac {d}{x^2}\right )^{5/2} x^7}{7 c}+\frac {(7 b c-2 a d) \int \left (c+\frac {d}{x^2}\right )^{3/2} x^4 \, dx}{7 c}\\ &=\frac {(7 b c-2 a d) \left (c+\frac {d}{x^2}\right )^{5/2} x^5}{35 c^2}+\frac {a \left (c+\frac {d}{x^2}\right )^{5/2} x^7}{7 c}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 44, normalized size = 0.83 \begin {gather*} \frac {x \sqrt {c+\frac {d}{x^2}} \left (c x^2+d\right )^2 \left (5 a c x^2-2 a d+7 b c\right )}{35 c^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.08, size = 44, normalized size = 0.83 \begin {gather*} \frac {x \sqrt {c+\frac {d}{x^2}} \left (c x^2+d\right )^2 \left (5 a c x^2-2 a d+7 b c\right )}{35 c^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 80, normalized size = 1.51 \begin {gather*} \frac {{\left (5 \, a c^{3} x^{7} + {\left (7 \, b c^{3} + 8 \, a c^{2} d\right )} x^{5} + {\left (14 \, b c^{2} d + a c d^{2}\right )} x^{3} + {\left (7 \, b c d^{2} - 2 \, a d^{3}\right )} x\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{35 \, c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 72, normalized size = 1.36 \begin {gather*} -\frac {{\left (7 \, b c d^{\frac {5}{2}} - 2 \, a d^{\frac {7}{2}}\right )} \mathrm {sgn}\relax (x)}{35 \, c^{2}} + \frac {5 \, {\left (c x^{2} + d\right )}^{\frac {7}{2}} a \mathrm {sgn}\relax (x) + 7 \, {\left (c x^{2} + d\right )}^{\frac {5}{2}} b c \mathrm {sgn}\relax (x) - 7 \, {\left (c x^{2} + d\right )}^{\frac {5}{2}} a d \mathrm {sgn}\relax (x)}{35 \, c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 45, normalized size = 0.85 \begin {gather*} \frac {\left (\frac {c \,x^{2}+d}{x^{2}}\right )^{\frac {3}{2}} \left (5 a \,x^{2} c -2 a d +7 b c \right ) \left (c \,x^{2}+d \right ) x^{3}}{35 c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.65, size = 55, normalized size = 1.04 \begin {gather*} \frac {b {\left (c + \frac {d}{x^{2}}\right )}^{\frac {5}{2}} x^{5}}{5 \, c} + \frac {{\left (5 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {7}{2}} x^{7} - 7 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {5}{2}} d x^{5}\right )} a}{35 \, c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.63, size = 77, normalized size = 1.45 \begin {gather*} \sqrt {c+\frac {d}{x^2}}\,\left (\frac {x^5\,\left (7\,b\,c^3+8\,a\,d\,c^2\right )}{35\,c^2}-\frac {x\,\left (2\,a\,d^3-7\,b\,c\,d^2\right )}{35\,c^2}+\frac {a\,c\,x^7}{7}+\frac {d\,x^3\,\left (a\,d+14\,b\,c\right )}{35\,c}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 5.95, size = 498, normalized size = 9.40 \begin {gather*} \frac {15 a c^{6} 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^{5} 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^{4} 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^{3} 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^{2} 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 c 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 {a d^{\frac {3}{2}} x^{4} \sqrt {\frac {c x^{2}}{d} + 1}}{5} + \frac {a d^{\frac {5}{2}} x^{2} \sqrt {\frac {c x^{2}}{d} + 1}}{15 c} - \frac {2 a d^{\frac {7}{2}} \sqrt {\frac {c x^{2}}{d} + 1}}{15 c^{2}} + \frac {b c \sqrt {d} x^{4} \sqrt {\frac {c x^{2}}{d} + 1}}{5} + \frac {2 b d^{\frac {3}{2}} x^{2} \sqrt {\frac {c x^{2}}{d} + 1}}{5} + \frac {b d^{\frac {5}{2}} \sqrt {\frac {c x^{2}}{d} + 1}}{5 c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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