Optimal. Leaf size=53 \[ \frac {x^3 \left (c+\frac {d}{x^2}\right )^{3/2} (5 b c-2 a d)}{15 c^2}+\frac {a x^5 \left (c+\frac {d}{x^2}\right )^{3/2}}{5 c} \]
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Rubi [A] time = 0.02, 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} \[ \frac {x^3 \left (c+\frac {d}{x^2}\right )^{3/2} (5 b c-2 a d)}{15 c^2}+\frac {a x^5 \left (c+\frac {d}{x^2}\right )^{3/2}}{5 c} \]
Antiderivative was successfully verified.
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Rule 264
Rule 453
Rubi steps
\begin {align*} \int \left (a+\frac {b}{x^2}\right ) \sqrt {c+\frac {d}{x^2}} x^4 \, dx &=\frac {a \left (c+\frac {d}{x^2}\right )^{3/2} x^5}{5 c}+\frac {(5 b c-2 a d) \int \sqrt {c+\frac {d}{x^2}} x^2 \, dx}{5 c}\\ &=\frac {(5 b c-2 a d) \left (c+\frac {d}{x^2}\right )^{3/2} x^3}{15 c^2}+\frac {a \left (c+\frac {d}{x^2}\right )^{3/2} x^5}{5 c}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 42, normalized size = 0.79 \[ \frac {x \sqrt {c+\frac {d}{x^2}} \left (c x^2+d\right ) \left (3 a c x^2-2 a d+5 b c\right )}{15 c^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 57, normalized size = 1.08 \[ \frac {{\left (3 \, a c^{2} x^{5} + {\left (5 \, b c^{2} + a c d\right )} x^{3} + {\left (5 \, b c d - 2 \, a d^{2}\right )} x\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{15 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 72, normalized size = 1.36 \[ -\frac {{\left (5 \, b c d^{\frac {3}{2}} - 2 \, a d^{\frac {5}{2}}\right )} \mathrm {sgn}\relax (x)}{15 \, c^{2}} + \frac {3 \, {\left (c x^{2} + d\right )}^{\frac {5}{2}} a \mathrm {sgn}\relax (x) + 5 \, {\left (c x^{2} + d\right )}^{\frac {3}{2}} b c \mathrm {sgn}\relax (x) - 5 \, {\left (c x^{2} + d\right )}^{\frac {3}{2}} a d \mathrm {sgn}\relax (x)}{15 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 43, normalized size = 0.81 \[ \frac {\sqrt {\frac {c \,x^{2}+d}{x^{2}}}\, \left (3 a \,x^{2} c -2 a d +5 b c \right ) \left (c \,x^{2}+d \right ) x}{15 c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.66, size = 55, normalized size = 1.04 \[ \frac {b {\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} x^{3}}{3 \, c} + \frac {{\left (3 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {5}{2}} x^{5} - 5 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} d x^{3}\right )} a}{15 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.44, size = 54, normalized size = 1.02 \[ \sqrt {c+\frac {d}{x^2}}\,\left (\frac {a\,x^5}{5}-\frac {x\,\left (2\,a\,d^2-5\,b\,c\,d\right )}{15\,c^2}+\frac {x^3\,\left (5\,b\,c^2+a\,d\,c\right )}{15\,c^2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 3.04, size = 119, normalized size = 2.25 \[ \frac {a \sqrt {d} x^{4} \sqrt {\frac {c x^{2}}{d} + 1}}{5} + \frac {a d^{\frac {3}{2}} x^{2} \sqrt {\frac {c x^{2}}{d} + 1}}{15 c} - \frac {2 a d^{\frac {5}{2}} \sqrt {\frac {c x^{2}}{d} + 1}}{15 c^{2}} + \frac {b \sqrt {d} x^{2} \sqrt {\frac {c x^{2}}{d} + 1}}{3} + \frac {b d^{\frac {3}{2}} \sqrt {\frac {c x^{2}}{d} + 1}}{3 c} \]
Verification of antiderivative is not currently implemented for this CAS.
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