Optimal. Leaf size=46 \[ \frac {\left (c+\frac {d}{x^2}\right )^{3/2} (b c-a d)}{3 d^2}-\frac {b \left (c+\frac {d}{x^2}\right )^{5/2}}{5 d^2} \]
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Rubi [A] time = 0.04, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {444, 43} \[ \frac {\left (c+\frac {d}{x^2}\right )^{3/2} (b c-a d)}{3 d^2}-\frac {b \left (c+\frac {d}{x^2}\right )^{5/2}}{5 d^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 444
Rubi steps
\begin {align*} \int \frac {\left (a+\frac {b}{x^2}\right ) \sqrt {c+\frac {d}{x^2}}}{x^3} \, dx &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int (a+b x) \sqrt {c+d x} \, dx,x,\frac {1}{x^2}\right )\right )\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {(-b c+a d) \sqrt {c+d x}}{d}+\frac {b (c+d x)^{3/2}}{d}\right ) \, dx,x,\frac {1}{x^2}\right )\right )\\ &=\frac {(b c-a d) \left (c+\frac {d}{x^2}\right )^{3/2}}{3 d^2}-\frac {b \left (c+\frac {d}{x^2}\right )^{5/2}}{5 d^2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 47, normalized size = 1.02 \[ -\frac {\sqrt {c+\frac {d}{x^2}} \left (c x^2+d\right ) \left (5 a d x^2-2 b c x^2+3 b d\right )}{15 d^2 x^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 60, normalized size = 1.30 \[ \frac {{\left ({\left (2 \, b c^{2} - 5 \, a c d\right )} x^{4} - 3 \, b d^{2} - {\left (b c d + 5 \, a d^{2}\right )} x^{2}\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{15 \, d^{2} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.92, size = 250, normalized size = 5.43 \[ \frac {2 \, {\left (15 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{8} a c^{\frac {3}{2}} \mathrm {sgn}\relax (x) + 30 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{6} b c^{\frac {5}{2}} \mathrm {sgn}\relax (x) - 30 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{6} a c^{\frac {3}{2}} d \mathrm {sgn}\relax (x) + 10 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{4} b c^{\frac {5}{2}} d \mathrm {sgn}\relax (x) + 20 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{4} a c^{\frac {3}{2}} d^{2} \mathrm {sgn}\relax (x) + 10 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} b c^{\frac {5}{2}} d^{2} \mathrm {sgn}\relax (x) - 10 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} a c^{\frac {3}{2}} d^{3} \mathrm {sgn}\relax (x) - 2 \, b c^{\frac {5}{2}} d^{3} \mathrm {sgn}\relax (x) + 5 \, a c^{\frac {3}{2}} d^{4} \mathrm {sgn}\relax (x)\right )}}{15 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + d}\right )}^{2} - d\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 48, normalized size = 1.04 \[ -\frac {\sqrt {\frac {c \,x^{2}+d}{x^{2}}}\, \left (5 a d \,x^{2}-2 b c \,x^{2}+3 b d \right ) \left (c \,x^{2}+d \right )}{15 d^{2} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.72, size = 49, normalized size = 1.07 \[ -\frac {1}{15} \, b {\left (\frac {3 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {5}{2}}}{d^{2}} - \frac {5 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} c}{d^{2}}\right )} - \frac {a {\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.82, size = 91, normalized size = 1.98 \[ \frac {\sqrt {c+\frac {d}{x^2}}\,\left (b\,c^2+a\,d\,c\right )}{5\,d^2}-\frac {b\,\sqrt {c+\frac {d}{x^2}}}{5\,x^4}-\frac {\sqrt {c+\frac {d}{x^2}}\,\left (5\,a\,d^2+b\,c\,d\right )}{15\,d^2\,x^2}-\frac {c\,\sqrt {c+\frac {d}{x^2}}\,\left (8\,a\,d+b\,c\right )}{15\,d^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.02, size = 58, normalized size = 1.26 \[ - \frac {a \left (\begin {cases} \frac {\sqrt {c}}{x^{2}} & \text {for}\: d = 0 \\\frac {2 \left (c + \frac {d}{x^{2}}\right )^{\frac {3}{2}}}{3 d} & \text {otherwise} \end {cases}\right )}{2} - \frac {b \left (- \frac {c \left (c + \frac {d}{x^{2}}\right )^{\frac {3}{2}}}{3} + \frac {\left (c + \frac {d}{x^{2}}\right )^{\frac {5}{2}}}{5}\right )}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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