Optimal. Leaf size=68 \[ \frac {\sqrt {c+\frac {d}{x^2}} (2 b c-a d)}{d^3}+\frac {c (b c-a d)}{d^3 \sqrt {c+\frac {d}{x^2}}}-\frac {b \left (c+\frac {d}{x^2}\right )^{3/2}}{3 d^3} \]
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Rubi [A] time = 0.05, antiderivative size = 68, 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 = {446, 77} \begin {gather*} \frac {\sqrt {c+\frac {d}{x^2}} (2 b c-a d)}{d^3}+\frac {c (b c-a d)}{d^3 \sqrt {c+\frac {d}{x^2}}}-\frac {b \left (c+\frac {d}{x^2}\right )^{3/2}}{3 d^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rule 446
Rubi steps
\begin {align*} \int \frac {a+\frac {b}{x^2}}{\left (c+\frac {d}{x^2}\right )^{3/2} x^5} \, dx &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {x (a+b x)}{(c+d x)^{3/2}} \, dx,x,\frac {1}{x^2}\right )\right )\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {c (b c-a d)}{d^2 (c+d x)^{3/2}}+\frac {-2 b c+a d}{d^2 \sqrt {c+d x}}+\frac {b \sqrt {c+d x}}{d^2}\right ) \, dx,x,\frac {1}{x^2}\right )\right )\\ &=\frac {c (b c-a d)}{d^3 \sqrt {c+\frac {d}{x^2}}}+\frac {(2 b c-a d) \sqrt {c+\frac {d}{x^2}}}{d^3}-\frac {b \left (c+\frac {d}{x^2}\right )^{3/2}}{3 d^3}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 60, normalized size = 0.88 \begin {gather*} \frac {b \left (8 c^2 x^4+4 c d x^2-d^2\right )-3 a d x^2 \left (2 c x^2+d\right )}{3 d^3 x^4 \sqrt {c+\frac {d}{x^2}}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.08, size = 75, normalized size = 1.10 \begin {gather*} \frac {\sqrt {\frac {c x^2+d}{x^2}} \left (-6 a c d x^4-3 a d^2 x^2+8 b c^2 x^4+4 b c d x^2-b d^2\right )}{3 d^3 x^2 \left (c x^2+d\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 73, normalized size = 1.07 \begin {gather*} \frac {{\left (2 \, {\left (4 \, b c^{2} - 3 \, a c d\right )} x^{4} - b d^{2} + {\left (4 \, b c d - 3 \, a d^{2}\right )} x^{2}\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{3 \, {\left (c d^{3} x^{4} + d^{4} x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a + \frac {b}{x^{2}}}{{\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} x^{5}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 69, normalized size = 1.01 \begin {gather*} -\frac {\left (6 a c d \,x^{4}-8 b \,c^{2} x^{4}+3 a \,d^{2} x^{2}-4 b c d \,x^{2}+b \,d^{2}\right ) \left (c \,x^{2}+d \right )}{3 \left (\frac {c \,x^{2}+d}{x^{2}}\right )^{\frac {3}{2}} d^{3} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.61, size = 81, normalized size = 1.19 \begin {gather*} -\frac {1}{3} \, b {\left (\frac {{\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}}}{d^{3}} - \frac {6 \, \sqrt {c + \frac {d}{x^{2}}} c}{d^{3}} - \frac {3 \, c^{2}}{\sqrt {c + \frac {d}{x^{2}}} d^{3}}\right )} - a {\left (\frac {\sqrt {c + \frac {d}{x^{2}}}}{d^{2}} + \frac {c}{\sqrt {c + \frac {d}{x^{2}}} d^{2}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.64, size = 66, normalized size = 0.97 \begin {gather*} -\frac {\sqrt {c+\frac {d}{x^2}}\,\left (-8\,b\,c^2\,x^4+6\,a\,c\,d\,x^4-4\,b\,c\,d\,x^2+3\,a\,d^2\,x^2+b\,d^2\right )}{3\,d^3\,x^2\,\left (c\,x^2+d\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 11.07, size = 61, normalized size = 0.90 \begin {gather*} - \frac {b \left (c + \frac {d}{x^{2}}\right )^{\frac {3}{2}}}{3 d^{3}} - \frac {c \left (a d - b c\right )}{d^{3} \sqrt {c + \frac {d}{x^{2}}}} - \frac {\sqrt {c + \frac {d}{x^{2}}} \left (a d - 2 b c\right )}{d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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