Optimal. Leaf size=72 \[ \frac {\left (c+\frac {d}{x^2}\right )^{3/2} (2 b c-a d)}{3 d^3}-\frac {c \sqrt {c+\frac {d}{x^2}} (b c-a d)}{d^3}-\frac {b \left (c+\frac {d}{x^2}\right )^{5/2}}{5 d^3} \]
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Rubi [A] time = 0.05, antiderivative size = 72, 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 {\left (c+\frac {d}{x^2}\right )^{3/2} (2 b c-a d)}{3 d^3}-\frac {c \sqrt {c+\frac {d}{x^2}} (b c-a d)}{d^3}-\frac {b \left (c+\frac {d}{x^2}\right )^{5/2}}{5 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}}{\sqrt {c+\frac {d}{x^2}} x^5} \, dx &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {x (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 {c (b c-a d)}{d^2 \sqrt {c+d x}}+\frac {(-2 b c+a d) \sqrt {c+d x}}{d^2}+\frac {b (c+d x)^{3/2}}{d^2}\right ) \, dx,x,\frac {1}{x^2}\right )\right )\\ &=-\frac {c (b c-a d) \sqrt {c+\frac {d}{x^2}}}{d^3}+\frac {(2 b c-a d) \left (c+\frac {d}{x^2}\right )^{3/2}}{3 d^3}-\frac {b \left (c+\frac {d}{x^2}\right )^{5/2}}{5 d^3}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 60, normalized size = 0.83 \begin {gather*} \frac {\sqrt {c+\frac {d}{x^2}} \left (b \left (-8 c^2 x^4+4 c d x^2-3 d^2\right )-5 a d x^2 \left (d-2 c x^2\right )\right )}{15 d^3 x^4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.07, size = 66, normalized size = 0.92 \begin {gather*} \frac {\sqrt {\frac {c x^2+d}{x^2}} \left (10 a c d x^4-5 a d^2 x^2-8 b c^2 x^4+4 b c d x^2-3 b d^2\right )}{15 d^3 x^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 62, normalized size = 0.86 \begin {gather*} -\frac {{\left (2 \, {\left (4 \, b c^{2} - 5 \, a c d\right )} x^{4} + 3 \, b d^{2} - {\left (4 \, b c d - 5 \, a d^{2}\right )} x^{2}\right )} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{15 \, d^{3} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.50, size = 153, normalized size = 2.12 \begin {gather*} \frac {15 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + d x^{2}}\right )}^{3} a \sqrt {c} + 20 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + d x^{2}}\right )}^{2} b c + 5 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + d x^{2}}\right )}^{2} a d + 15 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + d x^{2}}\right )} b \sqrt {c} d + 3 \, b d^{2}}{15 \, {\left (\sqrt {c} x^{2} - \sqrt {c x^{4} + d x^{2}}\right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 70, normalized size = 0.97 \begin {gather*} \frac {\left (10 a c d \,x^{4}-8 b \,c^{2} x^{4}-5 a \,d^{2} x^{2}+4 b c d \,x^{2}-3 b \,d^{2}\right ) \left (c \,x^{2}+d \right )}{15 \sqrt {\frac {c \,x^{2}+d}{x^{2}}}\, d^{3} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 83, normalized size = 1.15 \begin {gather*} -\frac {1}{15} \, b {\left (\frac {3 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {5}{2}}}{d^{3}} - \frac {10 \, {\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}} c}{d^{3}} + \frac {15 \, \sqrt {c + \frac {d}{x^{2}}} c^{2}}{d^{3}}\right )} - \frac {1}{3} \, a {\left (\frac {{\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}}}{d^{2}} - \frac {3 \, \sqrt {c + \frac {d}{x^{2}}} c}{d^{2}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.68, size = 58, normalized size = 0.81 \begin {gather*} -\frac {\sqrt {c+\frac {d}{x^2}}\,\left (8\,b\,c^2\,x^4-10\,a\,c\,d\,x^4-4\,b\,c\,d\,x^2+5\,a\,d^2\,x^2+3\,b\,d^2\right )}{15\,d^3\,x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 13.17, size = 204, normalized size = 2.83 \begin {gather*} \frac {\begin {cases} \frac {- \frac {a}{2 x^{4}} - \frac {b}{3 x^{6}}}{\sqrt {c}} & \text {for}\: d = 0 \\\frac {\frac {2 a c \left (- \frac {c}{\sqrt {c + \frac {d}{x^{2}}}} - \sqrt {c + \frac {d}{x^{2}}}\right )}{d} + \frac {2 a \left (\frac {c^{2}}{\sqrt {c + \frac {d}{x^{2}}}} + 2 c \sqrt {c + \frac {d}{x^{2}}} - \frac {\left (c + \frac {d}{x^{2}}\right )^{\frac {3}{2}}}{3}\right )}{d} + \frac {2 b c \left (\frac {c^{2}}{\sqrt {c + \frac {d}{x^{2}}}} + 2 c \sqrt {c + \frac {d}{x^{2}}} - \frac {\left (c + \frac {d}{x^{2}}\right )^{\frac {3}{2}}}{3}\right )}{d^{2}} + \frac {2 b \left (- \frac {c^{3}}{\sqrt {c + \frac {d}{x^{2}}}} - 3 c^{2} \sqrt {c + \frac {d}{x^{2}}} + c \left (c + \frac {d}{x^{2}}\right )^{\frac {3}{2}} - \frac {\left (c + \frac {d}{x^{2}}\right )^{\frac {5}{2}}}{5}\right )}{d^{2}}}{d} & \text {otherwise} \end {cases}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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