Optimal. Leaf size=56 \[ \frac {4 a \left (a+b \sqrt {\frac {c}{x}}\right )^{3/2}}{3 b^2 c}-\frac {4 \left (a+b \sqrt {\frac {c}{x}}\right )^{5/2}}{5 b^2 c} \]
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Rubi [A] time = 0.04, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {369, 266, 43} \begin {gather*} \frac {4 a \left (a+b \sqrt {\frac {c}{x}}\right )^{3/2}}{3 b^2 c}-\frac {4 \left (a+b \sqrt {\frac {c}{x}}\right )^{5/2}}{5 b^2 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 369
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b \sqrt {\frac {c}{x}}}}{x^2} \, dx &=\operatorname {Subst}\left (\int \frac {\sqrt {a+\frac {b \sqrt {c}}{\sqrt {x}}}}{x^2} \, dx,\sqrt {x},\frac {\sqrt {\frac {c}{x}} x}{\sqrt {c}}\right )\\ &=-\operatorname {Subst}\left (2 \operatorname {Subst}\left (\int x \sqrt {a+b \sqrt {c} x} \, dx,x,\frac {1}{\sqrt {x}}\right ),\sqrt {x},\frac {\sqrt {\frac {c}{x}} x}{\sqrt {c}}\right )\\ &=-\operatorname {Subst}\left (2 \operatorname {Subst}\left (\int \left (-\frac {a \sqrt {a+b \sqrt {c} x}}{b \sqrt {c}}+\frac {\left (a+b \sqrt {c} x\right )^{3/2}}{b \sqrt {c}}\right ) \, dx,x,\frac {1}{\sqrt {x}}\right ),\sqrt {x},\frac {\sqrt {\frac {c}{x}} x}{\sqrt {c}}\right )\\ &=\frac {4 a \left (a+b \sqrt {\frac {c}{x}}\right )^{3/2}}{3 b^2 c}-\frac {4 \left (a+b \sqrt {\frac {c}{x}}\right )^{5/2}}{5 b^2 c}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 43, normalized size = 0.77 \begin {gather*} \frac {4 \left (2 a-3 b \sqrt {\frac {c}{x}}\right ) \left (a+b \sqrt {\frac {c}{x}}\right )^{3/2}}{15 b^2 c} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.03, size = 55, normalized size = 0.98 \begin {gather*} \frac {4 \sqrt {a+b \sqrt {\frac {c}{x}}} \left (2 a^2-a b \sqrt {\frac {c}{x}}-\frac {3 b^2 c}{x}\right )}{15 b^2 c} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.29, size = 48, normalized size = 0.86 \begin {gather*} -\frac {4 \, {\left (a b x \sqrt {\frac {c}{x}} + 3 \, b^{2} c - 2 \, a^{2} x\right )} \sqrt {b \sqrt {\frac {c}{x}} + a}}{15 \, b^{2} c x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 70, normalized size = 1.25 \begin {gather*} -\frac {4 \sqrt {a +\sqrt {\frac {c}{x}}\, b}\, \left (a x +\sqrt {\frac {c}{x}}\, b x \right )^{\frac {3}{2}} \left (-2 a +3 \sqrt {\frac {c}{x}}\, b \right )}{15 \sqrt {\left (a +\sqrt {\frac {c}{x}}\, b \right ) x}\, b^{2} c x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 43, normalized size = 0.77 \begin {gather*} -\frac {4 \, {\left (\frac {3 \, {\left (b \sqrt {\frac {c}{x}} + a\right )}^{\frac {5}{2}}}{b^{2}} - \frac {5 \, {\left (b \sqrt {\frac {c}{x}} + a\right )}^{\frac {3}{2}} a}{b^{2}}\right )}}{15 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.39, size = 52, normalized size = 0.93 \begin {gather*} -\frac {\sqrt {a+b\,\sqrt {\frac {c}{x}}}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{2},2;\ 3;\ -\frac {b\,\sqrt {\frac {c}{x}}}{a}\right )}{x\,\sqrt {\frac {b\,\sqrt {\frac {c}{x}}}{a}+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a + b \sqrt {\frac {c}{x}}}}{x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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