Optimal. Leaf size=54 \[ \frac {4 a \sqrt {a+b \sqrt {\frac {c}{x}}}}{b^2 c}-\frac {4 \left (a+b \sqrt {\frac {c}{x}}\right )^{3/2}}{3 b^2 c} \]
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Rubi [A] time = 0.04, antiderivative size = 54, 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} \[ \frac {4 a \sqrt {a+b \sqrt {\frac {c}{x}}}}{b^2 c}-\frac {4 \left (a+b \sqrt {\frac {c}{x}}\right )^{3/2}}{3 b^2 c} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 369
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a+b \sqrt {\frac {c}{x}}} x^2} \, dx &=\operatorname {Subst}\left (\int \frac {1}{\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 \frac {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}{b \sqrt {c} \sqrt {a+b \sqrt {c} x}}+\frac {\sqrt {a+b \sqrt {c} x}}{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 \sqrt {a+b \sqrt {\frac {c}{x}}}}{b^2 c}-\frac {4 \left (a+b \sqrt {\frac {c}{x}}\right )^{3/2}}{3 b^2 c}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 42, normalized size = 0.78 \[ -\frac {4 \left (b \sqrt {\frac {c}{x}}-2 a\right ) \sqrt {a+b \sqrt {\frac {c}{x}}}}{3 b^2 c} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 34, normalized size = 0.63 \[ -\frac {4 \, \sqrt {b \sqrt {\frac {c}{x}} + a} {\left (b \sqrt {\frac {c}{x}} - 2 \, a\right )}}{3 \, b^{2} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 60, normalized size = 1.11 \[ -\frac {4 \, {\left ({\left (b \sqrt {\frac {c}{x}} + a\right )}^{\frac {3}{2}} b - 3 \, \sqrt {b \sqrt {\frac {c}{x}} + a} a b\right )} \mathrm {sgn}\left ({\left (b \sqrt {\frac {c}{x}} + a\right )} b - a b\right )}{3 \, b^{3} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.04, size = 274, normalized size = 5.07 \[ -\frac {\sqrt {a +\sqrt {\frac {c}{x}}\, b}\, \left (-3 \sqrt {\frac {c}{x}}\, a^{2} b \,x^{2} \ln \left (\frac {2 a \sqrt {x}+\sqrt {\frac {c}{x}}\, b \sqrt {x}+2 \sqrt {\left (a +\sqrt {\frac {c}{x}}\, b \right ) x}\, \sqrt {a}}{2 \sqrt {a}}\right )+3 \sqrt {\frac {c}{x}}\, a^{2} b \,x^{2} \ln \left (\frac {2 a \sqrt {x}+\sqrt {\frac {c}{x}}\, b \sqrt {x}+2 \sqrt {a x +\sqrt {\frac {c}{x}}\, b x}\, \sqrt {a}}{2 \sqrt {a}}\right )+6 \sqrt {\left (a +\sqrt {\frac {c}{x}}\, b \right ) x}\, a^{\frac {5}{2}} x^{\frac {3}{2}}+6 \sqrt {a x +\sqrt {\frac {c}{x}}\, b x}\, a^{\frac {5}{2}} x^{\frac {3}{2}}-12 \left (a x +\sqrt {\frac {c}{x}}\, b x \right )^{\frac {3}{2}} a^{\frac {3}{2}} \sqrt {x}+4 \sqrt {\frac {c}{x}}\, \left (a x +\sqrt {\frac {c}{x}}\, b x \right )^{\frac {3}{2}} \sqrt {a}\, b \sqrt {x}\right )}{3 \sqrt {\left (a +\sqrt {\frac {c}{x}}\, b \right ) x}\, \left (\frac {c}{x}\right )^{\frac {3}{2}} \sqrt {a}\, b^{3} x^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.56, size = 42, normalized size = 0.78 \[ -\frac {4 \, {\left (\frac {{\left (b \sqrt {\frac {c}{x}} + a\right )}^{\frac {3}{2}}}{b^{2}} - \frac {3 \, \sqrt {b \sqrt {\frac {c}{x}} + a} a}{b^{2}}\right )}}{3 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.44, size = 52, normalized size = 0.96 \[ -\frac {\sqrt {\frac {b\,\sqrt {\frac {c}{x}}}{a}+1}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},2;\ 3;\ -\frac {b\,\sqrt {\frac {c}{x}}}{a}\right )}{x\,\sqrt {a+b\,\sqrt {\frac {c}{x}}}} \]
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 \[ \int \frac {1}{x^{2} \sqrt {a + b \sqrt {\frac {c}{x}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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