Optimal. Leaf size=95 \[ \frac {3 b^2 c \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {\frac {c}{x}}}}{\sqrt {a}}\right )}{2 a^{5/2}}-\frac {3 b c \sqrt {a+b \sqrt {\frac {c}{x}}}}{2 a^2 \sqrt {\frac {c}{x}}}+\frac {x \sqrt {a+b \sqrt {\frac {c}{x}}}}{a} \]
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Rubi [A] time = 0.05, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {255, 190, 51, 63, 208} \[ \frac {3 b^2 c \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {\frac {c}{x}}}}{\sqrt {a}}\right )}{2 a^{5/2}}-\frac {3 b c \sqrt {a+b \sqrt {\frac {c}{x}}}}{2 a^2 \sqrt {\frac {c}{x}}}+\frac {x \sqrt {a+b \sqrt {\frac {c}{x}}}}{a} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 190
Rule 208
Rule 255
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a+b \sqrt {\frac {c}{x}}}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{\sqrt {a+\frac {b \sqrt {c}}{\sqrt {x}}}} \, dx,\sqrt {x},\frac {\sqrt {\frac {c}{x}} x}{\sqrt {c}}\right )\\ &=-\operatorname {Subst}\left (2 \operatorname {Subst}\left (\int \frac {1}{x^3 \sqrt {a+b \sqrt {c} x}} \, dx,x,\frac {1}{\sqrt {x}}\right ),\sqrt {x},\frac {\sqrt {\frac {c}{x}} x}{\sqrt {c}}\right )\\ &=\frac {\sqrt {a+b \sqrt {\frac {c}{x}}} x}{a}+\operatorname {Subst}\left (\frac {\left (3 b \sqrt {c}\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b \sqrt {c} x}} \, dx,x,\frac {1}{\sqrt {x}}\right )}{2 a},\sqrt {x},\frac {\sqrt {\frac {c}{x}} x}{\sqrt {c}}\right )\\ &=-\frac {3 b c \sqrt {a+b \sqrt {\frac {c}{x}}}}{2 a^2 \sqrt {\frac {c}{x}}}+\frac {\sqrt {a+b \sqrt {\frac {c}{x}}} x}{a}-\operatorname {Subst}\left (\frac {\left (3 b^2 c\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b \sqrt {c} x}} \, dx,x,\frac {1}{\sqrt {x}}\right )}{4 a^2},\sqrt {x},\frac {\sqrt {\frac {c}{x}} x}{\sqrt {c}}\right )\\ &=-\frac {3 b c \sqrt {a+b \sqrt {\frac {c}{x}}}}{2 a^2 \sqrt {\frac {c}{x}}}+\frac {\sqrt {a+b \sqrt {\frac {c}{x}}} x}{a}-\operatorname {Subst}\left (\frac {\left (3 b \sqrt {c}\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b \sqrt {c}}+\frac {x^2}{b \sqrt {c}}} \, dx,x,\sqrt {a+\frac {b \sqrt {c}}{\sqrt {x}}}\right )}{2 a^2},\sqrt {x},\frac {\sqrt {\frac {c}{x}} x}{\sqrt {c}}\right )\\ &=-\frac {3 b c \sqrt {a+b \sqrt {\frac {c}{x}}}}{2 a^2 \sqrt {\frac {c}{x}}}+\frac {\sqrt {a+b \sqrt {\frac {c}{x}}} x}{a}+\frac {3 b^2 c \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {\frac {c}{x}}}}{\sqrt {a}}\right )}{2 a^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 89, normalized size = 0.94 \[ \frac {3 b^2 c \tanh ^{-1}\left (\frac {\sqrt {a}}{\sqrt {a+b \sqrt {\frac {c}{x}}}}\right )}{2 a^{5/2}}+\frac {2 a^2 x-a b x \sqrt {\frac {c}{x}}-3 b^2 c}{2 a^2 \sqrt {a+b \sqrt {\frac {c}{x}}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.97, size = 165, normalized size = 1.74 \[ \left [\frac {3 \, \sqrt {a} b^{2} c \log \left (2 \, \sqrt {b \sqrt {\frac {c}{x}} + a} \sqrt {a} x \sqrt {\frac {c}{x}} + 2 \, a x \sqrt {\frac {c}{x}} + b c\right ) - 2 \, {\left (3 \, a b x \sqrt {\frac {c}{x}} - 2 \, a^{2} x\right )} \sqrt {b \sqrt {\frac {c}{x}} + a}}{4 \, a^{3}}, -\frac {3 \, \sqrt {-a} b^{2} c \arctan \left (\frac {\sqrt {b \sqrt {\frac {c}{x}} + a} \sqrt {-a}}{a}\right ) + {\left (3 \, a b x \sqrt {\frac {c}{x}} - 2 \, a^{2} x\right )} \sqrt {b \sqrt {\frac {c}{x}} + a}}{2 \, a^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.41, size = 139, normalized size = 1.46 \[ \frac {\frac {3 \, b^{2} c^{2} \log \left (c^{2} {\left | b \right |}\right )}{\sqrt {a c} a^{2}} - \frac {3 \, b^{2} c^{2} \log \left ({\left | -b c^{2} - 2 \, \sqrt {a c} {\left (\sqrt {a c} \sqrt {c x} - \sqrt {a c^{2} x + \sqrt {c x} b c^{2}}\right )} \right |}\right )}{\sqrt {a c} a^{2}} - 2 \, \sqrt {a c^{2} x + \sqrt {c x} b c^{2}} {\left (\frac {3 \, b}{a^{2}} - \frac {2 \, \sqrt {c x}}{a c}\right )}}{4 \, \sqrt {c} \mathrm {sgn}\relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 229, normalized size = 2.41 \[ -\frac {\sqrt {a +\sqrt {\frac {c}{x}}\, b}\, \left (-4 a \,b^{2} c \ln \left (\frac {2 a \sqrt {x}+\sqrt {\frac {c}{x}}\, b \sqrt {x}+2 \sqrt {\left (a +\sqrt {\frac {c}{x}}\, b \right ) x}\, \sqrt {a}}{2 \sqrt {a}}\right )+a \,b^{2} c \ln \left (\frac {2 a \sqrt {x}+\sqrt {\frac {c}{x}}\, b \sqrt {x}+2 \sqrt {a x +\sqrt {\frac {c}{x}}\, b x}\, \sqrt {a}}{2 \sqrt {a}}\right )-4 \sqrt {a x +\sqrt {\frac {c}{x}}\, b x}\, a^{\frac {5}{2}} \sqrt {x}+8 \sqrt {\left (a +\sqrt {\frac {c}{x}}\, b \right ) x}\, \sqrt {\frac {c}{x}}\, a^{\frac {3}{2}} b \sqrt {x}-2 \sqrt {a x +\sqrt {\frac {c}{x}}\, b x}\, \sqrt {\frac {c}{x}}\, a^{\frac {3}{2}} b \sqrt {x}\right ) \sqrt {x}}{4 \sqrt {\left (a +\sqrt {\frac {c}{x}}\, b \right ) x}\, a^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.16, size = 131, normalized size = 1.38 \[ -\frac {1}{4} \, c {\left (\frac {3 \, b^{2} \log \left (\frac {\sqrt {b \sqrt {\frac {c}{x}} + a} - \sqrt {a}}{\sqrt {b \sqrt {\frac {c}{x}} + a} + \sqrt {a}}\right )}{a^{\frac {5}{2}}} + \frac {2 \, {\left (3 \, {\left (b \sqrt {\frac {c}{x}} + a\right )}^{\frac {3}{2}} b^{2} - 5 \, \sqrt {b \sqrt {\frac {c}{x}} + a} a b^{2}\right )}}{{\left (b \sqrt {\frac {c}{x}} + a\right )}^{2} a^{2} - 2 \, {\left (b \sqrt {\frac {c}{x}} + a\right )} a^{3} + a^{4}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\sqrt {a+b\,\sqrt {\frac {c}{x}}}} \,d x \]
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 \[ \int \frac {1}{\sqrt {a + b \sqrt {\frac {c}{x}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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