Integrand size = 19, antiderivative size = 172 \[ \int \frac {x}{\sqrt {a+b \sqrt {\frac {c}{x}}}} \, dx=-\frac {7 b c^2 \sqrt {a+b \sqrt {\frac {c}{x}}}}{12 a^2 \left (\frac {c}{x}\right )^{3/2}}-\frac {35 b^3 c^2 \sqrt {a+b \sqrt {\frac {c}{x}}}}{32 a^4 \sqrt {\frac {c}{x}}}+\frac {35 b^2 c \sqrt {a+b \sqrt {\frac {c}{x}}} x}{48 a^3}+\frac {\sqrt {a+b \sqrt {\frac {c}{x}}} x^2}{2 a}+\frac {35 b^4 c^2 \text {arctanh}\left (\frac {\sqrt {a+b \sqrt {\frac {c}{x}}}}{\sqrt {a}}\right )}{32 a^{9/2}} \]
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Time = 0.07 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.02, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {376, 272, 44, 65, 214} \[ \int \frac {x}{\sqrt {a+b \sqrt {\frac {c}{x}}}} \, dx=\frac {35 b^4 c^2 \text {arctanh}\left (\frac {\sqrt {a+b \sqrt {\frac {c}{x}}}}{\sqrt {a}}\right )}{32 a^{9/2}}-\frac {35 b^3 c^2 \sqrt {a+b \sqrt {\frac {c}{x}}}}{32 a^4 \sqrt {\frac {c}{x}}}+\frac {35 b^2 c x \sqrt {a+b \sqrt {\frac {c}{x}}}}{48 a^3}-\frac {7 b x^3 \left (\frac {c}{x}\right )^{3/2} \sqrt {a+b \sqrt {\frac {c}{x}}}}{12 a^2 c}+\frac {x^2 \sqrt {a+b \sqrt {\frac {c}{x}}}}{2 a} \]
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Rule 44
Rule 65
Rule 214
Rule 272
Rule 376
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {x}{\sqrt {a+\frac {b \sqrt {c}}{\sqrt {x}}}} \, dx,\sqrt {x},\frac {\sqrt {\frac {c}{x}} x}{\sqrt {c}}\right ) \\ & = -\text {Subst}\left (2 \text {Subst}\left (\int \frac {1}{x^5 \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^2}{2 a}+\text {Subst}\left (\frac {\left (7 b \sqrt {c}\right ) \text {Subst}\left (\int \frac {1}{x^4 \sqrt {a+b \sqrt {c} x}} \, dx,x,\frac {1}{\sqrt {x}}\right )}{4 a},\sqrt {x},\frac {\sqrt {\frac {c}{x}} x}{\sqrt {c}}\right ) \\ & = \frac {\sqrt {a+b \sqrt {\frac {c}{x}}} x^2}{2 a}-\frac {7 b \sqrt {a+b \sqrt {\frac {c}{x}}} \left (\frac {c}{x}\right )^{3/2} x^3}{12 a^2 c}-\text {Subst}\left (\frac {\left (35 b^2 c\right ) \text {Subst}\left (\int \frac {1}{x^3 \sqrt {a+b \sqrt {c} x}} \, dx,x,\frac {1}{\sqrt {x}}\right )}{24 a^2},\sqrt {x},\frac {\sqrt {\frac {c}{x}} x}{\sqrt {c}}\right ) \\ & = \frac {35 b^2 c \sqrt {a+b \sqrt {\frac {c}{x}}} x}{48 a^3}+\frac {\sqrt {a+b \sqrt {\frac {c}{x}}} x^2}{2 a}-\frac {7 b \sqrt {a+b \sqrt {\frac {c}{x}}} \left (\frac {c}{x}\right )^{3/2} x^3}{12 a^2 c}+\text {Subst}\left (\frac {\left (35 b^3 c^{3/2}\right ) \text {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b \sqrt {c} x}} \, dx,x,\frac {1}{\sqrt {x}}\right )}{32 a^3},\sqrt {x},\frac {\sqrt {\frac {c}{x}} x}{\sqrt {c}}\right ) \\ & = -\frac {35 b^3 c^2 \sqrt {a+b \sqrt {\frac {c}{x}}}}{32 a^4 \sqrt {\frac {c}{x}}}+\frac {35 b^2 c \sqrt {a+b \sqrt {\frac {c}{x}}} x}{48 a^3}+\frac {\sqrt {a+b \sqrt {\frac {c}{x}}} x^2}{2 a}-\frac {7 b \sqrt {a+b \sqrt {\frac {c}{x}}} \left (\frac {c}{x}\right )^{3/2} x^3}{12 a^2 c}-\text {Subst}\left (\frac {\left (35 b^4 c^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b \sqrt {c} x}} \, dx,x,\frac {1}{\sqrt {x}}\right )}{64 a^4},\sqrt {x},\frac {\sqrt {\frac {c}{x}} x}{\sqrt {c}}\right ) \\ & = -\frac {35 b^3 c^2 \sqrt {a+b \sqrt {\frac {c}{x}}}}{32 a^4 \sqrt {\frac {c}{x}}}+\frac {35 b^2 c \sqrt {a+b \sqrt {\frac {c}{x}}} x}{48 a^3}+\frac {\sqrt {a+b \sqrt {\frac {c}{x}}} x^2}{2 a}-\frac {7 b \sqrt {a+b \sqrt {\frac {c}{x}}} \left (\frac {c}{x}\right )^{3/2} x^3}{12 a^2 c}-\text {Subst}\left (\frac {\left (35 b^3 c^{3/2}\right ) \text {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 )}{32 a^4},\sqrt {x},\frac {\sqrt {\frac {c}{x}} x}{\sqrt {c}}\right ) \\ & = -\frac {35 b^3 c^2 \sqrt {a+b \sqrt {\frac {c}{x}}}}{32 a^4 \sqrt {\frac {c}{x}}}+\frac {35 b^2 c \sqrt {a+b \sqrt {\frac {c}{x}}} x}{48 a^3}+\frac {\sqrt {a+b \sqrt {\frac {c}{x}}} x^2}{2 a}-\frac {7 b \sqrt {a+b \sqrt {\frac {c}{x}}} \left (\frac {c}{x}\right )^{3/2} x^3}{12 a^2 c}+\frac {35 b^4 c^2 \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {\frac {c}{x}}}}{\sqrt {a}}\right )}{32 a^{9/2}} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.65 \[ \int \frac {x}{\sqrt {a+b \sqrt {\frac {c}{x}}}} \, dx=\frac {\sqrt {a+b \sqrt {\frac {c}{x}}} \left (48 a^3-56 a^2 b \sqrt {\frac {c}{x}}-105 b^3 \left (\frac {c}{x}\right )^{3/2}+\frac {70 a b^2 c}{x}\right ) x^2}{96 a^4}+\frac {35 b^4 c^2 \text {arctanh}\left (\frac {\sqrt {a+b \sqrt {\frac {c}{x}}}}{\sqrt {a}}\right )}{32 a^{9/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(297\) vs. \(2(134)=268\).
Time = 4.40 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.73
method | result | size |
default | \(\frac {\sqrt {a +b \sqrt {\frac {c}{x}}}\, \sqrt {x}\, \left (174 a^{\frac {3}{2}} \sqrt {a x +b \sqrt {\frac {c}{x}}\, x}\, \left (\frac {c}{x}\right )^{\frac {3}{2}} x^{\frac {3}{2}} b^{3}-384 a^{\frac {3}{2}} \sqrt {x \left (a +b \sqrt {\frac {c}{x}}\right )}\, \left (\frac {c}{x}\right )^{\frac {3}{2}} x^{\frac {3}{2}} b^{3}+348 c \,a^{\frac {5}{2}} \sqrt {a x +b \sqrt {\frac {c}{x}}\, x}\, \sqrt {x}\, b^{2}-87 c^{2} \ln \left (\frac {b \sqrt {\frac {c}{x}}\, \sqrt {x}+2 \sqrt {a x +b \sqrt {\frac {c}{x}}\, x}\, \sqrt {a}+2 a \sqrt {x}}{2 \sqrt {a}}\right ) a \,b^{4}+192 a \,c^{2} \ln \left (\frac {b \sqrt {\frac {c}{x}}\, \sqrt {x}+2 \sqrt {x \left (a +b \sqrt {\frac {c}{x}}\right )}\, \sqrt {a}+2 a \sqrt {x}}{2 \sqrt {a}}\right ) b^{4}+96 \sqrt {x}\, \left (a x +b \sqrt {\frac {c}{x}}\, x \right )^{\frac {3}{2}} a^{\frac {7}{2}}-208 a^{\frac {5}{2}} \left (a x +b \sqrt {\frac {c}{x}}\, x \right )^{\frac {3}{2}} \sqrt {\frac {c}{x}}\, \sqrt {x}\, b \right )}{192 \sqrt {x \left (a +b \sqrt {\frac {c}{x}}\right )}\, a^{\frac {11}{2}}}\) | \(298\) |
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Time = 0.33 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.30 \[ \int \frac {x}{\sqrt {a+b \sqrt {\frac {c}{x}}}} \, dx=\left [\frac {105 \, \sqrt {a} b^{4} c^{2} \log \left (2 \, \sqrt {b \sqrt {\frac {c}{x}} + a} \sqrt {a} x \sqrt {\frac {c}{x}} + 2 \, a x \sqrt {\frac {c}{x}} + b c\right ) + 2 \, {\left (70 \, a^{2} b^{2} c x + 48 \, a^{4} x^{2} - 7 \, {\left (15 \, a b^{3} c x + 8 \, a^{3} b x^{2}\right )} \sqrt {\frac {c}{x}}\right )} \sqrt {b \sqrt {\frac {c}{x}} + a}}{192 \, a^{5}}, -\frac {105 \, \sqrt {-a} b^{4} c^{2} \arctan \left (\frac {\sqrt {b \sqrt {\frac {c}{x}} + a} \sqrt {-a}}{a}\right ) - {\left (70 \, a^{2} b^{2} c x + 48 \, a^{4} x^{2} - 7 \, {\left (15 \, a b^{3} c x + 8 \, a^{3} b x^{2}\right )} \sqrt {\frac {c}{x}}\right )} \sqrt {b \sqrt {\frac {c}{x}} + a}}{96 \, a^{5}}\right ] \]
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\[ \int \frac {x}{\sqrt {a+b \sqrt {\frac {c}{x}}}} \, dx=\int \frac {x}{\sqrt {a + b \sqrt {\frac {c}{x}}}}\, dx \]
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Time = 0.30 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.23 \[ \int \frac {x}{\sqrt {a+b \sqrt {\frac {c}{x}}}} \, dx=-\frac {1}{192} \, c^{2} {\left (\frac {105 \, b^{4} \log \left (\frac {\sqrt {b \sqrt {\frac {c}{x}} + a} - \sqrt {a}}{\sqrt {b \sqrt {\frac {c}{x}} + a} + \sqrt {a}}\right )}{a^{\frac {9}{2}}} + \frac {2 \, {\left (105 \, {\left (b \sqrt {\frac {c}{x}} + a\right )}^{\frac {7}{2}} b^{4} - 385 \, {\left (b \sqrt {\frac {c}{x}} + a\right )}^{\frac {5}{2}} a b^{4} + 511 \, {\left (b \sqrt {\frac {c}{x}} + a\right )}^{\frac {3}{2}} a^{2} b^{4} - 279 \, \sqrt {b \sqrt {\frac {c}{x}} + a} a^{3} b^{4}\right )}}{{\left (b \sqrt {\frac {c}{x}} + a\right )}^{4} a^{4} - 4 \, {\left (b \sqrt {\frac {c}{x}} + a\right )}^{3} a^{5} + 6 \, {\left (b \sqrt {\frac {c}{x}} + a\right )}^{2} a^{6} - 4 \, {\left (b \sqrt {\frac {c}{x}} + a\right )} a^{7} + a^{8}}\right )} \]
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Time = 0.35 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.02 \[ \int \frac {x}{\sqrt {a+b \sqrt {\frac {c}{x}}}} \, dx=\frac {\frac {105 \, b^{4} c^{4} \log \left (c^{2} {\left | b \right |}\right )}{\sqrt {a c} a^{4}} - \frac {105 \, b^{4} c^{4} \log \left ({\left | -b c^{2} - 2 \, \sqrt {a c} {\left (\sqrt {a c} \sqrt {c x} - \sqrt {a c^{2} x + \sqrt {c x} b c^{2}}\right )} \right |}\right )}{\sqrt {a c} a^{4}} - 2 \, \sqrt {a c^{2} x + \sqrt {c x} b c^{2}} {\left (2 \, \sqrt {c x} {\left (4 \, \sqrt {c x} {\left (\frac {7 \, b}{a^{2}} - \frac {6 \, \sqrt {c x}}{a c}\right )} - \frac {35 \, b^{2} c}{a^{3}}\right )} + \frac {105 \, b^{3} c^{2}}{a^{4}}\right )}}{192 \, c^{\frac {3}{2}} \mathrm {sgn}\left (x\right )} \]
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Timed out. \[ \int \frac {x}{\sqrt {a+b \sqrt {\frac {c}{x}}}} \, dx=\int \frac {x}{\sqrt {a+b\,\sqrt {\frac {c}{x}}}} \,d x \]
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