Integrand size = 21, antiderivative size = 174 \[ \int \frac {\sqrt {a+b \sqrt {\frac {c}{x}}}}{x^4} \, dx=\frac {4 a^5 \left (a+b \sqrt {\frac {c}{x}}\right )^{3/2}}{3 b^6 c^3}-\frac {4 a^4 \left (a+b \sqrt {\frac {c}{x}}\right )^{5/2}}{b^6 c^3}+\frac {40 a^3 \left (a+b \sqrt {\frac {c}{x}}\right )^{7/2}}{7 b^6 c^3}-\frac {40 a^2 \left (a+b \sqrt {\frac {c}{x}}\right )^{9/2}}{9 b^6 c^3}+\frac {20 a \left (a+b \sqrt {\frac {c}{x}}\right )^{11/2}}{11 b^6 c^3}-\frac {4 \left (a+b \sqrt {\frac {c}{x}}\right )^{13/2}}{13 b^6 c^3} \]
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Time = 0.07 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {376, 272, 45} \[ \int \frac {\sqrt {a+b \sqrt {\frac {c}{x}}}}{x^4} \, dx=\frac {4 a^5 \left (a+b \sqrt {\frac {c}{x}}\right )^{3/2}}{3 b^6 c^3}-\frac {4 a^4 \left (a+b \sqrt {\frac {c}{x}}\right )^{5/2}}{b^6 c^3}+\frac {40 a^3 \left (a+b \sqrt {\frac {c}{x}}\right )^{7/2}}{7 b^6 c^3}-\frac {40 a^2 \left (a+b \sqrt {\frac {c}{x}}\right )^{9/2}}{9 b^6 c^3}-\frac {4 \left (a+b \sqrt {\frac {c}{x}}\right )^{13/2}}{13 b^6 c^3}+\frac {20 a \left (a+b \sqrt {\frac {c}{x}}\right )^{11/2}}{11 b^6 c^3} \]
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Rule 45
Rule 272
Rule 376
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {\sqrt {a+\frac {b \sqrt {c}}{\sqrt {x}}}}{x^4} \, dx,\sqrt {x},\frac {\sqrt {\frac {c}{x}} x}{\sqrt {c}}\right ) \\ & = -\text {Subst}\left (2 \text {Subst}\left (\int 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 ) \\ & = -\text {Subst}\left (2 \text {Subst}\left (\int \left (-\frac {a^5 \sqrt {a+b \sqrt {c} x}}{b^5 c^{5/2}}+\frac {5 a^4 \left (a+b \sqrt {c} x\right )^{3/2}}{b^5 c^{5/2}}-\frac {10 a^3 \left (a+b \sqrt {c} x\right )^{5/2}}{b^5 c^{5/2}}+\frac {10 a^2 \left (a+b \sqrt {c} x\right )^{7/2}}{b^5 c^{5/2}}-\frac {5 a \left (a+b \sqrt {c} x\right )^{9/2}}{b^5 c^{5/2}}+\frac {\left (a+b \sqrt {c} x\right )^{11/2}}{b^5 c^{5/2}}\right ) \, dx,x,\frac {1}{\sqrt {x}}\right ),\sqrt {x},\frac {\sqrt {\frac {c}{x}} x}{\sqrt {c}}\right ) \\ & = \frac {4 a^5 \left (a+b \sqrt {\frac {c}{x}}\right )^{3/2}}{3 b^6 c^3}-\frac {4 a^4 \left (a+b \sqrt {\frac {c}{x}}\right )^{5/2}}{b^6 c^3}+\frac {40 a^3 \left (a+b \sqrt {\frac {c}{x}}\right )^{7/2}}{7 b^6 c^3}-\frac {40 a^2 \left (a+b \sqrt {\frac {c}{x}}\right )^{9/2}}{9 b^6 c^3}+\frac {20 a \left (a+b \sqrt {\frac {c}{x}}\right )^{11/2}}{11 b^6 c^3}-\frac {4 \left (a+b \sqrt {\frac {c}{x}}\right )^{13/2}}{13 b^6 c^3} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.59 \[ \int \frac {\sqrt {a+b \sqrt {\frac {c}{x}}}}{x^4} \, dx=-\frac {4 \left (a+b \sqrt {\frac {c}{x}}\right )^{3/2} \left (-256 a^5+384 a^4 b \sqrt {\frac {c}{x}}+560 a^2 b^3 \left (\frac {c}{x}\right )^{3/2}+693 b^5 \left (\frac {c}{x}\right )^{5/2}-\frac {630 a b^4 c^2}{x^2}-\frac {480 a^3 b^2 c}{x}\right )}{9009 b^6 c^3} \]
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Time = 4.01 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.76
method | result | size |
default | \(-\frac {4 \sqrt {a +b \sqrt {\frac {c}{x}}}\, \left (a x +b \sqrt {\frac {c}{x}}\, x \right )^{\frac {3}{2}} \left (693 x^{2} \left (\frac {c}{x}\right )^{\frac {5}{2}} b^{5}+560 x^{2} \left (\frac {c}{x}\right )^{\frac {3}{2}} a^{2} b^{3}+384 x^{2} \sqrt {\frac {c}{x}}\, a^{4} b -256 a^{5} x^{2}-480 c x \,a^{3} b^{2}-630 c^{2} a \,b^{4}\right )}{9009 c^{3} x^{3} \sqrt {x \left (a +b \sqrt {\frac {c}{x}}\right )}\, b^{6}}\) | \(133\) |
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Time = 0.36 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.60 \[ \int \frac {\sqrt {a+b \sqrt {\frac {c}{x}}}}{x^4} \, dx=-\frac {4 \, {\left (693 \, b^{6} c^{3} - 70 \, a^{2} b^{4} c^{2} x - 96 \, a^{4} b^{2} c x^{2} - 256 \, a^{6} x^{3} + {\left (63 \, a b^{5} c^{2} x + 80 \, a^{3} b^{3} c x^{2} + 128 \, a^{5} b x^{3}\right )} \sqrt {\frac {c}{x}}\right )} \sqrt {b \sqrt {\frac {c}{x}} + a}}{9009 \, b^{6} c^{3} x^{3}} \]
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\[ \int \frac {\sqrt {a+b \sqrt {\frac {c}{x}}}}{x^4} \, dx=\int \frac {\sqrt {a + b \sqrt {\frac {c}{x}}}}{x^{4}}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.73 \[ \int \frac {\sqrt {a+b \sqrt {\frac {c}{x}}}}{x^4} \, dx=-\frac {4 \, {\left (\frac {693 \, {\left (b \sqrt {\frac {c}{x}} + a\right )}^{\frac {13}{2}}}{b^{6}} - \frac {4095 \, {\left (b \sqrt {\frac {c}{x}} + a\right )}^{\frac {11}{2}} a}{b^{6}} + \frac {10010 \, {\left (b \sqrt {\frac {c}{x}} + a\right )}^{\frac {9}{2}} a^{2}}{b^{6}} - \frac {12870 \, {\left (b \sqrt {\frac {c}{x}} + a\right )}^{\frac {7}{2}} a^{3}}{b^{6}} + \frac {9009 \, {\left (b \sqrt {\frac {c}{x}} + a\right )}^{\frac {5}{2}} a^{4}}{b^{6}} - \frac {3003 \, {\left (b \sqrt {\frac {c}{x}} + a\right )}^{\frac {3}{2}} a^{5}}{b^{6}}\right )}}{9009 \, c^{3}} \]
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Timed out. \[ \int \frac {\sqrt {a+b \sqrt {\frac {c}{x}}}}{x^4} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\sqrt {a+b \sqrt {\frac {c}{x}}}}{x^4} \, dx=\int \frac {\sqrt {a+b\,\sqrt {\frac {c}{x}}}}{x^4} \,d x \]
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