Optimal. Leaf size=145 \[ -2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}+2 \sqrt {a} \tanh ^{-1}\left (\frac {2 a+b \sqrt {\frac {d}{x}}}{2 \sqrt {a} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )-\frac {b \sqrt {d} \tanh ^{-1}\left (\frac {b d+2 c \sqrt {\frac {d}{x}}}{2 \sqrt {c} \sqrt {d} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )}{\sqrt {c}} \]
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Rubi [A] time = 0.21, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {1970, 1357, 734, 843, 621, 206, 724} \[ -2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}+2 \sqrt {a} \tanh ^{-1}\left (\frac {2 a+b \sqrt {\frac {d}{x}}}{2 \sqrt {a} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )-\frac {b \sqrt {d} \tanh ^{-1}\left (\frac {b d+2 c \sqrt {\frac {d}{x}}}{2 \sqrt {c} \sqrt {d} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )}{\sqrt {c}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 724
Rule 734
Rule 843
Rule 1357
Rule 1970
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{x} \, dx &=-\operatorname {Subst}\left (\int \frac {\sqrt {a+b \sqrt {x}+\frac {c x}{d}}}{x} \, dx,x,\frac {d}{x}\right )\\ &=-\left (2 \operatorname {Subst}\left (\int \frac {\sqrt {a+b x+\frac {c x^2}{d}}}{x} \, dx,x,\sqrt {\frac {d}{x}}\right )\right )\\ &=-2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}+\operatorname {Subst}\left (\int \frac {-2 a-b x}{x \sqrt {a+b x+\frac {c x^2}{d}}} \, dx,x,\sqrt {\frac {d}{x}}\right )\\ &=-2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}-(2 a) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x+\frac {c x^2}{d}}} \, dx,x,\sqrt {\frac {d}{x}}\right )-b \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x+\frac {c x^2}{d}}} \, dx,x,\sqrt {\frac {d}{x}}\right )\\ &=-2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}+(4 a) \operatorname {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {2 a+b \sqrt {\frac {d}{x}}}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )-(2 b) \operatorname {Subst}\left (\int \frac {1}{\frac {4 c}{d}-x^2} \, dx,x,\frac {b+\frac {2 c \sqrt {\frac {d}{x}}}{d}}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )\\ &=-2 \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}+2 \sqrt {a} \tanh ^{-1}\left (\frac {2 a+b \sqrt {\frac {d}{x}}}{2 \sqrt {a} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )-\frac {b \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d} \left (b+\frac {2 c \sqrt {\frac {d}{x}}}{d}\right )}{2 \sqrt {c} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )}{\sqrt {c}}\\ \end {align*}
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Mathematica [F] time = 0.15, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{x} \, dx \]
Verification is Not applicable to the result.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.16, size = 237, normalized size = 1.63 \[ \frac {\sqrt {\frac {a x +\sqrt {\frac {d}{x}}\, b x +c}{x}}\, \left (-\sqrt {\frac {d}{x}}\, a^{\frac {3}{2}} b \sqrt {c}\, x \ln \left (\frac {\sqrt {\frac {d}{x}}\, b x +2 c +2 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, \sqrt {c}}{\sqrt {x}}\right )+2 a^{2} c \sqrt {x}\, \ln \left (\frac {2 a \sqrt {x}+\sqrt {\frac {d}{x}}\, b \sqrt {x}+2 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, \sqrt {a}}{2 \sqrt {a}}\right )+2 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, a^{\frac {5}{2}} x +2 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, \sqrt {\frac {d}{x}}\, a^{\frac {3}{2}} b x -2 \left (a x +\sqrt {\frac {d}{x}}\, b x +c \right )^{\frac {3}{2}} a^{\frac {3}{2}}\right )}{\sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, a^{\frac {3}{2}} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {b \sqrt {\frac {d}{x}} + a + \frac {c}{x}}}{x}\,{d x} \]
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 \[ \int \frac {\sqrt {a+\frac {c}{x}+b\,\sqrt {\frac {d}{x}}}}{x} \,d x \]
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 \[ \int \frac {\sqrt {a + b \sqrt {\frac {d}{x}} + \frac {c}{x}}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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