Optimal. Leaf size=104 \[ \frac {2 \sqrt {d} \sqrt {b c-a d} \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{c^2}+\frac {(b c-2 a d) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{\sqrt {a} c^2}+\frac {x \sqrt {a+\frac {b}{x}}}{c} \]
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Rubi [A] time = 0.11, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {375, 99, 156, 63, 208, 205} \[ \frac {2 \sqrt {d} \sqrt {b c-a d} \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{c^2}+\frac {(b c-2 a d) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{\sqrt {a} c^2}+\frac {x \sqrt {a+\frac {b}{x}}}{c} \]
Antiderivative was successfully verified.
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Rule 63
Rule 99
Rule 156
Rule 205
Rule 208
Rule 375
Rubi steps
\begin {align*} \int \frac {\sqrt {a+\frac {b}{x}}}{c+\frac {d}{x}} \, dx &=-\operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x^2 (c+d x)} \, dx,x,\frac {1}{x}\right )\\ &=\frac {\sqrt {a+\frac {b}{x}} x}{c}-\frac {\operatorname {Subst}\left (\int \frac {\frac {1}{2} (b c-2 a d)-\frac {b d x}{2}}{x \sqrt {a+b x} (c+d x)} \, dx,x,\frac {1}{x}\right )}{c}\\ &=\frac {\sqrt {a+\frac {b}{x}} x}{c}-\frac {(b c-2 a d) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x}\right )}{2 c^2}+\frac {(d (b c-a d)) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x} (c+d x)} \, dx,x,\frac {1}{x}\right )}{c^2}\\ &=\frac {\sqrt {a+\frac {b}{x}} x}{c}-\frac {(b c-2 a d) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x}}\right )}{b c^2}+\frac {(2 d (b c-a d)) \operatorname {Subst}\left (\int \frac {1}{c-\frac {a d}{b}+\frac {d x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x}}\right )}{b c^2}\\ &=\frac {\sqrt {a+\frac {b}{x}} x}{c}+\frac {2 \sqrt {d} \sqrt {b c-a d} \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{c^2}+\frac {(b c-2 a d) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{\sqrt {a} c^2}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 100, normalized size = 0.96 \[ \frac {2 \sqrt {d} \sqrt {b c-a d} \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )+\frac {(b c-2 a d) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{\sqrt {a}}+c x \sqrt {a+\frac {b}{x}}}{c^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.97, size = 482, normalized size = 4.63 \[ \left [\frac {2 \, a c x \sqrt {\frac {a x + b}{x}} - {\left (b c - 2 \, a d\right )} \sqrt {a} \log \left (2 \, a x - 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right ) + 2 \, \sqrt {-b c d + a d^{2}} a \log \left (\frac {b d - {\left (b c - 2 \, a d\right )} x + 2 \, \sqrt {-b c d + a d^{2}} x \sqrt {\frac {a x + b}{x}}}{c x + d}\right )}{2 \, a c^{2}}, \frac {2 \, a c x \sqrt {\frac {a x + b}{x}} - 4 \, \sqrt {b c d - a d^{2}} a \arctan \left (\frac {\sqrt {b c d - a d^{2}} x \sqrt {\frac {a x + b}{x}}}{a d x + b d}\right ) - {\left (b c - 2 \, a d\right )} \sqrt {a} \log \left (2 \, a x - 2 \, \sqrt {a} x \sqrt {\frac {a x + b}{x}} + b\right )}{2 \, a c^{2}}, \frac {a c x \sqrt {\frac {a x + b}{x}} - {\left (b c - 2 \, a d\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a x + b}{x}}}{a}\right ) + \sqrt {-b c d + a d^{2}} a \log \left (\frac {b d - {\left (b c - 2 \, a d\right )} x + 2 \, \sqrt {-b c d + a d^{2}} x \sqrt {\frac {a x + b}{x}}}{c x + d}\right )}{a c^{2}}, \frac {a c x \sqrt {\frac {a x + b}{x}} - 2 \, \sqrt {b c d - a d^{2}} a \arctan \left (\frac {\sqrt {b c d - a d^{2}} x \sqrt {\frac {a x + b}{x}}}{a d x + b d}\right ) - {\left (b c - 2 \, a d\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a} \sqrt {\frac {a x + b}{x}}}{a}\right )}{a c^{2}}\right ] \]
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 = 0.11, size = 287, normalized size = 2.76 \[ -\frac {\sqrt {\frac {a x +b}{x}}\, \left (2 a^{\frac {3}{2}} d^{2} \ln \left (\frac {-2 a d x +b c x -b d +2 \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {\left (a x +b \right ) x}\, c}{c x +d}\right )-2 \sqrt {a}\, b c d \ln \left (\frac {-2 a d x +b c x -b d +2 \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {\left (a x +b \right ) x}\, c}{c x +d}\right )+2 \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, a c d \ln \left (\frac {2 a x +b +2 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}}{2 \sqrt {a}}\right )-\sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, b \,c^{2} \ln \left (\frac {2 a x +b +2 \sqrt {\left (a x +b \right ) x}\, \sqrt {a}}{2 \sqrt {a}}\right )-2 \sqrt {\left (a x +b \right ) x}\, \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {a}\, c^{2}\right ) x}{2 \sqrt {\left (a x +b \right ) x}\, \sqrt {\frac {\left (a d -b c \right ) d}{c^{2}}}\, \sqrt {a}\, c^{3}} \]
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 \[ \int \frac {\sqrt {a + \frac {b}{x}}}{c + \frac {d}{x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.63, size = 149, normalized size = 1.43 \[ \frac {x\,\sqrt {a+\frac {b}{x}}}{c}+\frac {\ln \left (\sqrt {a+\frac {b}{x}}-\sqrt {a}\right )\,\left (a\,d-\frac {b\,c}{2}\right )}{\sqrt {a}\,c^2}-\frac {\ln \left (\sqrt {a+\frac {b}{x}}+\sqrt {a}\right )\,\left (2\,a\,d-b\,c\right )}{2\,\sqrt {a}\,c^2}-\frac {\mathrm {atan}\left (\frac {b^4\,d^3\,\sqrt {a+\frac {b}{x}}\,\sqrt {a\,d^2-b\,c\,d}\,4{}\mathrm {i}}{4\,a\,b^4\,d^4-4\,b^5\,c\,d^3}\right )\,\sqrt {a\,d^2-b\,c\,d}\,2{}\mathrm {i}}{c^2} \]
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 \[ \int \frac {x \sqrt {a + \frac {b}{x}}}{c x + d}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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