Optimal. Leaf size=93 \[ \frac {2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+\frac {b}{x}}}{\sqrt {a} \sqrt {c+\frac {d}{x}}}\right )}{\sqrt {a}}-\frac {2 \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b} \sqrt {c+\frac {d}{x}}}\right )}{\sqrt {b}} \]
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Rubi [A] time = 0.09, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {446, 105, 63, 217, 206, 93, 208} \begin {gather*} \frac {2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+\frac {b}{x}}}{\sqrt {a} \sqrt {c+\frac {d}{x}}}\right )}{\sqrt {a}}-\frac {2 \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b} \sqrt {c+\frac {d}{x}}}\right )}{\sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 93
Rule 105
Rule 206
Rule 208
Rule 217
Rule 446
Rubi steps
\begin {align*} \int \frac {\sqrt {c+\frac {d}{x}}}{\sqrt {a+\frac {b}{x}} x} \, dx &=-\operatorname {Subst}\left (\int \frac {\sqrt {c+d x}}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x}\right )\\ &=-\left (c \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}} \, dx,x,\frac {1}{x}\right )\right )-d \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx,x,\frac {1}{x}\right )\\ &=-\left ((2 c) \operatorname {Subst}\left (\int \frac {1}{-a+c x^2} \, dx,x,\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {c+\frac {d}{x}}}\right )\right )-\frac {(2 d) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+\frac {b}{x}}\right )}{b}\\ &=\frac {2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+\frac {b}{x}}}{\sqrt {a} \sqrt {c+\frac {d}{x}}}\right )}{\sqrt {a}}-\frac {(2 d) \operatorname {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {c+\frac {d}{x}}}\right )}{b}\\ &=\frac {2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+\frac {b}{x}}}{\sqrt {a} \sqrt {c+\frac {d}{x}}}\right )}{\sqrt {a}}-\frac {2 \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b} \sqrt {c+\frac {d}{x}}}\right )}{\sqrt {b}}\\ \end {align*}
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Mathematica [A] time = 0.68, size = 142, normalized size = 1.53 \begin {gather*} \frac {2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+\frac {b}{x}}}{\sqrt {a} \sqrt {c+\frac {d}{x}}}\right )}{\sqrt {a}}-\frac {2 \sqrt {d} x \sqrt {c+\frac {d}{x}} \sqrt {b c-a d} \sqrt {\frac {b (c x+d)}{x (b c-a d)}} \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{b c x+b d} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 2.67, size = 442, normalized size = 4.75 \begin {gather*} \frac {c \sqrt {\frac {a}{c}} \sqrt {a x+b} \sqrt {c+\frac {d}{x}} \left (\sqrt {a x+b}-\sqrt {\frac {a}{c}} \sqrt {c x+d}\right ) \left (-2 c \sqrt {\frac {a}{c}} \sqrt {a x+b} \sqrt {c x+d}+a (2 c x+d)+b c\right ) \left (-\frac {2 c \sqrt {\frac {a}{c}} \log \left (\sqrt {\frac {a (c x+d)}{c}-\frac {a d}{c}+b}-\sqrt {\frac {a}{c}} \sqrt {c x+d}\right )}{a}-\frac {2 \sqrt {c} \sqrt {d} \sqrt {\frac {a}{c}} \tanh ^{-1}\left (-\frac {\sqrt {a} (c x+d)}{\sqrt {b} \sqrt {c} \sqrt {d}}+\frac {\sqrt {c} \sqrt {\frac {a}{c}} \sqrt {c x+d} \sqrt {\frac {a (c x+d)}{c}-\frac {a d}{c}+b}}{\sqrt {a} \sqrt {b} \sqrt {d}}+\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}}\right )}{\sqrt {a} \sqrt {b}}\right )}{\sqrt {a+\frac {b}{x}} \sqrt {c x+d} \left (a \left (4 c^2 x \sqrt {\frac {a}{c}} \sqrt {a x+b}+3 c d \sqrt {\frac {a}{c}} \sqrt {a x+b}-4 a c x \sqrt {c x+d}-a d \sqrt {c x+d}\right )+b c \left (c \sqrt {\frac {a}{c}} \sqrt {a x+b}-3 a \sqrt {c x+d}\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.71, size = 757, normalized size = 8.14 \begin {gather*} \left [\frac {1}{2} \, \sqrt {\frac {c}{a}} \log \left (-8 \, a^{2} c^{2} x^{2} - b^{2} c^{2} - 6 \, a b c d - a^{2} d^{2} - 4 \, {\left (2 \, a^{2} c x^{2} + {\left (a b c + a^{2} d\right )} x\right )} \sqrt {\frac {c}{a}} \sqrt {\frac {a x + b}{x}} \sqrt {\frac {c x + d}{x}} - 8 \, {\left (a b c^{2} + a^{2} c d\right )} x\right ) + \frac {1}{2} \, \sqrt {\frac {d}{b}} \log \left (-\frac {8 \, b^{2} d^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, b^{2} d x + {\left (b^{2} c + a b d\right )} x^{2}\right )} \sqrt {\frac {d}{b}} \sqrt {\frac {a x + b}{x}} \sqrt {\frac {c x + d}{x}} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x}{x^{2}}\right ), -\sqrt {-\frac {c}{a}} \arctan \left (\frac {2 \, a x \sqrt {-\frac {c}{a}} \sqrt {\frac {a x + b}{x}} \sqrt {\frac {c x + d}{x}}}{2 \, a c x + b c + a d}\right ) + \frac {1}{2} \, \sqrt {\frac {d}{b}} \log \left (-\frac {8 \, b^{2} d^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, b^{2} d x + {\left (b^{2} c + a b d\right )} x^{2}\right )} \sqrt {\frac {d}{b}} \sqrt {\frac {a x + b}{x}} \sqrt {\frac {c x + d}{x}} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x}{x^{2}}\right ), \sqrt {-\frac {d}{b}} \arctan \left (\frac {{\left (2 \, b d x + {\left (b c + a d\right )} x^{2}\right )} \sqrt {-\frac {d}{b}} \sqrt {\frac {a x + b}{x}} \sqrt {\frac {c x + d}{x}}}{2 \, {\left (a c d x^{2} + b d^{2} + {\left (b c d + a d^{2}\right )} x\right )}}\right ) + \frac {1}{2} \, \sqrt {\frac {c}{a}} \log \left (-8 \, a^{2} c^{2} x^{2} - b^{2} c^{2} - 6 \, a b c d - a^{2} d^{2} - 4 \, {\left (2 \, a^{2} c x^{2} + {\left (a b c + a^{2} d\right )} x\right )} \sqrt {\frac {c}{a}} \sqrt {\frac {a x + b}{x}} \sqrt {\frac {c x + d}{x}} - 8 \, {\left (a b c^{2} + a^{2} c d\right )} x\right ), -\sqrt {-\frac {c}{a}} \arctan \left (\frac {2 \, a x \sqrt {-\frac {c}{a}} \sqrt {\frac {a x + b}{x}} \sqrt {\frac {c x + d}{x}}}{2 \, a c x + b c + a d}\right ) + \sqrt {-\frac {d}{b}} \arctan \left (\frac {{\left (2 \, b d x + {\left (b c + a d\right )} x^{2}\right )} \sqrt {-\frac {d}{b}} \sqrt {\frac {a x + b}{x}} \sqrt {\frac {c x + d}{x}}}{2 \, {\left (a c d x^{2} + b d^{2} + {\left (b c d + a d^{2}\right )} x\right )}}\right )\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {c + \frac {d}{x}}}{\sqrt {a + \frac {b}{x}} x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.10, size = 143, normalized size = 1.54 \begin {gather*} -\frac {\sqrt {\frac {a x +b}{x}}\, \sqrt {\frac {c x +d}{x}}\, \left (-\sqrt {b d}\, c \ln \left (\frac {2 a c x +a d +b c +2 \sqrt {\left (c x +d \right ) \left (a x +b \right )}\, \sqrt {a c}}{2 \sqrt {a c}}\right )+\sqrt {a c}\, d \ln \left (\frac {a d x +b c x +2 b d +2 \sqrt {b d}\, \sqrt {\left (c x +d \right ) \left (a x +b \right )}}{x}\right )\right ) x}{\sqrt {\left (c x +d \right ) \left (a x +b \right )}\, \sqrt {b d}\, \sqrt {a c}} \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {c + \frac {d}{x}}}{\sqrt {a + \frac {b}{x}} x}\,{d x} \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {c+\frac {d}{x}}}{x\,\sqrt {a+\frac {b}{x}}} \,d x \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {c + \frac {d}{x}}}{x \sqrt {a + \frac {b}{x}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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