Optimal. Leaf size=116 \[ 4 \sqrt {a+b \sqrt {c+d x}}-2 \sqrt {a-b \sqrt {c}} \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a-b \sqrt {c}}}\right )-2 \sqrt {a+b \sqrt {c}} \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a+b \sqrt {c}}}\right ) \]
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Rubi [A] time = 0.16, antiderivative size = 116, 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 = {371, 1398, 825, 827, 1166, 207} \[ 4 \sqrt {a+b \sqrt {c+d x}}-2 \sqrt {a-b \sqrt {c}} \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a-b \sqrt {c}}}\right )-2 \sqrt {a+b \sqrt {c}} \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a+b \sqrt {c}}}\right ) \]
Antiderivative was successfully verified.
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Rule 207
Rule 371
Rule 825
Rule 827
Rule 1166
Rule 1398
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b \sqrt {c+d x}}}{x} \, dx &=\operatorname {Subst}\left (\int \frac {\sqrt {a+b \sqrt {x}}}{-c+x} \, dx,x,c+d x\right )\\ &=2 \operatorname {Subst}\left (\int \frac {x \sqrt {a+b x}}{-c+x^2} \, dx,x,\sqrt {c+d x}\right )\\ &=4 \sqrt {a+b \sqrt {c+d x}}+2 \operatorname {Subst}\left (\int \frac {b c+a x}{\sqrt {a+b x} \left (-c+x^2\right )} \, dx,x,\sqrt {c+d x}\right )\\ &=4 \sqrt {a+b \sqrt {c+d x}}+4 \operatorname {Subst}\left (\int \frac {-a^2+b^2 c+a x^2}{a^2-b^2 c-2 a x^2+x^4} \, dx,x,\sqrt {a+b \sqrt {c+d x}}\right )\\ &=4 \sqrt {a+b \sqrt {c+d x}}+\left (2 \left (a-b \sqrt {c}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-a+b \sqrt {c}+x^2} \, dx,x,\sqrt {a+b \sqrt {c+d x}}\right )+\left (2 \left (a+b \sqrt {c}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-a-b \sqrt {c}+x^2} \, dx,x,\sqrt {a+b \sqrt {c+d x}}\right )\\ &=4 \sqrt {a+b \sqrt {c+d x}}-2 \sqrt {a-b \sqrt {c}} \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a-b \sqrt {c}}}\right )-2 \sqrt {a+b \sqrt {c}} \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a+b \sqrt {c}}}\right )\\ \end {align*}
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Mathematica [A] time = 0.13, size = 116, normalized size = 1.00 \[ 4 \sqrt {a+b \sqrt {c+d x}}-2 \sqrt {a-b \sqrt {c}} \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a-b \sqrt {c}}}\right )-2 \sqrt {a+b \sqrt {c}} \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a+b \sqrt {c}}}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.54, size = 194, normalized size = 1.67 \[ -\sqrt {a + \sqrt {b^{2} c}} \log \left (2 \, \sqrt {\sqrt {d x + c} b + a} + 2 \, \sqrt {a + \sqrt {b^{2} c}}\right ) + \sqrt {a + \sqrt {b^{2} c}} \log \left (2 \, \sqrt {\sqrt {d x + c} b + a} - 2 \, \sqrt {a + \sqrt {b^{2} c}}\right ) - \sqrt {a - \sqrt {b^{2} c}} \log \left (2 \, \sqrt {\sqrt {d x + c} b + a} + 2 \, \sqrt {a - \sqrt {b^{2} c}}\right ) + \sqrt {a - \sqrt {b^{2} c}} \log \left (2 \, \sqrt {\sqrt {d x + c} b + a} - 2 \, \sqrt {a - \sqrt {b^{2} c}}\right ) + 4 \, \sqrt {\sqrt {d x + c} b + a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.49, size = 150, normalized size = 1.29 \[ \frac {2 \, {\left (2 \, \sqrt {\sqrt {d x + c} b + a} b - \frac {{\left (b^{3} c - a^{2} b\right )} \arctan \left (\frac {\sqrt {\sqrt {d x + c} b + a}}{\sqrt {-a + \sqrt {b^{2} c}}}\right )}{{\left (b \sqrt {c} + a\right )} \sqrt {b \sqrt {c} - a}} + \frac {{\left (b^{3} c - a^{2} b\right )} \arctan \left (\frac {\sqrt {\sqrt {d x + c} b + a}}{\sqrt {-a - \sqrt {b^{2} c}}}\right )}{{\left (b \sqrt {c} - a\right )} \sqrt {-b \sqrt {c} - a}}\right )}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 221, normalized size = 1.91 \[ \frac {2 b^{2} c \arctan \left (\frac {\sqrt {a +\sqrt {d x +c}\, b}}{\sqrt {-a -\sqrt {b^{2} c}}}\right )}{\sqrt {b^{2} c}\, \sqrt {-a -\sqrt {b^{2} c}}}-\frac {2 b^{2} c \arctan \left (\frac {\sqrt {a +\sqrt {d x +c}\, b}}{\sqrt {-a +\sqrt {b^{2} c}}}\right )}{\sqrt {b^{2} c}\, \sqrt {-a +\sqrt {b^{2} c}}}+\frac {2 a \arctan \left (\frac {\sqrt {a +\sqrt {d x +c}\, b}}{\sqrt {-a -\sqrt {b^{2} c}}}\right )}{\sqrt {-a -\sqrt {b^{2} c}}}+\frac {2 a \arctan \left (\frac {\sqrt {a +\sqrt {d x +c}\, b}}{\sqrt {-a +\sqrt {b^{2} c}}}\right )}{\sqrt {-a +\sqrt {b^{2} c}}}+4 \sqrt {a +\sqrt {d x +c}\, b} \]
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 \[ \int \frac {\sqrt {\sqrt {d x + c} b + a}}{x}\,{d x} \]
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 \[ \int \frac {\sqrt {a+b\,\sqrt {c+d\,x}}}{x} \,d x \]
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 \[ \int \frac {\sqrt {a + b \sqrt {c + d x}}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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