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 = 0.10, 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 = {378, 1412, 839,
841, 1180, 213} \begin {gather*} 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 {gather*}
Antiderivative was successfully verified.
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Rule 213
Rule 378
Rule 839
Rule 841
Rule 1180
Rule 1412
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b \sqrt {c+d x}}}{x} \, dx &=\text {Subst}\left (\int \frac {\sqrt {a+b \sqrt {x}}}{-c+x} \, dx,x,c+d x\right )\\ &=2 \text {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 \text {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 \text {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 ) \text {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 ) \text {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 = 0.23, size = 124, normalized size = 1.07 \begin {gather*} 4 \sqrt {a+b \sqrt {c+d x}}-2 \sqrt {-a-b \sqrt {c}} \tan ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {-a-b \sqrt {c}}}\right )-2 \sqrt {-a+b \sqrt {c}} \tan ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {-a+b \sqrt {c}}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 152, normalized size = 1.31
method | result | size |
derivativedivides | \(4 \sqrt {a +b \sqrt {d x +c}}+\frac {2 \left (-b^{2} c +a \sqrt {b^{2} c}\right ) \arctan \left (\frac {\sqrt {a +b \sqrt {d x +c}}}{\sqrt {\sqrt {b^{2} c}-a}}\right )}{\sqrt {b^{2} c}\, \sqrt {\sqrt {b^{2} c}-a}}+\frac {2 \left (b^{2} c +a \sqrt {b^{2} c}\right ) \arctan \left (\frac {\sqrt {a +b \sqrt {d x +c}}}{\sqrt {-\sqrt {b^{2} c}-a}}\right )}{\sqrt {b^{2} c}\, \sqrt {-\sqrt {b^{2} c}-a}}\) | \(152\) |
default | \(4 \sqrt {a +b \sqrt {d x +c}}+\frac {2 \left (-b^{2} c +a \sqrt {b^{2} c}\right ) \arctan \left (\frac {\sqrt {a +b \sqrt {d x +c}}}{\sqrt {\sqrt {b^{2} c}-a}}\right )}{\sqrt {b^{2} c}\, \sqrt {\sqrt {b^{2} c}-a}}+\frac {2 \left (b^{2} c +a \sqrt {b^{2} c}\right ) \arctan \left (\frac {\sqrt {a +b \sqrt {d x +c}}}{\sqrt {-\sqrt {b^{2} c}-a}}\right )}{\sqrt {b^{2} c}\, \sqrt {-\sqrt {b^{2} c}-a}}\) | \(152\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 194 vs.
\(2 (88) = 176\).
time = 0.40, size = 194, normalized size = 1.67 \begin {gather*} -\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} \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 {a + b \sqrt {c + d x}}}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.94, size = 150, normalized size = 1.29 \begin {gather*} \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} \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 {a+b\,\sqrt {c+d\,x}}}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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