Optimal. Leaf size=54 \[ \frac {2 \tanh ^{-1}\left (\frac {2 a+b \sqrt {\frac {d}{x}}}{2 \sqrt {a} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )}{\sqrt {a}} \]
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Rubi [A]
time = 0.07, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1994, 1371,
738, 212} \begin {gather*} \frac {2 \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 )}{\sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 738
Rule 1371
Rule 1994
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}} x} \, dx &=-\text {Subst}\left (\int \frac {1}{x \sqrt {a+b \sqrt {x}+\frac {c x}{d}}} \, dx,x,\frac {d}{x}\right )\\ &=-\left (2 \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x+\frac {c x^2}{d}}} \, dx,x,\sqrt {\frac {d}{x}}\right )\right )\\ &=4 \text {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 )\\ &=\frac {2 \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 )}{\sqrt {a}}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(115\) vs. \(2(54)=108\).
time = 0.25, size = 115, normalized size = 2.13 \begin {gather*} \frac {4 \sqrt {\frac {d \left (c+\left (a+b \sqrt {\frac {d}{x}}\right ) x\right )}{x}} \tanh ^{-1}\left (\frac {-\sqrt {c} \sqrt {\frac {d}{x}}+\sqrt {\frac {d \left (c+a x+b \sqrt {\frac {d}{x}} x\right )}{x}}}{\sqrt {a} \sqrt {d}}\right )}{\sqrt {a} \sqrt {d} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(93\) vs.
\(2(42)=84\).
time = 0.07, size = 94, normalized size = 1.74
method | result | size |
default | \(\frac {2 \sqrt {\frac {b \sqrt {\frac {d}{x}}\, x +a x +c}{x}}\, \sqrt {x}\, \ln \left (\frac {b \sqrt {\frac {d}{x}}\, \sqrt {x}+2 \sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, \sqrt {a}+2 a \sqrt {x}}{2 \sqrt {a}}\right )}{\sqrt {b \sqrt {\frac {d}{x}}\, x +a x +c}\, \sqrt {a}}\) | \(94\) |
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 [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 {1}{x \sqrt {a + b \sqrt {\frac {d}{x}} + \frac {c}{x}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 110 vs.
\(2 (42) = 84\).
time = 1.30, size = 110, normalized size = 2.04 \begin {gather*} -\frac {2 \, \sqrt {d} {\left (\frac {\sqrt {a d} \log \left ({\left | -\sqrt {a d} b d - 2 \, {\left (\sqrt {a d} \sqrt {d x} - \sqrt {a d^{2} x + \sqrt {d x} b d^{2} + c d^{2}}\right )} a \right |}\right )}{a d} - \frac {\sqrt {a d} \log \left ({\left | -\sqrt {a d} b d + 2 \, \sqrt {c d^{2}} a \right |}\right )}{a d}\right )}}{\mathrm {sgn}\left (x\right )} \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 = -0.02 \begin {gather*} \int \frac {1}{x\,\sqrt {a+\frac {c}{x}+b\,\sqrt {\frac {d}{x}}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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