Optimal. Leaf size=55 \[ \frac {4}{3} \sqrt {a+b \sqrt {c x^3}}-\frac {4}{3} \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c x^3}}}{\sqrt {a}}\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {376, 272, 52,
65, 214} \begin {gather*} \frac {4}{3} \sqrt {a+b \sqrt {c x^3}}-\frac {4}{3} \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c x^3}}}{\sqrt {a}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 214
Rule 272
Rule 376
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b \sqrt {c x^3}}}{x} \, dx &=\text {Subst}\left (\int \frac {\sqrt {a+b \sqrt {c} x^{3/2}}}{x} \, dx,\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )\\ &=\text {Subst}\left (\frac {2}{3} \text {Subst}\left (\int \frac {\sqrt {a+b \sqrt {c} x}}{x} \, dx,x,x^{3/2}\right ),\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )\\ &=\frac {4}{3} \sqrt {a+b \sqrt {c x^3}}+\text {Subst}\left (\frac {1}{3} (2 a) \text {Subst}\left (\int \frac {1}{x \sqrt {a+b \sqrt {c} x}} \, dx,x,x^{3/2}\right ),\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )\\ &=\frac {4}{3} \sqrt {a+b \sqrt {c x^3}}+\text {Subst}\left (\frac {(4 a) \text {Subst}\left (\int \frac {1}{-\frac {a}{b \sqrt {c}}+\frac {x^2}{b \sqrt {c}}} \, dx,x,\sqrt {a+b \sqrt {c} x^{3/2}}\right )}{3 b \sqrt {c}},\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )\\ &=\frac {4}{3} \sqrt {a+b \sqrt {c x^3}}-\frac {4}{3} \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c x^3}}}{\sqrt {a}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 55, normalized size = 1.00 \begin {gather*} \frac {4}{3} \sqrt {a+b \sqrt {c x^3}}-\frac {4}{3} \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c x^3}}}{\sqrt {a}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.64, size = 40, normalized size = 0.73
method | result | size |
default | \(-\frac {4 \arctanh \left (\frac {\sqrt {a +b \sqrt {c \,x^{3}}}}{\sqrt {a}}\right ) \sqrt {a}}{3}+\frac {4 \sqrt {a +b \sqrt {c \,x^{3}}}}{3}\) | \(40\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.57, size = 61, normalized size = 1.11 \begin {gather*} \frac {2}{3} \, \sqrt {a} \log \left (\frac {\sqrt {\sqrt {c x^{3}} b + a} - \sqrt {a}}{\sqrt {\sqrt {c x^{3}} b + a} + \sqrt {a}}\right ) + \frac {4}{3} \, \sqrt {\sqrt {c x^{3}} b + a} \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 {\sqrt {a + b \sqrt {c x^{3}}}}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.31, size = 67, normalized size = 1.22 \begin {gather*} \frac {4 \, {\left (\frac {a \arctan \left (\frac {\sqrt {\sqrt {c x} b c^{2} x + a c^{2}}}{\sqrt {-a} c}\right )}{\sqrt {-a}} + \frac {\sqrt {\sqrt {c x} b c^{2} x + a c^{2}}}{c}\right )} {\left | c \right |}}{3 \, c} \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 {\sqrt {a+b\,\sqrt {c\,x^3}}}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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