Optimal. Leaf size=55 \[ \frac {4}{9} \sqrt {a+b \left (c x^3\right )^{3/2}}-\frac {4}{9} \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \left (c x^3\right )^{3/2}}}{\sqrt {a}}\right ) \]
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Rubi [A] time = 0.06, 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 = {369, 266, 50, 63, 208} \[ \frac {4}{9} \sqrt {a+b \left (c x^3\right )^{3/2}}-\frac {4}{9} \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \left (c x^3\right )^{3/2}}}{\sqrt {a}}\right ) \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 208
Rule 266
Rule 369
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b \left (c x^3\right )^{3/2}}}{x} \, dx &=\operatorname {Subst}\left (\int \frac {\sqrt {a+b c^{3/2} x^{9/2}}}{x} \, dx,\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )\\ &=\operatorname {Subst}\left (\frac {2}{9} \operatorname {Subst}\left (\int \frac {\sqrt {a+b c^{3/2} x}}{x} \, dx,x,x^{9/2}\right ),\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )\\ &=\frac {4}{9} \sqrt {a+b \left (c x^3\right )^{3/2}}+\operatorname {Subst}\left (\frac {1}{9} (2 a) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b c^{3/2} x}} \, dx,x,x^{9/2}\right ),\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )\\ &=\frac {4}{9} \sqrt {a+b \left (c x^3\right )^{3/2}}+\operatorname {Subst}\left (\frac {(4 a) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b c^{3/2}}+\frac {x^2}{b c^{3/2}}} \, dx,x,\sqrt {a+b c^{3/2} x^{9/2}}\right )}{9 b c^{3/2}},\sqrt {x},\frac {\sqrt {c x^3}}{\sqrt {c} x}\right )\\ &=\frac {4}{9} \sqrt {a+b \left (c x^3\right )^{3/2}}-\frac {4}{9} \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \left (c x^3\right )^{3/2}}}{\sqrt {a}}\right )\\ \end {align*}
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Mathematica [A] time = 0.07, size = 55, normalized size = 1.00 \[ \frac {4}{9} \sqrt {a+b \left (c x^3\right )^{3/2}}-\frac {4}{9} \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \left (c x^3\right )^{3/2}}}{\sqrt {a}}\right ) \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 74, normalized size = 1.35 \[ \frac {4 \, {\left (\frac {a c \arctan \left (\frac {\sqrt {\sqrt {c x} b c^{5} x^{4} + a c^{4}}}{\sqrt {-a} c^{2}}\right )}{\sqrt {-a}} + \frac {\sqrt {\sqrt {c x} b c^{5} x^{4} + a c^{4}}}{c}\right )} {\left | c \right |}^{2}}{9 \, c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.23, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a +\left (c \,x^{3}\right )^{\frac {3}{2}} b}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.24, size = 61, normalized size = 1.11 \[ \frac {2}{9} \, \sqrt {a} \log \left (\frac {\sqrt {\left (c x^{3}\right )^{\frac {3}{2}} b + a} - \sqrt {a}}{\sqrt {\left (c x^{3}\right )^{\frac {3}{2}} b + a} + \sqrt {a}}\right ) + \frac {4}{9} \, \sqrt {\left (c x^{3}\right )^{\frac {3}{2}} b + a} \]
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 \[ \int \frac {\sqrt {a+b\,{\left (c\,x^3\right )}^{3/2}}}{x} \,d x \]
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 \[ \int \frac {\sqrt {a + b \left (c x^{3}\right )^{\frac {3}{2}}}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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