Optimal. Leaf size=55 \[ \frac {2}{3} \sqrt {a+b \left (c x^2\right )^{3/2}}-\frac {2}{3} \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{\sqrt {a}}\right ) \]
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Rubi [A] time = 0.04, 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 = {368, 266, 50, 63, 208} \begin {gather*} \frac {2}{3} \sqrt {a+b \left (c x^2\right )^{3/2}}-\frac {2}{3} \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{\sqrt {a}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 208
Rule 266
Rule 368
Rubi steps
\begin {align*} \int \frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{x} \, dx &=\operatorname {Subst}\left (\int \frac {\sqrt {a+b x^3}}{x} \, dx,x,\sqrt {c x^2}\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {\sqrt {a+b x}}{x} \, dx,x,\left (c x^2\right )^{3/2}\right )\\ &=\frac {2}{3} \sqrt {a+b \left (c x^2\right )^{3/2}}+\frac {1}{3} a \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\left (c x^2\right )^{3/2}\right )\\ &=\frac {2}{3} \sqrt {a+b \left (c x^2\right )^{3/2}}+\frac {(2 a) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \left (c x^2\right )^{3/2}}\right )}{3 b}\\ &=\frac {2}{3} \sqrt {a+b \left (c x^2\right )^{3/2}}-\frac {2}{3} \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{\sqrt {a}}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 55, normalized size = 1.00 \begin {gather*} \frac {2}{3} \sqrt {a+b \left (c x^2\right )^{3/2}}-\frac {2}{3} \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \left (c x^2\right )^{3/2}}}{\sqrt {a}}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.34, size = 61, normalized size = 1.11 \begin {gather*} \frac {2}{3} \sqrt {a+b c^{3/2} \left (x^2\right )^{3/2}}-\frac {2}{3} \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b c^{3/2} \left (x^2\right )^{3/2}}}{\sqrt {a}}\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.85, size = 133, normalized size = 2.42 \begin {gather*} \left [\frac {1}{3} \, \sqrt {a} \log \left (\frac {b c^{2} x^{4} - 2 \, \sqrt {\sqrt {c x^{2}} b c x^{2} + a} \sqrt {c x^{2}} \sqrt {a} + 2 \, \sqrt {c x^{2}} a}{x^{4}}\right ) + \frac {2}{3} \, \sqrt {\sqrt {c x^{2}} b c x^{2} + a}, \frac {2}{3} \, \sqrt {-a} \arctan \left (\frac {\sqrt {\sqrt {c x^{2}} b c x^{2} + a} \sqrt {-a}}{a}\right ) + \frac {2}{3} \, \sqrt {\sqrt {c x^{2}} b c x^{2} + a}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 42, normalized size = 0.76 \begin {gather*} \frac {2 \, a \arctan \left (\frac {\sqrt {b c^{\frac {3}{2}} x^{3} + a}}{\sqrt {-a}}\right )}{3 \, \sqrt {-a}} + \frac {2}{3} \, \sqrt {b c^{\frac {3}{2}} x^{3} + a} \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {a +\left (c \,x^{2}\right )^{\frac {3}{2}} b}}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.15, size = 61, normalized size = 1.11 \begin {gather*} \frac {1}{3} \, \sqrt {a} \log \left (\frac {\sqrt {\left (c x^{2}\right )^{\frac {3}{2}} b + a} - \sqrt {a}}{\sqrt {\left (c x^{2}\right )^{\frac {3}{2}} b + a} + \sqrt {a}}\right ) + \frac {2}{3} \, \sqrt {\left (c x^{2}\right )^{\frac {3}{2}} b + a} \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\,{\left (c\,x^2\right )}^{3/2}}}{x} \,d x \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 \left (c x^{2}\right )^{\frac {3}{2}}}}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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