Optimal. Leaf size=71 \[ \frac {c \sqrt {\frac {c}{a+b x^2}}}{a}-\frac {c \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {c}{a+b x^2}} \tanh ^{-1}\left (\sqrt {\frac {b x^2}{a}+1}\right )}{a} \]
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Rubi [A] time = 0.14, antiderivative size = 73, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {6720, 266, 51, 63, 208} \[ \frac {c \sqrt {\frac {c}{a+b x^2}}}{a}-\frac {c \sqrt {a+b x^2} \sqrt {\frac {c}{a+b x^2}} \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{3/2}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rule 266
Rule 6720
Rubi steps
\begin {align*} \int \frac {\left (\frac {c}{a+b x^2}\right )^{3/2}}{x} \, dx &=\left (c \sqrt {\frac {c}{a+b x^2}} \sqrt {a+b x^2}\right ) \int \frac {1}{x \left (a+b x^2\right )^{3/2}} \, dx\\ &=\frac {1}{2} \left (c \sqrt {\frac {c}{a+b x^2}} \sqrt {a+b x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{x (a+b x)^{3/2}} \, dx,x,x^2\right )\\ &=\frac {c \sqrt {\frac {c}{a+b x^2}}}{a}+\frac {\left (c \sqrt {\frac {c}{a+b x^2}} \sqrt {a+b x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^2\right )}{2 a}\\ &=\frac {c \sqrt {\frac {c}{a+b x^2}}}{a}+\frac {\left (c \sqrt {\frac {c}{a+b x^2}} \sqrt {a+b x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^2}\right )}{a b}\\ &=\frac {c \sqrt {\frac {c}{a+b x^2}}}{a}-\frac {c \sqrt {\frac {c}{a+b x^2}} \sqrt {a+b x^2} \tanh ^{-1}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{3/2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 38, normalized size = 0.54 \[ \frac {c \sqrt {\frac {c}{a+b x^2}} \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {b x^2}{a}+1\right )}{a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 138, normalized size = 1.94 \[ \left [\frac {c \sqrt {\frac {c}{a}} \log \left (-\frac {b c x^{2} + 2 \, a c - 2 \, {\left (a b x^{2} + a^{2}\right )} \sqrt {\frac {c}{b x^{2} + a}} \sqrt {\frac {c}{a}}}{x^{2}}\right ) + 2 \, c \sqrt {\frac {c}{b x^{2} + a}}}{2 \, a}, \frac {c \sqrt {-\frac {c}{a}} \arctan \left (\frac {a \sqrt {\frac {c}{b x^{2} + a}} \sqrt {-\frac {c}{a}}}{c}\right ) + c \sqrt {\frac {c}{b x^{2} + a}}}{a}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 59, normalized size = 0.83 \[ c {\left (\frac {c \arctan \left (\frac {\sqrt {b c x^{2} + a c}}{\sqrt {-a c}}\right )}{\sqrt {-a c} a} + \frac {c}{\sqrt {b c x^{2} + a c} a}\right )} \mathrm {sgn}\left (b x^{2} + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 64, normalized size = 0.90 \[ -\frac {\left (\frac {c}{b \,x^{2}+a}\right )^{\frac {3}{2}} \left (b \,x^{2}+a \right ) \left (\sqrt {b \,x^{2}+a}\, a \ln \left (\frac {2 a +2 \sqrt {b \,x^{2}+a}\, \sqrt {a}}{x}\right )-a^{\frac {3}{2}}\right )}{a^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.95, size = 80, normalized size = 1.13 \[ \frac {1}{2} \, c {\left (\frac {c \log \left (\frac {a \sqrt {\frac {c}{b x^{2} + a}} - \sqrt {a c}}{a \sqrt {\frac {c}{b x^{2} + a}} + \sqrt {a c}}\right )}{\sqrt {a c} a} + \frac {2 \, \sqrt {\frac {c}{b x^{2} + a}}}{a}\right )} \]
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 \[ \int \frac {{\left (\frac {c}{b\,x^2+a}\right )}^{3/2}}{x} \,d x \]
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 \[ \int \frac {\left (\frac {c}{a + b x^{2}}\right )^{\frac {3}{2}}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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