Optimal. Leaf size=21 \[ -\frac {c \sqrt {\frac {c}{a+b x^2}}}{b} \]
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Rubi [A] time = 0.02, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {1591, 15, 30} \[ -\frac {c \sqrt {\frac {c}{a+b x^2}}}{b} \]
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rule 1591
Rubi steps
\begin {align*} \int x \left (\frac {c}{a+b x^2}\right )^{3/2} \, dx &=\frac {\operatorname {Subst}\left (\int \left (\frac {c}{x}\right )^{3/2} \, dx,x,a+b x^2\right )}{2 b}\\ &=\frac {\left (c \sqrt {\frac {c}{a+b x^2}} \sqrt {a+b x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{x^{3/2}} \, dx,x,a+b x^2\right )}{2 b}\\ &=-\frac {c \sqrt {\frac {c}{a+b x^2}}}{b}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 21, normalized size = 1.00 \[ -\frac {c \sqrt {\frac {c}{a+b x^2}}}{b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 19, normalized size = 0.90 \[ -\frac {c \sqrt {\frac {c}{b x^{2} + a}}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 28, normalized size = 1.33 \[ -\frac {c^{2} \mathrm {sgn}\left (b x^{2} + a\right )}{\sqrt {b c x^{2} + a c} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 26, normalized size = 1.24 \[ -\frac {\left (b \,x^{2}+a \right ) \left (\frac {c}{b \,x^{2}+a}\right )^{\frac {3}{2}}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.06, size = 19, normalized size = 0.90 \[ -\frac {c \sqrt {\frac {c}{b x^{2} + a}}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.64, size = 19, normalized size = 0.90 \[ -\frac {c\,\sqrt {\frac {c}{b\,x^2+a}}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.51, size = 53, normalized size = 2.52 \[ \begin {cases} - \frac {a c^{\frac {3}{2}} \left (\frac {1}{a + b x^{2}}\right )^{\frac {3}{2}}}{b} - c^{\frac {3}{2}} x^{2} \left (\frac {1}{a + b x^{2}}\right )^{\frac {3}{2}} & \text {for}\: b \neq 0 \\\frac {x^{2} \left (\frac {c}{a}\right )^{\frac {3}{2}}}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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