Optimal. Leaf size=28 \[ -\frac {c \sqrt {\frac {c}{(a+b x)^2}}}{2 b (a+b x)} \]
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Rubi [A] time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {247, 15, 30} \[ -\frac {c \sqrt {\frac {c}{(a+b x)^2}}}{2 b (a+b x)} \]
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rule 247
Rubi steps
\begin {align*} \int \left (\frac {c}{(a+b x)^2}\right )^{3/2} \, dx &=\frac {\operatorname {Subst}\left (\int \left (\frac {c}{x^2}\right )^{3/2} \, dx,x,a+b x\right )}{b}\\ &=\frac {\left (c \sqrt {\frac {c}{(a+b x)^2}} (a+b x)\right ) \operatorname {Subst}\left (\int \frac {1}{x^3} \, dx,x,a+b x\right )}{b}\\ &=-\frac {c \sqrt {\frac {c}{(a+b x)^2}}}{2 b (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 0.89 \[ -\frac {(a+b x) \left (\frac {c}{(a+b x)^2}\right )^{3/2}}{2 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 36, normalized size = 1.29 \[ -\frac {c \sqrt {\frac {c}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{2 \, {\left (b^{2} x + a b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 21, normalized size = 0.75 \[ -\frac {c^{\frac {3}{2}} \mathrm {sgn}\left (b x + a\right )}{2 \, {\left (b x + a\right )}^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 22, normalized size = 0.79 \[ -\frac {\left (b x +a \right ) \left (\frac {c}{\left (b x +a \right )^{2}}\right )^{\frac {3}{2}}}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.66, size = 27, normalized size = 0.96 \[ -\frac {c^{\frac {3}{2}}}{2 \, {\left (b^{3} x^{2} + 2 \, a b^{2} x + a^{2} b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.17, size = 24, normalized size = 0.86 \[ -\frac {c\,\sqrt {\frac {c}{{\left (a+b\,x\right )}^2}}}{2\,b\,\left (a+b\,x\right )} \]
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 \[ \int \left (\frac {c}{\left (a + b x\right )^{2}}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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