Optimal. Leaf size=32 \[ \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{4 b} \]
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Rubi [A] time = 0.00, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {609} \[ \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{4 b} \]
Antiderivative was successfully verified.
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Rule 609
Rubi steps
\begin {align*} \int \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx &=\frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{4 b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 23, normalized size = 0.72 \[ \frac {(a+b x) \left ((a+b x)^2\right )^{3/2}}{4 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.97, size = 31, normalized size = 0.97 \[ \frac {1}{4} \, b^{3} x^{4} + a b^{2} x^{3} + \frac {3}{2} \, a^{2} b x^{2} + a^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.23, size = 57, normalized size = 1.78 \[ \frac {1}{2} \, {\left (b x^{2} + 2 \, a x\right )} a^{2} \mathrm {sgn}\left (b x + a\right ) + \frac {a^{4} \mathrm {sgn}\left (b x + a\right )}{4 \, b} + \frac {1}{4} \, {\left (b x^{2} + 2 \, a x\right )}^{2} b \mathrm {sgn}\left (b x + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 49, normalized size = 1.53 \[ \frac {\left (b^{3} x^{3}+4 a \,b^{2} x^{2}+6 a^{2} b x +4 a^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}} x}{4 \left (b x +a \right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 46, normalized size = 1.44 \[ \frac {1}{4} \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} x + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} a}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.16, size = 32, normalized size = 1.00 \[ \frac {\left (x\,b^2+a\,b\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{4\,b^2} \]
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 \[ \int \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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