Optimal. Leaf size=27 \[ \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 b} \]
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Rubi [A] time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {629} \begin {gather*} \frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 629
Rubi steps
\begin {align*} \int (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} \, dx &=\frac {\left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{3 b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 18, normalized size = 0.67 \begin {gather*} \frac {\left ((a+b x)^2\right )^{3/2}}{3 b} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.02, size = 18, normalized size = 0.67 \begin {gather*} \frac {\left ((a+b x)^2\right )^{3/2}}{3 b} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 20, normalized size = 0.74 \begin {gather*} \frac {1}{3} \, b^{2} x^{3} + a b x^{2} + a^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 18, normalized size = 0.67 \begin {gather*} \frac {{\left (b x + a\right )}^{3} \mathrm {sgn}\left (b x + a\right )}{3 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 38, normalized size = 1.41 \begin {gather*} \frac {\left (b^{2} x^{2}+3 a b x +3 a^{2}\right ) \sqrt {\left (b x +a \right )^{2}}\, x}{3 b x +3 a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.70, size = 14, normalized size = 0.52 \begin {gather*} \frac {{\left ({\left (b x + a\right )}^{2}\right )}^{\frac {3}{2}}}{3 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.17, size = 76, normalized size = 2.81 \begin {gather*} \frac {\left (8\,b^2\,\left (a^2+b^2\,x^2\right )-12\,a^2\,b^2+4\,a\,b^3\,x\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{24\,b^3}+\frac {a\,\sqrt {{\left (a+b\,x\right )}^2}\,\left (a+b\,x\right )}{2\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 19, normalized size = 0.70 \begin {gather*} a^{2} x + a b x^{2} + \frac {b^{2} x^{3}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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