Optimal. Leaf size=55 \[ \frac {1}{2} a^2 x \sqrt {c x^2}+\frac {2}{3} a b x^2 \sqrt {c x^2}+\frac {1}{4} b^2 x^3 \sqrt {c x^2} \]
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Rubi [A] time = 0.01, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {15, 43} \[ \frac {1}{2} a^2 x \sqrt {c x^2}+\frac {2}{3} a b x^2 \sqrt {c x^2}+\frac {1}{4} b^2 x^3 \sqrt {c x^2} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin {align*} \int \sqrt {c x^2} (a+b x)^2 \, dx &=\frac {\sqrt {c x^2} \int x (a+b x)^2 \, dx}{x}\\ &=\frac {\sqrt {c x^2} \int \left (a^2 x+2 a b x^2+b^2 x^3\right ) \, dx}{x}\\ &=\frac {1}{2} a^2 x \sqrt {c x^2}+\frac {2}{3} a b x^2 \sqrt {c x^2}+\frac {1}{4} b^2 x^3 \sqrt {c x^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 33, normalized size = 0.60 \[ \frac {1}{12} x \sqrt {c x^2} \left (6 a^2+8 a b x+3 b^2 x^2\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 31, normalized size = 0.56 \[ \frac {1}{12} \, {\left (3 \, b^{2} x^{3} + 8 \, a b x^{2} + 6 \, a^{2} x\right )} \sqrt {c x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.95, size = 35, normalized size = 0.64 \[ \frac {1}{12} \, {\left (3 \, b^{2} x^{4} \mathrm {sgn}\relax (x) + 8 \, a b x^{3} \mathrm {sgn}\relax (x) + 6 \, a^{2} x^{2} \mathrm {sgn}\relax (x)\right )} \sqrt {c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 30, normalized size = 0.55 \[ \frac {\left (3 b^{2} x^{2}+8 a b x +6 a^{2}\right ) \sqrt {c \,x^{2}}\, x}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 44, normalized size = 0.80 \[ \frac {1}{2} \, \sqrt {c x^{2}} a^{2} x + \frac {\left (c x^{2}\right )^{\frac {3}{2}} b^{2} x}{4 \, c} + \frac {2 \, \left (c x^{2}\right )^{\frac {3}{2}} a b}{3 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \sqrt {c\,x^2}\,{\left (a+b\,x\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.30, size = 60, normalized size = 1.09 \[ \frac {a^{2} \sqrt {c} x \sqrt {x^{2}}}{2} + \frac {2 a b \sqrt {c} x^{2} \sqrt {x^{2}}}{3} + \frac {b^{2} \sqrt {c} x^{3} \sqrt {x^{2}}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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