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, 45}
\begin {gather*} \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 {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 45
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 \begin {gather*} \frac {1}{12} x \sqrt {c x^2} \left (6 a^2+8 a b x+3 b^2 x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 30, normalized size = 0.55
method | result | size |
gosper | \(\frac {x \left (3 x^{2} b^{2}+8 a b x +6 a^{2}\right ) \sqrt {c \,x^{2}}}{12}\) | \(30\) |
default | \(\frac {x \left (3 x^{2} b^{2}+8 a b x +6 a^{2}\right ) \sqrt {c \,x^{2}}}{12}\) | \(30\) |
risch | \(\frac {a^{2} x \sqrt {c \,x^{2}}}{2}+\frac {2 a b \,x^{2} \sqrt {c \,x^{2}}}{3}+\frac {b^{2} x^{3} \sqrt {c \,x^{2}}}{4}\) | \(44\) |
trager | \(\frac {\left (3 b^{2} x^{3}+8 a b \,x^{2}+3 x^{2} b^{2}+6 a^{2} x +8 a b x +3 b^{2} x +6 a^{2}+8 a b +3 b^{2}\right ) \left (-1+x \right ) \sqrt {c \,x^{2}}}{12 x}\) | \(71\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 44, normalized size = 0.80 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.51, size = 31, normalized size = 0.56 \begin {gather*} \frac {1}{12} \, {\left (3 \, b^{2} x^{3} + 8 \, a b x^{2} + 6 \, a^{2} x\right )} \sqrt {c x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.11, size = 49, normalized size = 0.89 \begin {gather*} \frac {a^{2} x \sqrt {c x^{2}}}{2} + \frac {2 a b x^{2} \sqrt {c x^{2}}}{3} + \frac {b^{2} x^{3} \sqrt {c x^{2}}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.05, size = 35, normalized size = 0.64 \begin {gather*} \frac {1}{12} \, {\left (3 \, b^{2} x^{4} \mathrm {sgn}\left (x\right ) + 8 \, a b x^{3} \mathrm {sgn}\left (x\right ) + 6 \, a^{2} x^{2} \mathrm {sgn}\left (x\right )\right )} \sqrt {c} \end {gather*}
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 \begin {gather*} \int \sqrt {c\,x^2}\,{\left (a+b\,x\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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