Optimal. Leaf size=57 \[ \frac {1}{3} a^2 x^2 \sqrt {c x^2}+\frac {1}{2} a b x^3 \sqrt {c x^2}+\frac {1}{5} b^2 x^4 \sqrt {c x^2} \]
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Rubi [A]
time = 0.01, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {15, 45}
\begin {gather*} \frac {1}{3} a^2 x^2 \sqrt {c x^2}+\frac {1}{2} a b x^3 \sqrt {c x^2}+\frac {1}{5} b^2 x^4 \sqrt {c x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 45
Rubi steps
\begin {align*} \int x \sqrt {c x^2} (a+b x)^2 \, dx &=\frac {\sqrt {c x^2} \int x^2 (a+b x)^2 \, dx}{x}\\ &=\frac {\sqrt {c x^2} \int \left (a^2 x^2+2 a b x^3+b^2 x^4\right ) \, dx}{x}\\ &=\frac {1}{3} a^2 x^2 \sqrt {c x^2}+\frac {1}{2} a b x^3 \sqrt {c x^2}+\frac {1}{5} b^2 x^4 \sqrt {c x^2}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 35, normalized size = 0.61 \begin {gather*} \frac {1}{30} x^2 \sqrt {c x^2} \left (10 a^2+15 a b x+6 b^2 x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 32, normalized size = 0.56
method | result | size |
gosper | \(\frac {x^{2} \left (6 x^{2} b^{2}+15 a b x +10 a^{2}\right ) \sqrt {c \,x^{2}}}{30}\) | \(32\) |
default | \(\frac {x^{2} \left (6 x^{2} b^{2}+15 a b x +10 a^{2}\right ) \sqrt {c \,x^{2}}}{30}\) | \(32\) |
risch | \(\frac {a^{2} x^{2} \sqrt {c \,x^{2}}}{3}+\frac {a b \,x^{3} \sqrt {c \,x^{2}}}{2}+\frac {b^{2} x^{4} \sqrt {c \,x^{2}}}{5}\) | \(46\) |
trager | \(\frac {\left (6 b^{2} x^{4}+15 a b \,x^{3}+6 b^{2} x^{3}+10 a^{2} x^{2}+15 a b \,x^{2}+6 x^{2} b^{2}+10 a^{2} x +15 a b x +6 b^{2} x +10 a^{2}+15 a b +6 b^{2}\right ) \left (-1+x \right ) \sqrt {c \,x^{2}}}{30 x}\) | \(94\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 49, normalized size = 0.86 \begin {gather*} \frac {\left (c x^{2}\right )^{\frac {3}{2}} b^{2} x^{2}}{5 \, c} + \frac {\left (c x^{2}\right )^{\frac {3}{2}} a b x}{2 \, c} + \frac {\left (c x^{2}\right )^{\frac {3}{2}} a^{2}}{3 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.71, size = 33, normalized size = 0.58 \begin {gather*} \frac {1}{30} \, {\left (6 \, b^{2} x^{4} + 15 \, a b x^{3} + 10 \, a^{2} x^{2}\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 = 0.86 \begin {gather*} \frac {a^{2} x^{2} \sqrt {c x^{2}}}{3} + \frac {a b x^{3} \sqrt {c x^{2}}}{2} + \frac {b^{2} x^{4} \sqrt {c x^{2}}}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.40, size = 35, normalized size = 0.61 \begin {gather*} \frac {1}{30} \, {\left (6 \, b^{2} x^{5} \mathrm {sgn}\left (x\right ) + 15 \, a b x^{4} \mathrm {sgn}\left (x\right ) + 10 \, a^{2} x^{3} \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 x\,\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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