Optimal. Leaf size=57 \[ \frac {a^2 x^3}{2 \sqrt {c x^2}}+\frac {2 a b x^4}{3 \sqrt {c x^2}}+\frac {b^2 x^5}{4 \sqrt {c x^2}} \]
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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 = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 45}
\begin {gather*} \frac {a^2 x^3}{2 \sqrt {c x^2}}+\frac {2 a b x^4}{3 \sqrt {c x^2}}+\frac {b^2 x^5}{4 \sqrt {c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 45
Rubi steps
\begin {align*} \int \frac {x^2 (a+b x)^2}{\sqrt {c x^2}} \, dx &=\frac {x \int x (a+b x)^2 \, dx}{\sqrt {c x^2}}\\ &=\frac {x \int \left (a^2 x+2 a b x^2+b^2 x^3\right ) \, dx}{\sqrt {c x^2}}\\ &=\frac {a^2 x^3}{2 \sqrt {c x^2}}+\frac {2 a b x^4}{3 \sqrt {c x^2}}+\frac {b^2 x^5}{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 {x^3 \left (6 a^2+8 a b x+3 b^2 x^2\right )}{12 \sqrt {c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 32, normalized size = 0.56
method | result | size |
gosper | \(\frac {x^{3} \left (3 x^{2} b^{2}+8 a b x +6 a^{2}\right )}{12 \sqrt {c \,x^{2}}}\) | \(32\) |
default | \(\frac {x^{3} \left (3 x^{2} b^{2}+8 a b x +6 a^{2}\right )}{12 \sqrt {c \,x^{2}}}\) | \(32\) |
risch | \(\frac {a^{2} x^{3}}{2 \sqrt {c \,x^{2}}}+\frac {2 a b \,x^{4}}{3 \sqrt {c \,x^{2}}}+\frac {b^{2} x^{5}}{4 \sqrt {c \,x^{2}}}\) | \(46\) |
trager | \(\frac {\left (3 b^{2} x^{3}+8 a b \,x^{2}+3 x^{2} b^{2}+6 a^{2} x +8 a b x +3 b^{2} x +6 a^{2}+8 a b +3 b^{2}\right ) \left (-1+x \right ) \sqrt {c \,x^{2}}}{12 c x}\) | \(74\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 47, normalized size = 0.82 \begin {gather*} \frac {\sqrt {c x^{2}} b^{2} x^{3}}{4 \, c} + \frac {2 \, \sqrt {c x^{2}} a b x^{2}}{3 \, c} + \frac {a^{2} x^{2}}{2 \, \sqrt {c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.44, size = 34, normalized size = 0.60 \begin {gather*} \frac {{\left (3 \, b^{2} x^{3} + 8 \, a b x^{2} + 6 \, a^{2} x\right )} \sqrt {c x^{2}}}{12 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.22, size = 51, normalized size = 0.89 \begin {gather*} \frac {a^{2} x^{3}}{2 \sqrt {c x^{2}}} + \frac {2 a b x^{4}}{3 \sqrt {c x^{2}}} + \frac {b^{2} x^{5}}{4 \sqrt {c x^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.87, size = 42, normalized size = 0.74 \begin {gather*} \frac {3 \, b^{2} \sqrt {c} x^{4} + 8 \, a b \sqrt {c} x^{3} + 6 \, a^{2} \sqrt {c} x^{2}}{12 \, c \mathrm {sgn}\left (x\right )} \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 \frac {x^2\,{\left (a+b\,x\right )}^2}{\sqrt {c\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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