Optimal. Leaf size=24 \[ \frac {x (a+b x)^3}{3 b \sqrt {c x^2}} \]
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Rubi [A]
time = 0.00, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {15, 32}
\begin {gather*} \frac {x (a+b x)^3}{3 b \sqrt {c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 32
Rubi steps
\begin {align*} \int \frac {x (a+b x)^2}{\sqrt {c x^2}} \, dx &=\frac {x \int (a+b x)^2 \, dx}{\sqrt {c x^2}}\\ &=\frac {x (a+b x)^3}{3 b \sqrt {c x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 24, normalized size = 1.00 \begin {gather*} \frac {x (a+b x)^3}{3 b \sqrt {c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 21, normalized size = 0.88
method | result | size |
default | \(\frac {x \left (b x +a \right )^{3}}{3 b \sqrt {c \,x^{2}}}\) | \(21\) |
risch | \(\frac {x \left (b x +a \right )^{3}}{3 b \sqrt {c \,x^{2}}}\) | \(21\) |
gosper | \(\frac {x^{2} \left (x^{2} b^{2}+3 a b x +3 a^{2}\right )}{3 \sqrt {c \,x^{2}}}\) | \(31\) |
trager | \(\frac {\left (x^{2} b^{2}+3 a b x +b^{2} x +3 a^{2}+3 a b +b^{2}\right ) \left (-1+x \right ) \sqrt {c \,x^{2}}}{3 c x}\) | \(49\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 42 vs.
\(2 (20) = 40\).
time = 0.26, size = 42, normalized size = 1.75 \begin {gather*} \frac {\sqrt {c x^{2}} b^{2} x^{2}}{3 \, c} + \frac {a b x^{2}}{\sqrt {c}} + \frac {\sqrt {c x^{2}} a^{2}}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 30, normalized size = 1.25 \begin {gather*} \frac {{\left (b^{2} x^{2} + 3 \, a b x + 3 \, a^{2}\right )} \sqrt {c x^{2}}}{3 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 46 vs.
\(2 (19) = 38\).
time = 0.20, size = 46, normalized size = 1.92 \begin {gather*} \frac {a^{2} x^{2}}{\sqrt {c x^{2}}} + \frac {a b x^{3}}{\sqrt {c x^{2}}} + \frac {b^{2} x^{4}}{3 \sqrt {c x^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.93, size = 39, normalized size = 1.62 \begin {gather*} \frac {b^{2} \sqrt {c} x^{3} + 3 \, a b \sqrt {c} x^{2} + 3 \, a^{2} \sqrt {c} x}{3 \, 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 = -0.04 \begin {gather*} \int \frac {x\,{\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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