Optimal. Leaf size=29 \[ -\frac {(a+b x)^3}{3 a c^2 x^2 \sqrt {c x^2}} \]
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Rubi [A]
time = 0.00, antiderivative size = 29, 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, 37}
\begin {gather*} -\frac {(a+b x)^3}{3 a c^2 x^2 \sqrt {c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 37
Rubi steps
\begin {align*} \int \frac {x (a+b x)^2}{\left (c x^2\right )^{5/2}} \, dx &=\frac {x \int \frac {(a+b x)^2}{x^4} \, dx}{c^2 \sqrt {c x^2}}\\ &=-\frac {(a+b x)^3}{3 a c^2 x^2 \sqrt {c x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 35, normalized size = 1.21 \begin {gather*} \frac {x^2 \left (-a^2-3 a b x-3 b^2 x^2\right )}{3 \left (c x^2\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 30, normalized size = 1.03
method | result | size |
gosper | \(-\frac {x^{2} \left (3 x^{2} b^{2}+3 a b x +a^{2}\right )}{3 \left (c \,x^{2}\right )^{\frac {5}{2}}}\) | \(30\) |
default | \(-\frac {x^{2} \left (3 x^{2} b^{2}+3 a b x +a^{2}\right )}{3 \left (c \,x^{2}\right )^{\frac {5}{2}}}\) | \(30\) |
risch | \(\frac {-x^{2} b^{2}-a b x -\frac {1}{3} a^{2}}{c^{2} x^{2} \sqrt {c \,x^{2}}}\) | \(34\) |
trager | \(\frac {\left (-1+x \right ) \left (a^{2} x^{2}+3 a b \,x^{2}+3 x^{2} b^{2}+a^{2} x +3 a b x +a^{2}\right ) \sqrt {c \,x^{2}}}{3 c^{3} x^{4}}\) | \(55\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 44, normalized size = 1.52 \begin {gather*} -\frac {b^{2} x^{2}}{\left (c x^{2}\right )^{\frac {3}{2}} c} - \frac {a^{2}}{3 \, \left (c x^{2}\right )^{\frac {3}{2}} c} - \frac {a b}{c^{\frac {5}{2}} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.16, size = 32, normalized size = 1.10 \begin {gather*} -\frac {{\left (3 \, b^{2} x^{2} + 3 \, a b x + a^{2}\right )} \sqrt {c x^{2}}}{3 \, c^{3} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.28, size = 48, normalized size = 1.66 \begin {gather*} - \frac {a^{2} x^{2}}{3 \left (c x^{2}\right )^{\frac {5}{2}}} - \frac {a b x^{3}}{\left (c x^{2}\right )^{\frac {5}{2}}} - \frac {b^{2} x^{4}}{\left (c x^{2}\right )^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.97, size = 39, normalized size = 1.34 \begin {gather*} -\frac {3 \, b^{2} \sqrt {c} x^{2} + 3 \, a b \sqrt {c} x + a^{2} \sqrt {c}}{3 \, c^{3} x^{3} \mathrm {sgn}\left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.18, size = 33, normalized size = 1.14 \begin {gather*} -\frac {a^2\,x^2+3\,a\,b\,x^3+3\,b^2\,x^4}{3\,c^{5/2}\,{\left (x^2\right )}^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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