Optimal. Leaf size=26 \[ -\frac {(a+b x)^3}{3 a x^2 \sqrt {c x^2}} \]
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Rubi [A]
time = 0.00, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 37}
\begin {gather*} -\frac {(a+b x)^3}{3 a 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 {(a+b x)^2}{x^3 \sqrt {c x^2}} \, dx &=\frac {x \int \frac {(a+b x)^2}{x^4} \, dx}{\sqrt {c x^2}}\\ &=-\frac {(a+b x)^3}{3 a x^2 \sqrt {c x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 33, normalized size = 1.27 \begin {gather*} \frac {c \left (-a^2-3 a b x-3 b^2 x^2\right )}{3 \left (c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 30, normalized size = 1.15
method | result | size |
gosper | \(-\frac {3 x^{2} b^{2}+3 a b x +a^{2}}{3 x^{2} \sqrt {c \,x^{2}}}\) | \(30\) |
default | \(-\frac {3 x^{2} b^{2}+3 a b x +a^{2}}{3 x^{2} \sqrt {c \,x^{2}}}\) | \(30\) |
risch | \(\frac {-x^{2} b^{2}-a b x -\frac {1}{3} a^{2}}{x^{2} \sqrt {c \,x^{2}}}\) | \(31\) |
trager | \(\frac {\left (-1+x \right ) \left (a^{2} x^{2}+3 a b \,x^{2}+3 x^{2} b^{2}+a^{2} x +3 a b x +a^{2}\right ) \sqrt {c \,x^{2}}}{3 c \,x^{4}}\) | \(55\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 33, normalized size = 1.27 \begin {gather*} -\frac {b^{2}}{\sqrt {c} x} - \frac {a b}{\sqrt {c} x^{2}} - \frac {a^{2}}{3 \, \sqrt {c} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 32, normalized size = 1.23 \begin {gather*} -\frac {{\left (3 \, b^{2} x^{2} + 3 \, a b x + a^{2}\right )} \sqrt {c x^{2}}}{3 \, c x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.23, size = 42, normalized size = 1.62 \begin {gather*} - \frac {a^{2}}{3 x^{2} \sqrt {c x^{2}}} - \frac {a b}{x \sqrt {c x^{2}}} - \frac {b^{2}}{\sqrt {c x^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.36, size = 39, normalized size = 1.50 \begin {gather*} -\frac {3 \, b^{2} \sqrt {c} x^{2} + 3 \, a b \sqrt {c} x + a^{2} \sqrt {c}}{3 \, c 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 = 1.27 \begin {gather*} -\frac {a^2\,x^2+3\,a\,b\,x^3+3\,b^2\,x^4}{3\,\sqrt {c}\,{\left (x^2\right )}^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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