Optimal. Leaf size=41 \[ -\frac {a}{4 c x^3 \sqrt {c x^2}}-\frac {b}{3 c x^2 \sqrt {c x^2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 41, 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 {a}{4 c x^3 \sqrt {c x^2}}-\frac {b}{3 c x^2 \sqrt {c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 45
Rubi steps
\begin {align*} \int \frac {a+b x}{x^2 \left (c x^2\right )^{3/2}} \, dx &=\frac {x \int \frac {a+b x}{x^5} \, dx}{c \sqrt {c x^2}}\\ &=\frac {x \int \left (\frac {a}{x^5}+\frac {b}{x^4}\right ) \, dx}{c \sqrt {c x^2}}\\ &=-\frac {a}{4 c x^3 \sqrt {c x^2}}-\frac {b}{3 c x^2 \sqrt {c x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 27, normalized size = 0.66 \begin {gather*} -\frac {\sqrt {c x^2} (3 a+4 b x)}{12 c^2 x^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 21, normalized size = 0.51
method | result | size |
gosper | \(-\frac {4 b x +3 a}{12 x \left (c \,x^{2}\right )^{\frac {3}{2}}}\) | \(21\) |
default | \(-\frac {4 b x +3 a}{12 x \left (c \,x^{2}\right )^{\frac {3}{2}}}\) | \(21\) |
risch | \(\frac {-\frac {b x}{3}-\frac {a}{4}}{c \,x^{3} \sqrt {c \,x^{2}}}\) | \(23\) |
trager | \(\frac {\left (-1+x \right ) \left (3 a \,x^{3}+4 b \,x^{3}+3 a \,x^{2}+4 x^{2} b +3 a x +4 b x +3 a \right ) \sqrt {c \,x^{2}}}{12 c^{2} x^{5}}\) | \(55\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 19, normalized size = 0.46 \begin {gather*} -\frac {b}{3 \, c^{\frac {3}{2}} x^{3}} - \frac {a}{4 \, c^{\frac {3}{2}} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.82, size = 23, normalized size = 0.56 \begin {gather*} -\frac {\sqrt {c x^{2}} {\left (4 \, b x + 3 \, a\right )}}{12 \, c^{2} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.24, size = 26, normalized size = 0.63 \begin {gather*} - \frac {a}{4 x \left (c x^{2}\right )^{\frac {3}{2}}} - \frac {b}{3 \left (c x^{2}\right )^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.63, size = 20, normalized size = 0.49 \begin {gather*} -\frac {4 \, b x + 3 \, a}{12 \, c^{\frac {3}{2}} x^{4} \mathrm {sgn}\left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.15, size = 26, normalized size = 0.63 \begin {gather*} -\frac {3\,a\,\sqrt {x^2}+4\,b\,x\,\sqrt {x^2}}{12\,c^{3/2}\,x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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