Optimal. Leaf size=41 \[ -\frac {a}{5 c^2 x^4 \sqrt {c x^2}}-\frac {b}{4 c^2 x^3 \sqrt {c x^2}} \]
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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, 43} \begin {gather*} -\frac {a}{5 c^2 x^4 \sqrt {c x^2}}-\frac {b}{4 c^2 x^3 \sqrt {c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin {align*} \int \frac {a+b x}{x \left (c x^2\right )^{5/2}} \, dx &=\frac {x \int \frac {a+b x}{x^6} \, dx}{c^2 \sqrt {c x^2}}\\ &=\frac {x \int \left (\frac {a}{x^6}+\frac {b}{x^5}\right ) \, dx}{c^2 \sqrt {c x^2}}\\ &=-\frac {a}{5 c^2 x^4 \sqrt {c x^2}}-\frac {b}{4 c^2 x^3 \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} (4 a+5 b x)}{20 c^3 x^6} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.02, size = 21, normalized size = 0.51 \begin {gather*} \frac {-4 a-5 b x}{20 \left (c x^2\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 23, normalized size = 0.56 \begin {gather*} -\frac {\sqrt {c x^{2}} {\left (5 \, b x + 4 \, a\right )}}{20 \, c^{3} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {b x + a}{\left (c x^{2}\right )^{\frac {5}{2}} x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 18, normalized size = 0.44 \begin {gather*} -\frac {5 b x +4 a}{20 \left (c \,x^{2}\right )^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 19, normalized size = 0.46 \begin {gather*} -\frac {b}{4 \, c^{\frac {5}{2}} x^{4}} - \frac {a}{5 \, c^{\frac {5}{2}} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.16, size = 26, normalized size = 0.63 \begin {gather*} -\frac {4\,a\,\sqrt {x^2}+5\,b\,x\,\sqrt {x^2}}{20\,c^{5/2}\,x^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.12, size = 32, normalized size = 0.78 \begin {gather*} - \frac {a}{5 c^{\frac {5}{2}} \left (x^{2}\right )^{\frac {5}{2}}} - \frac {b x}{4 c^{\frac {5}{2}} \left (x^{2}\right )^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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