Optimal. Leaf size=41 \[ -\frac {a}{7 c^2 x^6 \sqrt {c x^2}}-\frac {b}{6 c^2 x^5 \sqrt {c x^2}} \]
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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} \[ -\frac {a}{7 c^2 x^6 \sqrt {c x^2}}-\frac {b}{6 c^2 x^5 \sqrt {c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin {align*} \int \frac {a+b x}{x^3 \left (c x^2\right )^{5/2}} \, dx &=\frac {x \int \frac {a+b x}{x^8} \, dx}{c^2 \sqrt {c x^2}}\\ &=\frac {x \int \left (\frac {a}{x^8}+\frac {b}{x^7}\right ) \, dx}{c^2 \sqrt {c x^2}}\\ &=-\frac {a}{7 c^2 x^6 \sqrt {c x^2}}-\frac {b}{6 c^2 x^5 \sqrt {c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 22, normalized size = 0.54 \[ \frac {c (-6 a-7 b x)}{42 \left (c x^2\right )^{7/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 23, normalized size = 0.56 \[ -\frac {\sqrt {c x^{2}} {\left (7 \, b x + 6 \, a\right )}}{42 \, c^{3} x^{8}} \]
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 \[ \int \frac {b x + a}{\left (c x^{2}\right )^{\frac {5}{2}} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 21, normalized size = 0.51 \[ -\frac {7 b x +6 a}{42 \left (c \,x^{2}\right )^{\frac {5}{2}} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.37, size = 19, normalized size = 0.46 \[ -\frac {b}{6 \, c^{\frac {5}{2}} x^{6}} - \frac {a}{7 \, c^{\frac {5}{2}} x^{7}} \]
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 \[ -\frac {6\,a\,\sqrt {x^2}+7\,b\,x\,\sqrt {x^2}}{42\,c^{5/2}\,x^8} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.64, size = 36, normalized size = 0.88 \[ - \frac {a}{7 c^{\frac {5}{2}} x^{2} \left (x^{2}\right )^{\frac {5}{2}}} - \frac {b}{6 c^{\frac {5}{2}} x \left (x^{2}\right )^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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