Optimal. Leaf size=35 \[ -\frac {a}{4 x^3 \sqrt {c x^2}}-\frac {b}{3 x^2 \sqrt {c x^2}} \]
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Rubi [A] time = 0.01, antiderivative size = 35, 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} \[ -\frac {a}{4 x^3 \sqrt {c x^2}}-\frac {b}{3 x^2 \sqrt {c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin {align*} \int \frac {a+b x}{x^4 \sqrt {c x^2}} \, dx &=\frac {x \int \frac {a+b x}{x^5} \, dx}{\sqrt {c x^2}}\\ &=\frac {x \int \left (\frac {a}{x^5}+\frac {b}{x^4}\right ) \, dx}{\sqrt {c x^2}}\\ &=-\frac {a}{4 x^3 \sqrt {c x^2}}-\frac {b}{3 x^2 \sqrt {c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 24, normalized size = 0.69 \[ \frac {-3 a-4 b x}{12 x^3 \sqrt {c x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 23, normalized size = 0.66 \[ -\frac {\sqrt {c x^{2}} {\left (4 \, b x + 3 \, a\right )}}{12 \, c x^{5}} \]
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 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 21, normalized size = 0.60 \[ -\frac {4 b x +3 a}{12 \sqrt {c \,x^{2}}\, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.30, size = 19, normalized size = 0.54 \[ -\frac {b}{3 \, \sqrt {c} x^{3}} - \frac {a}{4 \, \sqrt {c} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.15, size = 26, normalized size = 0.74 \[ -\frac {3\,a\,\sqrt {x^2}+4\,b\,x\,\sqrt {x^2}}{12\,\sqrt {c}\,x^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.81, size = 37, normalized size = 1.06 \[ - \frac {a}{4 \sqrt {c} x^{3} \sqrt {x^{2}}} - \frac {b}{3 \sqrt {c} x^{2} \sqrt {x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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