Optimal. Leaf size=57 \[ -\frac {a^2}{4 x^3 \sqrt {c x^2}}-\frac {2 a b}{3 x^2 \sqrt {c x^2}}-\frac {b^2}{2 x \sqrt {c x^2}} \]
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Rubi [A] time = 0.01, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 43} \[ -\frac {a^2}{4 x^3 \sqrt {c x^2}}-\frac {2 a b}{3 x^2 \sqrt {c x^2}}-\frac {b^2}{2 x \sqrt {c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin {align*} \int \frac {(a+b x)^2}{x^4 \sqrt {c x^2}} \, dx &=\frac {x \int \frac {(a+b x)^2}{x^5} \, dx}{\sqrt {c x^2}}\\ &=\frac {x \int \left (\frac {a^2}{x^5}+\frac {2 a b}{x^4}+\frac {b^2}{x^3}\right ) \, dx}{\sqrt {c x^2}}\\ &=-\frac {a^2}{4 x^3 \sqrt {c x^2}}-\frac {2 a b}{3 x^2 \sqrt {c x^2}}-\frac {b^2}{2 x \sqrt {c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 35, normalized size = 0.61 \[ \frac {-3 a^2-8 a b x-6 b^2 x^2}{12 x^3 \sqrt {c x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 34, normalized size = 0.60 \[ -\frac {{\left (6 \, b^{2} x^{2} + 8 \, a b x + 3 \, a^{2}\right )} \sqrt {c x^{2}}}{12 \, c x^{5}} \]
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 = 0.01, size = 32, normalized size = 0.56 \[ -\frac {6 b^{2} x^{2}+8 a b x +3 a^{2}}{12 \sqrt {c \,x^{2}}\, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 33, normalized size = 0.58 \[ -\frac {b^{2}}{2 \, \sqrt {c} x^{2}} - \frac {2 \, a b}{3 \, \sqrt {c} x^{3}} - \frac {a^{2}}{4 \, \sqrt {c} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.19, size = 42, normalized size = 0.74 \[ -\frac {3\,a^2\,\sqrt {x^2}+6\,b^2\,x^2\,\sqrt {x^2}+8\,a\,b\,x\,\sqrt {x^2}}{12\,\sqrt {c}\,x^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.82, size = 61, normalized size = 1.07 \[ - \frac {a^{2}}{4 \sqrt {c} x^{3} \sqrt {x^{2}}} - \frac {2 a b}{3 \sqrt {c} x^{2} \sqrt {x^{2}}} - \frac {b^{2}}{2 \sqrt {c} x \sqrt {x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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