Optimal. Leaf size=66 \[ -\frac {a^2}{4 c x^3 \sqrt {c x^2}}-\frac {2 a b}{3 c x^2 \sqrt {c x^2}}-\frac {b^2}{2 c x \sqrt {c x^2}} \]
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Rubi [A] time = 0.01, antiderivative size = 66, 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} \[ -\frac {a^2}{4 c x^3 \sqrt {c x^2}}-\frac {2 a b}{3 c x^2 \sqrt {c x^2}}-\frac {b^2}{2 c x \sqrt {c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin {align*} \int \frac {(a+b x)^2}{x^2 \left (c x^2\right )^{3/2}} \, dx &=\frac {x \int \frac {(a+b x)^2}{x^5} \, dx}{c \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}{c \sqrt {c x^2}}\\ &=-\frac {a^2}{4 c x^3 \sqrt {c x^2}}-\frac {2 a b}{3 c x^2 \sqrt {c x^2}}-\frac {b^2}{2 c x \sqrt {c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 38, normalized size = 0.58 \[ -\frac {\sqrt {c x^2} \left (3 a^2+8 a b x+6 b^2 x^2\right )}{12 c^2 x^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 34, normalized size = 0.52 \[ -\frac {{\left (6 \, b^{2} x^{2} + 8 \, a b x + 3 \, a^{2}\right )} \sqrt {c x^{2}}}{12 \, c^{2} x^{5}} \]
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 = 0.48 \[ -\frac {6 b^{2} x^{2}+8 a b x +3 a^{2}}{12 \left (c \,x^{2}\right )^{\frac {3}{2}} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.31, size = 33, normalized size = 0.50 \[ -\frac {b^{2}}{2 \, c^{\frac {3}{2}} x^{2}} - \frac {2 \, a b}{3 \, c^{\frac {3}{2}} x^{3}} - \frac {a^{2}}{4 \, c^{\frac {3}{2}} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.19, size = 42, normalized size = 0.64 \[ -\frac {3\,a^2\,\sqrt {x^2}+6\,b^2\,x^2\,\sqrt {x^2}+8\,a\,b\,x\,\sqrt {x^2}}{12\,c^{3/2}\,x^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.81, size = 56, normalized size = 0.85 \[ - \frac {a^{2}}{4 c^{\frac {3}{2}} x \left (x^{2}\right )^{\frac {3}{2}}} - \frac {2 a b}{3 c^{\frac {3}{2}} \left (x^{2}\right )^{\frac {3}{2}}} - \frac {b^{2} x}{2 c^{\frac {3}{2}} \left (x^{2}\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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