Optimal. Leaf size=29 \[ -\frac {(a+b x)^3}{3 a c x^2 \sqrt {c x^2}} \]
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Rubi [A] time = 0.00, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 37} \begin {gather*} -\frac {(a+b x)^3}{3 a c x^2 \sqrt {c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 37
Rubi steps
\begin {align*} \int \frac {(a+b x)^2}{x \left (c x^2\right )^{3/2}} \, dx &=\frac {x \int \frac {(a+b x)^2}{x^4} \, dx}{c \sqrt {c x^2}}\\ &=-\frac {(a+b x)^3}{3 a c x^2 \sqrt {c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 36, normalized size = 1.24 \begin {gather*} \frac {c x^2 \left (-a^2-3 a b x-3 b^2 x^2\right )}{3 \left (c x^2\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.03, size = 32, normalized size = 1.10 \begin {gather*} \frac {-a^2-3 a b x-3 b^2 x^2}{3 \left (c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 32, normalized size = 1.10 \begin {gather*} -\frac {{\left (3 \, b^{2} x^{2} + 3 \, a b x + a^{2}\right )} \sqrt {c x^{2}}}{3 \, c^{2} x^{4}} \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 {{\left (b x + a\right )}^{2}}{\left (c x^{2}\right )^{\frac {3}{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 = 27, normalized size = 0.93 \begin {gather*} -\frac {3 b^{2} x^{2}+3 a b x +a^{2}}{3 \left (c \,x^{2}\right )^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 37, normalized size = 1.28 \begin {gather*} -\frac {b^{2}}{\sqrt {c x^{2}} c} - \frac {a b}{c^{\frac {3}{2}} x^{2}} - \frac {a^{2}}{3 \, c^{\frac {3}{2}} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.19, size = 33, normalized size = 1.14 \begin {gather*} -\frac {a^2\,x^2+3\,a\,b\,x^3+3\,b^2\,x^4}{3\,c^{3/2}\,{\left (x^2\right )}^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.67, size = 53, normalized size = 1.83 \begin {gather*} - \frac {a^{2}}{3 c^{\frac {3}{2}} \left (x^{2}\right )^{\frac {3}{2}}} - \frac {a b x}{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}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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