Optimal. Leaf size=66 \[ -\frac{a^2}{6 c x^5 \sqrt{c x^2}}-\frac{2 a b}{5 c x^4 \sqrt{c x^2}}-\frac{b^2}{4 c x^3 \sqrt{c x^2}} \]
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Rubi [A] time = 0.0139281, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {15, 43} \[ -\frac{a^2}{6 c x^5 \sqrt{c x^2}}-\frac{2 a b}{5 c x^4 \sqrt{c x^2}}-\frac{b^2}{4 c x^3 \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x)^2}{x^4 \left (c x^2\right )^{3/2}} \, dx &=\frac{x \int \frac{(a+b x)^2}{x^7} \, dx}{c \sqrt{c x^2}}\\ &=\frac{x \int \left (\frac{a^2}{x^7}+\frac{2 a b}{x^6}+\frac{b^2}{x^5}\right ) \, dx}{c \sqrt{c x^2}}\\ &=-\frac{a^2}{6 c x^5 \sqrt{c x^2}}-\frac{2 a b}{5 c x^4 \sqrt{c x^2}}-\frac{b^2}{4 c x^3 \sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0079686, size = 35, normalized size = 0.53 \[ \frac{-10 a^2-24 a b x-15 b^2 x^2}{60 x^3 \left (c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 32, normalized size = 0.5 \begin{align*} -{\frac{15\,{b}^{2}{x}^{2}+24\,abx+10\,{a}^{2}}{60\,{x}^{3}} \left ( c{x}^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01388, size = 45, normalized size = 0.68 \begin{align*} -\frac{b^{2}}{4 \, c^{\frac{3}{2}} x^{4}} - \frac{2 \, a b}{5 \, c^{\frac{3}{2}} x^{5}} - \frac{a^{2}}{6 \, c^{\frac{3}{2}} x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.46298, size = 84, normalized size = 1.27 \begin{align*} -\frac{{\left (15 \, b^{2} x^{2} + 24 \, a b x + 10 \, a^{2}\right )} \sqrt{c x^{2}}}{60 \, c^{2} x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.06204, size = 61, normalized size = 0.92 \begin{align*} - \frac{a^{2}}{6 c^{\frac{3}{2}} x^{3} \left (x^{2}\right )^{\frac{3}{2}}} - \frac{2 a b}{5 c^{\frac{3}{2}} x^{2} \left (x^{2}\right )^{\frac{3}{2}}} - \frac{b^{2}}{4 c^{\frac{3}{2}} x \left (x^{2}\right )^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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