Optimal. Leaf size=66 \[ -\frac{a^2}{5 c^2 x^4 \sqrt{c x^2}}-\frac{a b}{2 c^2 x^3 \sqrt{c x^2}}-\frac{b^2}{3 c^2 x^2 \sqrt{c x^2}} \]
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Rubi [A] time = 0.0124531, 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} \[ -\frac{a^2}{5 c^2 x^4 \sqrt{c x^2}}-\frac{a b}{2 c^2 x^3 \sqrt{c x^2}}-\frac{b^2}{3 c^2 x^2 \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin{align*} \int \frac{(a+b x)^2}{x \left (c x^2\right )^{5/2}} \, dx &=\frac{x \int \frac{(a+b x)^2}{x^6} \, dx}{c^2 \sqrt{c x^2}}\\ &=\frac{x \int \left (\frac{a^2}{x^6}+\frac{2 a b}{x^5}+\frac{b^2}{x^4}\right ) \, dx}{c^2 \sqrt{c x^2}}\\ &=-\frac{a^2}{5 c^2 x^4 \sqrt{c x^2}}-\frac{a b}{2 c^2 x^3 \sqrt{c x^2}}-\frac{b^2}{3 c^2 x^2 \sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0122521, size = 38, normalized size = 0.58 \[ -\frac{\sqrt{c x^2} \left (6 a^2+15 a b x+10 b^2 x^2\right )}{30 c^3 x^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 29, normalized size = 0.4 \begin{align*} -{\frac{10\,{b}^{2}{x}^{2}+15\,abx+6\,{a}^{2}}{30} \left ( c{x}^{2} \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08766, size = 50, normalized size = 0.76 \begin{align*} -\frac{b^{2}}{3 \, \left (c x^{2}\right )^{\frac{3}{2}} c} - \frac{a b}{2 \, c^{\frac{5}{2}} x^{4}} - \frac{a^{2}}{5 \, c^{\frac{5}{2}} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.47218, size = 82, normalized size = 1.24 \begin{align*} -\frac{{\left (10 \, b^{2} x^{2} + 15 \, a b x + 6 \, a^{2}\right )} \sqrt{c x^{2}}}{30 \, c^{3} x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.03065, size = 56, normalized size = 0.85 \begin{align*} - \frac{a^{2}}{5 c^{\frac{5}{2}} \left (x^{2}\right )^{\frac{5}{2}}} - \frac{a b x}{2 c^{\frac{5}{2}} \left (x^{2}\right )^{\frac{5}{2}}} - \frac{b^{2} x^{2}}{3 c^{\frac{5}{2}} \left (x^{2}\right )^{\frac{5}{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*} \int \frac{{\left (b x + a\right )}^{2}}{\left (c x^{2}\right )^{\frac{5}{2}} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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