Optimal. Leaf size=49 \[ -\frac{2 a^2}{3 b^3 (a+b x)^{3/2}}+\frac{4 a}{b^3 \sqrt{a+b x}}+\frac{2 \sqrt{a+b x}}{b^3} \]
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Rubi [A] time = 0.0130616, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {43} \[ -\frac{2 a^2}{3 b^3 (a+b x)^{3/2}}+\frac{4 a}{b^3 \sqrt{a+b x}}+\frac{2 \sqrt{a+b x}}{b^3} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{x^2}{(a+b x)^{5/2}} \, dx &=\int \left (\frac{a^2}{b^2 (a+b x)^{5/2}}-\frac{2 a}{b^2 (a+b x)^{3/2}}+\frac{1}{b^2 \sqrt{a+b x}}\right ) \, dx\\ &=-\frac{2 a^2}{3 b^3 (a+b x)^{3/2}}+\frac{4 a}{b^3 \sqrt{a+b x}}+\frac{2 \sqrt{a+b x}}{b^3}\\ \end{align*}
Mathematica [A] time = 0.0176182, size = 35, normalized size = 0.71 \[ \frac{2 \left (8 a^2+12 a b x+3 b^2 x^2\right )}{3 b^3 (a+b x)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 32, normalized size = 0.7 \begin{align*}{\frac{6\,{b}^{2}{x}^{2}+24\,abx+16\,{a}^{2}}{3\,{b}^{3}} \left ( bx+a \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14883, size = 55, normalized size = 1.12 \begin{align*} \frac{2 \, \sqrt{b x + a}}{b^{3}} + \frac{4 \, a}{\sqrt{b x + a} b^{3}} - \frac{2 \, a^{2}}{3 \,{\left (b x + a\right )}^{\frac{3}{2}} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55674, size = 111, normalized size = 2.27 \begin{align*} \frac{2 \,{\left (3 \, b^{2} x^{2} + 12 \, a b x + 8 \, a^{2}\right )} \sqrt{b x + a}}{3 \,{\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.54181, size = 121, normalized size = 2.47 \begin{align*} \begin{cases} \frac{16 a^{2}}{3 a b^{3} \sqrt{a + b x} + 3 b^{4} x \sqrt{a + b x}} + \frac{24 a b x}{3 a b^{3} \sqrt{a + b x} + 3 b^{4} x \sqrt{a + b x}} + \frac{6 b^{2} x^{2}}{3 a b^{3} \sqrt{a + b x} + 3 b^{4} x \sqrt{a + b x}} & \text{for}\: b \neq 0 \\\frac{x^{3}}{3 a^{\frac{5}{2}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15255, size = 53, normalized size = 1.08 \begin{align*} \frac{2 \, \sqrt{b x + a}}{b^{3}} + \frac{2 \,{\left (6 \,{\left (b x + a\right )} a - a^{2}\right )}}{3 \,{\left (b x + a\right )}^{\frac{3}{2}} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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