Optimal. Leaf size=54 \[ -\frac{a^2}{3 b^3 \left (a+b x^2\right )^{3/2}}+\frac{2 a}{b^3 \sqrt{a+b x^2}}+\frac{\sqrt{a+b x^2}}{b^3} \]
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Rubi [A] time = 0.0314975, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ -\frac{a^2}{3 b^3 \left (a+b x^2\right )^{3/2}}+\frac{2 a}{b^3 \sqrt{a+b x^2}}+\frac{\sqrt{a+b x^2}}{b^3} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^5}{\left (a+b x^2\right )^{5/2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2}{(a+b x)^{5/2}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\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,x,x^2\right )\\ &=-\frac{a^2}{3 b^3 \left (a+b x^2\right )^{3/2}}+\frac{2 a}{b^3 \sqrt{a+b x^2}}+\frac{\sqrt{a+b x^2}}{b^3}\\ \end{align*}
Mathematica [A] time = 0.0173704, size = 39, normalized size = 0.72 \[ \frac{8 a^2+12 a b x^2+3 b^2 x^4}{3 b^3 \left (a+b x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 36, normalized size = 0.7 \begin{align*}{\frac{3\,{b}^{2}{x}^{4}+12\,ab{x}^{2}+8\,{a}^{2}}{3\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.27047, size = 119, normalized size = 2.2 \begin{align*} \frac{{\left (3 \, b^{2} x^{4} + 12 \, a b x^{2} + 8 \, a^{2}\right )} \sqrt{b x^{2} + a}}{3 \,{\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.07568, size = 138, normalized size = 2.56 \begin{align*} \begin{cases} \frac{8 a^{2}}{3 a b^{3} \sqrt{a + b x^{2}} + 3 b^{4} x^{2} \sqrt{a + b x^{2}}} + \frac{12 a b x^{2}}{3 a b^{3} \sqrt{a + b x^{2}} + 3 b^{4} x^{2} \sqrt{a + b x^{2}}} + \frac{3 b^{2} x^{4}}{3 a b^{3} \sqrt{a + b x^{2}} + 3 b^{4} x^{2} \sqrt{a + b x^{2}}} & \text{for}\: b \neq 0 \\\frac{x^{6}}{6 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 = 2.35319, size = 58, normalized size = 1.07 \begin{align*} \frac{3 \, \sqrt{b x^{2} + a} + \frac{6 \,{\left (b x^{2} + a\right )} a - a^{2}}{{\left (b x^{2} + a\right )}^{\frac{3}{2}}}}{3 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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