Optimal. Leaf size=38 \[ \frac{a}{7 b^2 \left (a+b x^2\right )^{7/2}}-\frac{1}{5 b^2 \left (a+b x^2\right )^{5/2}} \]
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Rubi [A] time = 0.0232085, antiderivative size = 38, 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} \[ \frac{a}{7 b^2 \left (a+b x^2\right )^{7/2}}-\frac{1}{5 b^2 \left (a+b x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^3}{\left (a+b x^2\right )^{9/2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{(a+b x)^{9/2}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{a}{b (a+b x)^{9/2}}+\frac{1}{b (a+b x)^{7/2}}\right ) \, dx,x,x^2\right )\\ &=\frac{a}{7 b^2 \left (a+b x^2\right )^{7/2}}-\frac{1}{5 b^2 \left (a+b x^2\right )^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0138211, size = 28, normalized size = 0.74 \[ \frac{-2 a-7 b x^2}{35 b^2 \left (a+b x^2\right )^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 25, normalized size = 0.7 \begin{align*} -{\frac{7\,b{x}^{2}+2\,a}{35\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.01263, size = 45, normalized size = 1.18 \begin{align*} -\frac{x^{2}}{5 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b} - \frac{2 \, a}{35 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.33584, size = 142, normalized size = 3.74 \begin{align*} -\frac{{\left (7 \, b x^{2} + 2 \, a\right )} \sqrt{b x^{2} + a}}{35 \,{\left (b^{6} x^{8} + 4 \, a b^{5} x^{6} + 6 \, a^{2} b^{4} x^{4} + 4 \, a^{3} b^{3} x^{2} + a^{4} b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.25198, size = 180, normalized size = 4.74 \begin{align*} \begin{cases} - \frac{2 a}{35 a^{3} b^{2} \sqrt{a + b x^{2}} + 105 a^{2} b^{3} x^{2} \sqrt{a + b x^{2}} + 105 a b^{4} x^{4} \sqrt{a + b x^{2}} + 35 b^{5} x^{6} \sqrt{a + b x^{2}}} - \frac{7 b x^{2}}{35 a^{3} b^{2} \sqrt{a + b x^{2}} + 105 a^{2} b^{3} x^{2} \sqrt{a + b x^{2}} + 105 a b^{4} x^{4} \sqrt{a + b x^{2}} + 35 b^{5} x^{6} \sqrt{a + b x^{2}}} & \text{for}\: b \neq 0 \\\frac{x^{4}}{4 a^{\frac{9}{2}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.3914, size = 32, normalized size = 0.84 \begin{align*} -\frac{7 \, b x^{2} + 2 \, a}{35 \,{\left (b x^{2} + a\right )}^{\frac{7}{2}} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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