Optimal. Leaf size=93 \[ \frac{2 b (d x)^{11/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 d^3 \left (a+b x^2\right )}+\frac{2 a (d x)^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 d \left (a+b x^2\right )} \]
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Rubi [A] time = 0.0297022, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {1112, 14} \[ \frac{2 b (d x)^{11/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 d^3 \left (a+b x^2\right )}+\frac{2 a (d x)^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 d \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Rule 1112
Rule 14
Rubi steps
\begin{align*} \int (d x)^{5/2} \sqrt{a^2+2 a b x^2+b^2 x^4} \, dx &=\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \int (d x)^{5/2} \left (a b+b^2 x^2\right ) \, dx}{a b+b^2 x^2}\\ &=\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \int \left (a b (d x)^{5/2}+\frac{b^2 (d x)^{9/2}}{d^2}\right ) \, dx}{a b+b^2 x^2}\\ &=\frac{2 a (d x)^{7/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{7 d \left (a+b x^2\right )}+\frac{2 b (d x)^{11/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}{11 d^3 \left (a+b x^2\right )}\\ \end{align*}
Mathematica [A] time = 0.0207229, size = 44, normalized size = 0.47 \[ \frac{2 x (d x)^{5/2} \sqrt{\left (a+b x^2\right )^2} \left (11 a+7 b x^2\right )}{77 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 39, normalized size = 0.4 \begin{align*}{\frac{2\, \left ( 7\,b{x}^{2}+11\,a \right ) x}{77\,b{x}^{2}+77\,a} \left ( dx \right ) ^{{\frac{5}{2}}}\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01641, size = 30, normalized size = 0.32 \begin{align*} \frac{2}{77} \,{\left (7 \, b d^{\frac{5}{2}} x^{3} + 11 \, a d^{\frac{5}{2}} x\right )} x^{\frac{5}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.26214, size = 61, normalized size = 0.66 \begin{align*} \frac{2}{77} \,{\left (7 \, b d^{2} x^{5} + 11 \, a d^{2} x^{3}\right )} \sqrt{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16491, size = 61, normalized size = 0.66 \begin{align*} \frac{2}{11} \, \sqrt{d x} b d^{2} x^{5} \mathrm{sgn}\left (b x^{2} + a\right ) + \frac{2}{7} \, \sqrt{d x} a d^{2} x^{3} \mathrm{sgn}\left (b x^{2} + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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