Optimal. Leaf size=63 \[ \frac{2}{3} a^2 c x^{3/2}+\frac{2}{11} b x^{11/2} (2 a d+b c)+\frac{2}{7} a x^{7/2} (a d+2 b c)+\frac{2}{15} b^2 d x^{15/2} \]
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Rubi [A] time = 0.0285362, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {448} \[ \frac{2}{3} a^2 c x^{3/2}+\frac{2}{11} b x^{11/2} (2 a d+b c)+\frac{2}{7} a x^{7/2} (a d+2 b c)+\frac{2}{15} b^2 d x^{15/2} \]
Antiderivative was successfully verified.
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Rule 448
Rubi steps
\begin{align*} \int \sqrt{x} \left (a+b x^2\right )^2 \left (c+d x^2\right ) \, dx &=\int \left (a^2 c \sqrt{x}+a (2 b c+a d) x^{5/2}+b (b c+2 a d) x^{9/2}+b^2 d x^{13/2}\right ) \, dx\\ &=\frac{2}{3} a^2 c x^{3/2}+\frac{2}{7} a (2 b c+a d) x^{7/2}+\frac{2}{11} b (b c+2 a d) x^{11/2}+\frac{2}{15} b^2 d x^{15/2}\\ \end{align*}
Mathematica [A] time = 0.0295099, size = 53, normalized size = 0.84 \[ \frac{2 x^{3/2} \left (385 a^2 c+105 b x^4 (2 a d+b c)+165 a x^2 (a d+2 b c)+77 b^2 d x^6\right )}{1155} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 56, normalized size = 0.9 \begin{align*}{\frac{154\,{b}^{2}d{x}^{6}+420\,{x}^{4}abd+210\,{b}^{2}c{x}^{4}+330\,{x}^{2}{a}^{2}d+660\,abc{x}^{2}+770\,{a}^{2}c}{1155}{x}^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0643, size = 69, normalized size = 1.1 \begin{align*} \frac{2}{15} \, b^{2} d x^{\frac{15}{2}} + \frac{2}{11} \,{\left (b^{2} c + 2 \, a b d\right )} x^{\frac{11}{2}} + \frac{2}{3} \, a^{2} c x^{\frac{3}{2}} + \frac{2}{7} \,{\left (2 \, a b c + a^{2} d\right )} x^{\frac{7}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.707551, size = 136, normalized size = 2.16 \begin{align*} \frac{2}{1155} \,{\left (77 \, b^{2} d x^{7} + 105 \,{\left (b^{2} c + 2 \, a b d\right )} x^{5} + 385 \, a^{2} c x + 165 \,{\left (2 \, a b c + a^{2} d\right )} x^{3}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.49226, size = 66, normalized size = 1.05 \begin{align*} \frac{2 a^{2} c x^{\frac{3}{2}}}{3} + \frac{2 b^{2} d x^{\frac{15}{2}}}{15} + \frac{2 x^{\frac{11}{2}} \left (2 a b d + b^{2} c\right )}{11} + \frac{2 x^{\frac{7}{2}} \left (a^{2} d + 2 a b c\right )}{7} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17171, size = 72, normalized size = 1.14 \begin{align*} \frac{2}{15} \, b^{2} d x^{\frac{15}{2}} + \frac{2}{11} \, b^{2} c x^{\frac{11}{2}} + \frac{4}{11} \, a b d x^{\frac{11}{2}} + \frac{4}{7} \, a b c x^{\frac{7}{2}} + \frac{2}{7} \, a^{2} d x^{\frac{7}{2}} + \frac{2}{3} \, a^{2} c x^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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