Optimal. Leaf size=97 \[ \frac{2}{11} x^{11/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )+\frac{2}{3} a^2 c^2 x^{3/2}+\frac{4}{15} b d x^{15/2} (a d+b c)+\frac{4}{7} a c x^{7/2} (a d+b c)+\frac{2}{19} b^2 d^2 x^{19/2} \]
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Rubi [A] time = 0.0465173, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {448} \[ \frac{2}{11} x^{11/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )+\frac{2}{3} a^2 c^2 x^{3/2}+\frac{4}{15} b d x^{15/2} (a d+b c)+\frac{4}{7} a c x^{7/2} (a d+b c)+\frac{2}{19} b^2 d^2 x^{19/2} \]
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 )^2 \, dx &=\int \left (a^2 c^2 \sqrt{x}+2 a c (b c+a d) x^{5/2}+\left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^{9/2}+2 b d (b c+a d) x^{13/2}+b^2 d^2 x^{17/2}\right ) \, dx\\ &=\frac{2}{3} a^2 c^2 x^{3/2}+\frac{4}{7} a c (b c+a d) x^{7/2}+\frac{2}{11} \left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^{11/2}+\frac{4}{15} b d (b c+a d) x^{15/2}+\frac{2}{19} b^2 d^2 x^{19/2}\\ \end{align*}
Mathematica [A] time = 0.0342166, size = 83, normalized size = 0.86 \[ \frac{2 x^{3/2} \left (1995 x^4 \left (a^2 d^2+4 a b c d+b^2 c^2\right )+7315 a^2 c^2+2926 b d x^6 (a d+b c)+6270 a c x^2 (a d+b c)+1155 b^2 d^2 x^8\right )}{21945} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 97, normalized size = 1. \begin{align*}{\frac{2310\,{b}^{2}{d}^{2}{x}^{8}+5852\,{x}^{6}ab{d}^{2}+5852\,{x}^{6}{b}^{2}cd+3990\,{x}^{4}{a}^{2}{d}^{2}+15960\,{x}^{4}abcd+3990\,{x}^{4}{b}^{2}{c}^{2}+12540\,{x}^{2}{a}^{2}cd+12540\,a{c}^{2}b{x}^{2}+14630\,{a}^{2}{c}^{2}}{21945}{x}^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07824, size = 115, normalized size = 1.19 \begin{align*} \frac{2}{19} \, b^{2} d^{2} x^{\frac{19}{2}} + \frac{4}{15} \,{\left (b^{2} c d + a b d^{2}\right )} x^{\frac{15}{2}} + \frac{2}{11} \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{\frac{11}{2}} + \frac{2}{3} \, a^{2} c^{2} x^{\frac{3}{2}} + \frac{4}{7} \,{\left (a b c^{2} + a^{2} c 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 = 0.736918, size = 213, normalized size = 2.2 \begin{align*} \frac{2}{21945} \,{\left (1155 \, b^{2} d^{2} x^{9} + 2926 \,{\left (b^{2} c d + a b d^{2}\right )} x^{7} + 1995 \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{5} + 7315 \, a^{2} c^{2} x + 6270 \,{\left (a b c^{2} + a^{2} c 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 = 3.73765, size = 110, normalized size = 1.13 \begin{align*} \frac{2 a^{2} c^{2} x^{\frac{3}{2}}}{3} + \frac{2 b^{2} d^{2} x^{\frac{19}{2}}}{19} + \frac{2 x^{\frac{15}{2}} \left (2 a b d^{2} + 2 b^{2} c d\right )}{15} + \frac{2 x^{\frac{11}{2}} \left (a^{2} d^{2} + 4 a b c d + b^{2} c^{2}\right )}{11} + \frac{2 x^{\frac{7}{2}} \left (2 a^{2} c d + 2 a b c^{2}\right )}{7} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2185, size = 127, normalized size = 1.31 \begin{align*} \frac{2}{19} \, b^{2} d^{2} x^{\frac{19}{2}} + \frac{4}{15} \, b^{2} c d x^{\frac{15}{2}} + \frac{4}{15} \, a b d^{2} x^{\frac{15}{2}} + \frac{2}{11} \, b^{2} c^{2} x^{\frac{11}{2}} + \frac{8}{11} \, a b c d x^{\frac{11}{2}} + \frac{2}{11} \, a^{2} d^{2} x^{\frac{11}{2}} + \frac{4}{7} \, a b c^{2} x^{\frac{7}{2}} + \frac{4}{7} \, a^{2} c d x^{\frac{7}{2}} + \frac{2}{3} \, a^{2} c^{2} x^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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