Optimal. Leaf size=139 \[ \frac{2}{17} d x^{17/2} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{2}{13} c x^{13/2} \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+\frac{2}{5} a^2 c^3 x^{5/2}+\frac{2}{9} a c^2 x^{9/2} (3 a d+2 b c)+\frac{2}{21} b d^2 x^{21/2} (2 a d+3 b c)+\frac{2}{25} b^2 d^3 x^{25/2} \]
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Rubi [A] time = 0.0632243, antiderivative size = 139, 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} \[ \frac{2}{17} d x^{17/2} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{2}{13} c x^{13/2} \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+\frac{2}{5} a^2 c^3 x^{5/2}+\frac{2}{9} a c^2 x^{9/2} (3 a d+2 b c)+\frac{2}{21} b d^2 x^{21/2} (2 a d+3 b c)+\frac{2}{25} b^2 d^3 x^{25/2} \]
Antiderivative was successfully verified.
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Rule 448
Rubi steps
\begin{align*} \int x^{3/2} \left (a+b x^2\right )^2 \left (c+d x^2\right )^3 \, dx &=\int \left (a^2 c^3 x^{3/2}+a c^2 (2 b c+3 a d) x^{7/2}+c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^{11/2}+d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^{15/2}+b d^2 (3 b c+2 a d) x^{19/2}+b^2 d^3 x^{23/2}\right ) \, dx\\ &=\frac{2}{5} a^2 c^3 x^{5/2}+\frac{2}{9} a c^2 (2 b c+3 a d) x^{9/2}+\frac{2}{13} c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^{13/2}+\frac{2}{17} d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^{17/2}+\frac{2}{21} b d^2 (3 b c+2 a d) x^{21/2}+\frac{2}{25} b^2 d^3 x^{25/2}\\ \end{align*}
Mathematica [A] time = 0.037461, size = 139, normalized size = 1. \[ \frac{2}{17} d x^{17/2} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{2}{13} c x^{13/2} \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+\frac{2}{5} a^2 c^3 x^{5/2}+\frac{2}{9} a c^2 x^{9/2} (3 a d+2 b c)+\frac{2}{21} b d^2 x^{21/2} (2 a d+3 b c)+\frac{2}{25} b^2 d^3 x^{25/2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 138, normalized size = 1. \begin{align*}{\frac{27846\,{b}^{2}{d}^{3}{x}^{10}+66300\,{x}^{8}ab{d}^{3}+99450\,{x}^{8}{b}^{2}c{d}^{2}+40950\,{x}^{6}{a}^{2}{d}^{3}+245700\,{x}^{6}abc{d}^{2}+122850\,{x}^{6}{b}^{2}{c}^{2}d+160650\,{x}^{4}{a}^{2}c{d}^{2}+321300\,{x}^{4}ab{c}^{2}d+53550\,{x}^{4}{b}^{2}{c}^{3}+232050\,{x}^{2}{a}^{2}{c}^{2}d+154700\,{x}^{2}ab{c}^{3}+139230\,{a}^{2}{c}^{3}}{348075}{x}^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05972, size = 171, normalized size = 1.23 \begin{align*} \frac{2}{25} \, b^{2} d^{3} x^{\frac{25}{2}} + \frac{2}{21} \,{\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{\frac{21}{2}} + \frac{2}{17} \,{\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{\frac{17}{2}} + \frac{2}{5} \, a^{2} c^{3} x^{\frac{5}{2}} + \frac{2}{13} \,{\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{\frac{13}{2}} + \frac{2}{9} \,{\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{\frac{9}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.76166, size = 319, normalized size = 2.29 \begin{align*} \frac{2}{348075} \,{\left (13923 \, b^{2} d^{3} x^{12} + 16575 \,{\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{10} + 20475 \,{\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{8} + 69615 \, a^{2} c^{3} x^{2} + 26775 \,{\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{6} + 38675 \,{\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{4}\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 21.2537, size = 192, normalized size = 1.38 \begin{align*} \frac{2 a^{2} c^{3} x^{\frac{5}{2}}}{5} + \frac{2 a^{2} c^{2} d x^{\frac{9}{2}}}{3} + \frac{6 a^{2} c d^{2} x^{\frac{13}{2}}}{13} + \frac{2 a^{2} d^{3} x^{\frac{17}{2}}}{17} + \frac{4 a b c^{3} x^{\frac{9}{2}}}{9} + \frac{12 a b c^{2} d x^{\frac{13}{2}}}{13} + \frac{12 a b c d^{2} x^{\frac{17}{2}}}{17} + \frac{4 a b d^{3} x^{\frac{21}{2}}}{21} + \frac{2 b^{2} c^{3} x^{\frac{13}{2}}}{13} + \frac{6 b^{2} c^{2} d x^{\frac{17}{2}}}{17} + \frac{2 b^{2} c d^{2} x^{\frac{21}{2}}}{7} + \frac{2 b^{2} d^{3} x^{\frac{25}{2}}}{25} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17432, size = 182, normalized size = 1.31 \begin{align*} \frac{2}{25} \, b^{2} d^{3} x^{\frac{25}{2}} + \frac{2}{7} \, b^{2} c d^{2} x^{\frac{21}{2}} + \frac{4}{21} \, a b d^{3} x^{\frac{21}{2}} + \frac{6}{17} \, b^{2} c^{2} d x^{\frac{17}{2}} + \frac{12}{17} \, a b c d^{2} x^{\frac{17}{2}} + \frac{2}{17} \, a^{2} d^{3} x^{\frac{17}{2}} + \frac{2}{13} \, b^{2} c^{3} x^{\frac{13}{2}} + \frac{12}{13} \, a b c^{2} d x^{\frac{13}{2}} + \frac{6}{13} \, a^{2} c d^{2} x^{\frac{13}{2}} + \frac{4}{9} \, a b c^{3} x^{\frac{9}{2}} + \frac{2}{3} \, a^{2} c^{2} d x^{\frac{9}{2}} + \frac{2}{5} \, a^{2} c^{3} x^{\frac{5}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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