Optimal. Leaf size=139 \[ \frac{2}{15} d x^{15/2} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{2}{11} c x^{11/2} \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+\frac{2}{3} a^2 c^3 x^{3/2}+\frac{2}{7} a c^2 x^{7/2} (3 a d+2 b c)+\frac{2}{19} b d^2 x^{19/2} (2 a d+3 b c)+\frac{2}{23} b^2 d^3 x^{23/2} \]
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Rubi [A] time = 0.0635255, 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} \[ \frac{2}{15} d x^{15/2} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{2}{11} c x^{11/2} \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+\frac{2}{3} a^2 c^3 x^{3/2}+\frac{2}{7} a c^2 x^{7/2} (3 a d+2 b c)+\frac{2}{19} b d^2 x^{19/2} (2 a d+3 b c)+\frac{2}{23} b^2 d^3 x^{23/2} \]
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 )^3 \, dx &=\int \left (a^2 c^3 \sqrt{x}+a c^2 (2 b c+3 a d) x^{5/2}+c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^{9/2}+d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^{13/2}+b d^2 (3 b c+2 a d) x^{17/2}+b^2 d^3 x^{21/2}\right ) \, dx\\ &=\frac{2}{3} a^2 c^3 x^{3/2}+\frac{2}{7} a c^2 (2 b c+3 a d) x^{7/2}+\frac{2}{11} c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^{11/2}+\frac{2}{15} d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^{15/2}+\frac{2}{19} b d^2 (3 b c+2 a d) x^{19/2}+\frac{2}{23} b^2 d^3 x^{23/2}\\ \end{align*}
Mathematica [A] time = 0.0375953, size = 139, normalized size = 1. \[ \frac{2}{15} d x^{15/2} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{2}{11} c x^{11/2} \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+\frac{2}{3} a^2 c^3 x^{3/2}+\frac{2}{7} a c^2 x^{7/2} (3 a d+2 b c)+\frac{2}{19} b d^2 x^{19/2} (2 a d+3 b c)+\frac{2}{23} b^2 d^3 x^{23/2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 138, normalized size = 1. \begin{align*}{\frac{43890\,{b}^{2}{d}^{3}{x}^{10}+106260\,{x}^{8}ab{d}^{3}+159390\,{x}^{8}{b}^{2}c{d}^{2}+67298\,{x}^{6}{a}^{2}{d}^{3}+403788\,{x}^{6}abc{d}^{2}+201894\,{x}^{6}{b}^{2}{c}^{2}d+275310\,{x}^{4}{a}^{2}c{d}^{2}+550620\,{x}^{4}ab{c}^{2}d+91770\,{x}^{4}{b}^{2}{c}^{3}+432630\,{x}^{2}{a}^{2}{c}^{2}d+288420\,{x}^{2}ab{c}^{3}+336490\,{a}^{2}{c}^{3}}{504735}{x}^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04112, size = 171, normalized size = 1.23 \begin{align*} \frac{2}{23} \, b^{2} d^{3} x^{\frac{23}{2}} + \frac{2}{19} \,{\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{\frac{19}{2}} + \frac{2}{15} \,{\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{\frac{15}{2}} + \frac{2}{3} \, a^{2} c^{3} x^{\frac{3}{2}} + \frac{2}{11} \,{\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{\frac{11}{2}} + \frac{2}{7} \,{\left (2 \, a b c^{3} + 3 \, a^{2} c^{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 = 0.80791, size = 316, normalized size = 2.27 \begin{align*} \frac{2}{504735} \,{\left (21945 \, b^{2} d^{3} x^{11} + 26565 \,{\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{9} + 33649 \,{\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{7} + 168245 \, a^{2} c^{3} x + 45885 \,{\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{5} + 72105 \,{\left (2 \, a b c^{3} + 3 \, a^{2} c^{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 = 5.20731, size = 155, normalized size = 1.12 \begin{align*} \frac{2 a^{2} c^{3} x^{\frac{3}{2}}}{3} + \frac{2 b^{2} d^{3} x^{\frac{23}{2}}}{23} + \frac{2 x^{\frac{19}{2}} \left (2 a b d^{3} + 3 b^{2} c d^{2}\right )}{19} + \frac{2 x^{\frac{15}{2}} \left (a^{2} d^{3} + 6 a b c d^{2} + 3 b^{2} c^{2} d\right )}{15} + \frac{2 x^{\frac{11}{2}} \left (3 a^{2} c d^{2} + 6 a b c^{2} d + b^{2} c^{3}\right )}{11} + \frac{2 x^{\frac{7}{2}} \left (3 a^{2} c^{2} d + 2 a b c^{3}\right )}{7} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17829, size = 182, normalized size = 1.31 \begin{align*} \frac{2}{23} \, b^{2} d^{3} x^{\frac{23}{2}} + \frac{6}{19} \, b^{2} c d^{2} x^{\frac{19}{2}} + \frac{4}{19} \, a b d^{3} x^{\frac{19}{2}} + \frac{2}{5} \, b^{2} c^{2} d x^{\frac{15}{2}} + \frac{4}{5} \, a b c d^{2} x^{\frac{15}{2}} + \frac{2}{15} \, a^{2} d^{3} x^{\frac{15}{2}} + \frac{2}{11} \, b^{2} c^{3} x^{\frac{11}{2}} + \frac{12}{11} \, a b c^{2} d x^{\frac{11}{2}} + \frac{6}{11} \, a^{2} c d^{2} x^{\frac{11}{2}} + \frac{4}{7} \, a b c^{3} x^{\frac{7}{2}} + \frac{6}{7} \, a^{2} c^{2} d x^{\frac{7}{2}} + \frac{2}{3} \, a^{2} c^{3} x^{\frac{3}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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