Optimal. Leaf size=137 \[ \frac {2}{13} d x^{13/2} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac {2}{9} c x^{9/2} \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+2 a^2 c^3 \sqrt {x}+\frac {2}{5} a c^2 x^{5/2} (3 a d+2 b c)+\frac {2}{17} b d^2 x^{17/2} (2 a d+3 b c)+\frac {2}{21} b^2 d^3 x^{21/2} \]
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Rubi [A] time = 0.06, antiderivative size = 137, normalized size of antiderivative = 1.00, 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} \[ \frac {2}{13} d x^{13/2} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac {2}{9} c x^{9/2} \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+2 a^2 c^3 \sqrt {x}+\frac {2}{5} a c^2 x^{5/2} (3 a d+2 b c)+\frac {2}{17} b d^2 x^{17/2} (2 a d+3 b c)+\frac {2}{21} b^2 d^3 x^{21/2} \]
Antiderivative was successfully verified.
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Rule 448
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2 \left (c+d x^2\right )^3}{\sqrt {x}} \, dx &=\int \left (\frac {a^2 c^3}{\sqrt {x}}+a c^2 (2 b c+3 a d) x^{3/2}+c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^{7/2}+d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^{11/2}+b d^2 (3 b c+2 a d) x^{15/2}+b^2 d^3 x^{19/2}\right ) \, dx\\ &=2 a^2 c^3 \sqrt {x}+\frac {2}{5} a c^2 (2 b c+3 a d) x^{5/2}+\frac {2}{9} c \left (b^2 c^2+6 a b c d+3 a^2 d^2\right ) x^{9/2}+\frac {2}{13} d \left (3 b^2 c^2+6 a b c d+a^2 d^2\right ) x^{13/2}+\frac {2}{17} b d^2 (3 b c+2 a d) x^{17/2}+\frac {2}{21} b^2 d^3 x^{21/2}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 137, normalized size = 1.00 \[ \frac {2}{13} d x^{13/2} \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac {2}{9} c x^{9/2} \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+2 a^2 c^3 \sqrt {x}+\frac {2}{5} a c^2 x^{5/2} (3 a d+2 b c)+\frac {2}{17} b d^2 x^{17/2} (2 a d+3 b c)+\frac {2}{21} b^2 d^3 x^{21/2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 129, normalized size = 0.94 \[ \frac {2}{69615} \, {\left (3315 \, b^{2} d^{3} x^{10} + 4095 \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{8} + 5355 \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{6} + 69615 \, a^{2} c^{3} + 7735 \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{4} + 13923 \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{2}\right )} \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 135, normalized size = 0.99 \[ \frac {2}{21} \, b^{2} d^{3} x^{\frac {21}{2}} + \frac {6}{17} \, b^{2} c d^{2} x^{\frac {17}{2}} + \frac {4}{17} \, a b d^{3} x^{\frac {17}{2}} + \frac {6}{13} \, b^{2} c^{2} d x^{\frac {13}{2}} + \frac {12}{13} \, a b c d^{2} x^{\frac {13}{2}} + \frac {2}{13} \, a^{2} d^{3} x^{\frac {13}{2}} + \frac {2}{9} \, b^{2} c^{3} x^{\frac {9}{2}} + \frac {4}{3} \, a b c^{2} d x^{\frac {9}{2}} + \frac {2}{3} \, a^{2} c d^{2} x^{\frac {9}{2}} + \frac {4}{5} \, a b c^{3} x^{\frac {5}{2}} + \frac {6}{5} \, a^{2} c^{2} d x^{\frac {5}{2}} + 2 \, a^{2} c^{3} \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 138, normalized size = 1.01 \[ \frac {2 \left (3315 b^{2} d^{3} x^{10}+8190 a b \,d^{3} x^{8}+12285 b^{2} c \,d^{2} x^{8}+5355 a^{2} d^{3} x^{6}+32130 a b c \,d^{2} x^{6}+16065 b^{2} c^{2} d \,x^{6}+23205 a^{2} c \,d^{2} x^{4}+46410 a b \,c^{2} d \,x^{4}+7735 b^{2} c^{3} x^{4}+41769 a^{2} c^{2} d \,x^{2}+27846 a b \,c^{3} x^{2}+69615 a^{2} c^{3}\right ) \sqrt {x}}{69615} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.08, size = 127, normalized size = 0.93 \[ \frac {2}{21} \, b^{2} d^{3} x^{\frac {21}{2}} + \frac {2}{17} \, {\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{\frac {17}{2}} + \frac {2}{13} \, {\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{\frac {13}{2}} + 2 \, a^{2} c^{3} \sqrt {x} + \frac {2}{9} \, {\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{\frac {9}{2}} + \frac {2}{5} \, {\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{\frac {5}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 119, normalized size = 0.87 \[ x^{9/2}\,\left (\frac {2\,a^2\,c\,d^2}{3}+\frac {4\,a\,b\,c^2\,d}{3}+\frac {2\,b^2\,c^3}{9}\right )+x^{13/2}\,\left (\frac {2\,a^2\,d^3}{13}+\frac {12\,a\,b\,c\,d^2}{13}+\frac {6\,b^2\,c^2\,d}{13}\right )+2\,a^2\,c^3\,\sqrt {x}+\frac {2\,b^2\,d^3\,x^{21/2}}{21}+\frac {2\,a\,c^2\,x^{5/2}\,\left (3\,a\,d+2\,b\,c\right )}{5}+\frac {2\,b\,d^2\,x^{17/2}\,\left (2\,a\,d+3\,b\,c\right )}{17} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 9.68, size = 190, normalized size = 1.39 \[ 2 a^{2} c^{3} \sqrt {x} + \frac {6 a^{2} c^{2} d x^{\frac {5}{2}}}{5} + \frac {2 a^{2} c d^{2} x^{\frac {9}{2}}}{3} + \frac {2 a^{2} d^{3} x^{\frac {13}{2}}}{13} + \frac {4 a b c^{3} x^{\frac {5}{2}}}{5} + \frac {4 a b c^{2} d x^{\frac {9}{2}}}{3} + \frac {12 a b c d^{2} x^{\frac {13}{2}}}{13} + \frac {4 a b d^{3} x^{\frac {17}{2}}}{17} + \frac {2 b^{2} c^{3} x^{\frac {9}{2}}}{9} + \frac {6 b^{2} c^{2} d x^{\frac {13}{2}}}{13} + \frac {6 b^{2} c d^{2} x^{\frac {17}{2}}}{17} + \frac {2 b^{2} d^{3} x^{\frac {21}{2}}}{21} \]
Verification of antiderivative is not currently implemented for this CAS.
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