Optimal. Leaf size=63 \[ \frac {2}{9} a^2 c x^{9/2}+\frac {2}{17} b x^{17/2} (2 a d+b c)+\frac {2}{13} a x^{13/2} (a d+2 b c)+\frac {2}{21} b^2 d x^{21/2} \]
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Rubi [A] time = 0.03, antiderivative size = 63, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {2}{9} a^2 c x^{9/2}+\frac {2}{17} b x^{17/2} (2 a d+b c)+\frac {2}{13} a x^{13/2} (a d+2 b c)+\frac {2}{21} b^2 d x^{21/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 448
Rubi steps
\begin {align*} \int x^{7/2} \left (a+b x^2\right )^2 \left (c+d x^2\right ) \, dx &=\int \left (a^2 c x^{7/2}+a (2 b c+a d) x^{11/2}+b (b c+2 a d) x^{15/2}+b^2 d x^{19/2}\right ) \, dx\\ &=\frac {2}{9} a^2 c x^{9/2}+\frac {2}{13} a (2 b c+a d) x^{13/2}+\frac {2}{17} b (b c+2 a d) x^{17/2}+\frac {2}{21} b^2 d x^{21/2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 53, normalized size = 0.84 \begin {gather*} \frac {2 x^{9/2} \left (1547 a^2 c+819 b x^4 (2 a d+b c)+1071 a x^2 (a d+2 b c)+663 b^2 d x^6\right )}{13923} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.04, size = 69, normalized size = 1.10 \begin {gather*} \frac {2 \left (1547 a^2 c x^{9/2}+1071 a^2 d x^{13/2}+2142 a b c x^{13/2}+1638 a b d x^{17/2}+819 b^2 c x^{17/2}+663 b^2 d x^{21/2}\right )}{13923} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.05, size = 56, normalized size = 0.89 \begin {gather*} \frac {2}{13923} \, {\left (663 \, b^{2} d x^{10} + 819 \, {\left (b^{2} c + 2 \, a b d\right )} x^{8} + 1547 \, a^{2} c x^{4} + 1071 \, {\left (2 \, a b c + a^{2} d\right )} x^{6}\right )} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 53, normalized size = 0.84 \begin {gather*} \frac {2}{21} \, b^{2} d x^{\frac {21}{2}} + \frac {2}{17} \, b^{2} c x^{\frac {17}{2}} + \frac {4}{17} \, a b d x^{\frac {17}{2}} + \frac {4}{13} \, a b c x^{\frac {13}{2}} + \frac {2}{13} \, a^{2} d x^{\frac {13}{2}} + \frac {2}{9} \, a^{2} c x^{\frac {9}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 56, normalized size = 0.89 \begin {gather*} \frac {2 \left (663 b^{2} d \,x^{6}+1638 a b d \,x^{4}+819 b^{2} c \,x^{4}+1071 a^{2} d \,x^{2}+2142 a b c \,x^{2}+1547 a^{2} c \right ) x^{\frac {9}{2}}}{13923} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.00, size = 51, normalized size = 0.81 \begin {gather*} \frac {2}{21} \, b^{2} d x^{\frac {21}{2}} + \frac {2}{17} \, {\left (b^{2} c + 2 \, a b d\right )} x^{\frac {17}{2}} + \frac {2}{9} \, a^{2} c x^{\frac {9}{2}} + \frac {2}{13} \, {\left (2 \, a b c + a^{2} d\right )} x^{\frac {13}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 51, normalized size = 0.81 \begin {gather*} x^{13/2}\,\left (\frac {2\,d\,a^2}{13}+\frac {4\,b\,c\,a}{13}\right )+x^{17/2}\,\left (\frac {2\,c\,b^2}{17}+\frac {4\,a\,d\,b}{17}\right )+\frac {2\,a^2\,c\,x^{9/2}}{9}+\frac {2\,b^2\,d\,x^{21/2}}{21} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 19.90, size = 80, normalized size = 1.27 \begin {gather*} \frac {2 a^{2} c x^{\frac {9}{2}}}{9} + \frac {2 a^{2} d x^{\frac {13}{2}}}{13} + \frac {4 a b c x^{\frac {13}{2}}}{13} + \frac {4 a b d x^{\frac {17}{2}}}{17} + \frac {2 b^{2} c x^{\frac {17}{2}}}{17} + \frac {2 b^{2} d x^{\frac {21}{2}}}{21} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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