Optimal. Leaf size=39 \[ \frac {2}{17} x^{17/2} (A c+b B)+\frac {2}{13} A b x^{13/2}+\frac {2}{21} B c x^{21/2} \]
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Rubi [A] time = 0.02, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1584, 448} \[ \frac {2}{17} x^{17/2} (A c+b B)+\frac {2}{13} A b x^{13/2}+\frac {2}{21} B c x^{21/2} \]
Antiderivative was successfully verified.
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Rule 448
Rule 1584
Rubi steps
\begin {align*} \int x^{7/2} \left (A+B x^2\right ) \left (b x^2+c x^4\right ) \, dx &=\int x^{11/2} \left (A+B x^2\right ) \left (b+c x^2\right ) \, dx\\ &=\int \left (A b x^{11/2}+(b B+A c) x^{15/2}+B c x^{19/2}\right ) \, dx\\ &=\frac {2}{13} A b x^{13/2}+\frac {2}{17} (b B+A c) x^{17/2}+\frac {2}{21} B c x^{21/2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 33, normalized size = 0.85 \[ \frac {2 x^{13/2} \left (273 x^2 (A c+b B)+357 A b+221 B c x^4\right )}{4641} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.98, size = 32, normalized size = 0.82 \[ \frac {2}{4641} \, {\left (221 \, B c x^{10} + 273 \, {\left (B b + A c\right )} x^{8} + 357 \, A b x^{6}\right )} \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 29, normalized size = 0.74 \[ \frac {2}{21} \, B c x^{\frac {21}{2}} + \frac {2}{17} \, B b x^{\frac {17}{2}} + \frac {2}{17} \, A c x^{\frac {17}{2}} + \frac {2}{13} \, A b x^{\frac {13}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 32, normalized size = 0.82 \[ \frac {2 \left (221 B c \,x^{4}+273 A c \,x^{2}+273 B b \,x^{2}+357 A b \right ) x^{\frac {13}{2}}}{4641} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 27, normalized size = 0.69 \[ \frac {2}{21} \, B c x^{\frac {21}{2}} + \frac {2}{17} \, {\left (B b + A c\right )} x^{\frac {17}{2}} + \frac {2}{13} \, A b x^{\frac {13}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 31, normalized size = 0.79 \[ \frac {2\,x^{13/2}\,\left (357\,A\,b+273\,A\,c\,x^2+273\,B\,b\,x^2+221\,B\,c\,x^4\right )}{4641} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 20.14, size = 46, normalized size = 1.18 \[ \frac {2 A b x^{\frac {13}{2}}}{13} + \frac {2 A c x^{\frac {17}{2}}}{17} + \frac {2 B b x^{\frac {17}{2}}}{17} + \frac {2 B c x^{\frac {21}{2}}}{21} \]
Verification of antiderivative is not currently implemented for this CAS.
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