Optimal. Leaf size=63 \[ \frac {2}{13} A b^2 x^{13/2}+\frac {2}{17} b (b B+2 A c) x^{17/2}+\frac {2}{21} c (2 b B+A c) x^{21/2}+\frac {2}{25} B c^2 x^{25/2} \]
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Rubi [A]
time = 0.03, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1598, 459}
\begin {gather*} \frac {2}{13} A b^2 x^{13/2}+\frac {2}{21} c x^{21/2} (A c+2 b B)+\frac {2}{17} b x^{17/2} (2 A c+b B)+\frac {2}{25} B c^2 x^{25/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 459
Rule 1598
Rubi steps
\begin {align*} \int x^{3/2} \left (A+B x^2\right ) \left (b x^2+c x^4\right )^2 \, dx &=\int x^{11/2} \left (A+B x^2\right ) \left (b+c x^2\right )^2 \, dx\\ &=\int \left (A b^2 x^{11/2}+b (b B+2 A c) x^{15/2}+c (2 b B+A c) x^{19/2}+B c^2 x^{23/2}\right ) \, dx\\ &=\frac {2}{13} A b^2 x^{13/2}+\frac {2}{17} b (b B+2 A c) x^{17/2}+\frac {2}{21} c (2 b B+A c) x^{21/2}+\frac {2}{25} B c^2 x^{25/2}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 61, normalized size = 0.97 \begin {gather*} \frac {2 x^{13/2} \left (25 A \left (357 b^2+546 b c x^2+221 c^2 x^4\right )+13 B x^2 \left (525 b^2+850 b c x^2+357 c^2 x^4\right )\right )}{116025} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.35, size = 52, normalized size = 0.83
method | result | size |
derivativedivides | \(\frac {2 B \,c^{2} x^{\frac {25}{2}}}{25}+\frac {2 \left (A \,c^{2}+2 b B c \right ) x^{\frac {21}{2}}}{21}+\frac {2 \left (2 A b c +b^{2} B \right ) x^{\frac {17}{2}}}{17}+\frac {2 A \,b^{2} x^{\frac {13}{2}}}{13}\) | \(52\) |
default | \(\frac {2 B \,c^{2} x^{\frac {25}{2}}}{25}+\frac {2 \left (A \,c^{2}+2 b B c \right ) x^{\frac {21}{2}}}{21}+\frac {2 \left (2 A b c +b^{2} B \right ) x^{\frac {17}{2}}}{17}+\frac {2 A \,b^{2} x^{\frac {13}{2}}}{13}\) | \(52\) |
gosper | \(\frac {2 x^{\frac {13}{2}} \left (4641 B \,c^{2} x^{6}+5525 A \,c^{2} x^{4}+11050 x^{4} b B c +13650 A b c \,x^{2}+6825 b^{2} B \,x^{2}+8925 b^{2} A \right )}{116025}\) | \(56\) |
trager | \(\frac {2 x^{\frac {13}{2}} \left (4641 B \,c^{2} x^{6}+5525 A \,c^{2} x^{4}+11050 x^{4} b B c +13650 A b c \,x^{2}+6825 b^{2} B \,x^{2}+8925 b^{2} A \right )}{116025}\) | \(56\) |
risch | \(\frac {2 x^{\frac {13}{2}} \left (4641 B \,c^{2} x^{6}+5525 A \,c^{2} x^{4}+11050 x^{4} b B c +13650 A b c \,x^{2}+6825 b^{2} B \,x^{2}+8925 b^{2} A \right )}{116025}\) | \(56\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 51, normalized size = 0.81 \begin {gather*} \frac {2}{25} \, B c^{2} x^{\frac {25}{2}} + \frac {2}{21} \, {\left (2 \, B b c + A c^{2}\right )} x^{\frac {21}{2}} + \frac {2}{13} \, A b^{2} x^{\frac {13}{2}} + \frac {2}{17} \, {\left (B b^{2} + 2 \, A b c\right )} x^{\frac {17}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.55, size = 56, normalized size = 0.89 \begin {gather*} \frac {2}{116025} \, {\left (4641 \, B c^{2} x^{12} + 5525 \, {\left (2 \, B b c + A c^{2}\right )} x^{10} + 8925 \, A b^{2} x^{6} + 6825 \, {\left (B b^{2} + 2 \, A b c\right )} x^{8}\right )} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.10, size = 80, normalized size = 1.27 \begin {gather*} \frac {2 A b^{2} x^{\frac {13}{2}}}{13} + \frac {4 A b c x^{\frac {17}{2}}}{17} + \frac {2 A c^{2} x^{\frac {21}{2}}}{21} + \frac {2 B b^{2} x^{\frac {17}{2}}}{17} + \frac {4 B b c x^{\frac {21}{2}}}{21} + \frac {2 B c^{2} x^{\frac {25}{2}}}{25} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.47, size = 53, normalized size = 0.84 \begin {gather*} \frac {2}{25} \, B c^{2} x^{\frac {25}{2}} + \frac {4}{21} \, B b c x^{\frac {21}{2}} + \frac {2}{21} \, A c^{2} x^{\frac {21}{2}} + \frac {2}{17} \, B b^{2} x^{\frac {17}{2}} + \frac {4}{17} \, A b c x^{\frac {17}{2}} + \frac {2}{13} \, A b^{2} x^{\frac {13}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 51, normalized size = 0.81 \begin {gather*} x^{17/2}\,\left (\frac {2\,B\,b^2}{17}+\frac {4\,A\,c\,b}{17}\right )+x^{21/2}\,\left (\frac {2\,A\,c^2}{21}+\frac {4\,B\,b\,c}{21}\right )+\frac {2\,A\,b^2\,x^{13/2}}{13}+\frac {2\,B\,c^2\,x^{25/2}}{25} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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