Optimal. Leaf size=85 \[ \frac {2}{9} A b^3 x^{9/2}+\frac {2}{13} b^2 (b B+3 A c) x^{13/2}+\frac {6}{17} b c (b B+A c) x^{17/2}+\frac {2}{21} c^2 (3 b B+A c) x^{21/2}+\frac {2}{25} B c^3 x^{25/2} \]
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Rubi [A]
time = 0.03, antiderivative size = 85, 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}{9} A b^3 x^{9/2}+\frac {2}{13} b^2 x^{13/2} (3 A c+b B)+\frac {2}{21} c^2 x^{21/2} (A c+3 b B)+\frac {6}{17} b c x^{17/2} (A c+b B)+\frac {2}{25} B c^3 x^{25/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 459
Rule 1598
Rubi steps
\begin {align*} \int \frac {\left (A+B x^2\right ) \left (b x^2+c x^4\right )^3}{x^{5/2}} \, dx &=\int x^{7/2} \left (A+B x^2\right ) \left (b+c x^2\right )^3 \, dx\\ &=\int \left (A b^3 x^{7/2}+b^2 (b B+3 A c) x^{11/2}+3 b c (b B+A c) x^{15/2}+c^2 (3 b B+A c) x^{19/2}+B c^3 x^{23/2}\right ) \, dx\\ &=\frac {2}{9} A b^3 x^{9/2}+\frac {2}{13} b^2 (b B+3 A c) x^{13/2}+\frac {6}{17} b c (b B+A c) x^{17/2}+\frac {2}{21} c^2 (3 b B+A c) x^{21/2}+\frac {2}{25} B c^3 x^{25/2}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 83, normalized size = 0.98 \begin {gather*} \frac {2 x^{9/2} \left (25 A \left (1547 b^3+3213 b^2 c x^2+2457 b c^2 x^4+663 c^3 x^6\right )+9 B x^2 \left (2975 b^3+6825 b^2 c x^2+5525 b c^2 x^4+1547 c^3 x^6\right )\right )}{348075} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.35, size = 76, normalized size = 0.89
method | result | size |
derivativedivides | \(\frac {2 B \,c^{3} x^{\frac {25}{2}}}{25}+\frac {2 \left (A \,c^{3}+3 B b \,c^{2}\right ) x^{\frac {21}{2}}}{21}+\frac {2 \left (3 A b \,c^{2}+3 B \,b^{2} c \right ) x^{\frac {17}{2}}}{17}+\frac {2 \left (3 A \,b^{2} c +B \,b^{3}\right ) x^{\frac {13}{2}}}{13}+\frac {2 A \,b^{3} x^{\frac {9}{2}}}{9}\) | \(76\) |
default | \(\frac {2 B \,c^{3} x^{\frac {25}{2}}}{25}+\frac {2 \left (A \,c^{3}+3 B b \,c^{2}\right ) x^{\frac {21}{2}}}{21}+\frac {2 \left (3 A b \,c^{2}+3 B \,b^{2} c \right ) x^{\frac {17}{2}}}{17}+\frac {2 \left (3 A \,b^{2} c +B \,b^{3}\right ) x^{\frac {13}{2}}}{13}+\frac {2 A \,b^{3} x^{\frac {9}{2}}}{9}\) | \(76\) |
gosper | \(\frac {2 x^{\frac {9}{2}} \left (13923 B \,c^{3} x^{8}+16575 A \,c^{3} x^{6}+49725 x^{6} B b \,c^{2}+61425 A b \,c^{2} x^{4}+61425 x^{4} B \,b^{2} c +80325 A \,b^{2} c \,x^{2}+26775 x^{2} B \,b^{3}+38675 A \,b^{3}\right )}{348075}\) | \(80\) |
trager | \(\frac {2 x^{\frac {9}{2}} \left (13923 B \,c^{3} x^{8}+16575 A \,c^{3} x^{6}+49725 x^{6} B b \,c^{2}+61425 A b \,c^{2} x^{4}+61425 x^{4} B \,b^{2} c +80325 A \,b^{2} c \,x^{2}+26775 x^{2} B \,b^{3}+38675 A \,b^{3}\right )}{348075}\) | \(80\) |
risch | \(\frac {2 x^{\frac {9}{2}} \left (13923 B \,c^{3} x^{8}+16575 A \,c^{3} x^{6}+49725 x^{6} B b \,c^{2}+61425 A b \,c^{2} x^{4}+61425 x^{4} B \,b^{2} c +80325 A \,b^{2} c \,x^{2}+26775 x^{2} B \,b^{3}+38675 A \,b^{3}\right )}{348075}\) | \(80\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 73, normalized size = 0.86 \begin {gather*} \frac {2}{25} \, B c^{3} x^{\frac {25}{2}} + \frac {2}{21} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{\frac {21}{2}} + \frac {6}{17} \, {\left (B b^{2} c + A b c^{2}\right )} x^{\frac {17}{2}} + \frac {2}{9} \, A b^{3} x^{\frac {9}{2}} + \frac {2}{13} \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{\frac {13}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.57, size = 78, normalized size = 0.92 \begin {gather*} \frac {2}{348075} \, {\left (13923 \, B c^{3} x^{12} + 16575 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{10} + 61425 \, {\left (B b^{2} c + A b c^{2}\right )} x^{8} + 38675 \, A b^{3} x^{4} + 26775 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{6}\right )} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.02, size = 114, normalized size = 1.34 \begin {gather*} \frac {2 A b^{3} x^{\frac {9}{2}}}{9} + \frac {6 A b^{2} c x^{\frac {13}{2}}}{13} + \frac {6 A b c^{2} x^{\frac {17}{2}}}{17} + \frac {2 A c^{3} x^{\frac {21}{2}}}{21} + \frac {2 B b^{3} x^{\frac {13}{2}}}{13} + \frac {6 B b^{2} c x^{\frac {17}{2}}}{17} + \frac {2 B b c^{2} x^{\frac {21}{2}}}{7} + \frac {2 B c^{3} x^{\frac {25}{2}}}{25} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.46, size = 77, normalized size = 0.91 \begin {gather*} \frac {2}{25} \, B c^{3} x^{\frac {25}{2}} + \frac {2}{7} \, B b c^{2} x^{\frac {21}{2}} + \frac {2}{21} \, A c^{3} x^{\frac {21}{2}} + \frac {6}{17} \, B b^{2} c x^{\frac {17}{2}} + \frac {6}{17} \, A b c^{2} x^{\frac {17}{2}} + \frac {2}{13} \, B b^{3} x^{\frac {13}{2}} + \frac {6}{13} \, A b^{2} c x^{\frac {13}{2}} + \frac {2}{9} \, A b^{3} x^{\frac {9}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 69, normalized size = 0.81 \begin {gather*} x^{13/2}\,\left (\frac {2\,B\,b^3}{13}+\frac {6\,A\,c\,b^2}{13}\right )+x^{21/2}\,\left (\frac {2\,A\,c^3}{21}+\frac {2\,B\,b\,c^2}{7}\right )+\frac {2\,A\,b^3\,x^{9/2}}{9}+\frac {2\,B\,c^3\,x^{25/2}}{25}+\frac {6\,b\,c\,x^{17/2}\,\left (A\,c+B\,b\right )}{17} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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