Optimal. Leaf size=85 \[ \frac {2}{21} A b^3 x^{21/2}+\frac {2}{25} b^2 x^{25/2} (3 A c+b B)+\frac {2}{33} c^2 x^{33/2} (A c+3 b B)+\frac {6}{29} b c x^{29/2} (A c+b B)+\frac {2}{37} B c^3 x^{37/2} \]
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Rubi [A] time = 0.05, 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 = {1584, 448} \begin {gather*} \frac {2}{25} b^2 x^{25/2} (3 A c+b B)+\frac {2}{21} A b^3 x^{21/2}+\frac {2}{33} c^2 x^{33/2} (A c+3 b B)+\frac {6}{29} b c x^{29/2} (A c+b B)+\frac {2}{37} B c^3 x^{37/2} \end {gather*}
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 )^3 \, dx &=\int x^{19/2} \left (A+B x^2\right ) \left (b+c x^2\right )^3 \, dx\\ &=\int \left (A b^3 x^{19/2}+b^2 (b B+3 A c) x^{23/2}+3 b c (b B+A c) x^{27/2}+c^2 (3 b B+A c) x^{31/2}+B c^3 x^{35/2}\right ) \, dx\\ &=\frac {2}{21} A b^3 x^{21/2}+\frac {2}{25} b^2 (b B+3 A c) x^{25/2}+\frac {6}{29} b c (b B+A c) x^{29/2}+\frac {2}{33} c^2 (3 b B+A c) x^{33/2}+\frac {2}{37} B c^3 x^{37/2}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 85, normalized size = 1.00 \begin {gather*} \frac {2}{21} A b^3 x^{21/2}+\frac {2}{25} b^2 x^{25/2} (3 A c+b B)+\frac {2}{33} c^2 x^{33/2} (A c+3 b B)+\frac {6}{29} b c x^{29/2} (A c+b B)+\frac {2}{37} B c^3 x^{37/2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.06, size = 97, normalized size = 1.14 \begin {gather*} \frac {2 \left (295075 A b^3 x^{21/2}+743589 A b^2 c x^{25/2}+641025 A b c^2 x^{29/2}+187775 A c^3 x^{33/2}+247863 b^3 B x^{25/2}+641025 b^2 B c x^{29/2}+563325 b B c^2 x^{33/2}+167475 B c^3 x^{37/2}\right )}{6196575} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 78, normalized size = 0.92 \begin {gather*} \frac {2}{6196575} \, {\left (167475 \, B c^{3} x^{18} + 187775 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{16} + 641025 \, {\left (B b^{2} c + A b c^{2}\right )} x^{14} + 295075 \, A b^{3} x^{10} + 247863 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{12}\right )} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 77, normalized size = 0.91 \begin {gather*} \frac {2}{37} \, B c^{3} x^{\frac {37}{2}} + \frac {2}{11} \, B b c^{2} x^{\frac {33}{2}} + \frac {2}{33} \, A c^{3} x^{\frac {33}{2}} + \frac {6}{29} \, B b^{2} c x^{\frac {29}{2}} + \frac {6}{29} \, A b c^{2} x^{\frac {29}{2}} + \frac {2}{25} \, B b^{3} x^{\frac {25}{2}} + \frac {6}{25} \, A b^{2} c x^{\frac {25}{2}} + \frac {2}{21} \, A b^{3} x^{\frac {21}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 80, normalized size = 0.94 \begin {gather*} \frac {2 \left (167475 B \,c^{3} x^{8}+187775 A \,c^{3} x^{6}+563325 B b \,c^{2} x^{6}+641025 A b \,c^{2} x^{4}+641025 B \,b^{2} c \,x^{4}+743589 A \,b^{2} c \,x^{2}+247863 B \,b^{3} x^{2}+295075 A \,b^{3}\right ) x^{\frac {21}{2}}}{6196575} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 73, normalized size = 0.86 \begin {gather*} \frac {2}{37} \, B c^{3} x^{\frac {37}{2}} + \frac {2}{33} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{\frac {33}{2}} + \frac {6}{29} \, {\left (B b^{2} c + A b c^{2}\right )} x^{\frac {29}{2}} + \frac {2}{21} \, A b^{3} x^{\frac {21}{2}} + \frac {2}{25} \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x^{\frac {25}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 69, normalized size = 0.81 \begin {gather*} x^{25/2}\,\left (\frac {2\,B\,b^3}{25}+\frac {6\,A\,c\,b^2}{25}\right )+x^{33/2}\,\left (\frac {2\,A\,c^3}{33}+\frac {2\,B\,b\,c^2}{11}\right )+\frac {2\,A\,b^3\,x^{21/2}}{21}+\frac {2\,B\,c^3\,x^{37/2}}{37}+\frac {6\,b\,c\,x^{29/2}\,\left (A\,c+B\,b\right )}{29} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 134.77, size = 114, normalized size = 1.34 \begin {gather*} \frac {2 A b^{3} x^{\frac {21}{2}}}{21} + \frac {6 A b^{2} c x^{\frac {25}{2}}}{25} + \frac {6 A b c^{2} x^{\frac {29}{2}}}{29} + \frac {2 A c^{3} x^{\frac {33}{2}}}{33} + \frac {2 B b^{3} x^{\frac {25}{2}}}{25} + \frac {6 B b^{2} c x^{\frac {29}{2}}}{29} + \frac {2 B b c^{2} x^{\frac {33}{2}}}{11} + \frac {2 B c^{3} x^{\frac {37}{2}}}{37} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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