Optimal. Leaf size=51 \[ \frac {2}{19} b^3 x^{19/2}+\frac {6}{23} b^2 c x^{23/2}+\frac {2}{9} b c^2 x^{27/2}+\frac {2}{31} c^3 x^{31/2} \]
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Rubi [A] time = 0.02, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {1584, 270} \begin {gather*} \frac {6}{23} b^2 c x^{23/2}+\frac {2}{19} b^3 x^{19/2}+\frac {2}{9} b c^2 x^{27/2}+\frac {2}{31} c^3 x^{31/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rule 1584
Rubi steps
\begin {align*} \int x^{5/2} \left (b x^2+c x^4\right )^3 \, dx &=\int x^{17/2} \left (b+c x^2\right )^3 \, dx\\ &=\int \left (b^3 x^{17/2}+3 b^2 c x^{21/2}+3 b c^2 x^{25/2}+c^3 x^{29/2}\right ) \, dx\\ &=\frac {2}{19} b^3 x^{19/2}+\frac {6}{23} b^2 c x^{23/2}+\frac {2}{9} b c^2 x^{27/2}+\frac {2}{31} c^3 x^{31/2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 51, normalized size = 1.00 \begin {gather*} \frac {2}{19} b^3 x^{19/2}+\frac {6}{23} b^2 c x^{23/2}+\frac {2}{9} b c^2 x^{27/2}+\frac {2}{31} c^3 x^{31/2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.03, size = 47, normalized size = 0.92 \begin {gather*} \frac {2 \left (6417 b^3 x^{19/2}+15903 b^2 c x^{23/2}+13547 b c^2 x^{27/2}+3933 c^3 x^{31/2}\right )}{121923} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 40, normalized size = 0.78 \begin {gather*} \frac {2}{121923} \, {\left (3933 \, c^{3} x^{15} + 13547 \, b c^{2} x^{13} + 15903 \, b^{2} c x^{11} + 6417 \, b^{3} x^{9}\right )} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 35, normalized size = 0.69 \begin {gather*} \frac {2}{31} \, c^{3} x^{\frac {31}{2}} + \frac {2}{9} \, b c^{2} x^{\frac {27}{2}} + \frac {6}{23} \, b^{2} c x^{\frac {23}{2}} + \frac {2}{19} \, b^{3} x^{\frac {19}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 38, normalized size = 0.75 \begin {gather*} \frac {2 \left (3933 c^{3} x^{6}+13547 b \,c^{2} x^{4}+15903 b^{2} c \,x^{2}+6417 b^{3}\right ) x^{\frac {19}{2}}}{121923} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 35, normalized size = 0.69 \begin {gather*} \frac {2}{31} \, c^{3} x^{\frac {31}{2}} + \frac {2}{9} \, b c^{2} x^{\frac {27}{2}} + \frac {6}{23} \, b^{2} c x^{\frac {23}{2}} + \frac {2}{19} \, b^{3} x^{\frac {19}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 35, normalized size = 0.69 \begin {gather*} \frac {2\,b^3\,x^{19/2}}{19}+\frac {2\,c^3\,x^{31/2}}{31}+\frac {6\,b^2\,c\,x^{23/2}}{23}+\frac {2\,b\,c^2\,x^{27/2}}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 53.38, size = 49, normalized size = 0.96 \begin {gather*} \frac {2 b^{3} x^{\frac {19}{2}}}{19} + \frac {6 b^{2} c x^{\frac {23}{2}}}{23} + \frac {2 b c^{2} x^{\frac {27}{2}}}{9} + \frac {2 c^{3} x^{\frac {31}{2}}}{31} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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