Optimal. Leaf size=51 \[ \frac {2}{9} b^3 x^{9/2}+\frac {6}{13} b^2 c x^{13/2}+\frac {6}{17} b c^2 x^{17/2}+\frac {2}{21} c^3 x^{21/2} \]
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Rubi [A]
time = 0.01, 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 = {1598, 276}
\begin {gather*} \frac {2}{9} b^3 x^{9/2}+\frac {6}{13} b^2 c x^{13/2}+\frac {6}{17} b c^2 x^{17/2}+\frac {2}{21} c^3 x^{21/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 276
Rule 1598
Rubi steps
\begin {align*} \int \frac {\left (b x^2+c x^4\right )^3}{x^{5/2}} \, dx &=\int x^{7/2} \left (b+c x^2\right )^3 \, dx\\ &=\int \left (b^3 x^{7/2}+3 b^2 c x^{11/2}+3 b c^2 x^{15/2}+c^3 x^{19/2}\right ) \, dx\\ &=\frac {2}{9} b^3 x^{9/2}+\frac {6}{13} b^2 c x^{13/2}+\frac {6}{17} b c^2 x^{17/2}+\frac {2}{21} c^3 x^{21/2}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 41, normalized size = 0.80 \begin {gather*} \frac {2 x^{9/2} \left (1547 b^3+3213 b^2 c x^2+2457 b c^2 x^4+663 c^3 x^6\right )}{13923} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 36, normalized size = 0.71
method | result | size |
derivativedivides | \(\frac {2 b^{3} x^{\frac {9}{2}}}{9}+\frac {6 b^{2} c \,x^{\frac {13}{2}}}{13}+\frac {6 b \,c^{2} x^{\frac {17}{2}}}{17}+\frac {2 c^{3} x^{\frac {21}{2}}}{21}\) | \(36\) |
default | \(\frac {2 b^{3} x^{\frac {9}{2}}}{9}+\frac {6 b^{2} c \,x^{\frac {13}{2}}}{13}+\frac {6 b \,c^{2} x^{\frac {17}{2}}}{17}+\frac {2 c^{3} x^{\frac {21}{2}}}{21}\) | \(36\) |
gosper | \(\frac {2 x^{\frac {9}{2}} \left (663 c^{3} x^{6}+2457 b \,c^{2} x^{4}+3213 b^{2} c \,x^{2}+1547 b^{3}\right )}{13923}\) | \(38\) |
trager | \(\frac {2 x^{\frac {9}{2}} \left (663 c^{3} x^{6}+2457 b \,c^{2} x^{4}+3213 b^{2} c \,x^{2}+1547 b^{3}\right )}{13923}\) | \(38\) |
risch | \(\frac {2 x^{\frac {9}{2}} \left (663 c^{3} x^{6}+2457 b \,c^{2} x^{4}+3213 b^{2} c \,x^{2}+1547 b^{3}\right )}{13923}\) | \(38\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 35, normalized size = 0.69 \begin {gather*} \frac {2}{21} \, c^{3} x^{\frac {21}{2}} + \frac {6}{17} \, b c^{2} x^{\frac {17}{2}} + \frac {6}{13} \, b^{2} c x^{\frac {13}{2}} + \frac {2}{9} \, b^{3} x^{\frac {9}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 40, normalized size = 0.78 \begin {gather*} \frac {2}{13923} \, {\left (663 \, c^{3} x^{10} + 2457 \, b c^{2} x^{8} + 3213 \, b^{2} c x^{6} + 1547 \, b^{3} x^{4}\right )} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.62, size = 49, normalized size = 0.96 \begin {gather*} \frac {2 b^{3} x^{\frac {9}{2}}}{9} + \frac {6 b^{2} c x^{\frac {13}{2}}}{13} + \frac {6 b c^{2} x^{\frac {17}{2}}}{17} + \frac {2 c^{3} x^{\frac {21}{2}}}{21} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.00, size = 35, normalized size = 0.69 \begin {gather*} \frac {2}{21} \, c^{3} x^{\frac {21}{2}} + \frac {6}{17} \, b c^{2} x^{\frac {17}{2}} + \frac {6}{13} \, b^{2} c x^{\frac {13}{2}} + \frac {2}{9} \, 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 = 0.05, size = 35, normalized size = 0.69 \begin {gather*} \frac {2\,b^3\,x^{9/2}}{9}+\frac {2\,c^3\,x^{21/2}}{21}+\frac {6\,b^2\,c\,x^{13/2}}{13}+\frac {6\,b\,c^2\,x^{17/2}}{17} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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