Optimal. Leaf size=51 \[ \frac {2}{15} b^3 x^{15/2}+\frac {6}{19} b^2 c x^{19/2}+\frac {6}{23} b c^2 x^{23/2}+\frac {2}{27} c^3 x^{27/2} \]
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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} \[ \frac {6}{19} b^2 c x^{19/2}+\frac {2}{15} b^3 x^{15/2}+\frac {6}{23} b c^2 x^{23/2}+\frac {2}{27} c^3 x^{27/2} \]
Antiderivative was successfully verified.
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Rule 270
Rule 1584
Rubi steps
\begin {align*} \int \sqrt {x} \left (b x^2+c x^4\right )^3 \, dx &=\int x^{13/2} \left (b+c x^2\right )^3 \, dx\\ &=\int \left (b^3 x^{13/2}+3 b^2 c x^{17/2}+3 b c^2 x^{21/2}+c^3 x^{25/2}\right ) \, dx\\ &=\frac {2}{15} b^3 x^{15/2}+\frac {6}{19} b^2 c x^{19/2}+\frac {6}{23} b c^2 x^{23/2}+\frac {2}{27} c^3 x^{27/2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 41, normalized size = 0.80 \[ \frac {2 x^{15/2} \left (3933 b^3+9315 b^2 c x^2+7695 b c^2 x^4+2185 c^3 x^6\right )}{58995} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 40, normalized size = 0.78 \[ \frac {2}{58995} \, {\left (2185 \, c^{3} x^{13} + 7695 \, b c^{2} x^{11} + 9315 \, b^{2} c x^{9} + 3933 \, b^{3} x^{7}\right )} \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 35, normalized size = 0.69 \[ \frac {2}{27} \, c^{3} x^{\frac {27}{2}} + \frac {6}{23} \, b c^{2} x^{\frac {23}{2}} + \frac {6}{19} \, b^{2} c x^{\frac {19}{2}} + \frac {2}{15} \, b^{3} x^{\frac {15}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 38, normalized size = 0.75 \[ \frac {2 \left (2185 c^{3} x^{6}+7695 b \,c^{2} x^{4}+9315 b^{2} c \,x^{2}+3933 b^{3}\right ) x^{\frac {15}{2}}}{58995} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 35, normalized size = 0.69 \[ \frac {2}{27} \, c^{3} x^{\frac {27}{2}} + \frac {6}{23} \, b c^{2} x^{\frac {23}{2}} + \frac {6}{19} \, b^{2} c x^{\frac {19}{2}} + \frac {2}{15} \, b^{3} x^{\frac {15}{2}} \]
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 \[ \frac {2\,b^3\,x^{15/2}}{15}+\frac {2\,c^3\,x^{27/2}}{27}+\frac {6\,b^2\,c\,x^{19/2}}{19}+\frac {6\,b\,c^2\,x^{23/2}}{23} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.32, size = 49, normalized size = 0.96 \[ \frac {2 b^{3} x^{\frac {15}{2}}}{15} + \frac {6 b^{2} c x^{\frac {19}{2}}}{19} + \frac {6 b c^{2} x^{\frac {23}{2}}}{23} + \frac {2 c^{3} x^{\frac {27}{2}}}{27} \]
Verification of antiderivative is not currently implemented for this CAS.
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