Optimal. Leaf size=36 \[ \frac {2}{3} b^2 x^{3/2}+\frac {4}{7} b c x^{7/2}+\frac {2}{11} c^2 x^{11/2} \]
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Rubi [A] time = 0.02, antiderivative size = 36, 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 {2}{3} b^2 x^{3/2}+\frac {4}{7} b c x^{7/2}+\frac {2}{11} c^2 x^{11/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rule 1584
Rubi steps
\begin {align*} \int \frac {\left (b x^2+c x^4\right )^2}{x^{7/2}} \, dx &=\int \sqrt {x} \left (b+c x^2\right )^2 \, dx\\ &=\int \left (b^2 \sqrt {x}+2 b c x^{5/2}+c^2 x^{9/2}\right ) \, dx\\ &=\frac {2}{3} b^2 x^{3/2}+\frac {4}{7} b c x^{7/2}+\frac {2}{11} c^2 x^{11/2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 30, normalized size = 0.83 \begin {gather*} \frac {2}{231} x^{3/2} \left (77 b^2+66 b c x^2+21 c^2 x^4\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.02, size = 34, normalized size = 0.94 \begin {gather*} \frac {2}{231} \left (77 b^2 x^{3/2}+66 b c x^{7/2}+21 c^2 x^{11/2}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 27, normalized size = 0.75 \begin {gather*} \frac {2}{231} \, {\left (21 \, c^{2} x^{5} + 66 \, b c x^{3} + 77 \, b^{2} x\right )} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 24, normalized size = 0.67 \begin {gather*} \frac {2}{11} \, c^{2} x^{\frac {11}{2}} + \frac {4}{7} \, b c x^{\frac {7}{2}} + \frac {2}{3} \, b^{2} x^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 27, normalized size = 0.75 \begin {gather*} \frac {2 \left (21 c^{2} x^{4}+66 b c \,x^{2}+77 b^{2}\right ) x^{\frac {3}{2}}}{231} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 24, normalized size = 0.67 \begin {gather*} \frac {2}{11} \, c^{2} x^{\frac {11}{2}} + \frac {4}{7} \, b c x^{\frac {7}{2}} + \frac {2}{3} \, b^{2} x^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 26, normalized size = 0.72 \begin {gather*} \frac {2\,x^{3/2}\,\left (77\,b^2+66\,b\,c\,x^2+21\,c^2\,x^4\right )}{231} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 8.69, size = 34, normalized size = 0.94 \begin {gather*} \frac {2 b^{2} x^{\frac {3}{2}}}{3} + \frac {4 b c x^{\frac {7}{2}}}{7} + \frac {2 c^{2} x^{\frac {11}{2}}}{11} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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