Optimal. Leaf size=49 \[ \frac {2 x^{13/2}}{13}+\frac {8 x^{11/2}}{11}+\frac {2 x^{9/2}}{9}+\frac {8 x^{7/2}}{7}-\frac {2 x^6}{3}-x^4 \]
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Rubi [A] time = 0.06, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1584, 1820, 266, 43} \[ \frac {2 x^{13/2}}{13}-\frac {2 x^6}{3}+\frac {8 x^{11/2}}{11}+\frac {2 x^{9/2}}{9}-x^4+\frac {8 x^{7/2}}{7} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 1584
Rule 1820
Rubi steps
\begin {align*} \int \left (2 \sqrt {x}-x\right )^2 x^{3/2} \left (1+x^2\right ) \, dx &=\int \left (2-\sqrt {x}\right )^2 x^{5/2} \left (1+x^2\right ) \, dx\\ &=\int \left (\left (-2+\sqrt {x}\right )^2 x^{5/2}+\left (-2+\sqrt {x}\right )^2 x^{9/2}\right ) \, dx\\ &=\int \left (-2+\sqrt {x}\right )^2 x^{5/2} \, dx+\int \left (-2+\sqrt {x}\right )^2 x^{9/2} \, dx\\ &=2 \operatorname {Subst}\left (\int (-2+x)^2 x^6 \, dx,x,\sqrt {x}\right )+2 \operatorname {Subst}\left (\int (-2+x)^2 x^{10} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (4 x^6-4 x^7+x^8\right ) \, dx,x,\sqrt {x}\right )+2 \operatorname {Subst}\left (\int \left (4 x^{10}-4 x^{11}+x^{12}\right ) \, dx,x,\sqrt {x}\right )\\ &=\frac {8 x^{7/2}}{7}-x^4+\frac {2 x^{9/2}}{9}+\frac {8 x^{11/2}}{11}-\frac {2 x^6}{3}+\frac {2 x^{13/2}}{13}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 49, normalized size = 1.00 \[ \frac {2 x^{13/2}}{13}+\frac {8 x^{11/2}}{11}+\frac {2 x^{9/2}}{9}+\frac {8 x^{7/2}}{7}-\frac {2 x^6}{3}-x^4 \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.01, size = 49, normalized size = 1.00 \[ \frac {2 x^{13/2}}{13}+\frac {8 x^{11/2}}{11}+\frac {2 x^{9/2}}{9}+\frac {8 x^{7/2}}{7}-\frac {2 x^6}{3}-x^4 \]
Antiderivative was successfully verified.
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fricas [A] time = 1.63, size = 37, normalized size = 0.76 \[ -\frac {2}{3} \, x^{6} - x^{4} + \frac {2}{9009} \, {\left (693 \, x^{6} + 3276 \, x^{5} + 1001 \, x^{4} + 5148 \, x^{3}\right )} \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.99, size = 31, normalized size = 0.63 \[ \frac {2}{13} \, x^{\frac {13}{2}} - \frac {2}{3} \, x^{6} + \frac {8}{11} \, x^{\frac {11}{2}} + \frac {2}{9} \, x^{\frac {9}{2}} - x^{4} + \frac {8}{7} \, x^{\frac {7}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.29, size = 32, normalized size = 0.65
method | result | size |
derivativedivides | \(\frac {8 x^{\frac {7}{2}}}{7}-x^{4}+\frac {2 x^{\frac {9}{2}}}{9}+\frac {8 x^{\frac {11}{2}}}{11}-\frac {2 x^{6}}{3}+\frac {2 x^{\frac {13}{2}}}{13}\) | \(32\) |
default | \(\frac {8 x^{\frac {7}{2}}}{7}-x^{4}+\frac {2 x^{\frac {9}{2}}}{9}+\frac {8 x^{\frac {11}{2}}}{11}-\frac {2 x^{6}}{3}+\frac {2 x^{\frac {13}{2}}}{13}\) | \(32\) |
trager | \(-\frac {\left (2 x^{5}+2 x^{4}+5 x^{3}+5 x^{2}+5 x +5\right ) \left (-1+x \right )}{3}+\frac {2 x^{\frac {7}{2}} \left (693 x^{3}+3276 x^{2}+1001 x +5148\right )}{9009}\) | \(52\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 31, normalized size = 0.63 \[ \frac {2}{13} \, x^{\frac {13}{2}} - \frac {2}{3} \, x^{6} + \frac {8}{11} \, x^{\frac {11}{2}} + \frac {2}{9} \, x^{\frac {9}{2}} - x^{4} + \frac {8}{7} \, x^{\frac {7}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.19, size = 31, normalized size = 0.63 \[ \frac {8\,x^{7/2}}{7}-\frac {2\,x^6}{3}-x^4+\frac {2\,x^{9/2}}{9}+\frac {8\,x^{11/2}}{11}+\frac {2\,x^{13/2}}{13} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.60, size = 42, normalized size = 0.86 \[ \frac {2 x^{\frac {13}{2}}}{13} + \frac {8 x^{\frac {11}{2}}}{11} + \frac {2 x^{\frac {9}{2}}}{9} + \frac {8 x^{\frac {7}{2}}}{7} - \frac {2 x^{6}}{3} - x^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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