Optimal. Leaf size=582 \[ \frac {1+(-1+x)^2}{3 (4+a) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )^{3/2}}+\frac {2 \left (1+(-1+x)^2\right )}{3 (4+a)^2 \sqrt {3+a-2 (-1+x)^2-(-1+x)^4}}+\frac {(4+a) \left (2+(-1+x)^2\right ) (-1+x)}{6 \left (12+7 a+a^2\right ) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )^{3/2}}+\frac {\left (29+7 a+(13+3 a) (-1+x)^2\right ) (-1+x)}{12 (3+a)^2 (4+a) \sqrt {3+a-2 (-1+x)^2-(-1+x)^4}}-\frac {(13+3 a) \left (1-\sqrt {4+a}\right ) \left (1+\frac {(-1+x)^2}{1-\sqrt {4+a}}\right ) (-1+x)}{12 (3+a)^2 (4+a) \sqrt {3+a-2 (-1+x)^2-(-1+x)^4}}+\frac {(13+3 a) \left (1-\sqrt {4+a}\right ) \sqrt {1+\sqrt {4+a}} \left (1+\frac {(-1+x)^2}{1-\sqrt {4+a}}\right ) E\left (\tan ^{-1}\left (\frac {-1+x}{\sqrt {1+\sqrt {4+a}}}\right )|-\frac {2 \sqrt {4+a}}{1-\sqrt {4+a}}\right )}{12 (3+a)^2 (4+a) \sqrt {\frac {1+\frac {(-1+x)^2}{1-\sqrt {4+a}}}{1+\frac {(-1+x)^2}{1+\sqrt {4+a}}}} \sqrt {3+a-2 (-1+x)^2-(-1+x)^4}}+\frac {\sqrt {1+\sqrt {4+a}} \left (1+\frac {(-1+x)^2}{1-\sqrt {4+a}}\right ) F\left (\tan ^{-1}\left (\frac {-1+x}{\sqrt {1+\sqrt {4+a}}}\right )|-\frac {2 \sqrt {4+a}}{1-\sqrt {4+a}}\right )}{12 \left (12+7 a+a^2\right ) \sqrt {\frac {1+\frac {(-1+x)^2}{1-\sqrt {4+a}}}{1+\frac {(-1+x)^2}{1+\sqrt {4+a}}}} \sqrt {3+a-2 (-1+x)^2-(-1+x)^4}} \]
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Rubi [A]
time = 0.50, antiderivative size = 582, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 12, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {1694, 1687,
1192, 1216, 545, 429, 506, 422, 12, 1121, 628, 627} \begin {gather*} \frac {\sqrt {\sqrt {a+4}+1} \left (\frac {(x-1)^2}{1-\sqrt {a+4}}+1\right ) F\left (\text {ArcTan}\left (\frac {x-1}{\sqrt {\sqrt {a+4}+1}}\right )|-\frac {2 \sqrt {a+4}}{1-\sqrt {a+4}}\right )}{12 \left (a^2+7 a+12\right ) \sqrt {\frac {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}{\frac {(x-1)^2}{\sqrt {a+4}+1}+1}} \sqrt {a-(x-1)^4-2 (x-1)^2+3}}+\frac {(a+4) \left ((x-1)^2+2\right ) (x-1)}{6 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}}+\frac {(3 a+13) \left (1-\sqrt {a+4}\right ) \sqrt {\sqrt {a+4}+1} \left (\frac {(x-1)^2}{1-\sqrt {a+4}}+1\right ) E\left (\text {ArcTan}\left (\frac {x-1}{\sqrt {\sqrt {a+4}+1}}\right )|-\frac {2 \sqrt {a+4}}{1-\sqrt {a+4}}\right )}{12 (a+3)^2 (a+4) \sqrt {\frac {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}{\frac {(x-1)^2}{\sqrt {a+4}+1}+1}} \sqrt {a-(x-1)^4-2 (x-1)^2+3}}+\frac {2 \left ((x-1)^2+1\right )}{3 (a+4)^2 \sqrt {a-(x-1)^4-2 (x-1)^2+3}}+\frac {(x-1)^2+1}{3 (a+4) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}}+\frac {(x-1) \left ((3 a+13) (x-1)^2+7 a+29\right )}{12 (a+3)^2 (a+4) \sqrt {a-(x-1)^4-2 (x-1)^2+3}}-\frac {(3 a+13) \left (1-\sqrt {a+4}\right ) (x-1) \left (\frac {(x-1)^2}{1-\sqrt {a+4}}+1\right )}{12 (a+3)^2 (a+4) \sqrt {a-(x-1)^4-2 (x-1)^2+3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 422
Rule 429
Rule 506
Rule 545
Rule 627
Rule 628
Rule 1121
Rule 1192
Rule 1216
Rule 1687
Rule 1694
Rubi steps
\begin {align*} \int \frac {x^2}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^{5/2}} \, dx &=\text {Subst}\left (\int \frac {(1+x)^2}{\left (3+a-2 x^2-x^4\right )^{5/2}} \, dx,x,-1+x\right )\\ &=\text {Subst}\left (\int \frac {2 x}{\left (3+a-2 x^2-x^4\right )^{5/2}} \, dx,x,-1+x\right )+\text {Subst}\left (\int \frac {1+x^2}{\left (3+a-2 x^2-x^4\right )^{5/2}} \, dx,x,-1+x\right )\\ &=\frac {(4+a) \left (2+(-1+x)^2\right ) (-1+x)}{6 \left (12+7 a+a^2\right ) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )^{3/2}}+2 \text {Subst}\left (\int \frac {x}{\left (3+a-2 x^2-x^4\right )^{5/2}} \, dx,x,-1+x\right )-\frac {\text {Subst}\left (\int \frac {-8 (4+a)-6 (4+a) x^2}{\left (3+a-2 x^2-x^4\right )^{3/2}} \, dx,x,-1+x\right )}{12 \left (12+7 a+a^2\right )}\\ &=-\frac {\left (29+7 a+(13+3 a) (1-x)^2\right ) (1-x)}{12 (3+a)^2 (4+a) \sqrt {3+a-2 (1-x)^2-(1-x)^4}}+\frac {(4+a) \left (2+(-1+x)^2\right ) (-1+x)}{6 \left (12+7 a+a^2\right ) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )^{3/2}}+\frac {\text {Subst}\left (\int \frac {4 (3+a) (4+a)-4 (4+a) (13+3 a) x^2}{\sqrt {3+a-2 x^2-x^4}} \, dx,x,-1+x\right )}{48 \left (12+7 a+a^2\right )^2}+\text {Subst}\left (\int \frac {1}{\left (3+a-2 x-x^2\right )^{5/2}} \, dx,x,(-1+x)^2\right )\\ &=\frac {1+(-1+x)^2}{3 (4+a) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )^{3/2}}-\frac {\left (29+7 a+(13+3 a) (1-x)^2\right ) (1-x)}{12 (3+a)^2 (4+a) \sqrt {3+a-2 (1-x)^2-(1-x)^4}}+\frac {(4+a) \left (2+(-1+x)^2\right ) (-1+x)}{6 \left (12+7 a+a^2\right ) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )^{3/2}}+\frac {2 \text {Subst}\left (\int \frac {1}{\left (3+a-2 x-x^2\right )^{3/2}} \, dx,x,(-1+x)^2\right )}{3 (4+a)}+\frac {\left (\sqrt {1-\frac {2 (-1+x)^2}{-2-2 \sqrt {4+a}}} \sqrt {1-\frac {2 (-1+x)^2}{-2+2 \sqrt {4+a}}}\right ) \text {Subst}\left (\int \frac {4 (3+a) (4+a)-4 (4+a) (13+3 a) x^2}{\sqrt {1-\frac {2 x^2}{-2-2 \sqrt {4+a}}} \sqrt {1-\frac {2 x^2}{-2+2 \sqrt {4+a}}}} \, dx,x,-1+x\right )}{48 \left (12+7 a+a^2\right )^2 \sqrt {3+a-2 (-1+x)^2-(-1+x)^4}}\\ &=\frac {2 \left (1+(-1+x)^2\right )}{3 (4+a)^2 \sqrt {3+a-2 (1-x)^2-(1-x)^4}}+\frac {1+(-1+x)^2}{3 (4+a) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )^{3/2}}-\frac {\left (29+7 a+(13+3 a) (1-x)^2\right ) (1-x)}{12 (3+a)^2 (4+a) \sqrt {3+a-2 (1-x)^2-(1-x)^4}}+\frac {(4+a) \left (2+(-1+x)^2\right ) (-1+x)}{6 \left (12+7 a+a^2\right ) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )^{3/2}}+\frac {\left ((3+a) (4+a) \sqrt {1-\frac {2 (-1+x)^2}{-2-2 \sqrt {4+a}}} \sqrt {1-\frac {2 (-1+x)^2}{-2+2 \sqrt {4+a}}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {2 x^2}{-2-2 \sqrt {4+a}}} \sqrt {1-\frac {2 x^2}{-2+2 \sqrt {4+a}}}} \, dx,x,-1+x\right )}{12 \left (12+7 a+a^2\right )^2 \sqrt {3+a-2 (-1+x)^2-(-1+x)^4}}-\frac {\left ((4+a) (13+3 a) \sqrt {1-\frac {2 (-1+x)^2}{-2-2 \sqrt {4+a}}} \sqrt {1-\frac {2 (-1+x)^2}{-2+2 \sqrt {4+a}}}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1-\frac {2 x^2}{-2-2 \sqrt {4+a}}} \sqrt {1-\frac {2 x^2}{-2+2 \sqrt {4+a}}}} \, dx,x,-1+x\right )}{12 \left (12+7 a+a^2\right )^2 \sqrt {3+a-2 (-1+x)^2-(-1+x)^4}}\\ &=\frac {2 \left (1+(-1+x)^2\right )}{3 (4+a)^2 \sqrt {3+a-2 (1-x)^2-(1-x)^4}}+\frac {1+(-1+x)^2}{3 (4+a) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )^{3/2}}-\frac {\left (29+7 a+(13+3 a) (1-x)^2\right ) (1-x)}{12 (3+a)^2 (4+a) \sqrt {3+a-2 (1-x)^2-(1-x)^4}}+\frac {(13+3 a) \left (1-\sqrt {4+a}\right ) \left (1+\frac {(1-x)^2}{1-\sqrt {4+a}}\right ) (1-x)}{12 (3+a)^2 (4+a) \sqrt {3+a-2 (1-x)^2-(1-x)^4}}+\frac {(4+a) \left (2+(-1+x)^2\right ) (-1+x)}{6 \left (12+7 a+a^2\right ) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )^{3/2}}-\frac {\sqrt {1+\sqrt {4+a}} \left (1+\frac {(1-x)^2}{1-\sqrt {4+a}}\right ) F\left (\tan ^{-1}\left (\frac {1-x}{\sqrt {1+\sqrt {4+a}}}\right )|-\frac {2 \sqrt {4+a}}{1-\sqrt {4+a}}\right )}{12 \left (12+7 a+a^2\right ) \sqrt {\frac {1+\frac {(1-x)^2}{1-\sqrt {4+a}}}{1+\frac {(1-x)^2}{1+\sqrt {4+a}}}} \sqrt {3+a-2 (1-x)^2-(1-x)^4}}+\frac {\left ((4+a) (13+3 a) \left (1-\sqrt {4+a}\right ) \sqrt {1-\frac {2 (-1+x)^2}{-2-2 \sqrt {4+a}}} \sqrt {1-\frac {2 (-1+x)^2}{-2+2 \sqrt {4+a}}}\right ) \text {Subst}\left (\int \frac {\sqrt {1-\frac {2 x^2}{-2+2 \sqrt {4+a}}}}{\left (1-\frac {2 x^2}{-2-2 \sqrt {4+a}}\right )^{3/2}} \, dx,x,-1+x\right )}{12 \left (12+7 a+a^2\right )^2 \sqrt {3+a-2 (-1+x)^2-(-1+x)^4}}\\ &=\frac {2 \left (1+(-1+x)^2\right )}{3 (4+a)^2 \sqrt {3+a-2 (1-x)^2-(1-x)^4}}+\frac {1+(-1+x)^2}{3 (4+a) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )^{3/2}}-\frac {\left (29+7 a+(13+3 a) (1-x)^2\right ) (1-x)}{12 (3+a)^2 (4+a) \sqrt {3+a-2 (1-x)^2-(1-x)^4}}+\frac {(13+3 a) \left (1-\sqrt {4+a}\right ) \left (1+\frac {(1-x)^2}{1-\sqrt {4+a}}\right ) (1-x)}{12 (3+a)^2 (4+a) \sqrt {3+a-2 (1-x)^2-(1-x)^4}}+\frac {(4+a) \left (2+(-1+x)^2\right ) (-1+x)}{6 \left (12+7 a+a^2\right ) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )^{3/2}}-\frac {(13+3 a) \left (1-\sqrt {4+a}\right ) \sqrt {1+\sqrt {4+a}} \left (1+\frac {(1-x)^2}{1-\sqrt {4+a}}\right ) E\left (\tan ^{-1}\left (\frac {1-x}{\sqrt {1+\sqrt {4+a}}}\right )|-\frac {2 \sqrt {4+a}}{1-\sqrt {4+a}}\right )}{12 (3+a)^2 (4+a) \sqrt {\frac {1+\frac {(1-x)^2}{1-\sqrt {4+a}}}{1+\frac {(1-x)^2}{1+\sqrt {4+a}}}} \sqrt {3+a-2 (1-x)^2-(1-x)^4}}-\frac {\sqrt {1+\sqrt {4+a}} \left (1+\frac {(1-x)^2}{1-\sqrt {4+a}}\right ) F\left (\tan ^{-1}\left (\frac {1-x}{\sqrt {1+\sqrt {4+a}}}\right )|-\frac {2 \sqrt {4+a}}{1-\sqrt {4+a}}\right )}{12 \left (12+7 a+a^2\right ) \sqrt {\frac {1+\frac {(1-x)^2}{1-\sqrt {4+a}}}{1+\frac {(1-x)^2}{1+\sqrt {4+a}}}} \sqrt {3+a-2 (1-x)^2-(1-x)^4}}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(5812\) vs. \(2(582)=1164\).
time = 16.12, size = 5812, normalized size = 9.99 \begin {gather*} \text {Result too large to show} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2779\) vs.
\(2(628)=1256\).
time = 0.07, size = 2780, normalized size = 4.78
method | result | size |
default | \(\text {Expression too large to display}\) | \(2780\) |
elliptic | \(\text {Expression too large to display}\) | \(2780\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{\left (a - x^{4} + 4 x^{3} - 8 x^{2} + 8 x\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^2}{{\left (-x^4+4\,x^3-8\,x^2+8\,x+a\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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