Integrand size = 28, antiderivative size = 582 \[ \int \frac {x^2}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^{5/2}} \, dx=\frac {1+(-1+x)^2}{3 (4+a) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )^{3/2}}+\frac {2 \left (1+(-1+x)^2\right )}{3 (4+a)^2 \sqrt {3+a-2 (-1+x)^2-(-1+x)^4}}+\frac {(4+a) \left (2+(-1+x)^2\right ) (-1+x)}{6 \left (12+7 a+a^2\right ) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )^{3/2}}+\frac {\left (29+7 a+(13+3 a) (-1+x)^2\right ) (-1+x)}{12 (3+a)^2 (4+a) \sqrt {3+a-2 (-1+x)^2-(-1+x)^4}}-\frac {(13+3 a) \left (1-\sqrt {4+a}\right ) \left (1+\frac {(-1+x)^2}{1-\sqrt {4+a}}\right ) (-1+x)}{12 (3+a)^2 (4+a) \sqrt {3+a-2 (-1+x)^2-(-1+x)^4}}+\frac {(13+3 a) \left (1-\sqrt {4+a}\right ) \sqrt {1+\sqrt {4+a}} \left (1+\frac {(-1+x)^2}{1-\sqrt {4+a}}\right ) E\left (\arctan \left (\frac {-1+x}{\sqrt {1+\sqrt {4+a}}}\right )|-\frac {2 \sqrt {4+a}}{1-\sqrt {4+a}}\right )}{12 (3+a)^2 (4+a) \sqrt {\frac {1+\frac {(-1+x)^2}{1-\sqrt {4+a}}}{1+\frac {(-1+x)^2}{1+\sqrt {4+a}}}} \sqrt {3+a-2 (-1+x)^2-(-1+x)^4}}+\frac {\sqrt {1+\sqrt {4+a}} \left (1+\frac {(-1+x)^2}{1-\sqrt {4+a}}\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {-1+x}{\sqrt {1+\sqrt {4+a}}}\right ),-\frac {2 \sqrt {4+a}}{1-\sqrt {4+a}}\right )}{12 \left (12+7 a+a^2\right ) \sqrt {\frac {1+\frac {(-1+x)^2}{1-\sqrt {4+a}}}{1+\frac {(-1+x)^2}{1+\sqrt {4+a}}}} \sqrt {3+a-2 (-1+x)^2-(-1+x)^4}} \]
1/3*(1+(-1+x)^2)/(4+a)/(3+a-2*(-1+x)^2-(-1+x)^4)^(3/2)+1/6*(4+a)*(2+(-1+x) ^2)*(-1+x)/(a^2+7*a+12)/(3+a-2*(-1+x)^2-(-1+x)^4)^(3/2)+2/3*(1+(-1+x)^2)/( 4+a)^2/(3+a-2*(-1+x)^2-(-1+x)^4)^(1/2)+1/12*(29+7*a+(13+3*a)*(-1+x)^2)*(-1 +x)/(3+a)^2/(4+a)/(3+a-2*(-1+x)^2-(-1+x)^4)^(1/2)-1/12*(13+3*a)*(-1+x)*(1+ (-1+x)^2/(1-(4+a)^(1/2)))*(1-(4+a)^(1/2))/(3+a)^2/(4+a)/(3+a-2*(-1+x)^2-(- 1+x)^4)^(1/2)+1/12*(1/(1+(-1+x)^2/(1+(4+a)^(1/2))))^(1/2)*(1+(-1+x)^2/(1+( 4+a)^(1/2)))^(1/2)*EllipticF((-1+x)/(1+(4+a)^(1/2))^(1/2)/(1+(-1+x)^2/(1+( 4+a)^(1/2)))^(1/2),(-2*(4+a)^(1/2)/(1-(4+a)^(1/2)))^(1/2))*(1+(-1+x)^2/(1- (4+a)^(1/2)))*(1+(4+a)^(1/2))^(1/2)/(a^2+7*a+12)/(3+a-2*(-1+x)^2-(-1+x)^4) ^(1/2)/((1+(-1+x)^2/(1-(4+a)^(1/2)))/(1+(-1+x)^2/(1+(4+a)^(1/2))))^(1/2)+1 /12*(13+3*a)*(1/(1+(-1+x)^2/(1+(4+a)^(1/2))))^(1/2)*(1+(-1+x)^2/(1+(4+a)^( 1/2)))^(1/2)*EllipticE((-1+x)/(1+(4+a)^(1/2))^(1/2)/(1+(-1+x)^2/(1+(4+a)^( 1/2)))^(1/2),(-2*(4+a)^(1/2)/(1-(4+a)^(1/2)))^(1/2))*(1+(-1+x)^2/(1-(4+a)^ (1/2)))*(1-(4+a)^(1/2))*(1+(4+a)^(1/2))^(1/2)/(3+a)^2/(4+a)/(3+a-2*(-1+x)^ 2-(-1+x)^4)^(1/2)/((1+(-1+x)^2/(1-(4+a)^(1/2)))/(1+(-1+x)^2/(1+(4+a)^(1/2) )))^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(5812\) vs. \(2(582)=1164\).
Time = 17.14 (sec) , antiderivative size = 5812, normalized size of antiderivative = 9.99 \[ \int \frac {x^2}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^{5/2}} \, dx=\text {Result too large to show} \]
Time = 0.89 (sec) , antiderivative size = 653, normalized size of antiderivative = 1.12, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {2459, 2006, 2202, 27, 1432, 1089, 1088, 1492, 27, 1492, 27, 1514, 406, 320, 388, 313}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x^2}{\left (a-x^4+4 x^3-8 x^2+8 x\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 2459 |
| \(\displaystyle \int \frac {(x-1)^2+2 (x-1)+1}{\left (a-(x-1)^4-2 (x-1)^2+3\right )^{5/2}}d(x-1)\) |
| \(\Big \downarrow \) 2006 |
| \(\displaystyle \int \frac {x^2}{\left (a-(x-1)^4-2 (x-1)^2+3\right )^{5/2}}d(x-1)\) |
| \(\Big \downarrow \) 2202 |
| \(\displaystyle \int \frac {(x-1)^2+1}{\left (-(x-1)^4-2 (x-1)^2+a+3\right )^{5/2}}d(x-1)+\int \frac {2 (x-1)}{\left (-(x-1)^4-2 (x-1)^2+a+3\right )^{5/2}}d(x-1)\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(x-1)^2+1}{\left (-(x-1)^4-2 (x-1)^2+a+3\right )^{5/2}}d(x-1)+2 \int \frac {x-1}{\left (-(x-1)^4-2 (x-1)^2+a+3\right )^{5/2}}d(x-1)\) |
| \(\Big \downarrow \) 1432 |
| \(\displaystyle \int \frac {1}{\left (-(x-1)^4-2 (x-1)^2+a+3\right )^{5/2}}d(x-1)^2+\int \frac {(x-1)^2+1}{\left (-(x-1)^4-2 (x-1)^2+a+3\right )^{5/2}}d(x-1)\) |
| \(\Big \downarrow \) 1089 |
| \(\displaystyle \int \frac {(x-1)^2+1}{\left (-(x-1)^4-2 (x-1)^2+a+3\right )^{5/2}}d(x-1)+\frac {2 \int \frac {1}{\left (-(x-1)^4-2 (x-1)^2+a+3\right )^{3/2}}d(x-1)^2}{3 (a+4)}+\frac {x}{3 (a+4) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}}\) |
| \(\Big \downarrow \) 1088 |
| \(\displaystyle \int \frac {(x-1)^2+1}{\left (-(x-1)^4-2 (x-1)^2+a+3\right )^{5/2}}d(x-1)+\frac {2 x}{3 (a+4)^2 \sqrt {a-(x-1)^4-2 (x-1)^2+3}}+\frac {x}{3 (a+4) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}}\) |
| \(\Big \downarrow \) 1492 |
| \(\displaystyle -\frac {\int -\frac {2 (a+4) \left (3 (x-1)^2+4\right )}{\left (-(x-1)^4-2 (x-1)^2+a+3\right )^{3/2}}d(x-1)}{12 \left (a^2+7 a+12\right )}+\frac {(a+4) \left ((x-1)^2+2\right ) (x-1)}{6 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}}+\frac {2 x}{3 (a+4)^2 \sqrt {a-(x-1)^4-2 (x-1)^2+3}}+\frac {x}{3 (a+4) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(a+4) \int \frac {3 (x-1)^2+4}{\left (-(x-1)^4-2 (x-1)^2+a+3\right )^{3/2}}d(x-1)}{6 \left (a^2+7 a+12\right )}+\frac {(a+4) \left ((x-1)^2+2\right ) (x-1)}{6 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}}+\frac {2 x}{3 (a+4)^2 \sqrt {a-(x-1)^4-2 (x-1)^2+3}}+\frac {x}{3 (a+4) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}}\) |
| \(\Big \downarrow \) 1492 |
| \(\displaystyle \frac {(a+4) \left (\frac {(x-1) \left ((3 a+13) (x-1)^2+7 a+29\right )}{2 \left (a^2+7 a+12\right ) \sqrt {a-(x-1)^4-2 (x-1)^2+3}}-\frac {\int -\frac {2 \left (-\left ((3 a+13) (x-1)^2\right )+a+3\right )}{\sqrt {-(x-1)^4-2 (x-1)^2+a+3}}d(x-1)}{4 \left (a^2+7 a+12\right )}\right )}{6 \left (a^2+7 a+12\right )}+\frac {(a+4) \left ((x-1)^2+2\right ) (x-1)}{6 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}}+\frac {2 x}{3 (a+4)^2 \sqrt {a-(x-1)^4-2 (x-1)^2+3}}+\frac {x}{3 (a+4) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(a+4) \left (\frac {\int \frac {-\left ((3 a+13) (x-1)^2\right )+a+3}{\sqrt {-(x-1)^4-2 (x-1)^2+a+3}}d(x-1)}{2 \left (a^2+7 a+12\right )}+\frac {(x-1) \left ((3 a+13) (x-1)^2+7 a+29\right )}{2 \left (a^2+7 a+12\right ) \sqrt {a-(x-1)^4-2 (x-1)^2+3}}\right )}{6 \left (a^2+7 a+12\right )}+\frac {(a+4) \left ((x-1)^2+2\right ) (x-1)}{6 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}}+\frac {2 x}{3 (a+4)^2 \sqrt {a-(x-1)^4-2 (x-1)^2+3}}+\frac {x}{3 (a+4) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}}\) |
| \(\Big \downarrow \) 1514 |
| \(\displaystyle \frac {(a+4) \left (\frac {\sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1} \int \frac {-\left ((3 a+13) (x-1)^2\right )+a+3}{\sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}d(x-1)}{2 \left (a^2+7 a+12\right ) \sqrt {a-(x-1)^4-2 (x-1)^2+3}}+\frac {(x-1) \left ((3 a+13) (x-1)^2+7 a+29\right )}{2 \left (a^2+7 a+12\right ) \sqrt {a-(x-1)^4-2 (x-1)^2+3}}\right )}{6 \left (a^2+7 a+12\right )}+\frac {(a+4) \left ((x-1)^2+2\right ) (x-1)}{6 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}}+\frac {2 x}{3 (a+4)^2 \sqrt {a-(x-1)^4-2 (x-1)^2+3}}+\frac {x}{3 (a+4) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {(a+4) \left (\frac {\sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1} \left ((a+3) \int \frac {1}{\sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}d(x-1)-(3 a+13) \int \frac {(x-1)^2}{\sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}d(x-1)\right )}{2 \left (a^2+7 a+12\right ) \sqrt {a-(x-1)^4-2 (x-1)^2+3}}+\frac {(x-1) \left ((3 a+13) (x-1)^2+7 a+29\right )}{2 \left (a^2+7 a+12\right ) \sqrt {a-(x-1)^4-2 (x-1)^2+3}}\right )}{6 \left (a^2+7 a+12\right )}+\frac {(a+4) \left ((x-1)^2+2\right ) (x-1)}{6 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}}+\frac {2 x}{3 (a+4)^2 \sqrt {a-(x-1)^4-2 (x-1)^2+3}}+\frac {x}{3 (a+4) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {(a+4) \left (\frac {\sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1} \left (\frac {(a+3) \sqrt {\sqrt {a+4}+1} \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \operatorname {EllipticF}\left (\arctan \left (\frac {x-1}{\sqrt {\sqrt {a+4}+1}}\right ),-\frac {2 \sqrt {a+4}}{1-\sqrt {a+4}}\right )}{\sqrt {\frac {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}{\frac {(x-1)^2}{\sqrt {a+4}+1}+1}} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}-(3 a+13) \int \frac {(x-1)^2}{\sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}d(x-1)\right )}{2 \left (a^2+7 a+12\right ) \sqrt {a-(x-1)^4-2 (x-1)^2+3}}+\frac {(x-1) \left ((3 a+13) (x-1)^2+7 a+29\right )}{2 \left (a^2+7 a+12\right ) \sqrt {a-(x-1)^4-2 (x-1)^2+3}}\right )}{6 \left (a^2+7 a+12\right )}+\frac {(a+4) \left ((x-1)^2+2\right ) (x-1)}{6 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}}+\frac {2 x}{3 (a+4)^2 \sqrt {a-(x-1)^4-2 (x-1)^2+3}}+\frac {x}{3 (a+4) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {(a+4) \left (\frac {\sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1} \left (\frac {(a+3) \sqrt {\sqrt {a+4}+1} \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \operatorname {EllipticF}\left (\arctan \left (\frac {x-1}{\sqrt {\sqrt {a+4}+1}}\right ),-\frac {2 \sqrt {a+4}}{1-\sqrt {a+4}}\right )}{\sqrt {\frac {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}{\frac {(x-1)^2}{\sqrt {a+4}+1}+1}} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}-(3 a+13) \left (\frac {\left (1-\sqrt {a+4}\right ) (x-1) \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}}{\sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}-\left (1-\sqrt {a+4}\right ) \int \frac {\sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}}{\left (\frac {(x-1)^2}{\sqrt {a+4}+1}+1\right )^{3/2}}d(x-1)\right )\right )}{2 \left (a^2+7 a+12\right ) \sqrt {a-(x-1)^4-2 (x-1)^2+3}}+\frac {(x-1) \left ((3 a+13) (x-1)^2+7 a+29\right )}{2 \left (a^2+7 a+12\right ) \sqrt {a-(x-1)^4-2 (x-1)^2+3}}\right )}{6 \left (a^2+7 a+12\right )}+\frac {(a+4) \left ((x-1)^2+2\right ) (x-1)}{6 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}}+\frac {2 x}{3 (a+4)^2 \sqrt {a-(x-1)^4-2 (x-1)^2+3}}+\frac {x}{3 (a+4) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {(a+4) \left (\frac {\sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1} \left (\frac {(a+3) \sqrt {\sqrt {a+4}+1} \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \operatorname {EllipticF}\left (\arctan \left (\frac {x-1}{\sqrt {\sqrt {a+4}+1}}\right ),-\frac {2 \sqrt {a+4}}{1-\sqrt {a+4}}\right )}{\sqrt {\frac {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}{\frac {(x-1)^2}{\sqrt {a+4}+1}+1}} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}-(3 a+13) \left (\frac {\left (1-\sqrt {a+4}\right ) (x-1) \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}}{\sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}-\frac {\left (1-\sqrt {a+4}\right ) \sqrt {\sqrt {a+4}+1} \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} E\left (\arctan \left (\frac {x-1}{\sqrt {\sqrt {a+4}+1}}\right )|-\frac {2 \sqrt {a+4}}{1-\sqrt {a+4}}\right )}{\sqrt {\frac {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}{\frac {(x-1)^2}{\sqrt {a+4}+1}+1}} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}\right )\right )}{2 \left (a^2+7 a+12\right ) \sqrt {a-(x-1)^4-2 (x-1)^2+3}}+\frac {(x-1) \left ((3 a+13) (x-1)^2+7 a+29\right )}{2 \left (a^2+7 a+12\right ) \sqrt {a-(x-1)^4-2 (x-1)^2+3}}\right )}{6 \left (a^2+7 a+12\right )}+\frac {(a+4) \left ((x-1)^2+2\right ) (x-1)}{6 \left (a^2+7 a+12\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}}+\frac {2 x}{3 (a+4)^2 \sqrt {a-(x-1)^4-2 (x-1)^2+3}}+\frac {x}{3 (a+4) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}}\) |
((4 + a)*(2 + (-1 + x)^2)*(-1 + x))/(6*(12 + 7*a + a^2)*(3 + a - 2*(-1 + x )^2 - (-1 + x)^4)^(3/2)) + x/(3*(4 + a)*(3 + a - 2*(-1 + x)^2 - (-1 + x)^4 )^(3/2)) + (2*x)/(3*(4 + a)^2*Sqrt[3 + a - 2*(-1 + x)^2 - (-1 + x)^4]) + ( (4 + a)*(((29 + 7*a + (13 + 3*a)*(-1 + x)^2)*(-1 + x))/(2*(12 + 7*a + a^2) *Sqrt[3 + a - 2*(-1 + x)^2 - (-1 + x)^4]) + (Sqrt[1 + (-1 + x)^2/(1 - Sqrt [4 + a])]*Sqrt[1 + (-1 + x)^2/(1 + Sqrt[4 + a])]*(-((13 + 3*a)*(((1 - Sqrt [4 + a])*Sqrt[1 + (-1 + x)^2/(1 - Sqrt[4 + a])]*(-1 + x))/Sqrt[1 + (-1 + x )^2/(1 + Sqrt[4 + a])] - ((1 - Sqrt[4 + a])*Sqrt[1 + Sqrt[4 + a]]*Sqrt[1 + (-1 + x)^2/(1 - Sqrt[4 + a])]*EllipticE[ArcTan[(-1 + x)/Sqrt[1 + Sqrt[4 + a]]], (-2*Sqrt[4 + a])/(1 - Sqrt[4 + a])])/(Sqrt[(1 + (-1 + x)^2/(1 - Sqr t[4 + a]))/(1 + (-1 + x)^2/(1 + Sqrt[4 + a]))]*Sqrt[1 + (-1 + x)^2/(1 + Sq rt[4 + a])]))) + ((3 + a)*Sqrt[1 + Sqrt[4 + a]]*Sqrt[1 + (-1 + x)^2/(1 - S qrt[4 + a])]*EllipticF[ArcTan[(-1 + x)/Sqrt[1 + Sqrt[4 + a]]], (-2*Sqrt[4 + a])/(1 - Sqrt[4 + a])])/(Sqrt[(1 + (-1 + x)^2/(1 - Sqrt[4 + a]))/(1 + (- 1 + x)^2/(1 + Sqrt[4 + a]))]*Sqrt[1 + (-1 + x)^2/(1 + Sqrt[4 + a])])))/(2* (12 + 7*a + a^2)*Sqrt[3 + a - 2*(-1 + x)^2 - (-1 + x)^4])))/(6*(12 + 7*a + a^2))
3.8.95.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Sim p[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; FreeQ [{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*( c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; Fre eQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] && !SimplerSqrtQ[b/a, d/c]
Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt[c + d*x^2])), x] - Simp[c/b Int[Sqrt[ a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && PosQ[b/a] && PosQ[d/c] && !SimplerSqrtQ[b/a, d/c]
Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( x_)^2), x_Symbol] :> Simp[e Int[(a + b*x^2)^p*(c + d*x^2)^q, x], x] + Sim p[f Int[x^2*(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e, f, p, q}, x]
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[-2*((b + 2*c*x)/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2])), x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) *((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] - Simp[2*c*((2*p + 3)/((p + 1)*(b^2 - 4*a*c))) Int[(a + b*x + c*x^2)^(p + 1), x], x] /; Fre eQ[{a, b, c}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[3*p])
Int[(x_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[1/2 Subst[Int[(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x]
Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb ol] :> Simp[x*(a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 7)*(d*b - 2*a*e)*c*x^2, x]*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && IntegerQ[2*p]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[Sqrt[1 + 2*c*(x^2/(b - q))]*(Sqrt [1 + 2*c*(x^2/(b + q))]/Sqrt[a + b*x^2 + c*x^4]) Int[(d + e*x^2)/(Sqrt[1 + 2*c*(x^2/(b - q))]*Sqrt[1 + 2*c*(x^2/(b + q))]), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[c/a]
Int[(u_.)*(Px_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^Expon[Px , x], x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; PolyQ[Px, x] && GtQ[Expon[P x, x], 1] && NeQ[Coeff[Px, x, 0], 0] && !MatchQ[Px, (a_.)*(v_)^Expon[Px, x ] /; FreeQ[a, x] && LinearQ[v, x]]
Int[(Pn_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{n = Expon[Pn, x], k}, Int[Sum[Coeff[Pn, x, 2*k]*x^(2*k), {k, 0, n/2}]*(a + b *x^2 + c*x^4)^p, x] + Int[x*Sum[Coeff[Pn, x, 2*k + 1]*x^(2*k), {k, 0, (n - 1)/2}]*(a + b*x^2 + c*x^4)^p, x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pn, x] && !PolyQ[Pn, x^2]
Int[(Pn_)^(p_.)*(Qx_), x_Symbol] :> With[{S = Coeff[Pn, x, Expon[Pn, x] - 1 ]/(Expon[Pn, x]*Coeff[Pn, x, Expon[Pn, x]])}, Subst[Int[ExpandToSum[Pn /. x -> x - S, x]^p*ExpandToSum[Qx /. x -> x - S, x], x], x, x + S] /; Binomial Q[Pn /. x -> x - S, x] || (IntegerQ[Expon[Pn, x]/2] && TrinomialQ[Pn /. x - > x - S, x])] /; FreeQ[p, x] && PolyQ[Pn, x] && GtQ[Expon[Pn, x], 2] && NeQ [Coeff[Pn, x, Expon[Pn, x] - 1], 0] && PolyQ[Qx, x] && !(MonomialQ[Qx, x] && IGtQ[p, 0])
Leaf count of result is larger than twice the leaf count of optimal. \(2779\) vs. \(2(628)=1256\).
Time = 1.51 (sec) , antiderivative size = 2780, normalized size of antiderivative = 4.78
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(2780\) |
| elliptic | \(\text {Expression too large to display}\) | \(2780\) |
(1/6/(3+a)*x^3-1/6*(6+a)/(a^2+7*a+12)*x^2+1/6*(a+8)/(a^2+7*a+12)*x+1/6/(a^ 2+7*a+12)*a)*(-x^4+4*x^3-8*x^2+a+8*x)^(1/2)/(x^4-4*x^3+8*x^2-a-8*x)^2+2*(1 /24*(13+3*a)/(3+a)/(a^2+7*a+12)*x^3-1/24*(a^2+27*a+84)/(a^2+7*a+12)^2*x^2+ 1/6*(9*a+32)/(a^2+7*a+12)^2*x+1/12*(3*a^2+7*a-12)/(a^2+7*a+12)^2)/(-x^4+4* x^3-8*x^2+a+8*x)^(1/2)-(-1/6*(a^2-9*a-44)/(a^2+7*a+12)^2-1/3*(9*a+32)/(a^2 +7*a+12)^2)*((-1-(a+4)^(1/2))^(1/2)+(-1+(a+4)^(1/2))^(1/2))*((-(-1-(a+4)^( 1/2))^(1/2)+(-1+(a+4)^(1/2))^(1/2))*(x-1-(-1+(a+4)^(1/2))^(1/2))/(-(-1-(a+ 4)^(1/2))^(1/2)-(-1+(a+4)^(1/2))^(1/2))/(x-1+(-1+(a+4)^(1/2))^(1/2)))^(1/2 )*(x-1+(-1+(a+4)^(1/2))^(1/2))^2*(-2*(-1+(a+4)^(1/2))^(1/2)*(x-1-(-1-(a+4) ^(1/2))^(1/2))/((-1-(a+4)^(1/2))^(1/2)-(-1+(a+4)^(1/2))^(1/2))/(x-1+(-1+(a +4)^(1/2))^(1/2)))^(1/2)*(-2*(-1+(a+4)^(1/2))^(1/2)*(x-1+(-1-(a+4)^(1/2))^ (1/2))/(-(-1-(a+4)^(1/2))^(1/2)-(-1+(a+4)^(1/2))^(1/2))/(x-1+(-1+(a+4)^(1/ 2))^(1/2)))^(1/2)/(-(-1-(a+4)^(1/2))^(1/2)+(-1+(a+4)^(1/2))^(1/2))/(-1+(a+ 4)^(1/2))^(1/2)/(-(x-1-(-1+(a+4)^(1/2))^(1/2))*(x-1+(-1+(a+4)^(1/2))^(1/2) )*(x-1-(-1-(a+4)^(1/2))^(1/2))*(x-1+(-1-(a+4)^(1/2))^(1/2)))^(1/2)*Ellipti cF(((-(-1-(a+4)^(1/2))^(1/2)+(-1+(a+4)^(1/2))^(1/2))*(x-1-(-1+(a+4)^(1/2)) ^(1/2))/(-(-1-(a+4)^(1/2))^(1/2)-(-1+(a+4)^(1/2))^(1/2))/(x-1+(-1+(a+4)^(1 /2))^(1/2)))^(1/2),((-(-1-(a+4)^(1/2))^(1/2)-(-1+(a+4)^(1/2))^(1/2))*((-1- (a+4)^(1/2))^(1/2)+(-1+(a+4)^(1/2))^(1/2))/(-(-1-(a+4)^(1/2))^(1/2)+(-1+(a +4)^(1/2))^(1/2))/((-1-(a+4)^(1/2))^(1/2)-(-1+(a+4)^(1/2))^(1/2)))^(1/2...
\[ \int \frac {x^2}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^{5/2}} \, dx=\int { \frac {x^{2}}{{\left (-x^{4} + 4 \, x^{3} - 8 \, x^{2} + a + 8 \, x\right )}^{\frac {5}{2}}} \,d x } \]
integral(-sqrt(-x^4 + 4*x^3 - 8*x^2 + a + 8*x)*x^2/(x^12 - 12*x^11 + 72*x^ 10 - 3*(a - 256)*x^8 - 280*x^9 + 24*(a - 64)*x^7 - 32*(3*a - 70)*x^6 + 48* (5*a - 48)*x^5 + 3*(a^2 - 128*a + 512)*x^4 - 4*(3*a^2 - 96*a + 128)*x^3 - a^3 - 24*a^2*x + 24*(a^2 - 8*a)*x^2), x)
\[ \int \frac {x^2}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^{5/2}} \, dx=\int \frac {x^{2}}{\left (a - x^{4} + 4 x^{3} - 8 x^{2} + 8 x\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {x^2}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^{5/2}} \, dx=\int { \frac {x^{2}}{{\left (-x^{4} + 4 \, x^{3} - 8 \, x^{2} + a + 8 \, x\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {x^2}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^{5/2}} \, dx=\int { \frac {x^{2}}{{\left (-x^{4} + 4 \, x^{3} - 8 \, x^{2} + a + 8 \, x\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {x^2}{\left (a+8 x-8 x^2+4 x^3-x^4\right )^{5/2}} \, dx=\int \frac {x^2}{{\left (-x^4+4\,x^3-8\,x^2+8\,x+a\right )}^{5/2}} \,d x \]