Integrand size = 26, antiderivative size = 558 \[ \int x \left (a+8 x-8 x^2+4 x^3-x^4\right )^{3/2} \, dx=\frac {3}{16} (4+a) \left (1+(-1+x)^2\right ) \sqrt {3+a-2 (-1+x)^2-(-1+x)^4}+\frac {1}{8} \left (1+(-1+x)^2\right ) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )^{3/2}-\frac {16 (7+2 a) \left (1-\sqrt {4+a}\right ) \left (1+\frac {(-1+x)^2}{1-\sqrt {4+a}}\right ) (-1+x)}{35 \sqrt {3+a-2 (-1+x)^2-(-1+x)^4}}+\frac {2}{35} \left (13+5 a-3 (-1+x)^2\right ) \sqrt {3+a-2 (-1+x)^2-(-1+x)^4} (-1+x)+\frac {1}{7} \left (3+a-2 (-1+x)^2-(-1+x)^4\right )^{3/2} (-1+x)+\frac {3}{16} (4+a)^2 \arctan \left (\frac {1+(-1+x)^2}{\sqrt {3+a-2 (-1+x)^2-(-1+x)^4}}\right )+\frac {16 (7+2 a) \left (1-\sqrt {4+a}\right ) \sqrt {1+\sqrt {4+a}} \left (1+\frac {(-1+x)^2}{1-\sqrt {4+a}}\right ) E\left (\arctan \left (\frac {-1+x}{\sqrt {1+\sqrt {4+a}}}\right )|-\frac {2 \sqrt {4+a}}{1-\sqrt {4+a}}\right )}{35 \sqrt {\frac {1+\frac {(-1+x)^2}{1-\sqrt {4+a}}}{1+\frac {(-1+x)^2}{1+\sqrt {4+a}}}} \sqrt {3+a-2 (-1+x)^2-(-1+x)^4}}+\frac {4 (3+a) (16+5 a) \sqrt {1+\sqrt {4+a}} \left (1+\frac {(-1+x)^2}{1-\sqrt {4+a}}\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {-1+x}{\sqrt {1+\sqrt {4+a}}}\right ),-\frac {2 \sqrt {4+a}}{1-\sqrt {4+a}}\right )}{35 \sqrt {\frac {1+\frac {(-1+x)^2}{1-\sqrt {4+a}}}{1+\frac {(-1+x)^2}{1+\sqrt {4+a}}}} \sqrt {3+a-2 (-1+x)^2-(-1+x)^4}} \]
1/8*(1+(-1+x)^2)*(3+a-2*(-1+x)^2-(-1+x)^4)^(3/2)+1/7*(3+a-2*(-1+x)^2-(-1+x )^4)^(3/2)*(-1+x)+3/16*(4+a)^2*arctan((1+(-1+x)^2)/(3+a-2*(-1+x)^2-(-1+x)^ 4)^(1/2))-16/35*(7+2*a)*(-1+x)*(1+(-1+x)^2/(1-(4+a)^(1/2)))*(1-(4+a)^(1/2) )/(3+a-2*(-1+x)^2-(-1+x)^4)^(1/2)+3/16*(4+a)*(1+(-1+x)^2)*(3+a-2*(-1+x)^2- (-1+x)^4)^(1/2)+2/35*(13+5*a-3*(-1+x)^2)*(-1+x)*(3+a-2*(-1+x)^2-(-1+x)^4)^ (1/2)+4/35*(3+a)*(16+5*a)*(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)/(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)+16/35*( 7+2*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*(-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. \(7235\) vs. \(2(558)=1116\).
Time = 17.10 (sec) , antiderivative size = 7235, normalized size of antiderivative = 12.97 \[ \int x \left (a+8 x-8 x^2+4 x^3-x^4\right )^{3/2} \, dx=\text {Result too large to show} \]
Time = 0.84 (sec) , antiderivative size = 635, normalized size of antiderivative = 1.14, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {2459, 2202, 1404, 27, 1432, 1087, 1087, 1092, 217, 1490, 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.
| \(\displaystyle \int x \left (a-x^4+4 x^3-8 x^2+8 x\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 2459 |
| \(\displaystyle \int x \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}d(x-1)\) |
| \(\Big \downarrow \) 2202 |
| \(\displaystyle \int \left (-(x-1)^4-2 (x-1)^2+a+3\right )^{3/2}d(x-1)+\int \left (-(x-1)^4-2 (x-1)^2+a+3\right )^{3/2} (x-1)d(x-1)\) |
| \(\Big \downarrow \) 1404 |
| \(\displaystyle \frac {3}{7} \int 2 \left (-(x-1)^2+a+3\right ) \sqrt {-(x-1)^4-2 (x-1)^2+a+3}d(x-1)+\int \left (-(x-1)^4-2 (x-1)^2+a+3\right )^{3/2} (x-1)d(x-1)+\frac {1}{7} (x-1) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {6}{7} \int \left (-(x-1)^2+a+3\right ) \sqrt {-(x-1)^4-2 (x-1)^2+a+3}d(x-1)+\int \left (-(x-1)^4-2 (x-1)^2+a+3\right )^{3/2} (x-1)d(x-1)+\frac {1}{7} (x-1) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\) |
| \(\Big \downarrow \) 1432 |
| \(\displaystyle \frac {6}{7} \int \left (-(x-1)^2+a+3\right ) \sqrt {-(x-1)^4-2 (x-1)^2+a+3}d(x-1)+\frac {1}{2} \int \left (-(x-1)^4-2 (x-1)^2+a+3\right )^{3/2}d(x-1)^2+\frac {1}{7} (x-1) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {1}{2} \left (\frac {3}{4} (a+4) \int \sqrt {-(x-1)^4-2 (x-1)^2+a+3}d(x-1)^2+\frac {1}{4} x \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\right )+\frac {6}{7} \int \left (-(x-1)^2+a+3\right ) \sqrt {-(x-1)^4-2 (x-1)^2+a+3}d(x-1)+\frac {1}{7} (x-1) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {1}{2} \left (\frac {3}{4} (a+4) \left (\frac {1}{2} (a+4) \int \frac {1}{\sqrt {-(x-1)^4-2 (x-1)^2+a+3}}d(x-1)^2+\frac {1}{2} x \sqrt {a-(x-1)^4-2 (x-1)^2+3}\right )+\frac {1}{4} x \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\right )+\frac {6}{7} \int \left (-(x-1)^2+a+3\right ) \sqrt {-(x-1)^4-2 (x-1)^2+a+3}d(x-1)+\frac {1}{7} (x-1) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {1}{2} \left (\frac {3}{4} (a+4) \left ((a+4) \int \frac {1}{-(x-1)^4-4}d\left (-\frac {2 x}{\sqrt {-(x-1)^4-2 (x-1)^2+a+3}}\right )+\frac {1}{2} x \sqrt {a-(x-1)^4-2 (x-1)^2+3}\right )+\frac {1}{4} x \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\right )+\frac {6}{7} \int \left (-(x-1)^2+a+3\right ) \sqrt {-(x-1)^4-2 (x-1)^2+a+3}d(x-1)+\frac {1}{7} (x-1) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {6}{7} \int \left (-(x-1)^2+a+3\right ) \sqrt {-(x-1)^4-2 (x-1)^2+a+3}d(x-1)+\frac {1}{2} \left (\frac {3}{4} (a+4) \left (\frac {1}{2} (a+4) \arctan \left (\frac {x}{\sqrt {a-(x-1)^4-2 (x-1)^2+3}}\right )+\frac {1}{2} x \sqrt {a-(x-1)^4-2 (x-1)^2+3}\right )+\frac {1}{4} x \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\right )+\frac {1}{7} (x-1) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\) |
| \(\Big \downarrow \) 1490 |
| \(\displaystyle \frac {6}{7} \left (\frac {1}{15} (x-1) \left (5 a-3 (x-1)^2+13\right ) \sqrt {a-(x-1)^4-2 (x-1)^2+3}-\frac {1}{15} \int -\frac {2 \left ((a+3) (5 a+16)-4 (2 a+7) (x-1)^2\right )}{\sqrt {-(x-1)^4-2 (x-1)^2+a+3}}d(x-1)\right )+\frac {1}{2} \left (\frac {3}{4} (a+4) \left (\frac {1}{2} (a+4) \arctan \left (\frac {x}{\sqrt {a-(x-1)^4-2 (x-1)^2+3}}\right )+\frac {1}{2} x \sqrt {a-(x-1)^4-2 (x-1)^2+3}\right )+\frac {1}{4} x \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\right )+\frac {1}{7} (x-1) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {6}{7} \left (\frac {2}{15} \int \frac {(a+3) (5 a+16)-4 (2 a+7) (x-1)^2}{\sqrt {-(x-1)^4-2 (x-1)^2+a+3}}d(x-1)+\frac {1}{15} (x-1) \left (5 a-3 (x-1)^2+13\right ) \sqrt {a-(x-1)^4-2 (x-1)^2+3}\right )+\frac {1}{2} \left (\frac {3}{4} (a+4) \left (\frac {1}{2} (a+4) \arctan \left (\frac {x}{\sqrt {a-(x-1)^4-2 (x-1)^2+3}}\right )+\frac {1}{2} x \sqrt {a-(x-1)^4-2 (x-1)^2+3}\right )+\frac {1}{4} x \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\right )+\frac {1}{7} (x-1) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\) |
| \(\Big \downarrow \) 1514 |
| \(\displaystyle \frac {6}{7} \left (\frac {2 \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1} \int \frac {(a+3) (5 a+16)-4 (2 a+7) (x-1)^2}{\sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}d(x-1)}{15 \sqrt {a-(x-1)^4-2 (x-1)^2+3}}+\frac {1}{15} (x-1) \left (5 a-3 (x-1)^2+13\right ) \sqrt {a-(x-1)^4-2 (x-1)^2+3}\right )+\frac {1}{2} \left (\frac {3}{4} (a+4) \left (\frac {1}{2} (a+4) \arctan \left (\frac {x}{\sqrt {a-(x-1)^4-2 (x-1)^2+3}}\right )+\frac {1}{2} x \sqrt {a-(x-1)^4-2 (x-1)^2+3}\right )+\frac {1}{4} x \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\right )+\frac {1}{7} (x-1) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {6}{7} \left (\frac {2 \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1} \left ((a+3) (5 a+16) \int \frac {1}{\sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}d(x-1)-4 (2 a+7) \int \frac {(x-1)^2}{\sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}d(x-1)\right )}{15 \sqrt {a-(x-1)^4-2 (x-1)^2+3}}+\frac {1}{15} (x-1) \left (5 a-3 (x-1)^2+13\right ) \sqrt {a-(x-1)^4-2 (x-1)^2+3}\right )+\frac {1}{2} \left (\frac {3}{4} (a+4) \left (\frac {1}{2} (a+4) \arctan \left (\frac {x}{\sqrt {a-(x-1)^4-2 (x-1)^2+3}}\right )+\frac {1}{2} x \sqrt {a-(x-1)^4-2 (x-1)^2+3}\right )+\frac {1}{4} x \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\right )+\frac {1}{7} (x-1) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {6}{7} \left (\frac {2 \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1} \left (\frac {(a+3) (5 a+16) \sqrt {\sqrt {a+4}+1} \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \operatorname {EllipticF}\left (\arctan \left (\frac {x-1}{\sqrt {\sqrt {a+4}+1}}\right ),-\frac {2 \sqrt {a+4}}{1-\sqrt {a+4}}\right )}{\sqrt {\frac {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}{\frac {(x-1)^2}{\sqrt {a+4}+1}+1}} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}-4 (2 a+7) \int \frac {(x-1)^2}{\sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}d(x-1)\right )}{15 \sqrt {a-(x-1)^4-2 (x-1)^2+3}}+\frac {1}{15} (x-1) \left (5 a-3 (x-1)^2+13\right ) \sqrt {a-(x-1)^4-2 (x-1)^2+3}\right )+\frac {1}{2} \left (\frac {3}{4} (a+4) \left (\frac {1}{2} (a+4) \arctan \left (\frac {x}{\sqrt {a-(x-1)^4-2 (x-1)^2+3}}\right )+\frac {1}{2} x \sqrt {a-(x-1)^4-2 (x-1)^2+3}\right )+\frac {1}{4} x \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\right )+\frac {1}{7} (x-1) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {6}{7} \left (\frac {2 \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1} \left (\frac {(a+3) (5 a+16) \sqrt {\sqrt {a+4}+1} \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \operatorname {EllipticF}\left (\arctan \left (\frac {x-1}{\sqrt {\sqrt {a+4}+1}}\right ),-\frac {2 \sqrt {a+4}}{1-\sqrt {a+4}}\right )}{\sqrt {\frac {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}{\frac {(x-1)^2}{\sqrt {a+4}+1}+1}} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}-4 (2 a+7) \left (\frac {\left (1-\sqrt {a+4}\right ) (x-1) \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}}{\sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}-\left (1-\sqrt {a+4}\right ) \int \frac {\sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}}{\left (\frac {(x-1)^2}{\sqrt {a+4}+1}+1\right )^{3/2}}d(x-1)\right )\right )}{15 \sqrt {a-(x-1)^4-2 (x-1)^2+3}}+\frac {1}{15} (x-1) \left (5 a-3 (x-1)^2+13\right ) \sqrt {a-(x-1)^4-2 (x-1)^2+3}\right )+\frac {1}{2} \left (\frac {3}{4} (a+4) \left (\frac {1}{2} (a+4) \arctan \left (\frac {x}{\sqrt {a-(x-1)^4-2 (x-1)^2+3}}\right )+\frac {1}{2} x \sqrt {a-(x-1)^4-2 (x-1)^2+3}\right )+\frac {1}{4} x \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\right )+\frac {1}{7} (x-1) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {1}{2} \left (\frac {3}{4} (a+4) \left (\frac {1}{2} (a+4) \arctan \left (\frac {x}{\sqrt {a-(x-1)^4-2 (x-1)^2+3}}\right )+\frac {1}{2} x \sqrt {a-(x-1)^4-2 (x-1)^2+3}\right )+\frac {1}{4} x \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\right )+\frac {6}{7} \left (\frac {2 \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1} \left (\frac {(a+3) (5 a+16) \sqrt {\sqrt {a+4}+1} \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \operatorname {EllipticF}\left (\arctan \left (\frac {x-1}{\sqrt {\sqrt {a+4}+1}}\right ),-\frac {2 \sqrt {a+4}}{1-\sqrt {a+4}}\right )}{\sqrt {\frac {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}{\frac {(x-1)^2}{\sqrt {a+4}+1}+1}} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}-4 (2 a+7) \left (\frac {\left (1-\sqrt {a+4}\right ) (x-1) \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}}{\sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}-\frac {\left (1-\sqrt {a+4}\right ) \sqrt {\sqrt {a+4}+1} \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} E\left (\arctan \left (\frac {x-1}{\sqrt {\sqrt {a+4}+1}}\right )|-\frac {2 \sqrt {a+4}}{1-\sqrt {a+4}}\right )}{\sqrt {\frac {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}{\frac {(x-1)^2}{\sqrt {a+4}+1}+1}} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}\right )\right )}{15 \sqrt {a-(x-1)^4-2 (x-1)^2+3}}+\frac {1}{15} (x-1) \left (5 a-3 (x-1)^2+13\right ) \sqrt {a-(x-1)^4-2 (x-1)^2+3}\right )+\frac {1}{7} (x-1) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\) |
((3 + a - 2*(-1 + x)^2 - (-1 + x)^4)^(3/2)*(-1 + x))/7 + (((3 + a - 2*(-1 + x)^2 - (-1 + x)^4)^(3/2)*x)/4 + (3*(4 + a)*((Sqrt[3 + a - 2*(-1 + x)^2 - (-1 + x)^4]*x)/2 + ((4 + a)*ArcTan[x/Sqrt[3 + a - 2*(-1 + x)^2 - (-1 + x) ^4]])/2))/4)/2 + (6*(((13 + 5*a - 3*(-1 + x)^2)*Sqrt[3 + a - 2*(-1 + x)^2 - (-1 + x)^4]*(-1 + x))/15 + (2*Sqrt[1 + (-1 + x)^2/(1 - Sqrt[4 + a])]*Sqr t[1 + (-1 + x)^2/(1 + Sqrt[4 + a])]*(-4*(7 + 2*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*Sq rt[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])])) + ((3 + a)*(16 + 5*a)*Sqrt[1 + Sqrt[4 + a]]*Sqrt[1 + (-1 + x)^2/(1 - Sqrt [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])])))/(15*Sq rt[3 + a - 2*(-1 + x)^2 - (-1 + x)^4])))/7
3.8.86.3.1 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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
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)^(p_), x_Symbol] :> Simp[(b + 2*c*x) *((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* p + 1))) Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
Int[((a_.) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[x*((a + b *x^2 + c*x^4)^p/(4*p + 1)), x] + Simp[2*(p/(4*p + 1)) Int[(2*a + b*x^2)*( a + b*x^2 + c*x^4)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a* c, 0] && GtQ[p, 0] && IntegerQ[2*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*(2*b*e*p + c*d*(4*p + 3) + c*e*(4*p + 1)*x^2)*((a + b*x^2 + c *x^4)^p/(c*(4*p + 1)*(4*p + 3))), x] + Simp[2*(p/(c*(4*p + 1)*(4*p + 3))) Int[Simp[2*a*c*d*(4*p + 3) - a*b*e + (2*a*c*e*(4*p + 1) + b*c*d*(4*p + 3) - b^2*e*(2*p + 1))*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] && GtQ[p, 0] && FractionQ[p] && 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[(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. \(2693\) vs. \(2(600)=1200\).
Time = 4.90 (sec) , antiderivative size = 2694, normalized size of antiderivative = 4.83
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(2694\) |
| elliptic | \(\text {Expression too large to display}\) | \(2694\) |
| risch | \(\text {Expression too large to display}\) | \(3609\) |
-1/8*x^6*(-x^4+4*x^3-8*x^2+a+8*x)^(1/2)+17/28*x^5*(-x^4+4*x^3-8*x^2+a+8*x) ^(1/2)-43/28*x^4*(-x^4+4*x^3-8*x^2+a+8*x)^(1/2)+74/35*x^3*(-x^4+4*x^3-8*x^ 2+a+8*x)^(1/2)+(5/16*a-9/20)*x^2*(-x^4+4*x^3-8*x^2+a+8*x)^(1/2)+(-11/56*a- 29/70)*x*(-x^4+4*x^3-8*x^2+a+8*x)^(1/2)+(11/56*a+13/14)*(-x^4+4*x^3-8*x^2+ a+8*x)^(1/2)-(-(-11/56*a-29/70)*a-11/14*a-26/7)*((-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)*EllipticF(((-(-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))-(a^2-2*(5/16*a-9/20)*a+55/14*a+62/5...
\[ \int x \left (a+8 x-8 x^2+4 x^3-x^4\right )^{3/2} \, dx=\int { {\left (-x^{4} + 4 \, x^{3} - 8 \, x^{2} + a + 8 \, x\right )}^{\frac {3}{2}} x \,d x } \]
\[ \int x \left (a+8 x-8 x^2+4 x^3-x^4\right )^{3/2} \, dx=\int x \left (a - x^{4} + 4 x^{3} - 8 x^{2} + 8 x\right )^{\frac {3}{2}}\, dx \]
\[ \int x \left (a+8 x-8 x^2+4 x^3-x^4\right )^{3/2} \, dx=\int { {\left (-x^{4} + 4 \, x^{3} - 8 \, x^{2} + a + 8 \, x\right )}^{\frac {3}{2}} x \,d x } \]
\[ \int x \left (a+8 x-8 x^2+4 x^3-x^4\right )^{3/2} \, dx=\int { {\left (-x^{4} + 4 \, x^{3} - 8 \, x^{2} + a + 8 \, x\right )}^{\frac {3}{2}} x \,d x } \]
Timed out. \[ \int x \left (a+8 x-8 x^2+4 x^3-x^4\right )^{3/2} \, dx=\int x\,{\left (-x^4+4\,x^3-8\,x^2+8\,x+a\right )}^{3/2} \,d x \]