Integrand size = 26, antiderivative size = 520 \[ \int x \left (a+8 x-8 x^2+4 x^3-x^4\right )^{3/2} \, dx=\frac {3}{16} (4+a) \sqrt {3+a-2 (1-x)^2-(1-x)^4} \left (1+(-1+x)^2\right )+\frac {1}{8} \left (3+a-2 (1-x)^2-(1-x)^4\right )^{3/2} \left (1+(-1+x)^2\right )-\frac {2}{35} \left (13+5 a-3 (1-x)^2\right ) \sqrt {3+a-2 (1-x)^2-(1-x)^4} (1-x)-\frac {1}{7} \left (3+a-2 (1-x)^2-(1-x)^4\right )^{3/2} (1-x)+\frac {3}{16} (4+a)^2 \arctan \left (\frac {1+(-1+x)^2}{\sqrt {3+a-2 (1-x)^2-(1-x)^4}}\right )+\frac {16 (7+2 a) \sqrt {-1+\sqrt {4+a}} \left (1+\sqrt {4+a}\right ) \sqrt {1+\frac {(1-x)^2}{1-\sqrt {4+a}}} \sqrt {1+\frac {(1-x)^2}{1+\sqrt {4+a}}} E\left (\arcsin \left (\frac {1-x}{\sqrt {-1+\sqrt {4+a}}}\right )|\frac {1-\sqrt {4+a}}{1+\sqrt {4+a}}\right )}{35 \sqrt {3+a-2 (1-x)^2-(1-x)^4}}-\frac {4 \sqrt {-1+\sqrt {4+a}} \left (76+5 a^2+28 \sqrt {4+a}+a \left (39+8 \sqrt {4+a}\right )\right ) \sqrt {1+\frac {(1-x)^2}{1-\sqrt {4+a}}} \sqrt {1+\frac {(1-x)^2}{1+\sqrt {4+a}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-x}{\sqrt {-1+\sqrt {4+a}}}\right ),\frac {1-\sqrt {4+a}}{1+\sqrt {4+a}}\right )}{35 \sqrt {3+a-2 (1-x)^2-(1-x)^4}} \] Output:
3/16*(4+a)*(3+a-2*(1-x)^2-(1-x)^4)^(1/2)*(1+(-1+x)^2)+1/8*(3+a-2*(1-x)^2-( 1-x)^4)^(3/2)*(1+(-1+x)^2)-2/35*(13+5*a-3*(1-x)^2)*(3+a-2*(1-x)^2-(1-x)^4) ^(1/2)*(1-x)-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+(4+a)^(1/2))^ (1/2)*(1+(4+a)^(1/2))*(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-(4+a)^(1/2))/(1 +(4+a)^(1/2)))^(1/2))/(3+a-2*(1-x)^2-(1-x)^4)^(1/2)-4/35*(-1+(4+a)^(1/2))^ (1/2)*(76+5*a^2+28*(4+a)^(1/2)+a*(39+8*(4+a)^(1/2)))*(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-(4+a)^(1/2))/(1+(4+a)^(1/2)))^(1/2))/(3+a-2*(1-x)^2-(1-x)^ 4)^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(7235\) vs. \(2(520)=1040\).
Time = 17.25 (sec) , antiderivative size = 7235, normalized size of antiderivative = 13.91 \[ \int x \left (a+8 x-8 x^2+4 x^3-x^4\right )^{3/2} \, dx=\text {Result too large to show} \] Input:
Integrate[x*(a + 8*x - 8*x^2 + 4*x^3 - x^4)^(3/2),x]
Output:
Result too large to show
Time = 1.50 (sec) , antiderivative size = 635, normalized size of antiderivative = 1.22, 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 |
| \(\displaystyle \frac {1}{2} \left (\frac {3}{4} (a+4) \left (\frac {1}{2} (a+4) \arctan \left (\frac {x}{\sqrt {a-(x-1)^4-2 (x-1)^2+3}}\right )+\frac {1}{2} x \sqrt {a-(x-1)^4-2 (x-1)^2+3}\right )+\frac {1}{4} x \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\right )+\frac {6}{7} \left (\frac {2 \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1} \left (\frac {(a+3) (5 a+16) \sqrt {\sqrt {a+4}+1} \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \operatorname {EllipticF}\left (\arctan \left (\frac {x-1}{\sqrt {\sqrt {a+4}+1}}\right ),-\frac {2 \sqrt {a+4}}{1-\sqrt {a+4}}\right )}{\sqrt {\frac {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}{\frac {(x-1)^2}{\sqrt {a+4}+1}+1}} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}-4 (2 a+7) \left (\frac {\left (1-\sqrt {a+4}\right ) (x-1) \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}}{\sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}-\frac {\left (1-\sqrt {a+4}\right ) \sqrt {\sqrt {a+4}+1} \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} E\left (\arctan \left (\frac {x-1}{\sqrt {\sqrt {a+4}+1}}\right )|-\frac {2 \sqrt {a+4}}{1-\sqrt {a+4}}\right )}{\sqrt {\frac {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}{\frac {(x-1)^2}{\sqrt {a+4}+1}+1}} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}\right )\right )}{15 \sqrt {a-(x-1)^4-2 (x-1)^2+3}}+\frac {1}{15} (x-1) \left (5 a-3 (x-1)^2+13\right ) \sqrt {a-(x-1)^4-2 (x-1)^2+3}\right )+\frac {1}{7} (x-1) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\) |
Input:
Int[x*(a + 8*x - 8*x^2 + 4*x^3 - x^4)^(3/2),x]
Output:
((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
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. \(2(448)=896\).
Time = 7.90 (sec) , antiderivative size = 2694, normalized size of antiderivative = 5.18
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(2694\) |
| elliptic | \(\text {Expression too large to display}\) | \(2694\) |
| risch | \(\text {Expression too large to display}\) | \(3609\) |
Input:
int(x*(-x^4+4*x^3-8*x^2+a+8*x)^(3/2),x,method=_RETURNVERBOSE)
Output:
-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...
\[ \int x \left (a+8 x-8 x^2+4 x^3-x^4\right )^{3/2} \, dx=\int { {\left (-x^{4} + 4 \, x^{3} - 8 \, x^{2} + a + 8 \, x\right )}^{\frac {3}{2}} x \,d x } \] Input:
integrate(x*(-x^4+4*x^3-8*x^2+a+8*x)^(3/2),x, algorithm="fricas")
Output:
integral(-(x^5 - 4*x^4 + 8*x^3 - a*x - 8*x^2)*sqrt(-x^4 + 4*x^3 - 8*x^2 + a + 8*x), x)
\[ \int x \left (a+8 x-8 x^2+4 x^3-x^4\right )^{3/2} \, dx=\int x \left (a - x^{4} + 4 x^{3} - 8 x^{2} + 8 x\right )^{\frac {3}{2}}\, dx \] Input:
integrate(x*(-x**4+4*x**3-8*x**2+a+8*x)**(3/2),x)
Output:
Integral(x*(a - x**4 + 4*x**3 - 8*x**2 + 8*x)**(3/2), x)
\[ \int x \left (a+8 x-8 x^2+4 x^3-x^4\right )^{3/2} \, dx=\int { {\left (-x^{4} + 4 \, x^{3} - 8 \, x^{2} + a + 8 \, x\right )}^{\frac {3}{2}} x \,d x } \] Input:
integrate(x*(-x^4+4*x^3-8*x^2+a+8*x)^(3/2),x, algorithm="maxima")
Output:
integrate((-x^4 + 4*x^3 - 8*x^2 + a + 8*x)^(3/2)*x, x)
\[ \int x \left (a+8 x-8 x^2+4 x^3-x^4\right )^{3/2} \, dx=\int { {\left (-x^{4} + 4 \, x^{3} - 8 \, x^{2} + a + 8 \, x\right )}^{\frac {3}{2}} x \,d x } \] Input:
integrate(x*(-x^4+4*x^3-8*x^2+a+8*x)^(3/2),x, algorithm="giac")
Output:
integrate((-x^4 + 4*x^3 - 8*x^2 + a + 8*x)^(3/2)*x, x)
Timed out. \[ \int x \left (a+8 x-8 x^2+4 x^3-x^4\right )^{3/2} \, dx=\int x\,{\left (-x^4+4\,x^3-8\,x^2+8\,x+a\right )}^{3/2} \,d x \] Input:
int(x*(a + 8*x - 8*x^2 + 4*x^3 - x^4)^(3/2),x)
Output:
int(x*(a + 8*x - 8*x^2 + 4*x^3 - x^4)^(3/2), x)
\[ \int x \left (a+8 x-8 x^2+4 x^3-x^4\right )^{3/2} \, dx=\frac {5 \sqrt {-x^{4}+4 x^{3}-8 x^{2}+a +8 x}\, a \,x^{2}}{16}-\frac {11 \sqrt {-x^{4}+4 x^{3}-8 x^{2}+a +8 x}\, a x}{56}+\frac {37 \sqrt {-x^{4}+4 x^{3}-8 x^{2}+a +8 x}\, a}{840}-\frac {\sqrt {-x^{4}+4 x^{3}-8 x^{2}+a +8 x}\, x^{6}}{8}+\frac {17 \sqrt {-x^{4}+4 x^{3}-8 x^{2}+a +8 x}\, x^{5}}{28}-\frac {43 \sqrt {-x^{4}+4 x^{3}-8 x^{2}+a +8 x}\, x^{4}}{28}+\frac {74 \sqrt {-x^{4}+4 x^{3}-8 x^{2}+a +8 x}\, x^{3}}{35}-\frac {9 \sqrt {-x^{4}+4 x^{3}-8 x^{2}+a +8 x}\, x^{2}}{20}-\frac {29 \sqrt {-x^{4}+4 x^{3}-8 x^{2}+a +8 x}\, x}{70}+\frac {83 \sqrt {-x^{4}+4 x^{3}-8 x^{2}+a +8 x}}{210}+\frac {11 \left (\int \frac {\sqrt {-x^{4}+4 x^{3}-8 x^{2}+a +8 x}}{-x^{4}+4 x^{3}-8 x^{2}+a +8 x}d x \right ) a^{2}}{56}+\frac {5 \left (\int \frac {\sqrt {-x^{4}+4 x^{3}-8 x^{2}+a +8 x}}{-x^{4}+4 x^{3}-8 x^{2}+a +8 x}d x \right ) a}{21}-\frac {166 \left (\int \frac {\sqrt {-x^{4}+4 x^{3}-8 x^{2}+a +8 x}}{-x^{4}+4 x^{3}-8 x^{2}+a +8 x}d x \right )}{105}-\frac {32 \left (\int \frac {\sqrt {-x^{4}+4 x^{3}-8 x^{2}+a +8 x}\, x^{3}}{-x^{4}+4 x^{3}-8 x^{2}+a +8 x}d x \right ) a}{105}-\frac {16 \left (\int \frac {\sqrt {-x^{4}+4 x^{3}-8 x^{2}+a +8 x}\, x^{3}}{-x^{4}+4 x^{3}-8 x^{2}+a +8 x}d x \right )}{15}+\frac {3 \left (\int \frac {\sqrt {-x^{4}+4 x^{3}-8 x^{2}+a +8 x}\, x}{-x^{4}+4 x^{3}-8 x^{2}+a +8 x}d x \right ) a^{2}}{8}+\frac {379 \left (\int \frac {\sqrt {-x^{4}+4 x^{3}-8 x^{2}+a +8 x}\, x}{-x^{4}+4 x^{3}-8 x^{2}+a +8 x}d x \right ) a}{105}+\frac {122 \left (\int \frac {\sqrt {-x^{4}+4 x^{3}-8 x^{2}+a +8 x}\, x}{-x^{4}+4 x^{3}-8 x^{2}+a +8 x}d x \right )}{15} \] Input:
int(x*(-x^4+4*x^3-8*x^2+a+8*x)^(3/2),x)
Output:
(525*sqrt(a - x**4 + 4*x**3 - 8*x**2 + 8*x)*a*x**2 - 330*sqrt(a - x**4 + 4 *x**3 - 8*x**2 + 8*x)*a*x + 74*sqrt(a - x**4 + 4*x**3 - 8*x**2 + 8*x)*a - 210*sqrt(a - x**4 + 4*x**3 - 8*x**2 + 8*x)*x**6 + 1020*sqrt(a - x**4 + 4*x **3 - 8*x**2 + 8*x)*x**5 - 2580*sqrt(a - x**4 + 4*x**3 - 8*x**2 + 8*x)*x** 4 + 3552*sqrt(a - x**4 + 4*x**3 - 8*x**2 + 8*x)*x**3 - 756*sqrt(a - x**4 + 4*x**3 - 8*x**2 + 8*x)*x**2 - 696*sqrt(a - x**4 + 4*x**3 - 8*x**2 + 8*x)* x + 664*sqrt(a - x**4 + 4*x**3 - 8*x**2 + 8*x) + 330*int(sqrt(a - x**4 + 4 *x**3 - 8*x**2 + 8*x)/(a - x**4 + 4*x**3 - 8*x**2 + 8*x),x)*a**2 + 400*int (sqrt(a - x**4 + 4*x**3 - 8*x**2 + 8*x)/(a - x**4 + 4*x**3 - 8*x**2 + 8*x) ,x)*a - 2656*int(sqrt(a - x**4 + 4*x**3 - 8*x**2 + 8*x)/(a - x**4 + 4*x**3 - 8*x**2 + 8*x),x) - 512*int((sqrt(a - x**4 + 4*x**3 - 8*x**2 + 8*x)*x**3 )/(a - x**4 + 4*x**3 - 8*x**2 + 8*x),x)*a - 1792*int((sqrt(a - x**4 + 4*x* *3 - 8*x**2 + 8*x)*x**3)/(a - x**4 + 4*x**3 - 8*x**2 + 8*x),x) + 630*int(( sqrt(a - x**4 + 4*x**3 - 8*x**2 + 8*x)*x)/(a - x**4 + 4*x**3 - 8*x**2 + 8* x),x)*a**2 + 6064*int((sqrt(a - x**4 + 4*x**3 - 8*x**2 + 8*x)*x)/(a - x**4 + 4*x**3 - 8*x**2 + 8*x),x)*a + 13664*int((sqrt(a - x**4 + 4*x**3 - 8*x** 2 + 8*x)*x)/(a - x**4 + 4*x**3 - 8*x**2 + 8*x),x))/1680