3.8.91 \(\int x^2 (a+8 x-8 x^2+4 x^3-x^4)^{3/2} \, dx\) [791]

3.8.91.1 Optimal result

Integrand size = 28, antiderivative size = 585 \[ \int x^2 \left (a+8 x-8 x^2+4 x^3-x^4\right )^{3/2} \, dx=\frac {3}{8} (4+a) \left (1+(-1+x)^2\right ) \sqrt {3+a-2 (-1+x)^2-(-1+x)^4}+\frac {1}{4} \left (1+(-1+x)^2\right ) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )^{3/2}+\frac {4 \left (140+111 a+21 a^2\right ) \left (1-\sqrt {4+a}\right ) \left (1+\frac {(-1+x)^2}{1-\sqrt {4+a}}\right ) (-1+x)}{315 \sqrt {3+a-2 (-1+x)^2-(-1+x)^4}}+\frac {2}{315} \left (2 (80+27 a)+3 (20+7 a) (-1+x)^2\right ) \sqrt {3+a-2 (-1+x)^2-(-1+x)^4} (-1+x)+\frac {1}{63} \left (15+7 (-1+x)^2\right ) \left (3+a-2 (-1+x)^2-(-1+x)^4\right )^{3/2} (-1+x)+\frac {3}{8} (4+a)^2 \arctan \left (\frac {1+(-1+x)^2}{\sqrt {3+a-2 (-1+x)^2-(-1+x)^4}}\right )-\frac {4 \left (140+111 a+21 a^2\right ) \left (1-\sqrt {4+a}\right ) \sqrt {1+\sqrt {4+a}} \left (1+\frac {(-1+x)^2}{1-\sqrt {4+a}}\right ) E\left (\arctan \left (\frac {-1+x}{\sqrt {1+\sqrt {4+a}}}\right )|-\frac {2 \sqrt {4+a}}{1-\sqrt {4+a}}\right )}{315 \sqrt {\frac {1+\frac {(-1+x)^2}{1-\sqrt {4+a}}}{1+\frac {(-1+x)^2}{1+\sqrt {4+a}}}} \sqrt {3+a-2 (-1+x)^2-(-1+x)^4}}+\frac {4 (3+a) (100+33 a) \sqrt {1+\sqrt {4+a}} \left (1+\frac {(-1+x)^2}{1-\sqrt {4+a}}\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {-1+x}{\sqrt {1+\sqrt {4+a}}}\right ),-\frac {2 \sqrt {4+a}}{1-\sqrt {4+a}}\right )}{315 \sqrt {\frac {1+\frac {(-1+x)^2}{1-\sqrt {4+a}}}{1+\frac {(-1+x)^2}{1+\sqrt {4+a}}}} \sqrt {3+a-2 (-1+x)^2-(-1+x)^4}} \]

output

1/4*(1+(-1+x)^2)*(3+a-2*(-1+x)^2-(-1+x)^4)^(3/2)+1/63*(15+7*(-1+x)^2)*(3+a 
-2*(-1+x)^2-(-1+x)^4)^(3/2)*(-1+x)+3/8*(4+a)^2*arctan((1+(-1+x)^2)/(3+a-2* 
(-1+x)^2-(-1+x)^4)^(1/2))+4/315*(21*a^2+111*a+140)*(-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)+3/8*(4+a)*(1+ 
(-1+x)^2)*(3+a-2*(-1+x)^2-(-1+x)^4)^(1/2)+2/315*(160+54*a+3*(20+7*a)*(-1+x 
)^2)*(-1+x)*(3+a-2*(-1+x)^2-(-1+x)^4)^(1/2)+4/315*(3+a)*(100+33*a)*(1/(1+( 
-1+x)^2/(1+(4+a)^(1/2))))^(1/2)*(1+(-1+x)^2/(1+(4+a)^(1/2)))^(1/2)*Ellipti 
cF((-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)-4/315*(21*a^2+111*a+140)*(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)
 

3.8.91.2 Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(8500\) vs. \(2(585)=1170\).

Time = 17.05 (sec) , antiderivative size = 8500, normalized size of antiderivative = 14.53 \[ \int x^2 \left (a+8 x-8 x^2+4 x^3-x^4\right )^{3/2} \, dx=\text {Result too large to show} \]

input

Integrate[x^2*(a + 8*x - 8*x^2 + 4*x^3 - x^4)^(3/2),x]
 

output

Result too large to show
 

3.8.91.3 Rubi [A] (warning: unable to verify)

Time = 0.92 (sec) , antiderivative size = 651, normalized size of antiderivative = 1.11, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {2459, 2006, 2202, 27, 1432, 1087, 1087, 1092, 217, 1490, 27, 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.

\(\displaystyle \int x^2 \left (a-x^4+4 x^3-8 x^2+8 x\right )^{3/2} \, dx\)

\(\Big \downarrow \) 2459

\(\displaystyle \int \left ((x-1)^2+2 (x-1)+1\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}d(x-1)\)

\(\Big \downarrow \) 2006

\(\displaystyle \int x^2 \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}d(x-1)\)

\(\Big \downarrow \) 2202

\(\displaystyle \int \left ((x-1)^2+1\right ) \left (-(x-1)^4-2 (x-1)^2+a+3\right )^{3/2}d(x-1)+\int 2 \left (-(x-1)^4-2 (x-1)^2+a+3\right )^{3/2} (x-1)d(x-1)\)

\(\Big \downarrow \) 27

\(\displaystyle \int \left ((x-1)^2+1\right ) \left (-(x-1)^4-2 (x-1)^2+a+3\right )^{3/2}d(x-1)+2 \int \left (-(x-1)^4-2 (x-1)^2+a+3\right )^{3/2} (x-1)d(x-1)\)

\(\Big \downarrow \) 1432

\(\displaystyle \int \left (-(x-1)^4-2 (x-1)^2+a+3\right )^{3/2}d(x-1)^2+\int \left ((x-1)^2+1\right ) \left (-(x-1)^4-2 (x-1)^2+a+3\right )^{3/2}d(x-1)\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {3}{4} (a+4) \int \sqrt {-(x-1)^4-2 (x-1)^2+a+3}d(x-1)^2+\int \left ((x-1)^2+1\right ) \left (-(x-1)^4-2 (x-1)^2+a+3\right )^{3/2}d(x-1)+\frac {1}{4} x \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {3}{4} (a+4) \left (\frac {1}{2} (a+4) \int \frac {1}{\sqrt {-(x-1)^4-2 (x-1)^2+a+3}}d(x-1)^2+\frac {1}{2} x \sqrt {a-(x-1)^4-2 (x-1)^2+3}\right )+\int \left ((x-1)^2+1\right ) \left (-(x-1)^4-2 (x-1)^2+a+3\right )^{3/2}d(x-1)+\frac {1}{4} x \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {3}{4} (a+4) \left ((a+4) \int \frac {1}{-(x-1)^4-4}d\left (-\frac {2 x}{\sqrt {-(x-1)^4-2 (x-1)^2+a+3}}\right )+\frac {1}{2} x \sqrt {a-(x-1)^4-2 (x-1)^2+3}\right )+\int \left ((x-1)^2+1\right ) \left (-(x-1)^4-2 (x-1)^2+a+3\right )^{3/2}d(x-1)+\frac {1}{4} x \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\)

\(\Big \downarrow \) 217

\(\displaystyle \int \left ((x-1)^2+1\right ) \left (-(x-1)^4-2 (x-1)^2+a+3\right )^{3/2}d(x-1)+\frac {3}{4} (a+4) \left (\frac {1}{2} (a+4) \arctan \left (\frac {x}{\sqrt {a-(x-1)^4-2 (x-1)^2+3}}\right )+\frac {1}{2} x \sqrt {a-(x-1)^4-2 (x-1)^2+3}\right )+\frac {1}{4} x \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\)

\(\Big \downarrow \) 1490

\(\displaystyle -\frac {1}{21} \int -2 \left ((7 a+20) (x-1)^2+8 (a+3)\right ) \sqrt {-(x-1)^4-2 (x-1)^2+a+3}d(x-1)+\frac {3}{4} (a+4) \left (\frac {1}{2} (a+4) \arctan \left (\frac {x}{\sqrt {a-(x-1)^4-2 (x-1)^2+3}}\right )+\frac {1}{2} x \sqrt {a-(x-1)^4-2 (x-1)^2+3}\right )+\frac {1}{63} \left (7 (x-1)^2+15\right ) (x-1) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}+\frac {1}{4} x \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2}{21} \int \left ((7 a+20) (x-1)^2+8 (a+3)\right ) \sqrt {-(x-1)^4-2 (x-1)^2+a+3}d(x-1)+\frac {3}{4} (a+4) \left (\frac {1}{2} (a+4) \arctan \left (\frac {x}{\sqrt {a-(x-1)^4-2 (x-1)^2+3}}\right )+\frac {1}{2} x \sqrt {a-(x-1)^4-2 (x-1)^2+3}\right )+\frac {1}{63} \left (7 (x-1)^2+15\right ) (x-1) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}+\frac {1}{4} x \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\)

\(\Big \downarrow \) 1490

\(\displaystyle \frac {2}{21} \left (\frac {1}{15} (x-1) \left (3 (7 a+20) (x-1)^2+2 (27 a+80)\right ) \sqrt {a-(x-1)^4-2 (x-1)^2+3}-\frac {1}{15} \int -\frac {2 \left (\left (21 a^2+111 a+140\right ) (x-1)^2+(a+3) (33 a+100)\right )}{\sqrt {-(x-1)^4-2 (x-1)^2+a+3}}d(x-1)\right )+\frac {3}{4} (a+4) \left (\frac {1}{2} (a+4) \arctan \left (\frac {x}{\sqrt {a-(x-1)^4-2 (x-1)^2+3}}\right )+\frac {1}{2} x \sqrt {a-(x-1)^4-2 (x-1)^2+3}\right )+\frac {1}{63} \left (7 (x-1)^2+15\right ) (x-1) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}+\frac {1}{4} x \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2}{21} \left (\frac {2}{15} \int \frac {\left (21 a^2+111 a+140\right ) (x-1)^2+(a+3) (33 a+100)}{\sqrt {-(x-1)^4-2 (x-1)^2+a+3}}d(x-1)+\frac {1}{15} (x-1) \left (3 (7 a+20) (x-1)^2+2 (27 a+80)\right ) \sqrt {a-(x-1)^4-2 (x-1)^2+3}\right )+\frac {3}{4} (a+4) \left (\frac {1}{2} (a+4) \arctan \left (\frac {x}{\sqrt {a-(x-1)^4-2 (x-1)^2+3}}\right )+\frac {1}{2} x \sqrt {a-(x-1)^4-2 (x-1)^2+3}\right )+\frac {1}{63} \left (7 (x-1)^2+15\right ) (x-1) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}+\frac {1}{4} x \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\)

\(\Big \downarrow \) 1514

\(\displaystyle \frac {2}{21} \left (\frac {2 \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1} \int \frac {\left (21 a^2+111 a+140\right ) (x-1)^2+(a+3) (33 a+100)}{\sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}d(x-1)}{15 \sqrt {a-(x-1)^4-2 (x-1)^2+3}}+\frac {1}{15} (x-1) \left (3 (7 a+20) (x-1)^2+2 (27 a+80)\right ) \sqrt {a-(x-1)^4-2 (x-1)^2+3}\right )+\frac {3}{4} (a+4) \left (\frac {1}{2} (a+4) \arctan \left (\frac {x}{\sqrt {a-(x-1)^4-2 (x-1)^2+3}}\right )+\frac {1}{2} x \sqrt {a-(x-1)^4-2 (x-1)^2+3}\right )+\frac {1}{63} \left (7 (x-1)^2+15\right ) (x-1) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}+\frac {1}{4} x \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\)

\(\Big \downarrow \) 406

\(\displaystyle \frac {2}{21} \left (\frac {2 \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1} \left (\left (21 a^2+111 a+140\right ) \int \frac {(x-1)^2}{\sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}d(x-1)+(a+3) (33 a+100) \int \frac {1}{\sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}d(x-1)\right )}{15 \sqrt {a-(x-1)^4-2 (x-1)^2+3}}+\frac {1}{15} (x-1) \left (3 (7 a+20) (x-1)^2+2 (27 a+80)\right ) \sqrt {a-(x-1)^4-2 (x-1)^2+3}\right )+\frac {3}{4} (a+4) \left (\frac {1}{2} (a+4) \arctan \left (\frac {x}{\sqrt {a-(x-1)^4-2 (x-1)^2+3}}\right )+\frac {1}{2} x \sqrt {a-(x-1)^4-2 (x-1)^2+3}\right )+\frac {1}{63} \left (7 (x-1)^2+15\right ) (x-1) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}+\frac {1}{4} x \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\)

\(\Big \downarrow \) 320

\(\displaystyle \frac {2}{21} \left (\frac {2 \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1} \left (\left (21 a^2+111 a+140\right ) \int \frac {(x-1)^2}{\sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}d(x-1)+\frac {(a+3) (33 a+100) \sqrt {\sqrt {a+4}+1} \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \operatorname {EllipticF}\left (\arctan \left (\frac {x-1}{\sqrt {\sqrt {a+4}+1}}\right ),-\frac {2 \sqrt {a+4}}{1-\sqrt {a+4}}\right )}{\sqrt {\frac {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}{\frac {(x-1)^2}{\sqrt {a+4}+1}+1}} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}\right )}{15 \sqrt {a-(x-1)^4-2 (x-1)^2+3}}+\frac {1}{15} (x-1) \left (3 (7 a+20) (x-1)^2+2 (27 a+80)\right ) \sqrt {a-(x-1)^4-2 (x-1)^2+3}\right )+\frac {3}{4} (a+4) \left (\frac {1}{2} (a+4) \arctan \left (\frac {x}{\sqrt {a-(x-1)^4-2 (x-1)^2+3}}\right )+\frac {1}{2} x \sqrt {a-(x-1)^4-2 (x-1)^2+3}\right )+\frac {1}{63} \left (7 (x-1)^2+15\right ) (x-1) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}+\frac {1}{4} x \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\)

\(\Big \downarrow \) 388

\(\displaystyle \frac {2}{21} \left (\frac {2 \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1} \left (\left (21 a^2+111 a+140\right ) \left (\frac {\left (1-\sqrt {a+4}\right ) (x-1) \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}}{\sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}-\left (1-\sqrt {a+4}\right ) \int \frac {\sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}}{\left (\frac {(x-1)^2}{\sqrt {a+4}+1}+1\right )^{3/2}}d(x-1)\right )+\frac {(a+3) (33 a+100) \sqrt {\sqrt {a+4}+1} \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \operatorname {EllipticF}\left (\arctan \left (\frac {x-1}{\sqrt {\sqrt {a+4}+1}}\right ),-\frac {2 \sqrt {a+4}}{1-\sqrt {a+4}}\right )}{\sqrt {\frac {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}{\frac {(x-1)^2}{\sqrt {a+4}+1}+1}} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}\right )}{15 \sqrt {a-(x-1)^4-2 (x-1)^2+3}}+\frac {1}{15} (x-1) \left (3 (7 a+20) (x-1)^2+2 (27 a+80)\right ) \sqrt {a-(x-1)^4-2 (x-1)^2+3}\right )+\frac {3}{4} (a+4) \left (\frac {1}{2} (a+4) \arctan \left (\frac {x}{\sqrt {a-(x-1)^4-2 (x-1)^2+3}}\right )+\frac {1}{2} x \sqrt {a-(x-1)^4-2 (x-1)^2+3}\right )+\frac {1}{63} \left (7 (x-1)^2+15\right ) (x-1) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}+\frac {1}{4} x \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\)

\(\Big \downarrow \) 313

\(\displaystyle \frac {2}{21} \left (\frac {2 \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1} \left (\left (21 a^2+111 a+140\right ) \left (\frac {\left (1-\sqrt {a+4}\right ) (x-1) \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}}{\sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}-\frac {\left (1-\sqrt {a+4}\right ) \sqrt {\sqrt {a+4}+1} \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} E\left (\arctan \left (\frac {x-1}{\sqrt {\sqrt {a+4}+1}}\right )|-\frac {2 \sqrt {a+4}}{1-\sqrt {a+4}}\right )}{\sqrt {\frac {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}{\frac {(x-1)^2}{\sqrt {a+4}+1}+1}} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}\right )+\frac {(a+3) (33 a+100) \sqrt {\sqrt {a+4}+1} \sqrt {\frac {(x-1)^2}{1-\sqrt {a+4}}+1} \operatorname {EllipticF}\left (\arctan \left (\frac {x-1}{\sqrt {\sqrt {a+4}+1}}\right ),-\frac {2 \sqrt {a+4}}{1-\sqrt {a+4}}\right )}{\sqrt {\frac {\frac {(x-1)^2}{1-\sqrt {a+4}}+1}{\frac {(x-1)^2}{\sqrt {a+4}+1}+1}} \sqrt {\frac {(x-1)^2}{\sqrt {a+4}+1}+1}}\right )}{15 \sqrt {a-(x-1)^4-2 (x-1)^2+3}}+\frac {1}{15} (x-1) \left (3 (7 a+20) (x-1)^2+2 (27 a+80)\right ) \sqrt {a-(x-1)^4-2 (x-1)^2+3}\right )+\frac {3}{4} (a+4) \left (\frac {1}{2} (a+4) \arctan \left (\frac {x}{\sqrt {a-(x-1)^4-2 (x-1)^2+3}}\right )+\frac {1}{2} x \sqrt {a-(x-1)^4-2 (x-1)^2+3}\right )+\frac {1}{63} \left (7 (x-1)^2+15\right ) (x-1) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}+\frac {1}{4} x \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}\)

input

Int[x^2*(a + 8*x - 8*x^2 + 4*x^3 - x^4)^(3/2),x]
 

output

((15 + 7*(-1 + x)^2)*(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)^(3/2)*(-1 + x))/6 
3 + ((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*(((2*(80 + 27*a) + 3*(20 + 7*a)*(-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])]*Sqrt[1 + (-1 + x)^2/(1 + Sqrt[4 + a])]*((140 
 + 111*a + 21*a^2)*(((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[Arc 
Tan[(-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])])) + ((3 + a)*(100 + 33*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]))]*Sqr 
t[1 + (-1 + x)^2/(1 + Sqrt[4 + a])])))/(15*Sqrt[3 + a - 2*(-1 + x)^2 - (-1 
 + x)^4])))/21
 

3.8.91.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

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])
 

rule 313

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]
 

rule 320

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]
 

rule 388

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]
 

rule 406

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]
 

rule 1087

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])
 

rule 1092

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]
 

rule 1432

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]
 

rule 1490

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]
 

rule 1514

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]
 

rule 2006

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]]
 

rule 2202

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]
 

rule 2459

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])
 

3.8.91.4 Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(2732\) vs. \(2(624)=1248\).

Time = 4.15 (sec) , antiderivative size = 2733, normalized size of antiderivative = 4.67

method	result	size
default	\(\text {Expression too large to display}\)	\(2733\)
elliptic	\(\text {Expression too large to display}\)	\(2733\)
risch	\(\text {Expression too large to display}\)	\(3625\)

input

int(x^2*(-x^4+4*x^3-8*x^2+a+8*x)^(3/2),x,method=_RETURNVERBOSE)
 

output

-1/9*x^7*(-x^4+4*x^3-8*x^2+a+8*x)^(1/2)+19/36*x^6*(-x^4+4*x^3-8*x^2+a+8*x) 
^(1/2)-163/126*x^5*(-x^4+4*x^3-8*x^2+a+8*x)^(1/2)+71/42*x^4*(-x^4+4*x^3-8* 
x^2+a+8*x)^(1/2)+(11/45*a-16/63)*x^3*(-x^4+4*x^3-8*x^2+a+8*x)^(1/2)+(-13/1 
20*a-5/18)*x^2*(-x^4+4*x^3-8*x^2+a+8*x)^(1/2)+(9/140*a+23/63)*x*(-x^4+4*x^ 
3-8*x^2+a+8*x)^(1/2)+(107/252*a+101/63)*(-x^4+4*x^3-8*x^2+a+8*x)^(1/2)-(-( 
9/140*a+23/63)*a-107/63*a-404/63)*((-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+...
 

3.8.91.5 Fricas [F]

\[ \int x^2 \left (a+8 x-8 x^2+4 x^3-x^4\right )^{3/2} \, dx=\int { {\left (-x^{4} + 4 \, x^{3} - 8 \, x^{2} + a + 8 \, x\right )}^{\frac {3}{2}} x^{2} \,d x } \]

input

integrate(x^2*(-x^4+4*x^3-8*x^2+a+8*x)^(3/2),x, algorithm="fricas")
 

output

integral(-(x^6 - 4*x^5 + 8*x^4 - a*x^2 - 8*x^3)*sqrt(-x^4 + 4*x^3 - 8*x^2 
+ a + 8*x), x)
 

3.8.91.6 Sympy [F]

\[ \int x^2 \left (a+8 x-8 x^2+4 x^3-x^4\right )^{3/2} \, dx=\int x^{2} \left (a - x^{4} + 4 x^{3} - 8 x^{2} + 8 x\right )^{\frac {3}{2}}\, dx \]

input

integrate(x**2*(-x**4+4*x**3-8*x**2+a+8*x)**(3/2),x)
 

output

Integral(x**2*(a - x**4 + 4*x**3 - 8*x**2 + 8*x)**(3/2), x)
 

3.8.91.7 Maxima [F]

\[ \int x^2 \left (a+8 x-8 x^2+4 x^3-x^4\right )^{3/2} \, dx=\int { {\left (-x^{4} + 4 \, x^{3} - 8 \, x^{2} + a + 8 \, x\right )}^{\frac {3}{2}} x^{2} \,d x } \]

input

integrate(x^2*(-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^2, x)
 

3.8.91.8 Giac [F]

\[ \int x^2 \left (a+8 x-8 x^2+4 x^3-x^4\right )^{3/2} \, dx=\int { {\left (-x^{4} + 4 \, x^{3} - 8 \, x^{2} + a + 8 \, x\right )}^{\frac {3}{2}} x^{2} \,d x } \]

input

integrate(x^2*(-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^2, x)
 

3.8.91.9 Mupad [F(-1)]

Timed out. \[ \int x^2 \left (a+8 x-8 x^2+4 x^3-x^4\right )^{3/2} \, dx=\int x^2\,{\left (-x^4+4\,x^3-8\,x^2+8\,x+a\right )}^{3/2} \,d x \]

input

int(x^2*(a + 8*x - 8*x^2 + 4*x^3 - x^4)^(3/2),x)
 

output

int(x^2*(a + 8*x - 8*x^2 + 4*x^3 - x^4)^(3/2), x)