\(\int (c+d x)^2 (c^2-d^2 x^2)^{5/2} (A+B x+C x^2+D x^3) \, dx\) [154]

Optimal result

Integrand size = 39, antiderivative size = 416 \[ \int (c+d x)^2 \left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {c^6 \left (13 c^2 C d+20 B c d^2+90 A d^3+6 c^3 D\right ) x \sqrt {c^2-d^2 x^2}}{256 d^3}+\frac {c^4 \left (13 c^2 C d+20 B c d^2+90 A d^3+6 c^3 D\right ) x \left (c^2-d^2 x^2\right )^{3/2}}{384 d^3}+\frac {c^2 \left (13 c^2 C d+20 B c d^2+90 A d^3+6 c^3 D\right ) x \left (c^2-d^2 x^2\right )^{5/2}}{480 d^3}-\frac {2 c \left (c^2 C d+B c d^2+A d^3+c^3 D\right ) \left (c^2-d^2 x^2\right )^{7/2}}{7 d^4}-\frac {\left (13 c^2 C d+20 B c d^2+10 A d^3+6 c^3 D\right ) x \left (c^2-d^2 x^2\right )^{7/2}}{80 d^3}-\frac {(C d+2 c D) x^3 \left (c^2-d^2 x^2\right )^{7/2}}{10 d}+\frac {\left (2 c C d+B d^2+3 c^2 D\right ) \left (c^2-d^2 x^2\right )^{9/2}}{9 d^4}-\frac {D \left (c^2-d^2 x^2\right )^{11/2}}{11 d^4}+\frac {c^8 \left (13 c^2 C d+20 B c d^2+90 A d^3+6 c^3 D\right ) \arctan \left (\frac {d x}{\sqrt {c^2-d^2 x^2}}\right )}{256 d^4} \] Output:

1/256*c^6*(90*A*d^3+20*B*c*d^2+13*C*c^2*d+6*D*c^3)*x*(-d^2*x^2+c^2)^(1/2)/ 
d^3+1/384*c^4*(90*A*d^3+20*B*c*d^2+13*C*c^2*d+6*D*c^3)*x*(-d^2*x^2+c^2)^(3 
/2)/d^3+1/480*c^2*(90*A*d^3+20*B*c*d^2+13*C*c^2*d+6*D*c^3)*x*(-d^2*x^2+c^2 
)^(5/2)/d^3-2/7*c*(A*d^3+B*c*d^2+C*c^2*d+D*c^3)*(-d^2*x^2+c^2)^(7/2)/d^4-1 
/80*(10*A*d^3+20*B*c*d^2+13*C*c^2*d+6*D*c^3)*x*(-d^2*x^2+c^2)^(7/2)/d^3-1/ 
10*(C*d+2*D*c)*x^3*(-d^2*x^2+c^2)^(7/2)/d+1/9*(B*d^2+2*C*c*d+3*D*c^2)*(-d^ 
2*x^2+c^2)^(9/2)/d^4-1/11*D*(-d^2*x^2+c^2)^(11/2)/d^4+1/256*c^8*(90*A*d^3+ 
20*B*c*d^2+13*C*c^2*d+6*D*c^3)*arctan(d*x/(-d^2*x^2+c^2)^(1/2))/d^4
 

Mathematica [A] (verified)

Time = 3.61 (sec) , antiderivative size = 377, normalized size of antiderivative = 0.91 \[ \int (c+d x)^2 \left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right ) \, dx=-\frac {\sqrt {c^2-d^2 x^2} \left (38400 c^{10} D+110 c^9 d (512 C+189 D x)+5 c^8 d^2 \left (30976 B+9009 C x+3840 D x^2\right )-224 d^{10} x^7 \left (495 A+4 x \left (110 B+99 C x+90 D x^2\right )\right )-704 c d^9 x^6 (360 A+7 x (45 B+4 x (10 C+9 D x)))+220 c^7 d^3 (1152 A+x (315 B+x (128 C+63 D x)))+16 c^2 d^8 x^5 (10395 A+x (8800 B+21 x (363 C+320 D x)))-264 c^5 d^5 x^2 (2880 A+x (2065 B+2 x (800 C+651 D x)))+352 c^3 d^7 x^4 (2160 A+x (1785 B+x (1520 C+1323 D x)))+12 c^4 d^6 x^3 (17325 A+2 x (7040 B+x (5929 C+5120 D x)))-10 c^6 d^4 x (57519 A+x (36608 B+3 x (8855 C+6912 D x)))\right )+6930 c^8 \left (13 c^2 C d+20 B c d^2+90 A d^3+6 c^3 D\right ) \arctan \left (\frac {d x}{\sqrt {c^2}-\sqrt {c^2-d^2 x^2}}\right )}{887040 d^4} \] Input:

Integrate[(c + d*x)^2*(c^2 - d^2*x^2)^(5/2)*(A + B*x + C*x^2 + D*x^3),x]
 

Output:

-1/887040*(Sqrt[c^2 - d^2*x^2]*(38400*c^10*D + 110*c^9*d*(512*C + 189*D*x) 
 + 5*c^8*d^2*(30976*B + 9009*C*x + 3840*D*x^2) - 224*d^10*x^7*(495*A + 4*x 
*(110*B + 99*C*x + 90*D*x^2)) - 704*c*d^9*x^6*(360*A + 7*x*(45*B + 4*x*(10 
*C + 9*D*x))) + 220*c^7*d^3*(1152*A + x*(315*B + x*(128*C + 63*D*x))) + 16 
*c^2*d^8*x^5*(10395*A + x*(8800*B + 21*x*(363*C + 320*D*x))) - 264*c^5*d^5 
*x^2*(2880*A + x*(2065*B + 2*x*(800*C + 651*D*x))) + 352*c^3*d^7*x^4*(2160 
*A + x*(1785*B + x*(1520*C + 1323*D*x))) + 12*c^4*d^6*x^3*(17325*A + 2*x*( 
7040*B + x*(5929*C + 5120*D*x))) - 10*c^6*d^4*x*(57519*A + x*(36608*B + 3* 
x*(8855*C + 6912*D*x)))) + 6930*c^8*(13*c^2*C*d + 20*B*c*d^2 + 90*A*d^3 + 
6*c^3*D)*ArcTan[(d*x)/(Sqrt[c^2] - Sqrt[c^2 - d^2*x^2])])/d^4
 

Rubi [A] (verified)

Time = 1.82 (sec) , antiderivative size = 375, normalized size of antiderivative = 0.90, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2346, 25, 2346, 25, 2346, 25, 2346, 25, 27, 455, 211, 211, 211, 224, 216}

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.

\(\displaystyle \int (c+d x)^2 \left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right ) \, dx\)

\(\Big \downarrow \) 2346

\(\displaystyle -\frac {\int -\left (c^2-d^2 x^2\right )^{5/2} \left (11 d^3 (C d+2 c D) x^4+d^2 \left (15 D c^2+22 C d c+11 B d^2\right ) x^3+11 d^2 \left (C c^2+2 B d c+A d^2\right ) x^2+11 c d^2 (B c+2 A d) x+11 A c^2 d^2\right )dx}{11 d^2}-\frac {1}{11} D x^4 \left (c^2-d^2 x^2\right )^{7/2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \left (c^2-d^2 x^2\right )^{5/2} \left (11 d^3 (C d+2 c D) x^4+d^2 \left (15 D c^2+22 C d c+11 B d^2\right ) x^3+11 d^2 \left (C c^2+2 B d c+A d^2\right ) x^2+11 c d^2 (B c+2 A d) x+11 A c^2 d^2\right )dx}{11 d^2}-\frac {1}{11} D x^4 \left (c^2-d^2 x^2\right )^{7/2}\)

\(\Big \downarrow \) 2346

\(\displaystyle \frac {-\frac {\int -\left (c^2-d^2 x^2\right )^{5/2} \left (10 \left (15 D c^2+22 C d c+11 B d^2\right ) x^3 d^4+110 A c^2 d^4+110 c (B c+2 A d) x d^4+11 \left (6 D c^3+13 C d c^2+20 B d^2 c+10 A d^3\right ) x^2 d^3\right )dx}{10 d^2}-\frac {11}{10} d x^3 \left (c^2-d^2 x^2\right )^{7/2} (2 c D+C d)}{11 d^2}-\frac {1}{11} D x^4 \left (c^2-d^2 x^2\right )^{7/2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \left (c^2-d^2 x^2\right )^{5/2} \left (10 \left (15 D c^2+22 C d c+11 B d^2\right ) x^3 d^4+110 A c^2 d^4+110 c (B c+2 A d) x d^4+11 \left (6 D c^3+13 C d c^2+20 B d^2 c+10 A d^3\right ) x^2 d^3\right )dx}{10 d^2}-\frac {11}{10} d x^3 \left (c^2-d^2 x^2\right )^{7/2} (2 c D+C d)}{11 d^2}-\frac {1}{11} D x^4 \left (c^2-d^2 x^2\right )^{7/2}\)

\(\Big \downarrow \) 2346

\(\displaystyle \frac {\frac {-\frac {\int -\left (c^2-d^2 x^2\right )^{5/2} \left (990 A c^2 d^6+99 \left (6 D c^3+13 C d c^2+20 B d^2 c+10 A d^3\right ) x^2 d^5+10 c \left (30 D c^3+44 C d c^2+121 B d^2 c+198 A d^3\right ) x d^4\right )dx}{9 d^2}-\frac {10}{9} d^2 x^2 \left (c^2-d^2 x^2\right )^{7/2} \left (11 B d^2+15 c^2 D+22 c C d\right )}{10 d^2}-\frac {11}{10} d x^3 \left (c^2-d^2 x^2\right )^{7/2} (2 c D+C d)}{11 d^2}-\frac {1}{11} D x^4 \left (c^2-d^2 x^2\right )^{7/2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\frac {\int \left (c^2-d^2 x^2\right )^{5/2} \left (990 A c^2 d^6+99 \left (6 D c^3+13 C d c^2+20 B d^2 c+10 A d^3\right ) x^2 d^5+10 c \left (30 D c^3+44 C d c^2+121 B d^2 c+198 A d^3\right ) x d^4\right )dx}{9 d^2}-\frac {10}{9} d^2 x^2 \left (c^2-d^2 x^2\right )^{7/2} \left (11 B d^2+15 c^2 D+22 c C d\right )}{10 d^2}-\frac {11}{10} d x^3 \left (c^2-d^2 x^2\right )^{7/2} (2 c D+C d)}{11 d^2}-\frac {1}{11} D x^4 \left (c^2-d^2 x^2\right )^{7/2}\)

\(\Big \downarrow \) 2346

\(\displaystyle \frac {\frac {\frac {-\frac {\int -c d^5 \left (99 c \left (6 D c^3+13 C d c^2+20 B d^2 c+90 A d^3\right )+80 d \left (30 D c^3+44 C d c^2+121 B d^2 c+198 A d^3\right ) x\right ) \left (c^2-d^2 x^2\right )^{5/2}dx}{8 d^2}-\frac {99}{8} d^3 x \left (c^2-d^2 x^2\right )^{7/2} \left (10 A d^3+20 B c d^2+6 c^3 D+13 c^2 C d\right )}{9 d^2}-\frac {10}{9} d^2 x^2 \left (c^2-d^2 x^2\right )^{7/2} \left (11 B d^2+15 c^2 D+22 c C d\right )}{10 d^2}-\frac {11}{10} d x^3 \left (c^2-d^2 x^2\right )^{7/2} (2 c D+C d)}{11 d^2}-\frac {1}{11} D x^4 \left (c^2-d^2 x^2\right )^{7/2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\frac {\frac {\int c d^5 \left (99 c \left (6 D c^3+13 C d c^2+20 B d^2 c+90 A d^3\right )+80 d \left (30 D c^3+44 C d c^2+121 B d^2 c+198 A d^3\right ) x\right ) \left (c^2-d^2 x^2\right )^{5/2}dx}{8 d^2}-\frac {99}{8} d^3 x \left (c^2-d^2 x^2\right )^{7/2} \left (10 A d^3+20 B c d^2+6 c^3 D+13 c^2 C d\right )}{9 d^2}-\frac {10}{9} d^2 x^2 \left (c^2-d^2 x^2\right )^{7/2} \left (11 B d^2+15 c^2 D+22 c C d\right )}{10 d^2}-\frac {11}{10} d x^3 \left (c^2-d^2 x^2\right )^{7/2} (2 c D+C d)}{11 d^2}-\frac {1}{11} D x^4 \left (c^2-d^2 x^2\right )^{7/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\frac {1}{8} c d^3 \int \left (99 c \left (6 D c^3+13 C d c^2+20 B d^2 c+90 A d^3\right )+80 d \left (30 D c^3+44 C d c^2+121 B d^2 c+198 A d^3\right ) x\right ) \left (c^2-d^2 x^2\right )^{5/2}dx-\frac {99}{8} d^3 x \left (c^2-d^2 x^2\right )^{7/2} \left (10 A d^3+20 B c d^2+6 c^3 D+13 c^2 C d\right )}{9 d^2}-\frac {10}{9} d^2 x^2 \left (c^2-d^2 x^2\right )^{7/2} \left (11 B d^2+15 c^2 D+22 c C d\right )}{10 d^2}-\frac {11}{10} d x^3 \left (c^2-d^2 x^2\right )^{7/2} (2 c D+C d)}{11 d^2}-\frac {1}{11} D x^4 \left (c^2-d^2 x^2\right )^{7/2}\)

\(\Big \downarrow \) 455

\(\displaystyle \frac {\frac {\frac {\frac {1}{8} c d^3 \left (99 c \left (90 A d^3+20 B c d^2+6 c^3 D+13 c^2 C d\right ) \int \left (c^2-d^2 x^2\right )^{5/2}dx-\frac {80 \left (c^2-d^2 x^2\right )^{7/2} \left (198 A d^3+121 B c d^2+30 c^3 D+44 c^2 C d\right )}{7 d}\right )-\frac {99}{8} d^3 x \left (c^2-d^2 x^2\right )^{7/2} \left (10 A d^3+20 B c d^2+6 c^3 D+13 c^2 C d\right )}{9 d^2}-\frac {10}{9} d^2 x^2 \left (c^2-d^2 x^2\right )^{7/2} \left (11 B d^2+15 c^2 D+22 c C d\right )}{10 d^2}-\frac {11}{10} d x^3 \left (c^2-d^2 x^2\right )^{7/2} (2 c D+C d)}{11 d^2}-\frac {1}{11} D x^4 \left (c^2-d^2 x^2\right )^{7/2}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\frac {\frac {\frac {1}{8} c d^3 \left (99 c \left (90 A d^3+20 B c d^2+6 c^3 D+13 c^2 C d\right ) \left (\frac {5}{6} c^2 \int \left (c^2-d^2 x^2\right )^{3/2}dx+\frac {1}{6} x \left (c^2-d^2 x^2\right )^{5/2}\right )-\frac {80 \left (c^2-d^2 x^2\right )^{7/2} \left (198 A d^3+121 B c d^2+30 c^3 D+44 c^2 C d\right )}{7 d}\right )-\frac {99}{8} d^3 x \left (c^2-d^2 x^2\right )^{7/2} \left (10 A d^3+20 B c d^2+6 c^3 D+13 c^2 C d\right )}{9 d^2}-\frac {10}{9} d^2 x^2 \left (c^2-d^2 x^2\right )^{7/2} \left (11 B d^2+15 c^2 D+22 c C d\right )}{10 d^2}-\frac {11}{10} d x^3 \left (c^2-d^2 x^2\right )^{7/2} (2 c D+C d)}{11 d^2}-\frac {1}{11} D x^4 \left (c^2-d^2 x^2\right )^{7/2}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\frac {\frac {\frac {1}{8} c d^3 \left (99 c \left (90 A d^3+20 B c d^2+6 c^3 D+13 c^2 C d\right ) \left (\frac {5}{6} c^2 \left (\frac {3}{4} c^2 \int \sqrt {c^2-d^2 x^2}dx+\frac {1}{4} x \left (c^2-d^2 x^2\right )^{3/2}\right )+\frac {1}{6} x \left (c^2-d^2 x^2\right )^{5/2}\right )-\frac {80 \left (c^2-d^2 x^2\right )^{7/2} \left (198 A d^3+121 B c d^2+30 c^3 D+44 c^2 C d\right )}{7 d}\right )-\frac {99}{8} d^3 x \left (c^2-d^2 x^2\right )^{7/2} \left (10 A d^3+20 B c d^2+6 c^3 D+13 c^2 C d\right )}{9 d^2}-\frac {10}{9} d^2 x^2 \left (c^2-d^2 x^2\right )^{7/2} \left (11 B d^2+15 c^2 D+22 c C d\right )}{10 d^2}-\frac {11}{10} d x^3 \left (c^2-d^2 x^2\right )^{7/2} (2 c D+C d)}{11 d^2}-\frac {1}{11} D x^4 \left (c^2-d^2 x^2\right )^{7/2}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\frac {\frac {\frac {1}{8} c d^3 \left (99 c \left (90 A d^3+20 B c d^2+6 c^3 D+13 c^2 C d\right ) \left (\frac {5}{6} c^2 \left (\frac {3}{4} c^2 \left (\frac {1}{2} c^2 \int \frac {1}{\sqrt {c^2-d^2 x^2}}dx+\frac {1}{2} x \sqrt {c^2-d^2 x^2}\right )+\frac {1}{4} x \left (c^2-d^2 x^2\right )^{3/2}\right )+\frac {1}{6} x \left (c^2-d^2 x^2\right )^{5/2}\right )-\frac {80 \left (c^2-d^2 x^2\right )^{7/2} \left (198 A d^3+121 B c d^2+30 c^3 D+44 c^2 C d\right )}{7 d}\right )-\frac {99}{8} d^3 x \left (c^2-d^2 x^2\right )^{7/2} \left (10 A d^3+20 B c d^2+6 c^3 D+13 c^2 C d\right )}{9 d^2}-\frac {10}{9} d^2 x^2 \left (c^2-d^2 x^2\right )^{7/2} \left (11 B d^2+15 c^2 D+22 c C d\right )}{10 d^2}-\frac {11}{10} d x^3 \left (c^2-d^2 x^2\right )^{7/2} (2 c D+C d)}{11 d^2}-\frac {1}{11} D x^4 \left (c^2-d^2 x^2\right )^{7/2}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\frac {\frac {\frac {1}{8} c d^3 \left (99 c \left (90 A d^3+20 B c d^2+6 c^3 D+13 c^2 C d\right ) \left (\frac {5}{6} c^2 \left (\frac {3}{4} c^2 \left (\frac {1}{2} c^2 \int \frac {1}{\frac {d^2 x^2}{c^2-d^2 x^2}+1}d\frac {x}{\sqrt {c^2-d^2 x^2}}+\frac {1}{2} x \sqrt {c^2-d^2 x^2}\right )+\frac {1}{4} x \left (c^2-d^2 x^2\right )^{3/2}\right )+\frac {1}{6} x \left (c^2-d^2 x^2\right )^{5/2}\right )-\frac {80 \left (c^2-d^2 x^2\right )^{7/2} \left (198 A d^3+121 B c d^2+30 c^3 D+44 c^2 C d\right )}{7 d}\right )-\frac {99}{8} d^3 x \left (c^2-d^2 x^2\right )^{7/2} \left (10 A d^3+20 B c d^2+6 c^3 D+13 c^2 C d\right )}{9 d^2}-\frac {10}{9} d^2 x^2 \left (c^2-d^2 x^2\right )^{7/2} \left (11 B d^2+15 c^2 D+22 c C d\right )}{10 d^2}-\frac {11}{10} d x^3 \left (c^2-d^2 x^2\right )^{7/2} (2 c D+C d)}{11 d^2}-\frac {1}{11} D x^4 \left (c^2-d^2 x^2\right )^{7/2}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {\frac {\frac {\frac {1}{8} c d^3 \left (99 c \left (\frac {5}{6} c^2 \left (\frac {3}{4} c^2 \left (\frac {c^2 \arctan \left (\frac {d x}{\sqrt {c^2-d^2 x^2}}\right )}{2 d}+\frac {1}{2} x \sqrt {c^2-d^2 x^2}\right )+\frac {1}{4} x \left (c^2-d^2 x^2\right )^{3/2}\right )+\frac {1}{6} x \left (c^2-d^2 x^2\right )^{5/2}\right ) \left (90 A d^3+20 B c d^2+6 c^3 D+13 c^2 C d\right )-\frac {80 \left (c^2-d^2 x^2\right )^{7/2} \left (198 A d^3+121 B c d^2+30 c^3 D+44 c^2 C d\right )}{7 d}\right )-\frac {99}{8} d^3 x \left (c^2-d^2 x^2\right )^{7/2} \left (10 A d^3+20 B c d^2+6 c^3 D+13 c^2 C d\right )}{9 d^2}-\frac {10}{9} d^2 x^2 \left (c^2-d^2 x^2\right )^{7/2} \left (11 B d^2+15 c^2 D+22 c C d\right )}{10 d^2}-\frac {11}{10} d x^3 \left (c^2-d^2 x^2\right )^{7/2} (2 c D+C d)}{11 d^2}-\frac {1}{11} D x^4 \left (c^2-d^2 x^2\right )^{7/2}\)

Input:

Int[(c + d*x)^2*(c^2 - d^2*x^2)^(5/2)*(A + B*x + C*x^2 + D*x^3),x]
 

Output:

-1/11*(D*x^4*(c^2 - d^2*x^2)^(7/2)) + ((-11*d*(C*d + 2*c*D)*x^3*(c^2 - d^2 
*x^2)^(7/2))/10 + ((-10*d^2*(22*c*C*d + 11*B*d^2 + 15*c^2*D)*x^2*(c^2 - d^ 
2*x^2)^(7/2))/9 + ((-99*d^3*(13*c^2*C*d + 20*B*c*d^2 + 10*A*d^3 + 6*c^3*D) 
*x*(c^2 - d^2*x^2)^(7/2))/8 + (c*d^3*((-80*(44*c^2*C*d + 121*B*c*d^2 + 198 
*A*d^3 + 30*c^3*D)*(c^2 - d^2*x^2)^(7/2))/(7*d) + 99*c*(13*c^2*C*d + 20*B* 
c*d^2 + 90*A*d^3 + 6*c^3*D)*((x*(c^2 - d^2*x^2)^(5/2))/6 + (5*c^2*((x*(c^2 
 - d^2*x^2)^(3/2))/4 + (3*c^2*((x*Sqrt[c^2 - d^2*x^2])/2 + (c^2*ArcTan[(d* 
x)/Sqrt[c^2 - d^2*x^2]])/(2*d)))/4))/6)))/8)/(9*d^2))/(10*d^2))/(11*d^2)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

rule 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 0]
 

rule 455

Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*(( 
a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] + Simp[c   Int[(a + b*x^2)^p, x], x] 
/; FreeQ[{a, b, c, d, p}, x] &&  !LeQ[p, -1]
 

rule 2346

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], 
e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x^2)^(p + 1)/(b*( 
q + 2*p + 1))), x] + Simp[1/(b*(q + 2*p + 1))   Int[(a + b*x^2)^p*ExpandToS 
um[b*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + 2*p + 1)*x^q, x], 
x], x]] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] &&  !LeQ[p, -1]
 

Maple [A] (verified)

Time = 0.52 (sec) , antiderivative size = 582, normalized size of antiderivative = 1.40

method	result	size
default	\(A \,c^{2} \left (\frac {x \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 c^{2} \left (\frac {x \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 c^{2} \left (\frac {x \sqrt {-d^{2} x^{2}+c^{2}}}{2}+\frac {c^{2} \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+c^{2}}}\right )}{2 \sqrt {d^{2}}}\right )}{4}\right )}{6}\right )-\frac {c \left (2 A d +B c \right ) \left (-d^{2} x^{2}+c^{2}\right )^{\frac {7}{2}}}{7 d^{2}}+d \left (C d +2 D c \right ) \left (-\frac {x^{3} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {7}{2}}}{10 d^{2}}+\frac {3 c^{2} \left (-\frac {x \left (-d^{2} x^{2}+c^{2}\right )^{\frac {7}{2}}}{8 d^{2}}+\frac {c^{2} \left (\frac {x \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 c^{2} \left (\frac {x \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 c^{2} \left (\frac {x \sqrt {-d^{2} x^{2}+c^{2}}}{2}+\frac {c^{2} \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+c^{2}}}\right )}{2 \sqrt {d^{2}}}\right )}{4}\right )}{6}\right )}{8 d^{2}}\right )}{10 d^{2}}\right )+\left (A \,d^{2}+2 B c d +C \,c^{2}\right ) \left (-\frac {x \left (-d^{2} x^{2}+c^{2}\right )^{\frac {7}{2}}}{8 d^{2}}+\frac {c^{2} \left (\frac {x \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}{6}+\frac {5 c^{2} \left (\frac {x \left (-d^{2} x^{2}+c^{2}\right )^{\frac {3}{2}}}{4}+\frac {3 c^{2} \left (\frac {x \sqrt {-d^{2} x^{2}+c^{2}}}{2}+\frac {c^{2} \arctan \left (\frac {\sqrt {d^{2}}\, x}{\sqrt {-d^{2} x^{2}+c^{2}}}\right )}{2 \sqrt {d^{2}}}\right )}{4}\right )}{6}\right )}{8 d^{2}}\right )+\left (B \,d^{2}+2 C c d +D c^{2}\right ) \left (-\frac {x^{2} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {7}{2}}}{9 d^{2}}-\frac {2 c^{2} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {7}{2}}}{63 d^{4}}\right )+D d^{2} \left (-\frac {x^{4} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {7}{2}}}{11 d^{2}}+\frac {4 c^{2} \left (-\frac {x^{2} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {7}{2}}}{9 d^{2}}-\frac {2 c^{2} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {7}{2}}}{63 d^{4}}\right )}{11 d^{2}}\right )\)	\(582\)

Input:

int((d*x+c)^2*(-d^2*x^2+c^2)^(5/2)*(D*x^3+C*x^2+B*x+A),x,method=_RETURNVER 
BOSE)
 

Output:

A*c^2*(1/6*x*(-d^2*x^2+c^2)^(5/2)+5/6*c^2*(1/4*x*(-d^2*x^2+c^2)^(3/2)+3/4* 
c^2*(1/2*x*(-d^2*x^2+c^2)^(1/2)+1/2*c^2/(d^2)^(1/2)*arctan((d^2)^(1/2)*x/( 
-d^2*x^2+c^2)^(1/2)))))-1/7*c*(2*A*d+B*c)/d^2*(-d^2*x^2+c^2)^(7/2)+d*(C*d+ 
2*D*c)*(-1/10*x^3*(-d^2*x^2+c^2)^(7/2)/d^2+3/10*c^2/d^2*(-1/8*x*(-d^2*x^2+ 
c^2)^(7/2)/d^2+1/8*c^2/d^2*(1/6*x*(-d^2*x^2+c^2)^(5/2)+5/6*c^2*(1/4*x*(-d^ 
2*x^2+c^2)^(3/2)+3/4*c^2*(1/2*x*(-d^2*x^2+c^2)^(1/2)+1/2*c^2/(d^2)^(1/2)*a 
rctan((d^2)^(1/2)*x/(-d^2*x^2+c^2)^(1/2)))))))+(A*d^2+2*B*c*d+C*c^2)*(-1/8 
*x*(-d^2*x^2+c^2)^(7/2)/d^2+1/8*c^2/d^2*(1/6*x*(-d^2*x^2+c^2)^(5/2)+5/6*c^ 
2*(1/4*x*(-d^2*x^2+c^2)^(3/2)+3/4*c^2*(1/2*x*(-d^2*x^2+c^2)^(1/2)+1/2*c^2/ 
(d^2)^(1/2)*arctan((d^2)^(1/2)*x/(-d^2*x^2+c^2)^(1/2))))))+(B*d^2+2*C*c*d+ 
D*c^2)*(-1/9*x^2*(-d^2*x^2+c^2)^(7/2)/d^2-2/63*c^2*(-d^2*x^2+c^2)^(7/2)/d^ 
4)+D*d^2*(-1/11*x^4*(-d^2*x^2+c^2)^(7/2)/d^2+4/11*c^2/d^2*(-1/9*x^2*(-d^2* 
x^2+c^2)^(7/2)/d^2-2/63*c^2*(-d^2*x^2+c^2)^(7/2)/d^4))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 453, normalized size of antiderivative = 1.09 \[ \int (c+d x)^2 \left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right ) \, dx=-\frac {6930 \, {\left (6 \, D c^{11} + 13 \, C c^{10} d + 20 \, B c^{9} d^{2} + 90 \, A c^{8} d^{3}\right )} \arctan \left (-\frac {c - \sqrt {-d^{2} x^{2} + c^{2}}}{d x}\right ) - {\left (80640 \, D d^{10} x^{10} - 38400 \, D c^{10} - 56320 \, C c^{9} d - 154880 \, B c^{8} d^{2} - 253440 \, A c^{7} d^{3} + 88704 \, {\left (2 \, D c d^{9} + C d^{10}\right )} x^{9} - 8960 \, {\left (12 \, D c^{2} d^{8} - 22 \, C c d^{9} - 11 \, B d^{10}\right )} x^{8} - 11088 \, {\left (42 \, D c^{3} d^{7} + 11 \, C c^{2} d^{8} - 20 \, B c d^{9} - 10 \, A d^{10}\right )} x^{7} - 2560 \, {\left (48 \, D c^{4} d^{6} + 209 \, C c^{3} d^{7} + 55 \, B c^{2} d^{8} - 99 \, A c d^{9}\right )} x^{6} + 1848 \, {\left (186 \, D c^{5} d^{5} - 77 \, C c^{4} d^{6} - 340 \, B c^{3} d^{7} - 90 \, A c^{2} d^{8}\right )} x^{5} + 7680 \, {\left (27 \, D c^{6} d^{4} + 55 \, C c^{5} d^{5} - 22 \, B c^{4} d^{6} - 99 \, A c^{3} d^{7}\right )} x^{4} - 2310 \, {\left (6 \, D c^{7} d^{3} - 115 \, C c^{6} d^{4} - 236 \, B c^{5} d^{5} + 90 \, A c^{4} d^{6}\right )} x^{3} - 1280 \, {\left (15 \, D c^{8} d^{2} + 22 \, C c^{7} d^{3} - 286 \, B c^{6} d^{4} - 594 \, A c^{5} d^{5}\right )} x^{2} - 3465 \, {\left (6 \, D c^{9} d + 13 \, C c^{8} d^{2} + 20 \, B c^{7} d^{3} - 166 \, A c^{6} d^{4}\right )} x\right )} \sqrt {-d^{2} x^{2} + c^{2}}}{887040 \, d^{4}} \] Input:

integrate((d*x+c)^2*(-d^2*x^2+c^2)^(5/2)*(D*x^3+C*x^2+B*x+A),x, algorithm= 
"fricas")
 

Output:

-1/887040*(6930*(6*D*c^11 + 13*C*c^10*d + 20*B*c^9*d^2 + 90*A*c^8*d^3)*arc 
tan(-(c - sqrt(-d^2*x^2 + c^2))/(d*x)) - (80640*D*d^10*x^10 - 38400*D*c^10 
 - 56320*C*c^9*d - 154880*B*c^8*d^2 - 253440*A*c^7*d^3 + 88704*(2*D*c*d^9 
+ C*d^10)*x^9 - 8960*(12*D*c^2*d^8 - 22*C*c*d^9 - 11*B*d^10)*x^8 - 11088*( 
42*D*c^3*d^7 + 11*C*c^2*d^8 - 20*B*c*d^9 - 10*A*d^10)*x^7 - 2560*(48*D*c^4 
*d^6 + 209*C*c^3*d^7 + 55*B*c^2*d^8 - 99*A*c*d^9)*x^6 + 1848*(186*D*c^5*d^ 
5 - 77*C*c^4*d^6 - 340*B*c^3*d^7 - 90*A*c^2*d^8)*x^5 + 7680*(27*D*c^6*d^4 
+ 55*C*c^5*d^5 - 22*B*c^4*d^6 - 99*A*c^3*d^7)*x^4 - 2310*(6*D*c^7*d^3 - 11 
5*C*c^6*d^4 - 236*B*c^5*d^5 + 90*A*c^4*d^6)*x^3 - 1280*(15*D*c^8*d^2 + 22* 
C*c^7*d^3 - 286*B*c^6*d^4 - 594*A*c^5*d^5)*x^2 - 3465*(6*D*c^9*d + 13*C*c^ 
8*d^2 + 20*B*c^7*d^3 - 166*A*c^6*d^4)*x)*sqrt(-d^2*x^2 + c^2))/d^4
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1457 vs. \(2 (406) = 812\).

Time = 0.90 (sec) , antiderivative size = 1457, normalized size of antiderivative = 3.50 \[ \int (c+d x)^2 \left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)**2*(-d**2*x**2+c**2)**(5/2)*(D*x**3+C*x**2+B*x+A),x)
 

Output:

Piecewise((sqrt(c**2 - d**2*x**2)*(D*d**6*x**10/11 - x**9*(-C*d**8 - 2*D*c 
*d**7)/(10*d**2) - x**8*(-B*d**8 - 2*C*c*d**7 + 12*D*c**2*d**6/11)/(9*d**2 
) - x**7*(-A*d**8 - 2*B*c*d**7 + 2*C*c**2*d**6 + 6*D*c**3*d**5 + 9*c**2*(- 
C*d**8 - 2*D*c*d**7)/(10*d**2))/(8*d**2) - x**6*(-2*A*c*d**7 + 2*B*c**2*d* 
*6 + 6*C*c**3*d**5 + 8*c**2*(-B*d**8 - 2*C*c*d**7 + 12*D*c**2*d**6/11)/(9* 
d**2))/(7*d**2) - x**5*(2*A*c**2*d**6 + 6*B*c**3*d**5 - 6*D*c**5*d**3 + 7* 
c**2*(-A*d**8 - 2*B*c*d**7 + 2*C*c**2*d**6 + 6*D*c**3*d**5 + 9*c**2*(-C*d* 
*8 - 2*D*c*d**7)/(10*d**2))/(8*d**2))/(6*d**2) - x**4*(6*A*c**3*d**5 - 6*C 
*c**5*d**3 - 2*D*c**6*d**2 + 6*c**2*(-2*A*c*d**7 + 2*B*c**2*d**6 + 6*C*c** 
3*d**5 + 8*c**2*(-B*d**8 - 2*C*c*d**7 + 12*D*c**2*d**6/11)/(9*d**2))/(7*d* 
*2))/(5*d**2) - x**3*(-6*B*c**5*d**3 - 2*C*c**6*d**2 + 2*D*c**7*d + 5*c**2 
*(2*A*c**2*d**6 + 6*B*c**3*d**5 - 6*D*c**5*d**3 + 7*c**2*(-A*d**8 - 2*B*c* 
d**7 + 2*C*c**2*d**6 + 6*D*c**3*d**5 + 9*c**2*(-C*d**8 - 2*D*c*d**7)/(10*d 
**2))/(8*d**2))/(6*d**2))/(4*d**2) - x**2*(-6*A*c**5*d**3 - 2*B*c**6*d**2 
+ 2*C*c**7*d + D*c**8 + 4*c**2*(6*A*c**3*d**5 - 6*C*c**5*d**3 - 2*D*c**6*d 
**2 + 6*c**2*(-2*A*c*d**7 + 2*B*c**2*d**6 + 6*C*c**3*d**5 + 8*c**2*(-B*d** 
8 - 2*C*c*d**7 + 12*D*c**2*d**6/11)/(9*d**2))/(7*d**2))/(5*d**2))/(3*d**2) 
 - x*(-2*A*c**6*d**2 + 2*B*c**7*d + C*c**8 + 3*c**2*(-6*B*c**5*d**3 - 2*C* 
c**6*d**2 + 2*D*c**7*d + 5*c**2*(2*A*c**2*d**6 + 6*B*c**3*d**5 - 6*D*c**5* 
d**3 + 7*c**2*(-A*d**8 - 2*B*c*d**7 + 2*C*c**2*d**6 + 6*D*c**3*d**5 + 9...
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 648, normalized size of antiderivative = 1.56 \[ \int (c+d x)^2 \left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right ) \, dx =\text {Too large to display} \] Input:

integrate((d*x+c)^2*(-d^2*x^2+c^2)^(5/2)*(D*x^3+C*x^2+B*x+A),x, algorithm= 
"maxima")
 

Output:

5/16*A*c^8*arcsin(d*x/c)/d + 5/16*sqrt(-d^2*x^2 + c^2)*A*c^6*x - 1/11*(-d^ 
2*x^2 + c^2)^(7/2)*D*x^4 + 5/24*(-d^2*x^2 + c^2)^(3/2)*A*c^4*x + 3/256*(2* 
D*c*d + C*d^2)*c^10*arcsin(d*x/c)/d^5 + 5/128*(C*c^2 + 2*B*c*d + A*d^2)*c^ 
8*arcsin(d*x/c)/d^3 + 1/6*(-d^2*x^2 + c^2)^(5/2)*A*c^2*x + 3/256*sqrt(-d^2 
*x^2 + c^2)*(2*D*c*d + C*d^2)*c^8*x/d^4 + 5/128*sqrt(-d^2*x^2 + c^2)*(C*c^ 
2 + 2*B*c*d + A*d^2)*c^6*x/d^2 - 4/99*(-d^2*x^2 + c^2)^(7/2)*D*c^2*x^2/d^2 
 + 1/128*(-d^2*x^2 + c^2)^(3/2)*(2*D*c*d + C*d^2)*c^6*x/d^4 + 5/192*(-d^2* 
x^2 + c^2)^(3/2)*(C*c^2 + 2*B*c*d + A*d^2)*c^4*x/d^2 - 1/10*(-d^2*x^2 + c^ 
2)^(7/2)*(2*D*c*d + C*d^2)*x^3/d^2 - 8/693*(-d^2*x^2 + c^2)^(7/2)*D*c^4/d^ 
4 - 1/7*(-d^2*x^2 + c^2)^(7/2)*B*c^2/d^2 - 2/7*(-d^2*x^2 + c^2)^(7/2)*A*c/ 
d + 1/160*(-d^2*x^2 + c^2)^(5/2)*(2*D*c*d + C*d^2)*c^4*x/d^4 + 1/48*(-d^2* 
x^2 + c^2)^(5/2)*(C*c^2 + 2*B*c*d + A*d^2)*c^2*x/d^2 - 1/9*(-d^2*x^2 + c^2 
)^(7/2)*(D*c^2 + 2*C*c*d + B*d^2)*x^2/d^2 - 3/80*(-d^2*x^2 + c^2)^(7/2)*(2 
*D*c*d + C*d^2)*c^2*x/d^4 - 1/8*(-d^2*x^2 + c^2)^(7/2)*(C*c^2 + 2*B*c*d + 
A*d^2)*x/d^2 - 2/63*(-d^2*x^2 + c^2)^(7/2)*(D*c^2 + 2*C*c*d + B*d^2)*c^2/d 
^4
 

Giac [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 485, normalized size of antiderivative = 1.17 \[ \int (c+d x)^2 \left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {1}{887040} \, \sqrt {-d^{2} x^{2} + c^{2}} {\left ({\left (2 \, {\left ({\left (4 \, {\left ({\left (2 \, {\left (7 \, {\left (8 \, {\left (9 \, {\left (10 \, D d^{6} x + \frac {11 \, {\left (2 \, D c d^{23} + C d^{24}\right )}}{d^{18}}\right )} x - \frac {10 \, {\left (12 \, D c^{2} d^{22} - 22 \, C c d^{23} - 11 \, B d^{24}\right )}}{d^{18}}\right )} x - \frac {99 \, {\left (42 \, D c^{3} d^{21} + 11 \, C c^{2} d^{22} - 20 \, B c d^{23} - 10 \, A d^{24}\right )}}{d^{18}}\right )} x - \frac {160 \, {\left (48 \, D c^{4} d^{20} + 209 \, C c^{3} d^{21} + 55 \, B c^{2} d^{22} - 99 \, A c d^{23}\right )}}{d^{18}}\right )} x + \frac {231 \, {\left (186 \, D c^{5} d^{19} - 77 \, C c^{4} d^{20} - 340 \, B c^{3} d^{21} - 90 \, A c^{2} d^{22}\right )}}{d^{18}}\right )} x + \frac {960 \, {\left (27 \, D c^{6} d^{18} + 55 \, C c^{5} d^{19} - 22 \, B c^{4} d^{20} - 99 \, A c^{3} d^{21}\right )}}{d^{18}}\right )} x - \frac {1155 \, {\left (6 \, D c^{7} d^{17} - 115 \, C c^{6} d^{18} - 236 \, B c^{5} d^{19} + 90 \, A c^{4} d^{20}\right )}}{d^{18}}\right )} x - \frac {640 \, {\left (15 \, D c^{8} d^{16} + 22 \, C c^{7} d^{17} - 286 \, B c^{6} d^{18} - 594 \, A c^{5} d^{19}\right )}}{d^{18}}\right )} x - \frac {3465 \, {\left (6 \, D c^{9} d^{15} + 13 \, C c^{8} d^{16} + 20 \, B c^{7} d^{17} - 166 \, A c^{6} d^{18}\right )}}{d^{18}}\right )} x - \frac {1280 \, {\left (30 \, D c^{10} d^{14} + 44 \, C c^{9} d^{15} + 121 \, B c^{8} d^{16} + 198 \, A c^{7} d^{17}\right )}}{d^{18}}\right )} + \frac {{\left (6 \, D c^{11} + 13 \, C c^{10} d + 20 \, B c^{9} d^{2} + 90 \, A c^{8} d^{3}\right )} \arcsin \left (\frac {d x}{c}\right ) \mathrm {sgn}\left (c\right ) \mathrm {sgn}\left (d\right )}{256 \, d^{3} {\left | d \right |}} \] Input:

integrate((d*x+c)^2*(-d^2*x^2+c^2)^(5/2)*(D*x^3+C*x^2+B*x+A),x, algorithm= 
"giac")
 

Output:

1/887040*sqrt(-d^2*x^2 + c^2)*((2*((4*((2*(7*(8*(9*(10*D*d^6*x + 11*(2*D*c 
*d^23 + C*d^24)/d^18)*x - 10*(12*D*c^2*d^22 - 22*C*c*d^23 - 11*B*d^24)/d^1 
8)*x - 99*(42*D*c^3*d^21 + 11*C*c^2*d^22 - 20*B*c*d^23 - 10*A*d^24)/d^18)* 
x - 160*(48*D*c^4*d^20 + 209*C*c^3*d^21 + 55*B*c^2*d^22 - 99*A*c*d^23)/d^1 
8)*x + 231*(186*D*c^5*d^19 - 77*C*c^4*d^20 - 340*B*c^3*d^21 - 90*A*c^2*d^2 
2)/d^18)*x + 960*(27*D*c^6*d^18 + 55*C*c^5*d^19 - 22*B*c^4*d^20 - 99*A*c^3 
*d^21)/d^18)*x - 1155*(6*D*c^7*d^17 - 115*C*c^6*d^18 - 236*B*c^5*d^19 + 90 
*A*c^4*d^20)/d^18)*x - 640*(15*D*c^8*d^16 + 22*C*c^7*d^17 - 286*B*c^6*d^18 
 - 594*A*c^5*d^19)/d^18)*x - 3465*(6*D*c^9*d^15 + 13*C*c^8*d^16 + 20*B*c^7 
*d^17 - 166*A*c^6*d^18)/d^18)*x - 1280*(30*D*c^10*d^14 + 44*C*c^9*d^15 + 1 
21*B*c^8*d^16 + 198*A*c^7*d^17)/d^18) + 1/256*(6*D*c^11 + 13*C*c^10*d + 20 
*B*c^9*d^2 + 90*A*c^8*d^3)*arcsin(d*x/c)*sgn(c)*sgn(d)/(d^3*abs(d))
 

Mupad [F(-1)]

Timed out. \[ \int (c+d x)^2 \left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\int {\left (c^2-d^2\,x^2\right )}^{5/2}\,{\left (c+d\,x\right )}^2\,\left (A+B\,x+C\,x^2+x^3\,D\right ) \,d x \] Input:

int((c^2 - d^2*x^2)^(5/2)*(c + d*x)^2*(A + B*x + C*x^2 + x^3*D),x)
 

Output:

int((c^2 - d^2*x^2)^(5/2)*(c + d*x)^2*(A + B*x + C*x^2 + x^3*D), x)
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 721, normalized size of antiderivative = 1.73 \[ \int (c+d x)^2 \left (c^2-d^2 x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {311850 \mathit {asin} \left (\frac {d x}{c}\right ) a \,c^{8} d^{2}+69300 \mathit {asin} \left (\frac {d x}{c}\right ) b \,c^{9} d -253440 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{7} d^{2}+110880 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,d^{9} x^{7}-154880 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{8} d +98560 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,d^{9} x^{8}-65835 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{9} d x -47360 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{8} d^{2} x^{2}+251790 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{7} d^{3} x^{3}+629760 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{6} d^{4} x^{4}+201432 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{5} d^{5} x^{5}-657920 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{4} d^{6} x^{6}-587664 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{3} d^{7} x^{7}+89600 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{2} d^{8} x^{8}+266112 \sqrt {-d^{2} x^{2}+c^{2}}\, c \,d^{9} x^{9}-628320 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{3} d^{6} x^{5}-140800 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{2} d^{7} x^{6}+221760 \sqrt {-d^{2} x^{2}+c^{2}}\, b c \,d^{8} x^{7}+80640 \sqrt {-d^{2} x^{2}+c^{2}}\, d^{10} x^{10}+253440 a \,c^{8} d^{2}+154880 b \,c^{9} d +65835 \mathit {asin} \left (\frac {d x}{c}\right ) c^{11}-94720 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{10}+575190 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{6} d^{3} x +760320 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{5} d^{4} x^{2}-207900 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{4} d^{5} x^{3}-760320 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{3} d^{6} x^{4}-166320 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{2} d^{7} x^{5}+253440 \sqrt {-d^{2} x^{2}+c^{2}}\, a c \,d^{8} x^{6}-69300 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{7} d^{2} x +366080 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{6} d^{3} x^{2}+545160 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{5} d^{4} x^{3}-168960 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{4} d^{5} x^{4}+94720 c^{11}}{887040 d^{3}} \] Input:

int((d*x+c)^2*(-d^2*x^2+c^2)^(5/2)*(D*x^3+C*x^2+B*x+A),x)
 

Output:

(311850*asin((d*x)/c)*a*c**8*d**2 + 69300*asin((d*x)/c)*b*c**9*d + 65835*a 
sin((d*x)/c)*c**11 - 253440*sqrt(c**2 - d**2*x**2)*a*c**7*d**2 + 575190*sq 
rt(c**2 - d**2*x**2)*a*c**6*d**3*x + 760320*sqrt(c**2 - d**2*x**2)*a*c**5* 
d**4*x**2 - 207900*sqrt(c**2 - d**2*x**2)*a*c**4*d**5*x**3 - 760320*sqrt(c 
**2 - d**2*x**2)*a*c**3*d**6*x**4 - 166320*sqrt(c**2 - d**2*x**2)*a*c**2*d 
**7*x**5 + 253440*sqrt(c**2 - d**2*x**2)*a*c*d**8*x**6 + 110880*sqrt(c**2 
- d**2*x**2)*a*d**9*x**7 - 154880*sqrt(c**2 - d**2*x**2)*b*c**8*d - 69300* 
sqrt(c**2 - d**2*x**2)*b*c**7*d**2*x + 366080*sqrt(c**2 - d**2*x**2)*b*c** 
6*d**3*x**2 + 545160*sqrt(c**2 - d**2*x**2)*b*c**5*d**4*x**3 - 168960*sqrt 
(c**2 - d**2*x**2)*b*c**4*d**5*x**4 - 628320*sqrt(c**2 - d**2*x**2)*b*c**3 
*d**6*x**5 - 140800*sqrt(c**2 - d**2*x**2)*b*c**2*d**7*x**6 + 221760*sqrt( 
c**2 - d**2*x**2)*b*c*d**8*x**7 + 98560*sqrt(c**2 - d**2*x**2)*b*d**9*x**8 
 - 94720*sqrt(c**2 - d**2*x**2)*c**10 - 65835*sqrt(c**2 - d**2*x**2)*c**9* 
d*x - 47360*sqrt(c**2 - d**2*x**2)*c**8*d**2*x**2 + 251790*sqrt(c**2 - d** 
2*x**2)*c**7*d**3*x**3 + 629760*sqrt(c**2 - d**2*x**2)*c**6*d**4*x**4 + 20 
1432*sqrt(c**2 - d**2*x**2)*c**5*d**5*x**5 - 657920*sqrt(c**2 - d**2*x**2) 
*c**4*d**6*x**6 - 587664*sqrt(c**2 - d**2*x**2)*c**3*d**7*x**7 + 89600*sqr 
t(c**2 - d**2*x**2)*c**2*d**8*x**8 + 266112*sqrt(c**2 - d**2*x**2)*c*d**9* 
x**9 + 80640*sqrt(c**2 - d**2*x**2)*d**10*x**10 + 253440*a*c**8*d**2 + 154 
880*b*c**9*d + 94720*c**11)/(887040*d**3)