\(\int (c+d x)^3 (c^2-d^2 x^2)^{3/2} (A+B x+C x^2+D x^3) \, dx\) [140]

Optimal result

Integrand size = 39, antiderivative size = 416 \[ \int (c+d x)^3 \left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {c^5 \left (34 c^2 C d+54 B c d^2+144 A d^3+21 c^3 D\right ) x \sqrt {c^2-d^2 x^2}}{256 d^3}+\frac {c^3 \left (34 c^2 C d+54 B c d^2+144 A d^3+21 c^3 D\right ) x \left (c^2-d^2 x^2\right )^{3/2}}{384 d^3}-\frac {4 c^2 \left (c^2 C d+B c d^2+A d^3+c^3 D\right ) \left (c^2-d^2 x^2\right )^{5/2}}{5 d^4}-\frac {c \left (34 c^2 C d+54 B c d^2+48 A d^3+21 c^3 D\right ) x \left (c^2-d^2 x^2\right )^{5/2}}{96 d^3}-\frac {\left (6 c C d+2 B d^2+7 c^2 D\right ) x^3 \left (c^2-d^2 x^2\right )^{5/2}}{16 d}-\frac {1}{10} d D x^5 \left (c^2-d^2 x^2\right )^{5/2}+\frac {\left (5 c^2 C d+3 B c d^2+A d^3+7 c^3 D\right ) \left (c^2-d^2 x^2\right )^{7/2}}{7 d^4}-\frac {(C d+3 c D) \left (c^2-d^2 x^2\right )^{9/2}}{9 d^4}+\frac {c^7 \left (34 c^2 C d+54 B c d^2+144 A d^3+21 c^3 D\right ) \arctan \left (\frac {d x}{\sqrt {c^2-d^2 x^2}}\right )}{256 d^4} \] Output:

1/256*c^5*(144*A*d^3+54*B*c*d^2+34*C*c^2*d+21*D*c^3)*x*(-d^2*x^2+c^2)^(1/2 
)/d^3+1/384*c^3*(144*A*d^3+54*B*c*d^2+34*C*c^2*d+21*D*c^3)*x*(-d^2*x^2+c^2 
)^(3/2)/d^3-4/5*c^2*(A*d^3+B*c*d^2+C*c^2*d+D*c^3)*(-d^2*x^2+c^2)^(5/2)/d^4 
-1/96*c*(48*A*d^3+54*B*c*d^2+34*C*c^2*d+21*D*c^3)*x*(-d^2*x^2+c^2)^(5/2)/d 
^3-1/16*(2*B*d^2+6*C*c*d+7*D*c^2)*x^3*(-d^2*x^2+c^2)^(5/2)/d-1/10*d*D*x^5* 
(-d^2*x^2+c^2)^(5/2)+1/7*(A*d^3+3*B*c*d^2+5*C*c^2*d+7*D*c^3)*(-d^2*x^2+c^2 
)^(7/2)/d^4-1/9*(C*d+3*D*c)*(-d^2*x^2+c^2)^(9/2)/d^4+1/256*c^7*(144*A*d^3+ 
54*B*c*d^2+34*C*c^2*d+21*D*c^3)*arctan(d*x/(-d^2*x^2+c^2)^(1/2))/d^4
 

Mathematica [A] (verified)

Time = 3.16 (sec) , antiderivative size = 344, normalized size of antiderivative = 0.83 \[ \int (c+d x)^3 \left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=-\frac {\sqrt {c^2-d^2 x^2} \left (10752 c^9 D+c^8 d (15872 C+6615 D x)+6 c^7 d^2 (4992 B+7 x (255 C+128 D x))-12 c^4 d^5 x^2 \left (7872 A+x \left (5775 B+4544 C x+3738 D x^2\right )\right )+480 c d^8 x^5 (84 A+x (72 B+7 x (9 C+8 D x)))+32 d^9 x^6 (360 A+7 x (45 B+4 x (10 C+9 D x)))-48 c^3 d^6 x^3 (1050 A+x (816 B+35 x (19 C+16 D x)))-12 c^5 d^4 x (2940 A+x (2112 B+7 x (235 C+192 D x)))+16 c^2 d^7 x^4 (1872 A+x (1575 B+x (1360 C+1197 D x)))+2 c^6 d^3 (26496 A+x (8505 B+x (3968 C+2205 D x)))\right )+630 c^7 \left (34 c^2 C d+54 B c d^2+144 A d^3+21 c^3 D\right ) \arctan \left (\frac {d x}{\sqrt {c^2}-\sqrt {c^2-d^2 x^2}}\right )}{80640 d^4} \] Input:

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

Output:

-1/80640*(Sqrt[c^2 - d^2*x^2]*(10752*c^9*D + c^8*d*(15872*C + 6615*D*x) + 
6*c^7*d^2*(4992*B + 7*x*(255*C + 128*D*x)) - 12*c^4*d^5*x^2*(7872*A + x*(5 
775*B + 4544*C*x + 3738*D*x^2)) + 480*c*d^8*x^5*(84*A + x*(72*B + 7*x*(9*C 
 + 8*D*x))) + 32*d^9*x^6*(360*A + 7*x*(45*B + 4*x*(10*C + 9*D*x))) - 48*c^ 
3*d^6*x^3*(1050*A + x*(816*B + 35*x*(19*C + 16*D*x))) - 12*c^5*d^4*x*(2940 
*A + x*(2112*B + 7*x*(235*C + 192*D*x))) + 16*c^2*d^7*x^4*(1872*A + x*(157 
5*B + x*(1360*C + 1197*D*x))) + 2*c^6*d^3*(26496*A + x*(8505*B + x*(3968*C 
 + 2205*D*x)))) + 630*c^7*(34*c^2*C*d + 54*B*c*d^2 + 144*A*d^3 + 21*c^3*D) 
*ArcTan[(d*x)/(Sqrt[c^2] - Sqrt[c^2 - d^2*x^2])])/d^4
 

Rubi [A] (verified)

Time = 2.49 (sec) , antiderivative size = 413, normalized size of antiderivative = 0.99, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2346, 27, 2346, 25, 2346, 25, 2346, 25, 2346, 27, 455, 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.

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

\(\Big \downarrow \) 2346

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2346

\(\displaystyle \frac {-\frac {\int -\left (c^2-d^2 x^2\right )^{3/2} \left (9 \left (7 D c^2+6 C d c+2 B d^2\right ) x^4 d^5+18 A c^3 d^4+2 \left (21 D c^3+31 C d c^2+27 B d^2 c+9 A d^3\right ) x^3 d^4+18 c \left (C c^2+3 B d c+3 A d^2\right ) x^2 d^4+18 c^2 (B c+3 A d) x d^4\right )dx}{9 d^2}-\frac {2}{9} d^2 x^4 \left (c^2-d^2 x^2\right )^{5/2} (3 c D+C d)}{2 d^2}-\frac {1}{10} d D x^5 \left (c^2-d^2 x^2\right )^{5/2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \left (c^2-d^2 x^2\right )^{3/2} \left (9 \left (7 D c^2+6 C d c+2 B d^2\right ) x^4 d^5+18 A c^3 d^4+2 \left (21 D c^3+31 C d c^2+27 B d^2 c+9 A d^3\right ) x^3 d^4+18 c \left (C c^2+3 B d c+3 A d^2\right ) x^2 d^4+18 c^2 (B c+3 A d) x d^4\right )dx}{9 d^2}-\frac {2}{9} d^2 x^4 \left (c^2-d^2 x^2\right )^{5/2} (3 c D+C d)}{2 d^2}-\frac {1}{10} d D x^5 \left (c^2-d^2 x^2\right )^{5/2}\)

\(\Big \downarrow \) 2346

\(\displaystyle \frac {\frac {-\frac {\int -\left (c^2-d^2 x^2\right )^{3/2} \left (144 A c^3 d^6+16 \left (21 D c^3+31 C d c^2+27 B d^2 c+9 A d^3\right ) x^3 d^6+144 c^2 (B c+3 A d) x d^6+9 c \left (21 D c^3+34 C d c^2+54 B d^2 c+48 A d^3\right ) x^2 d^5\right )dx}{8 d^2}-\frac {9}{8} d^3 x^3 \left (c^2-d^2 x^2\right )^{5/2} \left (2 B d^2+7 c^2 D+6 c C d\right )}{9 d^2}-\frac {2}{9} d^2 x^4 \left (c^2-d^2 x^2\right )^{5/2} (3 c D+C d)}{2 d^2}-\frac {1}{10} d D x^5 \left (c^2-d^2 x^2\right )^{5/2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\frac {\int \left (c^2-d^2 x^2\right )^{3/2} \left (144 A c^3 d^6+16 \left (21 D c^3+31 C d c^2+27 B d^2 c+9 A d^3\right ) x^3 d^6+144 c^2 (B c+3 A d) x d^6+9 c \left (21 D c^3+34 C d c^2+54 B d^2 c+48 A d^3\right ) x^2 d^5\right )dx}{8 d^2}-\frac {9}{8} d^3 x^3 \left (c^2-d^2 x^2\right )^{5/2} \left (2 B d^2+7 c^2 D+6 c C d\right )}{9 d^2}-\frac {2}{9} d^2 x^4 \left (c^2-d^2 x^2\right )^{5/2} (3 c D+C d)}{2 d^2}-\frac {1}{10} d D x^5 \left (c^2-d^2 x^2\right )^{5/2}\)

\(\Big \downarrow \) 2346

\(\displaystyle \frac {\frac {\frac {-\frac {\int -\left (c^2-d^2 x^2\right )^{3/2} \left (1008 A c^3 d^8+63 c \left (21 D c^3+34 C d c^2+54 B d^2 c+48 A d^3\right ) x^2 d^7+16 c^2 \left (42 D c^3+62 C d c^2+117 B d^2 c+207 A d^3\right ) x d^6\right )dx}{7 d^2}-\frac {16}{7} d^4 x^2 \left (c^2-d^2 x^2\right )^{5/2} \left (9 A d^3+27 B c d^2+21 c^3 D+31 c^2 C d\right )}{8 d^2}-\frac {9}{8} d^3 x^3 \left (c^2-d^2 x^2\right )^{5/2} \left (2 B d^2+7 c^2 D+6 c C d\right )}{9 d^2}-\frac {2}{9} d^2 x^4 \left (c^2-d^2 x^2\right )^{5/2} (3 c D+C d)}{2 d^2}-\frac {1}{10} d D x^5 \left (c^2-d^2 x^2\right )^{5/2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\frac {\frac {\int \left (c^2-d^2 x^2\right )^{3/2} \left (1008 A c^3 d^8+63 c \left (21 D c^3+34 C d c^2+54 B d^2 c+48 A d^3\right ) x^2 d^7+16 c^2 \left (42 D c^3+62 C d c^2+117 B d^2 c+207 A d^3\right ) x d^6\right )dx}{7 d^2}-\frac {16}{7} d^4 x^2 \left (c^2-d^2 x^2\right )^{5/2} \left (9 A d^3+27 B c d^2+21 c^3 D+31 c^2 C d\right )}{8 d^2}-\frac {9}{8} d^3 x^3 \left (c^2-d^2 x^2\right )^{5/2} \left (2 B d^2+7 c^2 D+6 c C d\right )}{9 d^2}-\frac {2}{9} d^2 x^4 \left (c^2-d^2 x^2\right )^{5/2} (3 c D+C d)}{2 d^2}-\frac {1}{10} d D x^5 \left (c^2-d^2 x^2\right )^{5/2}\)

\(\Big \downarrow \) 2346

\(\displaystyle \frac {\frac {\frac {\frac {-\frac {\int -3 c^2 d^7 \left (21 c \left (21 D c^3+34 C d c^2+54 B d^2 c+144 A d^3\right )+32 d \left (42 D c^3+62 C d c^2+117 B d^2 c+207 A d^3\right ) x\right ) \left (c^2-d^2 x^2\right )^{3/2}dx}{6 d^2}-\frac {21}{2} c d^5 x \left (c^2-d^2 x^2\right )^{5/2} \left (48 A d^3+54 B c d^2+21 c^3 D+34 c^2 C d\right )}{7 d^2}-\frac {16}{7} d^4 x^2 \left (c^2-d^2 x^2\right )^{5/2} \left (9 A d^3+27 B c d^2+21 c^3 D+31 c^2 C d\right )}{8 d^2}-\frac {9}{8} d^3 x^3 \left (c^2-d^2 x^2\right )^{5/2} \left (2 B d^2+7 c^2 D+6 c C d\right )}{9 d^2}-\frac {2}{9} d^2 x^4 \left (c^2-d^2 x^2\right )^{5/2} (3 c D+C d)}{2 d^2}-\frac {1}{10} d D x^5 \left (c^2-d^2 x^2\right )^{5/2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\frac {\frac {1}{2} c^2 d^5 \int \left (21 c \left (21 D c^3+34 C d c^2+54 B d^2 c+144 A d^3\right )+32 d \left (42 D c^3+62 C d c^2+117 B d^2 c+207 A d^3\right ) x\right ) \left (c^2-d^2 x^2\right )^{3/2}dx-\frac {21}{2} c d^5 x \left (c^2-d^2 x^2\right )^{5/2} \left (48 A d^3+54 B c d^2+21 c^3 D+34 c^2 C d\right )}{7 d^2}-\frac {16}{7} d^4 x^2 \left (c^2-d^2 x^2\right )^{5/2} \left (9 A d^3+27 B c d^2+21 c^3 D+31 c^2 C d\right )}{8 d^2}-\frac {9}{8} d^3 x^3 \left (c^2-d^2 x^2\right )^{5/2} \left (2 B d^2+7 c^2 D+6 c C d\right )}{9 d^2}-\frac {2}{9} d^2 x^4 \left (c^2-d^2 x^2\right )^{5/2} (3 c D+C d)}{2 d^2}-\frac {1}{10} d D x^5 \left (c^2-d^2 x^2\right )^{5/2}\)

\(\Big \downarrow \) 455

\(\displaystyle \frac {\frac {\frac {\frac {\frac {1}{2} c^2 d^5 \left (21 c \left (144 A d^3+54 B c d^2+21 c^3 D+34 c^2 C d\right ) \int \left (c^2-d^2 x^2\right )^{3/2}dx-\frac {32 \left (c^2-d^2 x^2\right )^{5/2} \left (207 A d^3+117 B c d^2+42 c^3 D+62 c^2 C d\right )}{5 d}\right )-\frac {21}{2} c d^5 x \left (c^2-d^2 x^2\right )^{5/2} \left (48 A d^3+54 B c d^2+21 c^3 D+34 c^2 C d\right )}{7 d^2}-\frac {16}{7} d^4 x^2 \left (c^2-d^2 x^2\right )^{5/2} \left (9 A d^3+27 B c d^2+21 c^3 D+31 c^2 C d\right )}{8 d^2}-\frac {9}{8} d^3 x^3 \left (c^2-d^2 x^2\right )^{5/2} \left (2 B d^2+7 c^2 D+6 c C d\right )}{9 d^2}-\frac {2}{9} d^2 x^4 \left (c^2-d^2 x^2\right )^{5/2} (3 c D+C d)}{2 d^2}-\frac {1}{10} d D x^5 \left (c^2-d^2 x^2\right )^{5/2}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\frac {\frac {\frac {\frac {1}{2} c^2 d^5 \left (21 c \left (144 A d^3+54 B c d^2+21 c^3 D+34 c^2 C d\right ) \left (\frac {3}{4} c^2 \int \sqrt {c^2-d^2 x^2}dx+\frac {1}{4} x \left (c^2-d^2 x^2\right )^{3/2}\right )-\frac {32 \left (c^2-d^2 x^2\right )^{5/2} \left (207 A d^3+117 B c d^2+42 c^3 D+62 c^2 C d\right )}{5 d}\right )-\frac {21}{2} c d^5 x \left (c^2-d^2 x^2\right )^{5/2} \left (48 A d^3+54 B c d^2+21 c^3 D+34 c^2 C d\right )}{7 d^2}-\frac {16}{7} d^4 x^2 \left (c^2-d^2 x^2\right )^{5/2} \left (9 A d^3+27 B c d^2+21 c^3 D+31 c^2 C d\right )}{8 d^2}-\frac {9}{8} d^3 x^3 \left (c^2-d^2 x^2\right )^{5/2} \left (2 B d^2+7 c^2 D+6 c C d\right )}{9 d^2}-\frac {2}{9} d^2 x^4 \left (c^2-d^2 x^2\right )^{5/2} (3 c D+C d)}{2 d^2}-\frac {1}{10} d D x^5 \left (c^2-d^2 x^2\right )^{5/2}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\frac {\frac {\frac {\frac {1}{2} c^2 d^5 \left (21 c \left (144 A d^3+54 B c d^2+21 c^3 D+34 c^2 C d\right ) \left (\frac {3}{4} c^2 \left (\frac {1}{2} c^2 \int \frac {1}{\sqrt {c^2-d^2 x^2}}dx+\frac {1}{2} x \sqrt {c^2-d^2 x^2}\right )+\frac {1}{4} x \left (c^2-d^2 x^2\right )^{3/2}\right )-\frac {32 \left (c^2-d^2 x^2\right )^{5/2} \left (207 A d^3+117 B c d^2+42 c^3 D+62 c^2 C d\right )}{5 d}\right )-\frac {21}{2} c d^5 x \left (c^2-d^2 x^2\right )^{5/2} \left (48 A d^3+54 B c d^2+21 c^3 D+34 c^2 C d\right )}{7 d^2}-\frac {16}{7} d^4 x^2 \left (c^2-d^2 x^2\right )^{5/2} \left (9 A d^3+27 B c d^2+21 c^3 D+31 c^2 C d\right )}{8 d^2}-\frac {9}{8} d^3 x^3 \left (c^2-d^2 x^2\right )^{5/2} \left (2 B d^2+7 c^2 D+6 c C d\right )}{9 d^2}-\frac {2}{9} d^2 x^4 \left (c^2-d^2 x^2\right )^{5/2} (3 c D+C d)}{2 d^2}-\frac {1}{10} d D x^5 \left (c^2-d^2 x^2\right )^{5/2}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\frac {\frac {\frac {\frac {1}{2} c^2 d^5 \left (21 c \left (144 A d^3+54 B c d^2+21 c^3 D+34 c^2 C d\right ) \left (\frac {3}{4} c^2 \left (\frac {1}{2} c^2 \int \frac {1}{\frac {d^2 x^2}{c^2-d^2 x^2}+1}d\frac {x}{\sqrt {c^2-d^2 x^2}}+\frac {1}{2} x \sqrt {c^2-d^2 x^2}\right )+\frac {1}{4} x \left (c^2-d^2 x^2\right )^{3/2}\right )-\frac {32 \left (c^2-d^2 x^2\right )^{5/2} \left (207 A d^3+117 B c d^2+42 c^3 D+62 c^2 C d\right )}{5 d}\right )-\frac {21}{2} c d^5 x \left (c^2-d^2 x^2\right )^{5/2} \left (48 A d^3+54 B c d^2+21 c^3 D+34 c^2 C d\right )}{7 d^2}-\frac {16}{7} d^4 x^2 \left (c^2-d^2 x^2\right )^{5/2} \left (9 A d^3+27 B c d^2+21 c^3 D+31 c^2 C d\right )}{8 d^2}-\frac {9}{8} d^3 x^3 \left (c^2-d^2 x^2\right )^{5/2} \left (2 B d^2+7 c^2 D+6 c C d\right )}{9 d^2}-\frac {2}{9} d^2 x^4 \left (c^2-d^2 x^2\right )^{5/2} (3 c D+C d)}{2 d^2}-\frac {1}{10} d D x^5 \left (c^2-d^2 x^2\right )^{5/2}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {\frac {\frac {\frac {\frac {1}{2} c^2 d^5 \left (21 c \left (\frac {3}{4} c^2 \left (\frac {c^2 \arctan \left (\frac {d x}{\sqrt {c^2-d^2 x^2}}\right )}{2 d}+\frac {1}{2} x \sqrt {c^2-d^2 x^2}\right )+\frac {1}{4} x \left (c^2-d^2 x^2\right )^{3/2}\right ) \left (144 A d^3+54 B c d^2+21 c^3 D+34 c^2 C d\right )-\frac {32 \left (c^2-d^2 x^2\right )^{5/2} \left (207 A d^3+117 B c d^2+42 c^3 D+62 c^2 C d\right )}{5 d}\right )-\frac {21}{2} c d^5 x \left (c^2-d^2 x^2\right )^{5/2} \left (48 A d^3+54 B c d^2+21 c^3 D+34 c^2 C d\right )}{7 d^2}-\frac {16}{7} d^4 x^2 \left (c^2-d^2 x^2\right )^{5/2} \left (9 A d^3+27 B c d^2+21 c^3 D+31 c^2 C d\right )}{8 d^2}-\frac {9}{8} d^3 x^3 \left (c^2-d^2 x^2\right )^{5/2} \left (2 B d^2+7 c^2 D+6 c C d\right )}{9 d^2}-\frac {2}{9} d^2 x^4 \left (c^2-d^2 x^2\right )^{5/2} (3 c D+C d)}{2 d^2}-\frac {1}{10} d D x^5 \left (c^2-d^2 x^2\right )^{5/2}\)

Input:

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

Output:

-1/10*(d*D*x^5*(c^2 - d^2*x^2)^(5/2)) + ((-2*d^2*(C*d + 3*c*D)*x^4*(c^2 - 
d^2*x^2)^(5/2))/9 + ((-9*d^3*(6*c*C*d + 2*B*d^2 + 7*c^2*D)*x^3*(c^2 - d^2* 
x^2)^(5/2))/8 + ((-16*d^4*(31*c^2*C*d + 27*B*c*d^2 + 9*A*d^3 + 21*c^3*D)*x 
^2*(c^2 - d^2*x^2)^(5/2))/7 + ((-21*c*d^5*(34*c^2*C*d + 54*B*c*d^2 + 48*A* 
d^3 + 21*c^3*D)*x*(c^2 - d^2*x^2)^(5/2))/2 + (c^2*d^5*((-32*(62*c^2*C*d + 
117*B*c*d^2 + 207*A*d^3 + 42*c^3*D)*(c^2 - d^2*x^2)^(5/2))/(5*d) + 21*c*(3 
4*c^2*C*d + 54*B*c*d^2 + 144*A*d^3 + 21*c^3*D)*((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)))/2)/(7*d^2))/(8*d^2))/(9*d^2))/(2*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 = 0.63 (sec) , antiderivative size = 711, normalized size of antiderivative = 1.71

method	result	size
default	\(A \,c^{3} \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 )-\frac {c^{2} \left (3 A d +B c \right ) \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}{5 d^{2}}+d^{2} \left (C d +3 D c \right ) \left (-\frac {x^{4} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}{9 d^{2}}+\frac {4 c^{2} \left (-\frac {x^{2} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}{7 d^{2}}-\frac {2 c^{2} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}{35 d^{4}}\right )}{9 d^{2}}\right )+c \left (3 A \,d^{2}+3 B c d +C \,c^{2}\right ) \left (-\frac {x \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}{6 d^{2}}+\frac {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 d^{2}}\right )+d \left (B \,d^{2}+3 C c d +3 D c^{2}\right ) \left (-\frac {x^{3} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}{8 d^{2}}+\frac {3 c^{2} \left (-\frac {x \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}{6 d^{2}}+\frac {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 d^{2}}\right )}{8 d^{2}}\right )+\left (A \,d^{3}+3 B c \,d^{2}+3 C \,c^{2} d +D c^{3}\right ) \left (-\frac {x^{2} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}{7 d^{2}}-\frac {2 c^{2} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}{35 d^{4}}\right )+d^{3} D \left (-\frac {x^{5} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}{10 d^{2}}+\frac {c^{2} \left (-\frac {x^{3} \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}{8 d^{2}}+\frac {3 c^{2} \left (-\frac {x \left (-d^{2} x^{2}+c^{2}\right )^{\frac {5}{2}}}{6 d^{2}}+\frac {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 d^{2}}\right )}{8 d^{2}}\right )}{2 d^{2}}\right )\)	\(711\)

Input:

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

Output:

A*c^3*(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))))-1/5*c^2*(3*A* 
d+B*c)*(-d^2*x^2+c^2)^(5/2)/d^2+d^2*(C*d+3*D*c)*(-1/9*x^4*(-d^2*x^2+c^2)^( 
5/2)/d^2+4/9*c^2/d^2*(-1/7*x^2*(-d^2*x^2+c^2)^(5/2)/d^2-2/35*c^2*(-d^2*x^2 
+c^2)^(5/2)/d^4))+c*(3*A*d^2+3*B*c*d+C*c^2)*(-1/6*x*(-d^2*x^2+c^2)^(5/2)/d 
^2+1/6*c^2/d^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)))))+d*( 
B*d^2+3*C*c*d+3*D*c^2)*(-1/8*x^3*(-d^2*x^2+c^2)^(5/2)/d^2+3/8*c^2/d^2*(-1/ 
6*x*(-d^2*x^2+c^2)^(5/2)/d^2+1/6*c^2/d^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))))))+(A*d^3+3*B*c*d^2+3*C*c^2*d+D*c^3)*(-1/7*x^2*(-d^2* 
x^2+c^2)^(5/2)/d^2-2/35*c^2*(-d^2*x^2+c^2)^(5/2)/d^4)+d^3*D*(-1/10*x^5*(-d 
^2*x^2+c^2)^(5/2)/d^2+1/2*c^2/d^2*(-1/8*x^3*(-d^2*x^2+c^2)^(5/2)/d^2+3/8*c 
^2/d^2*(-1/6*x*(-d^2*x^2+c^2)^(5/2)/d^2+1/6*c^2/d^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)))))))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 410, normalized size of antiderivative = 0.99 \[ \int (c+d x)^3 \left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=-\frac {630 \, {\left (21 \, D c^{10} + 34 \, C c^{9} d + 54 \, B c^{8} d^{2} + 144 \, A c^{7} d^{3}\right )} \arctan \left (-\frac {c - \sqrt {-d^{2} x^{2} + c^{2}}}{d x}\right ) + {\left (8064 \, D d^{9} x^{9} + 10752 \, D c^{9} + 15872 \, C c^{8} d + 29952 \, B c^{7} d^{2} + 52992 \, A c^{6} d^{3} + 8960 \, {\left (3 \, D c d^{8} + C d^{9}\right )} x^{8} + 1008 \, {\left (19 \, D c^{2} d^{7} + 30 \, C c d^{8} + 10 \, B d^{9}\right )} x^{7} - 1280 \, {\left (21 \, D c^{3} d^{6} - 17 \, C c^{2} d^{7} - 27 \, B c d^{8} - 9 \, A d^{9}\right )} x^{6} - 168 \, {\left (267 \, D c^{4} d^{5} + 190 \, C c^{3} d^{6} - 150 \, B c^{2} d^{7} - 240 \, A c d^{8}\right )} x^{5} - 768 \, {\left (21 \, D c^{5} d^{4} + 71 \, C c^{4} d^{5} + 51 \, B c^{3} d^{6} - 39 \, A c^{2} d^{7}\right )} x^{4} + 210 \, {\left (21 \, D c^{6} d^{3} - 94 \, C c^{5} d^{4} - 330 \, B c^{4} d^{5} - 240 \, A c^{3} d^{6}\right )} x^{3} + 256 \, {\left (21 \, D c^{7} d^{2} + 31 \, C c^{6} d^{3} - 99 \, B c^{5} d^{4} - 369 \, A c^{4} d^{5}\right )} x^{2} + 315 \, {\left (21 \, D c^{8} d + 34 \, C c^{7} d^{2} + 54 \, B c^{6} d^{3} - 112 \, A c^{5} d^{4}\right )} x\right )} \sqrt {-d^{2} x^{2} + c^{2}}}{80640 \, d^{4}} \] Input:

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

Output:

-1/80640*(630*(21*D*c^10 + 34*C*c^9*d + 54*B*c^8*d^2 + 144*A*c^7*d^3)*arct 
an(-(c - sqrt(-d^2*x^2 + c^2))/(d*x)) + (8064*D*d^9*x^9 + 10752*D*c^9 + 15 
872*C*c^8*d + 29952*B*c^7*d^2 + 52992*A*c^6*d^3 + 8960*(3*D*c*d^8 + C*d^9) 
*x^8 + 1008*(19*D*c^2*d^7 + 30*C*c*d^8 + 10*B*d^9)*x^7 - 1280*(21*D*c^3*d^ 
6 - 17*C*c^2*d^7 - 27*B*c*d^8 - 9*A*d^9)*x^6 - 168*(267*D*c^4*d^5 + 190*C* 
c^3*d^6 - 150*B*c^2*d^7 - 240*A*c*d^8)*x^5 - 768*(21*D*c^5*d^4 + 71*C*c^4* 
d^5 + 51*B*c^3*d^6 - 39*A*c^2*d^7)*x^4 + 210*(21*D*c^6*d^3 - 94*C*c^5*d^4 
- 330*B*c^4*d^5 - 240*A*c^3*d^6)*x^3 + 256*(21*D*c^7*d^2 + 31*C*c^6*d^3 - 
99*B*c^5*d^4 - 369*A*c^4*d^5)*x^2 + 315*(21*D*c^8*d + 34*C*c^7*d^2 + 54*B* 
c^6*d^3 - 112*A*c^5*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. 1331 vs. \(2 (406) = 812\).

Time = 0.88 (sec) , antiderivative size = 1331, normalized size of antiderivative = 3.20 \[ \int (c+d x)^3 \left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

Piecewise((sqrt(c**2 - d**2*x**2)*(-D*d**5*x**9/10 - x**8*(C*d**7 + 3*D*c* 
d**6)/(9*d**2) - x**7*(B*d**7 + 3*C*c*d**6 + 19*D*c**2*d**5/10)/(8*d**2) - 
 x**6*(A*d**7 + 3*B*c*d**6 + C*c**2*d**5 - 5*D*c**3*d**4 + 8*c**2*(C*d**7 
+ 3*D*c*d**6)/(9*d**2))/(7*d**2) - x**5*(3*A*c*d**6 + B*c**2*d**5 - 5*C*c* 
*3*d**4 - 5*D*c**4*d**3 + 7*c**2*(B*d**7 + 3*C*c*d**6 + 19*D*c**2*d**5/10) 
/(8*d**2))/(6*d**2) - x**4*(A*c**2*d**5 - 5*B*c**3*d**4 - 5*C*c**4*d**3 + 
D*c**5*d**2 + 6*c**2*(A*d**7 + 3*B*c*d**6 + C*c**2*d**5 - 5*D*c**3*d**4 + 
8*c**2*(C*d**7 + 3*D*c*d**6)/(9*d**2))/(7*d**2))/(5*d**2) - x**3*(-5*A*c** 
3*d**4 - 5*B*c**4*d**3 + C*c**5*d**2 + 3*D*c**6*d + 5*c**2*(3*A*c*d**6 + B 
*c**2*d**5 - 5*C*c**3*d**4 - 5*D*c**4*d**3 + 7*c**2*(B*d**7 + 3*C*c*d**6 + 
 19*D*c**2*d**5/10)/(8*d**2))/(6*d**2))/(4*d**2) - x**2*(-5*A*c**4*d**3 + 
B*c**5*d**2 + 3*C*c**6*d + D*c**7 + 4*c**2*(A*c**2*d**5 - 5*B*c**3*d**4 - 
5*C*c**4*d**3 + D*c**5*d**2 + 6*c**2*(A*d**7 + 3*B*c*d**6 + C*c**2*d**5 - 
5*D*c**3*d**4 + 8*c**2*(C*d**7 + 3*D*c*d**6)/(9*d**2))/(7*d**2))/(5*d**2)) 
/(3*d**2) - x*(A*c**5*d**2 + 3*B*c**6*d + C*c**7 + 3*c**2*(-5*A*c**3*d**4 
- 5*B*c**4*d**3 + C*c**5*d**2 + 3*D*c**6*d + 5*c**2*(3*A*c*d**6 + B*c**2*d 
**5 - 5*C*c**3*d**4 - 5*D*c**4*d**3 + 7*c**2*(B*d**7 + 3*C*c*d**6 + 19*D*c 
**2*d**5/10)/(8*d**2))/(6*d**2))/(4*d**2))/(2*d**2) - (3*A*c**6*d + B*c**7 
 + 2*c**2*(-5*A*c**4*d**3 + B*c**5*d**2 + 3*C*c**6*d + D*c**7 + 4*c**2*(A* 
c**2*d**5 - 5*B*c**3*d**4 - 5*C*c**4*d**3 + D*c**5*d**2 + 6*c**2*(A*d**...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 809 vs. \(2 (380) = 760\).

Time = 0.12 (sec) , antiderivative size = 809, normalized size of antiderivative = 1.94 \[ \int (c+d x)^3 \left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

-1/10*(-d^2*x^2 + c^2)^(5/2)*D*d*x^5 + 3/256*D*c^10*arcsin(d*x/c)/d^4 + 3/ 
8*A*c^7*arcsin(d*x/c)/d + 3/8*sqrt(-d^2*x^2 + c^2)*A*c^5*x + 3/256*sqrt(-d 
^2*x^2 + c^2)*D*c^8*x/d^3 - 1/16*(-d^2*x^2 + c^2)^(5/2)*D*c^2*x^3/d + 1/4* 
(-d^2*x^2 + c^2)^(3/2)*A*c^3*x + 1/128*(-d^2*x^2 + c^2)^(3/2)*D*c^6*x/d^3 
- 1/32*(-d^2*x^2 + c^2)^(5/2)*D*c^4*x/d^3 - 1/9*(3*D*c*d^2 + C*d^3)*(-d^2* 
x^2 + c^2)^(5/2)*x^4/d^2 + 3/128*(3*D*c^2*d + 3*C*c*d^2 + B*d^3)*c^8*arcsi 
n(d*x/c)/d^5 + 1/16*(C*c^3 + 3*B*c^2*d + 3*A*c*d^2)*c^6*arcsin(d*x/c)/d^3 
- 1/5*(-d^2*x^2 + c^2)^(5/2)*B*c^3/d^2 - 3/5*(-d^2*x^2 + c^2)^(5/2)*A*c^2/ 
d + 3/128*(3*D*c^2*d + 3*C*c*d^2 + B*d^3)*sqrt(-d^2*x^2 + c^2)*c^6*x/d^4 + 
 1/16*(C*c^3 + 3*B*c^2*d + 3*A*c*d^2)*sqrt(-d^2*x^2 + c^2)*c^4*x/d^2 - 1/8 
*(3*D*c^2*d + 3*C*c*d^2 + B*d^3)*(-d^2*x^2 + c^2)^(5/2)*x^3/d^2 + 1/64*(3* 
D*c^2*d + 3*C*c*d^2 + B*d^3)*(-d^2*x^2 + c^2)^(3/2)*c^4*x/d^4 + 1/24*(C*c^ 
3 + 3*B*c^2*d + 3*A*c*d^2)*(-d^2*x^2 + c^2)^(3/2)*c^2*x/d^2 - 4/63*(3*D*c* 
d^2 + C*d^3)*(-d^2*x^2 + c^2)^(5/2)*c^2*x^2/d^4 - 1/7*(D*c^3 + 3*C*c^2*d + 
 3*B*c*d^2 + A*d^3)*(-d^2*x^2 + c^2)^(5/2)*x^2/d^2 - 1/16*(3*D*c^2*d + 3*C 
*c*d^2 + B*d^3)*(-d^2*x^2 + c^2)^(5/2)*c^2*x/d^4 - 1/6*(C*c^3 + 3*B*c^2*d 
+ 3*A*c*d^2)*(-d^2*x^2 + c^2)^(5/2)*x/d^2 - 8/315*(3*D*c*d^2 + C*d^3)*(-d^ 
2*x^2 + c^2)^(5/2)*c^4/d^6 - 2/35*(D*c^3 + 3*C*c^2*d + 3*B*c*d^2 + A*d^3)* 
(-d^2*x^2 + c^2)^(5/2)*c^2/d^4
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 439, normalized size of antiderivative = 1.06 \[ \int (c+d x)^3 \left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=-\frac {1}{80640} \, \sqrt {-d^{2} x^{2} + c^{2}} {\left ({\left (2 \, {\left ({\left (4 \, {\left ({\left (2 \, {\left (7 \, {\left (8 \, {\left (9 \, D d^{5} x + \frac {10 \, {\left (3 \, D c d^{20} + C d^{21}\right )}}{d^{16}}\right )} x + \frac {9 \, {\left (19 \, D c^{2} d^{19} + 30 \, C c d^{20} + 10 \, B d^{21}\right )}}{d^{16}}\right )} x - \frac {80 \, {\left (21 \, D c^{3} d^{18} - 17 \, C c^{2} d^{19} - 27 \, B c d^{20} - 9 \, A d^{21}\right )}}{d^{16}}\right )} x - \frac {21 \, {\left (267 \, D c^{4} d^{17} + 190 \, C c^{3} d^{18} - 150 \, B c^{2} d^{19} - 240 \, A c d^{20}\right )}}{d^{16}}\right )} x - \frac {96 \, {\left (21 \, D c^{5} d^{16} + 71 \, C c^{4} d^{17} + 51 \, B c^{3} d^{18} - 39 \, A c^{2} d^{19}\right )}}{d^{16}}\right )} x + \frac {105 \, {\left (21 \, D c^{6} d^{15} - 94 \, C c^{5} d^{16} - 330 \, B c^{4} d^{17} - 240 \, A c^{3} d^{18}\right )}}{d^{16}}\right )} x + \frac {128 \, {\left (21 \, D c^{7} d^{14} + 31 \, C c^{6} d^{15} - 99 \, B c^{5} d^{16} - 369 \, A c^{4} d^{17}\right )}}{d^{16}}\right )} x + \frac {315 \, {\left (21 \, D c^{8} d^{13} + 34 \, C c^{7} d^{14} + 54 \, B c^{6} d^{15} - 112 \, A c^{5} d^{16}\right )}}{d^{16}}\right )} x + \frac {256 \, {\left (42 \, D c^{9} d^{12} + 62 \, C c^{8} d^{13} + 117 \, B c^{7} d^{14} + 207 \, A c^{6} d^{15}\right )}}{d^{16}}\right )} + \frac {{\left (21 \, D c^{10} + 34 \, C c^{9} d + 54 \, B c^{8} d^{2} + 144 \, A c^{7} d^{3}\right )} \arcsin \left (\frac {d x}{c}\right ) \mathrm {sgn}\left (c\right ) \mathrm {sgn}\left (d\right )}{256 \, d^{3} {\left | d \right |}} \] Input:

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

Output:

-1/80640*sqrt(-d^2*x^2 + c^2)*((2*((4*((2*(7*(8*(9*D*d^5*x + 10*(3*D*c*d^2 
0 + C*d^21)/d^16)*x + 9*(19*D*c^2*d^19 + 30*C*c*d^20 + 10*B*d^21)/d^16)*x 
- 80*(21*D*c^3*d^18 - 17*C*c^2*d^19 - 27*B*c*d^20 - 9*A*d^21)/d^16)*x - 21 
*(267*D*c^4*d^17 + 190*C*c^3*d^18 - 150*B*c^2*d^19 - 240*A*c*d^20)/d^16)*x 
 - 96*(21*D*c^5*d^16 + 71*C*c^4*d^17 + 51*B*c^3*d^18 - 39*A*c^2*d^19)/d^16 
)*x + 105*(21*D*c^6*d^15 - 94*C*c^5*d^16 - 330*B*c^4*d^17 - 240*A*c^3*d^18 
)/d^16)*x + 128*(21*D*c^7*d^14 + 31*C*c^6*d^15 - 99*B*c^5*d^16 - 369*A*c^4 
*d^17)/d^16)*x + 315*(21*D*c^8*d^13 + 34*C*c^7*d^14 + 54*B*c^6*d^15 - 112* 
A*c^5*d^16)/d^16)*x + 256*(42*D*c^9*d^12 + 62*C*c^8*d^13 + 117*B*c^7*d^14 
+ 207*A*c^6*d^15)/d^16) + 1/256*(21*D*c^10 + 34*C*c^9*d + 54*B*c^8*d^2 + 1 
44*A*c^7*d^3)*arcsin(d*x/c)*sgn(c)*sgn(d)/(d^3*abs(d))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 647, normalized size of antiderivative = 1.56 \[ \int (c+d x)^3 \left (c^2-d^2 x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {45360 \mathit {asin} \left (\frac {d x}{c}\right ) a \,c^{7} d^{2}+17010 \mathit {asin} \left (\frac {d x}{c}\right ) b \,c^{8} d -52992 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{6} d^{2}-11520 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,d^{8} x^{6}-29952 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{7} d -10080 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,d^{8} x^{7}-17325 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{8} d x -13312 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{7} d^{2} x^{2}+15330 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{6} d^{3} x^{3}+70656 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{5} d^{4} x^{4}+76776 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{4} d^{5} x^{5}+5120 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{3} d^{6} x^{6}-49392 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{2} d^{7} x^{7}-35840 \sqrt {-d^{2} x^{2}+c^{2}}\, c \,d^{8} x^{8}-8064 \sqrt {-d^{2} x^{2}+c^{2}}\, d^{9} x^{9}+52992 a \,c^{7} d^{2}+29952 b \,c^{8} d +35280 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{5} d^{3} x +94464 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{4} d^{4} x^{2}+50400 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{3} d^{5} x^{3}-29952 \sqrt {-d^{2} x^{2}+c^{2}}\, a \,c^{2} d^{6} x^{4}-40320 \sqrt {-d^{2} x^{2}+c^{2}}\, a c \,d^{7} x^{5}-17010 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{6} d^{2} x +25344 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{5} d^{3} x^{2}+69300 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{4} d^{4} x^{3}+39168 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{3} d^{5} x^{4}-25200 \sqrt {-d^{2} x^{2}+c^{2}}\, b \,c^{2} d^{6} x^{5}-34560 \sqrt {-d^{2} x^{2}+c^{2}}\, b c \,d^{7} x^{6}+17325 \mathit {asin} \left (\frac {d x}{c}\right ) c^{10}-26624 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{9}+26624 c^{10}}{80640 d^{3}} \] Input:

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

Output:

(45360*asin((d*x)/c)*a*c**7*d**2 + 17010*asin((d*x)/c)*b*c**8*d + 17325*as 
in((d*x)/c)*c**10 - 52992*sqrt(c**2 - d**2*x**2)*a*c**6*d**2 + 35280*sqrt( 
c**2 - d**2*x**2)*a*c**5*d**3*x + 94464*sqrt(c**2 - d**2*x**2)*a*c**4*d**4 
*x**2 + 50400*sqrt(c**2 - d**2*x**2)*a*c**3*d**5*x**3 - 29952*sqrt(c**2 - 
d**2*x**2)*a*c**2*d**6*x**4 - 40320*sqrt(c**2 - d**2*x**2)*a*c*d**7*x**5 - 
 11520*sqrt(c**2 - d**2*x**2)*a*d**8*x**6 - 29952*sqrt(c**2 - d**2*x**2)*b 
*c**7*d - 17010*sqrt(c**2 - d**2*x**2)*b*c**6*d**2*x + 25344*sqrt(c**2 - d 
**2*x**2)*b*c**5*d**3*x**2 + 69300*sqrt(c**2 - d**2*x**2)*b*c**4*d**4*x**3 
 + 39168*sqrt(c**2 - d**2*x**2)*b*c**3*d**5*x**4 - 25200*sqrt(c**2 - d**2* 
x**2)*b*c**2*d**6*x**5 - 34560*sqrt(c**2 - d**2*x**2)*b*c*d**7*x**6 - 1008 
0*sqrt(c**2 - d**2*x**2)*b*d**8*x**7 - 26624*sqrt(c**2 - d**2*x**2)*c**9 - 
 17325*sqrt(c**2 - d**2*x**2)*c**8*d*x - 13312*sqrt(c**2 - d**2*x**2)*c**7 
*d**2*x**2 + 15330*sqrt(c**2 - d**2*x**2)*c**6*d**3*x**3 + 70656*sqrt(c**2 
 - d**2*x**2)*c**5*d**4*x**4 + 76776*sqrt(c**2 - d**2*x**2)*c**4*d**5*x**5 
 + 5120*sqrt(c**2 - d**2*x**2)*c**3*d**6*x**6 - 49392*sqrt(c**2 - d**2*x** 
2)*c**2*d**7*x**7 - 35840*sqrt(c**2 - d**2*x**2)*c*d**8*x**8 - 8064*sqrt(c 
**2 - d**2*x**2)*d**9*x**9 + 52992*a*c**7*d**2 + 29952*b*c**8*d + 26624*c* 
*10)/(80640*d**3)