\(\int (c+d x)^3 (c^2-d^2 x^2)^p (A+B x+C x^2+D x^3) \, dx\) [234]

Optimal result

Integrand size = 37, antiderivative size = 495 \[ \int (c+d x)^3 \left (c^2-d^2 x^2\right )^p \left (A+B x+C x^2+D x^3\right ) \, dx=-\frac {2 c^2 \left (c^2 C d+B c d^2+A d^3+c^3 D\right ) \left (c^2-d^2 x^2\right )^{1+p}}{d^4 (1+p)}-\frac {c \left (6 c^3 D (13+3 p)+6 B c d^2 \left (21+13 p+2 p^2\right )+2 c^2 C d \left (49+21 p+2 p^2\right )+3 A d^3 \left (35+24 p+4 p^2\right )\right ) x \left (c^2-d^2 x^2\right )^{1+p}}{d^3 (3+2 p) (5+2 p) (7+2 p)}-\frac {\left (3 c C d (7+2 p)+B d^2 (7+2 p)+2 c^2 D (13+3 p)\right ) x^3 \left (c^2-d^2 x^2\right )^{1+p}}{d (5+2 p) (7+2 p)}-\frac {d D x^5 \left (c^2-d^2 x^2\right )^{1+p}}{7+2 p}+\frac {\left (5 c^2 C d+3 B c d^2+A d^3+7 c^3 D\right ) \left (c^2-d^2 x^2\right )^{2+p}}{2 d^4 (2+p)}-\frac {(C d+3 c D) \left (c^2-d^2 x^2\right )^{3+p}}{2 d^4 (3+p)}+\frac {2 c^3 \left (3 c^3 D (13+3 p)+3 B c d^2 \left (21+13 p+2 p^2\right )+c^2 C d \left (49+21 p+2 p^2\right )+A d^3 \left (105+107 p+36 p^2+4 p^3\right )\right ) x \left (c^2-d^2 x^2\right )^p \left (1-\frac {d^2 x^2}{c^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},\frac {d^2 x^2}{c^2}\right )}{d^3 (3+2 p) (5+2 p) (7+2 p)} \] Output:

-2*c^2*(A*d^3+B*c*d^2+C*c^2*d+D*c^3)*(-d^2*x^2+c^2)^(p+1)/d^4/(p+1)-c*(6*c 
^3*D*(13+3*p)+6*B*c*d^2*(2*p^2+13*p+21)+2*c^2*C*d*(2*p^2+21*p+49)+3*A*d^3* 
(4*p^2+24*p+35))*x*(-d^2*x^2+c^2)^(p+1)/d^3/(3+2*p)/(5+2*p)/(7+2*p)-(3*c*C 
*d*(7+2*p)+B*d^2*(7+2*p)+2*c^2*D*(13+3*p))*x^3*(-d^2*x^2+c^2)^(p+1)/d/(5+2 
*p)/(7+2*p)-d*D*x^5*(-d^2*x^2+c^2)^(p+1)/(7+2*p)+1/2*(A*d^3+3*B*c*d^2+5*C* 
c^2*d+7*D*c^3)*(-d^2*x^2+c^2)^(2+p)/d^4/(2+p)-1/2*(C*d+3*D*c)*(-d^2*x^2+c^ 
2)^(3+p)/d^4/(3+p)+2*c^3*(3*c^3*D*(13+3*p)+3*B*c*d^2*(2*p^2+13*p+21)+c^2*C 
*d*(2*p^2+21*p+49)+A*d^3*(4*p^3+36*p^2+107*p+105))*x*(-d^2*x^2+c^2)^p*hype 
rgeom([1/2, -p],[3/2],d^2*x^2/c^2)/d^3/(3+2*p)/(5+2*p)/(7+2*p)/((1-d^2*x^2 
/c^2)^p)
 

Mathematica [A] (verified)

Time = 4.00 (sec) , antiderivative size = 397, normalized size of antiderivative = 0.80 \[ \int (c+d x)^3 \left (c^2-d^2 x^2\right )^p \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {1}{210} \left (c^2-d^2 x^2\right )^p \left (-\frac {105 \left (c^2-d^2 x^2\right ) \left (c^5 D (9+p)+c^4 C d (11+3 p)+d^5 (1+p) x^2 \left (A (3+p)+C (2+p) x^2\right )+3 c d^4 (1+p) x^2 \left (B (3+p)+D (2+p) x^2\right )+c^3 d^2 \left (B \left (15+8 p+p^2\right )+D \left (9+10 p+p^2\right ) x^2\right )+c^2 d^3 \left (A \left (21+16 p+3 p^2\right )+C \left (11+14 p+3 p^2\right ) x^2\right )\right )}{d^4 (1+p) (2+p) (3+p)}+210 A c^3 x \left (1-\frac {d^2 x^2}{c^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},\frac {d^2 x^2}{c^2}\right )+70 c \left (c^2 C+3 B c d+3 A d^2\right ) x^3 \left (1-\frac {d^2 x^2}{c^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-p,\frac {5}{2},\frac {d^2 x^2}{c^2}\right )+42 d \left (3 c C d+B d^2+3 c^2 D\right ) x^5 \left (1-\frac {d^2 x^2}{c^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},-p,\frac {7}{2},\frac {d^2 x^2}{c^2}\right )+30 d^3 D x^7 \left (1-\frac {d^2 x^2}{c^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {7}{2},-p,\frac {9}{2},\frac {d^2 x^2}{c^2}\right )\right ) \] Input:

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

Output:

((c^2 - d^2*x^2)^p*((-105*(c^2 - d^2*x^2)*(c^5*D*(9 + p) + c^4*C*d*(11 + 3 
*p) + d^5*(1 + p)*x^2*(A*(3 + p) + C*(2 + p)*x^2) + 3*c*d^4*(1 + p)*x^2*(B 
*(3 + p) + D*(2 + p)*x^2) + c^3*d^2*(B*(15 + 8*p + p^2) + D*(9 + 10*p + p^ 
2)*x^2) + c^2*d^3*(A*(21 + 16*p + 3*p^2) + C*(11 + 14*p + 3*p^2)*x^2)))/(d 
^4*(1 + p)*(2 + p)*(3 + p)) + (210*A*c^3*x*Hypergeometric2F1[1/2, -p, 3/2, 
 (d^2*x^2)/c^2])/(1 - (d^2*x^2)/c^2)^p + (70*c*(c^2*C + 3*B*c*d + 3*A*d^2) 
*x^3*Hypergeometric2F1[3/2, -p, 5/2, (d^2*x^2)/c^2])/(1 - (d^2*x^2)/c^2)^p 
 + (42*d*(3*c*C*d + B*d^2 + 3*c^2*D)*x^5*Hypergeometric2F1[5/2, -p, 7/2, ( 
d^2*x^2)/c^2])/(1 - (d^2*x^2)/c^2)^p + (30*d^3*D*x^7*Hypergeometric2F1[7/2 
, -p, 9/2, (d^2*x^2)/c^2])/(1 - (d^2*x^2)/c^2)^p))/210
 

Rubi [A] (verified)

Time = 6.41 (sec) , antiderivative size = 599, normalized size of antiderivative = 1.21, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.378, Rules used = {2346, 25, 2346, 27, 2346, 25, 2346, 27, 2346, 25, 27, 455, 238, 237}

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.

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

\(\Big \downarrow \) 2346

\(\displaystyle -\frac {\int -\left (c^2-d^2 x^2\right )^p \left (d^4 (C d+3 c D) (2 p+7) x^5+d^3 \left (2 D (3 p+13) c^2+3 C d (2 p+7) c+B d^2 (2 p+7)\right ) x^4+d^2 \left (D c^3+3 C d c^2+3 B d^2 c+A d^3\right ) (2 p+7) x^3+c d^2 \left (C c^2+3 B d c+3 A d^2\right ) (2 p+7) x^2+c^2 d^2 (B c+3 A d) (2 p+7) x+A c^3 d^2 (2 p+7)\right )dx}{d^2 (2 p+7)}-\frac {d D x^5 \left (c^2-d^2 x^2\right )^{p+1}}{2 p+7}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \left (c^2-d^2 x^2\right )^p \left (d^4 (C d+3 c D) (2 p+7) x^5+d^3 \left (2 D (3 p+13) c^2+3 C d (2 p+7) c+B d^2 (2 p+7)\right ) x^4+d^2 \left (D c^3+3 C d c^2+3 B d^2 c+A d^3\right ) (2 p+7) x^3+c d^2 \left (C c^2+3 B d c+3 A d^2\right ) (2 p+7) x^2+c^2 d^2 (B c+3 A d) (2 p+7) x+A c^3 d^2 (2 p+7)\right )dx}{d^2 (2 p+7)}-\frac {d D x^5 \left (c^2-d^2 x^2\right )^{p+1}}{2 p+7}\)

\(\Big \downarrow \) 2346

\(\displaystyle \frac {-\frac {\int -2 \left (c^2-d^2 x^2\right )^p \left ((p+3) \left (2 D (3 p+13) c^2+3 C d (2 p+7) c+B d^2 (2 p+7)\right ) x^4 d^5+(2 p+7) \left (D (p+9) c^3+C d (3 p+11) c^2+3 B d^2 (p+3) c+A d^3 (p+3)\right ) x^3 d^4+c \left (C c^2+3 B d c+3 A d^2\right ) (p+3) (2 p+7) x^2 d^4+A c^3 (p+3) (2 p+7) d^4+c^2 (B c+3 A d) (p+3) (2 p+7) x d^4\right )dx}{2 d^2 (p+3)}-\frac {d^2 (2 p+7) x^4 (3 c D+C d) \left (c^2-d^2 x^2\right )^{p+1}}{2 (p+3)}}{d^2 (2 p+7)}-\frac {d D x^5 \left (c^2-d^2 x^2\right )^{p+1}}{2 p+7}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \left (c^2-d^2 x^2\right )^p \left ((p+3) \left (2 D (3 p+13) c^2+3 C d (2 p+7) c+B d^2 (2 p+7)\right ) x^4 d^5+(2 p+7) \left (D (p+9) c^3+C d (3 p+11) c^2+3 B d^2 (p+3) c+A d^3 (p+3)\right ) x^3 d^4+c \left (C c^2+3 B d c+3 A d^2\right ) (p+3) (2 p+7) x^2 d^4+A c^3 (p+3) (2 p+7) d^4+c^2 (B c+3 A d) (p+3) (2 p+7) x d^4\right )dx}{d^2 (p+3)}-\frac {d^2 (2 p+7) x^4 (3 c D+C d) \left (c^2-d^2 x^2\right )^{p+1}}{2 (p+3)}}{d^2 (2 p+7)}-\frac {d D x^5 \left (c^2-d^2 x^2\right )^{p+1}}{2 p+7}\)

\(\Big \downarrow \) 2346

\(\displaystyle \frac {\frac {-\frac {\int -\left (c^2-d^2 x^2\right )^p \left (\left (4 p^2+24 p+35\right ) \left (D (p+9) c^3+C d (3 p+11) c^2+3 B d^2 (p+3) c+A d^3 (p+3)\right ) x^3 d^6+A c^3 (p+3) (2 p+5) (2 p+7) d^6+c^2 (B c+3 A d) (p+3) (2 p+5) (2 p+7) x d^6+c (p+3) \left (6 D (3 p+13) c^3+2 C d \left (2 p^2+21 p+49\right ) c^2+6 B d^2 \left (2 p^2+13 p+21\right ) c+3 A d^3 \left (4 p^2+24 p+35\right )\right ) x^2 d^5\right )dx}{d^2 (2 p+5)}-\frac {d^3 (p+3) x^3 \left (c^2-d^2 x^2\right )^{p+1} \left (B d^2 (2 p+7)+2 c^2 D (3 p+13)+3 c C d (2 p+7)\right )}{2 p+5}}{d^2 (p+3)}-\frac {d^2 (2 p+7) x^4 (3 c D+C d) \left (c^2-d^2 x^2\right )^{p+1}}{2 (p+3)}}{d^2 (2 p+7)}-\frac {d D x^5 \left (c^2-d^2 x^2\right )^{p+1}}{2 p+7}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\frac {\int \left (c^2-d^2 x^2\right )^p \left (\left (4 p^2+24 p+35\right ) \left (D (p+9) c^3+C d (3 p+11) c^2+3 B d^2 (p+3) c+A d^3 (p+3)\right ) x^3 d^6+A c^3 (p+3) (2 p+5) (2 p+7) d^6+c^2 (B c+3 A d) (p+3) (2 p+5) (2 p+7) x d^6+c (p+3) \left (6 D (3 p+13) c^3+2 C d \left (2 p^2+21 p+49\right ) c^2+6 B d^2 \left (2 p^2+13 p+21\right ) c+3 A d^3 \left (4 p^2+24 p+35\right )\right ) x^2 d^5\right )dx}{d^2 (2 p+5)}-\frac {d^3 (p+3) x^3 \left (c^2-d^2 x^2\right )^{p+1} \left (B d^2 (2 p+7)+2 c^2 D (3 p+13)+3 c C d (2 p+7)\right )}{2 p+5}}{d^2 (p+3)}-\frac {d^2 (2 p+7) x^4 (3 c D+C d) \left (c^2-d^2 x^2\right )^{p+1}}{2 (p+3)}}{d^2 (2 p+7)}-\frac {d D x^5 \left (c^2-d^2 x^2\right )^{p+1}}{2 p+7}\)

\(\Big \downarrow \) 2346

\(\displaystyle \frac {\frac {\frac {-\frac {\int -2 \left (c^2-d^2 x^2\right )^p \left (A c^3 (p+2) (p+3) (2 p+5) (2 p+7) d^8+c \left (p^2+5 p+6\right ) \left (6 D (3 p+13) c^3+2 C d \left (2 p^2+21 p+49\right ) c^2+6 B d^2 \left (2 p^2+13 p+21\right ) c+3 A d^3 \left (4 p^2+24 p+35\right )\right ) x^2 d^7+c^2 \left (4 p^2+24 p+35\right ) \left (D (p+9) c^3+C d (3 p+11) c^2+B d^2 \left (p^2+8 p+15\right ) c+A d^3 \left (3 p^2+16 p+21\right )\right ) x d^6\right )dx}{2 d^2 (p+2)}-\frac {d^4 \left (4 p^2+24 p+35\right ) x^2 \left (c^2-d^2 x^2\right )^{p+1} \left (A d^3 (p+3)+3 B c d^2 (p+3)+c^3 D (p+9)+c^2 C d (3 p+11)\right )}{2 (p+2)}}{d^2 (2 p+5)}-\frac {d^3 (p+3) x^3 \left (c^2-d^2 x^2\right )^{p+1} \left (B d^2 (2 p+7)+2 c^2 D (3 p+13)+3 c C d (2 p+7)\right )}{2 p+5}}{d^2 (p+3)}-\frac {d^2 (2 p+7) x^4 (3 c D+C d) \left (c^2-d^2 x^2\right )^{p+1}}{2 (p+3)}}{d^2 (2 p+7)}-\frac {d D x^5 \left (c^2-d^2 x^2\right )^{p+1}}{2 p+7}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\frac {\int \left (c^2-d^2 x^2\right )^p \left (A c^3 (p+2) (p+3) (2 p+5) (2 p+7) d^8+c \left (p^2+5 p+6\right ) \left (6 D (3 p+13) c^3+2 C d \left (2 p^2+21 p+49\right ) c^2+6 B d^2 \left (2 p^2+13 p+21\right ) c+3 A d^3 \left (4 p^2+24 p+35\right )\right ) x^2 d^7+c^2 \left (4 p^2+24 p+35\right ) \left (D (p+9) c^3+C d (3 p+11) c^2+B d^2 \left (p^2+8 p+15\right ) c+A d^3 \left (3 p^2+16 p+21\right )\right ) x d^6\right )dx}{d^2 (p+2)}-\frac {d^4 \left (4 p^2+24 p+35\right ) x^2 \left (c^2-d^2 x^2\right )^{p+1} \left (A d^3 (p+3)+3 B c d^2 (p+3)+c^3 D (p+9)+c^2 C d (3 p+11)\right )}{2 (p+2)}}{d^2 (2 p+5)}-\frac {d^3 (p+3) x^3 \left (c^2-d^2 x^2\right )^{p+1} \left (B d^2 (2 p+7)+2 c^2 D (3 p+13)+3 c C d (2 p+7)\right )}{2 p+5}}{d^2 (p+3)}-\frac {d^2 (2 p+7) x^4 (3 c D+C d) \left (c^2-d^2 x^2\right )^{p+1}}{2 (p+3)}}{d^2 (2 p+7)}-\frac {d D x^5 \left (c^2-d^2 x^2\right )^{p+1}}{2 p+7}\)

\(\Big \downarrow \) 2346

\(\displaystyle \frac {\frac {\frac {\frac {-\frac {\int -c^2 d^7 \left (2 c \left (p^2+5 p+6\right ) \left (3 D (3 p+13) c^3+C d \left (2 p^2+21 p+49\right ) c^2+3 B d^2 \left (2 p^2+13 p+21\right ) c+A d^3 \left (4 p^3+36 p^2+107 p+105\right )\right )+d \left (8 p^3+60 p^2+142 p+105\right ) \left (D (p+9) c^3+C d (3 p+11) c^2+B d^2 \left (p^2+8 p+15\right ) c+A d^3 \left (3 p^2+16 p+21\right )\right ) x\right ) \left (c^2-d^2 x^2\right )^pdx}{d^2 (2 p+3)}-\frac {c d^5 \left (p^2+5 p+6\right ) x \left (c^2-d^2 x^2\right )^{p+1} \left (3 A d^3 \left (4 p^2+24 p+35\right )+6 B c d^2 \left (2 p^2+13 p+21\right )+6 c^3 D (3 p+13)+2 c^2 C d \left (2 p^2+21 p+49\right )\right )}{2 p+3}}{d^2 (p+2)}-\frac {d^4 \left (4 p^2+24 p+35\right ) x^2 \left (c^2-d^2 x^2\right )^{p+1} \left (A d^3 (p+3)+3 B c d^2 (p+3)+c^3 D (p+9)+c^2 C d (3 p+11)\right )}{2 (p+2)}}{d^2 (2 p+5)}-\frac {d^3 (p+3) x^3 \left (c^2-d^2 x^2\right )^{p+1} \left (B d^2 (2 p+7)+2 c^2 D (3 p+13)+3 c C d (2 p+7)\right )}{2 p+5}}{d^2 (p+3)}-\frac {d^2 (2 p+7) x^4 (3 c D+C d) \left (c^2-d^2 x^2\right )^{p+1}}{2 (p+3)}}{d^2 (2 p+7)}-\frac {d D x^5 \left (c^2-d^2 x^2\right )^{p+1}}{2 p+7}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\frac {\frac {\frac {\int c^2 d^7 \left (2 c \left (p^2+5 p+6\right ) \left (3 D (3 p+13) c^3+C d \left (2 p^2+21 p+49\right ) c^2+3 B d^2 \left (2 p^2+13 p+21\right ) c+A d^3 \left (4 p^3+36 p^2+107 p+105\right )\right )+d \left (8 p^3+60 p^2+142 p+105\right ) \left (D (p+9) c^3+C d (3 p+11) c^2+B d^2 \left (p^2+8 p+15\right ) c+A d^3 \left (3 p^2+16 p+21\right )\right ) x\right ) \left (c^2-d^2 x^2\right )^pdx}{d^2 (2 p+3)}-\frac {c d^5 \left (p^2+5 p+6\right ) x \left (c^2-d^2 x^2\right )^{p+1} \left (3 A d^3 \left (4 p^2+24 p+35\right )+6 B c d^2 \left (2 p^2+13 p+21\right )+6 c^3 D (3 p+13)+2 c^2 C d \left (2 p^2+21 p+49\right )\right )}{2 p+3}}{d^2 (p+2)}-\frac {d^4 \left (4 p^2+24 p+35\right ) x^2 \left (c^2-d^2 x^2\right )^{p+1} \left (A d^3 (p+3)+3 B c d^2 (p+3)+c^3 D (p+9)+c^2 C d (3 p+11)\right )}{2 (p+2)}}{d^2 (2 p+5)}-\frac {d^3 (p+3) x^3 \left (c^2-d^2 x^2\right )^{p+1} \left (B d^2 (2 p+7)+2 c^2 D (3 p+13)+3 c C d (2 p+7)\right )}{2 p+5}}{d^2 (p+3)}-\frac {d^2 (2 p+7) x^4 (3 c D+C d) \left (c^2-d^2 x^2\right )^{p+1}}{2 (p+3)}}{d^2 (2 p+7)}-\frac {d D x^5 \left (c^2-d^2 x^2\right )^{p+1}}{2 p+7}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\frac {\frac {c^2 d^5 \int \left (2 c \left (p^2+5 p+6\right ) \left (3 D (3 p+13) c^3+C d \left (2 p^2+21 p+49\right ) c^2+3 B d^2 \left (2 p^2+13 p+21\right ) c+A d^3 \left (4 p^3+36 p^2+107 p+105\right )\right )+d \left (8 p^3+60 p^2+142 p+105\right ) \left (D (p+9) c^3+C d (3 p+11) c^2+B d^2 \left (p^2+8 p+15\right ) c+A d^3 \left (3 p^2+16 p+21\right )\right ) x\right ) \left (c^2-d^2 x^2\right )^pdx}{2 p+3}-\frac {c d^5 \left (p^2+5 p+6\right ) x \left (c^2-d^2 x^2\right )^{p+1} \left (3 A d^3 \left (4 p^2+24 p+35\right )+6 B c d^2 \left (2 p^2+13 p+21\right )+6 c^3 D (3 p+13)+2 c^2 C d \left (2 p^2+21 p+49\right )\right )}{2 p+3}}{d^2 (p+2)}-\frac {d^4 \left (4 p^2+24 p+35\right ) x^2 \left (c^2-d^2 x^2\right )^{p+1} \left (A d^3 (p+3)+3 B c d^2 (p+3)+c^3 D (p+9)+c^2 C d (3 p+11)\right )}{2 (p+2)}}{d^2 (2 p+5)}-\frac {d^3 (p+3) x^3 \left (c^2-d^2 x^2\right )^{p+1} \left (B d^2 (2 p+7)+2 c^2 D (3 p+13)+3 c C d (2 p+7)\right )}{2 p+5}}{d^2 (p+3)}-\frac {d^2 (2 p+7) x^4 (3 c D+C d) \left (c^2-d^2 x^2\right )^{p+1}}{2 (p+3)}}{d^2 (2 p+7)}-\frac {d D x^5 \left (c^2-d^2 x^2\right )^{p+1}}{2 p+7}\)

\(\Big \downarrow \) 455

\(\displaystyle \frac {\frac {\frac {\frac {\frac {c^2 d^5 \left (2 c \left (p^2+5 p+6\right ) \left (A d^3 \left (4 p^3+36 p^2+107 p+105\right )+3 B c d^2 \left (2 p^2+13 p+21\right )+3 c^3 D (3 p+13)+c^2 C d \left (2 p^2+21 p+49\right )\right ) \int \left (c^2-d^2 x^2\right )^pdx-\frac {\left (8 p^3+60 p^2+142 p+105\right ) \left (c^2-d^2 x^2\right )^{p+1} \left (A d^3 \left (3 p^2+16 p+21\right )+B c d^2 \left (p^2+8 p+15\right )+c^3 D (p+9)+c^2 C d (3 p+11)\right )}{2 d (p+1)}\right )}{2 p+3}-\frac {c d^5 \left (p^2+5 p+6\right ) x \left (c^2-d^2 x^2\right )^{p+1} \left (3 A d^3 \left (4 p^2+24 p+35\right )+6 B c d^2 \left (2 p^2+13 p+21\right )+6 c^3 D (3 p+13)+2 c^2 C d \left (2 p^2+21 p+49\right )\right )}{2 p+3}}{d^2 (p+2)}-\frac {d^4 \left (4 p^2+24 p+35\right ) x^2 \left (c^2-d^2 x^2\right )^{p+1} \left (A d^3 (p+3)+3 B c d^2 (p+3)+c^3 D (p+9)+c^2 C d (3 p+11)\right )}{2 (p+2)}}{d^2 (2 p+5)}-\frac {d^3 (p+3) x^3 \left (c^2-d^2 x^2\right )^{p+1} \left (B d^2 (2 p+7)+2 c^2 D (3 p+13)+3 c C d (2 p+7)\right )}{2 p+5}}{d^2 (p+3)}-\frac {d^2 (2 p+7) x^4 (3 c D+C d) \left (c^2-d^2 x^2\right )^{p+1}}{2 (p+3)}}{d^2 (2 p+7)}-\frac {d D x^5 \left (c^2-d^2 x^2\right )^{p+1}}{2 p+7}\)

\(\Big \downarrow \) 238

\(\displaystyle \frac {\frac {\frac {\frac {\frac {c^2 d^5 \left (2 c \left (p^2+5 p+6\right ) \left (c^2-d^2 x^2\right )^p \left (1-\frac {d^2 x^2}{c^2}\right )^{-p} \left (A d^3 \left (4 p^3+36 p^2+107 p+105\right )+3 B c d^2 \left (2 p^2+13 p+21\right )+3 c^3 D (3 p+13)+c^2 C d \left (2 p^2+21 p+49\right )\right ) \int \left (1-\frac {d^2 x^2}{c^2}\right )^pdx-\frac {\left (8 p^3+60 p^2+142 p+105\right ) \left (c^2-d^2 x^2\right )^{p+1} \left (A d^3 \left (3 p^2+16 p+21\right )+B c d^2 \left (p^2+8 p+15\right )+c^3 D (p+9)+c^2 C d (3 p+11)\right )}{2 d (p+1)}\right )}{2 p+3}-\frac {c d^5 \left (p^2+5 p+6\right ) x \left (c^2-d^2 x^2\right )^{p+1} \left (3 A d^3 \left (4 p^2+24 p+35\right )+6 B c d^2 \left (2 p^2+13 p+21\right )+6 c^3 D (3 p+13)+2 c^2 C d \left (2 p^2+21 p+49\right )\right )}{2 p+3}}{d^2 (p+2)}-\frac {d^4 \left (4 p^2+24 p+35\right ) x^2 \left (c^2-d^2 x^2\right )^{p+1} \left (A d^3 (p+3)+3 B c d^2 (p+3)+c^3 D (p+9)+c^2 C d (3 p+11)\right )}{2 (p+2)}}{d^2 (2 p+5)}-\frac {d^3 (p+3) x^3 \left (c^2-d^2 x^2\right )^{p+1} \left (B d^2 (2 p+7)+2 c^2 D (3 p+13)+3 c C d (2 p+7)\right )}{2 p+5}}{d^2 (p+3)}-\frac {d^2 (2 p+7) x^4 (3 c D+C d) \left (c^2-d^2 x^2\right )^{p+1}}{2 (p+3)}}{d^2 (2 p+7)}-\frac {d D x^5 \left (c^2-d^2 x^2\right )^{p+1}}{2 p+7}\)

\(\Big \downarrow \) 237

\(\displaystyle \frac {\frac {\frac {\frac {\frac {c^2 d^5 \left (2 c \left (p^2+5 p+6\right ) x \left (c^2-d^2 x^2\right )^p \left (1-\frac {d^2 x^2}{c^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},\frac {d^2 x^2}{c^2}\right ) \left (A d^3 \left (4 p^3+36 p^2+107 p+105\right )+3 B c d^2 \left (2 p^2+13 p+21\right )+3 c^3 D (3 p+13)+c^2 C d \left (2 p^2+21 p+49\right )\right )-\frac {\left (8 p^3+60 p^2+142 p+105\right ) \left (c^2-d^2 x^2\right )^{p+1} \left (A d^3 \left (3 p^2+16 p+21\right )+B c d^2 \left (p^2+8 p+15\right )+c^3 D (p+9)+c^2 C d (3 p+11)\right )}{2 d (p+1)}\right )}{2 p+3}-\frac {c d^5 \left (p^2+5 p+6\right ) x \left (c^2-d^2 x^2\right )^{p+1} \left (3 A d^3 \left (4 p^2+24 p+35\right )+6 B c d^2 \left (2 p^2+13 p+21\right )+6 c^3 D (3 p+13)+2 c^2 C d \left (2 p^2+21 p+49\right )\right )}{2 p+3}}{d^2 (p+2)}-\frac {d^4 \left (4 p^2+24 p+35\right ) x^2 \left (c^2-d^2 x^2\right )^{p+1} \left (A d^3 (p+3)+3 B c d^2 (p+3)+c^3 D (p+9)+c^2 C d (3 p+11)\right )}{2 (p+2)}}{d^2 (2 p+5)}-\frac {d^3 (p+3) x^3 \left (c^2-d^2 x^2\right )^{p+1} \left (B d^2 (2 p+7)+2 c^2 D (3 p+13)+3 c C d (2 p+7)\right )}{2 p+5}}{d^2 (p+3)}-\frac {d^2 (2 p+7) x^4 (3 c D+C d) \left (c^2-d^2 x^2\right )^{p+1}}{2 (p+3)}}{d^2 (2 p+7)}-\frac {d D x^5 \left (c^2-d^2 x^2\right )^{p+1}}{2 p+7}\)

Input:

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

Output:

-((d*D*x^5*(c^2 - d^2*x^2)^(1 + p))/(7 + 2*p)) + (-1/2*(d^2*(C*d + 3*c*D)* 
(7 + 2*p)*x^4*(c^2 - d^2*x^2)^(1 + p))/(3 + p) + (-((d^3*(3 + p)*(3*c*C*d* 
(7 + 2*p) + B*d^2*(7 + 2*p) + 2*c^2*D*(13 + 3*p))*x^3*(c^2 - d^2*x^2)^(1 + 
 p))/(5 + 2*p)) + (-1/2*(d^4*(35 + 24*p + 4*p^2)*(3*B*c*d^2*(3 + p) + A*d^ 
3*(3 + p) + c^3*D*(9 + p) + c^2*C*d*(11 + 3*p))*x^2*(c^2 - d^2*x^2)^(1 + p 
))/(2 + p) + (-((c*d^5*(6 + 5*p + p^2)*(6*c^3*D*(13 + 3*p) + 6*B*c*d^2*(21 
 + 13*p + 2*p^2) + 2*c^2*C*d*(49 + 21*p + 2*p^2) + 3*A*d^3*(35 + 24*p + 4* 
p^2))*x*(c^2 - d^2*x^2)^(1 + p))/(3 + 2*p)) + (c^2*d^5*(-1/2*((105 + 142*p 
 + 60*p^2 + 8*p^3)*(c^3*D*(9 + p) + c^2*C*d*(11 + 3*p) + B*c*d^2*(15 + 8*p 
 + p^2) + A*d^3*(21 + 16*p + 3*p^2))*(c^2 - d^2*x^2)^(1 + p))/(d*(1 + p)) 
+ (2*c*(6 + 5*p + p^2)*(3*c^3*D*(13 + 3*p) + 3*B*c*d^2*(21 + 13*p + 2*p^2) 
 + c^2*C*d*(49 + 21*p + 2*p^2) + A*d^3*(105 + 107*p + 36*p^2 + 4*p^3))*x*( 
c^2 - d^2*x^2)^p*Hypergeometric2F1[1/2, -p, 3/2, (d^2*x^2)/c^2])/(1 - (d^2 
*x^2)/c^2)^p))/(3 + 2*p))/(d^2*(2 + p)))/(d^2*(5 + 2*p)))/(d^2*(3 + p)))/( 
d^2*(7 + 2*p))
 

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 237

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F1[- 
p, 1/2, 1/2 + 1, (-b)*(x^2/a)], x] /; FreeQ[{a, b, p}, x] &&  !IntegerQ[2*p 
] && GtQ[a, 0]
 

rule 238

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x^2) 
^FracPart[p]/(1 + b*(x^2/a))^FracPart[p])   Int[(1 + b*(x^2/a))^p, x], x] / 
; FreeQ[{a, b, p}, x] &&  !IntegerQ[2*p] &&  !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 [F]

\[\int \left (d x +c \right )^{3} \left (-d^{2} x^{2}+c^{2}\right )^{p} \left (D x^{3}+C \,x^{2}+B x +A \right )d x\]

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((D*d^3*x^6 + (3*D*c*d^2 + C*d^3)*x^5 + (3*D*c^2*d + 3*C*c*d^2 + B 
*d^3)*x^4 + A*c^3 + (D*c^3 + 3*C*c^2*d + 3*B*c*d^2 + A*d^3)*x^3 + (C*c^3 + 
 3*B*c^2*d + 3*A*c*d^2)*x^2 + (B*c^3 + 3*A*c^2*d)*x)*(-d^2*x^2 + c^2)^p, x 
)
 

Sympy [A] (verification not implemented)

Time = 7.28 (sec) , antiderivative size = 3728, normalized size of antiderivative = 7.53 \[ \int (c+d x)^3 \left (c^2-d^2 x^2\right )^p \left (A+B x+C x^2+D x^3\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

A*c**3*c**(2*p)*x*hyper((1/2, -p), (3/2,), d**2*x**2*exp_polar(2*I*pi)/c** 
2) + 3*A*c**2*d*Piecewise((x**2*(c**2)**p/2, Eq(d**2, 0)), (-Piecewise(((c 
**2 - d**2*x**2)**(p + 1)/(p + 1), Ne(p, -1)), (log(c**2 - d**2*x**2), Tru 
e))/(2*d**2), True)) + A*c*c**(2*p)*d**2*x**3*hyper((3/2, -p), (5/2,), d** 
2*x**2*exp_polar(2*I*pi)/c**2) + A*d**3*Piecewise((x**4*(c**2)**p/4, Eq(d, 
 0)), (-c**2*log(-c/d + x)/(-2*c**2*d**4 + 2*d**6*x**2) - c**2*log(c/d + x 
)/(-2*c**2*d**4 + 2*d**6*x**2) - c**2/(-2*c**2*d**4 + 2*d**6*x**2) + d**2* 
x**2*log(-c/d + x)/(-2*c**2*d**4 + 2*d**6*x**2) + d**2*x**2*log(c/d + x)/( 
-2*c**2*d**4 + 2*d**6*x**2), Eq(p, -2)), (-c**2*log(-c/d + x)/(2*d**4) - c 
**2*log(c/d + x)/(2*d**4) - x**2/(2*d**2), Eq(p, -1)), (-c**4*(c**2 - d**2 
*x**2)**p/(2*d**4*p**2 + 6*d**4*p + 4*d**4) - c**2*d**2*p*x**2*(c**2 - d** 
2*x**2)**p/(2*d**4*p**2 + 6*d**4*p + 4*d**4) + d**4*p*x**4*(c**2 - d**2*x* 
*2)**p/(2*d**4*p**2 + 6*d**4*p + 4*d**4) + d**4*x**4*(c**2 - d**2*x**2)**p 
/(2*d**4*p**2 + 6*d**4*p + 4*d**4), True)) + B*c**3*Piecewise((x**2*(c**2) 
**p/2, Eq(d**2, 0)), (-Piecewise(((c**2 - d**2*x**2)**(p + 1)/(p + 1), Ne( 
p, -1)), (log(c**2 - d**2*x**2), True))/(2*d**2), True)) + B*c**2*c**(2*p) 
*d*x**3*hyper((3/2, -p), (5/2,), d**2*x**2*exp_polar(2*I*pi)/c**2) + 3*B*c 
*d**2*Piecewise((x**4*(c**2)**p/4, Eq(d, 0)), (-c**2*log(-c/d + x)/(-2*c** 
2*d**4 + 2*d**6*x**2) - c**2*log(c/d + x)/(-2*c**2*d**4 + 2*d**6*x**2) - c 
**2/(-2*c**2*d**4 + 2*d**6*x**2) + d**2*x**2*log(-c/d + x)/(-2*c**2*d**...
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (c+d x)^3 \left (c^2-d^2 x^2\right )^p \left (A+B x+C x^2+D x^3\right ) \, dx=\text {too large to display} \] Input:

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

Output:

( - 48*(c**2 - d**2*x**2)**p*a*c**4*d**2*p**6 - 640*(c**2 - d**2*x**2)**p* 
a*c**4*d**2*p**5 - 3416*(c**2 - d**2*x**2)**p*a*c**4*d**2*p**4 - 9248*(c** 
2 - d**2*x**2)**p*a*c**4*d**2*p**3 - 13171*(c**2 - d**2*x**2)**p*a*c**4*d* 
*2*p**2 - 9072*(c**2 - d**2*x**2)**p*a*c**4*d**2*p - 2205*(c**2 - d**2*x** 
2)**p*a*c**4*d**2 - 32*(c**2 - d**2*x**2)**p*a*c**3*d**3*p**6*x - 360*(c** 
2 - d**2*x**2)**p*a*c**3*d**3*p**5*x - 1496*(c**2 - d**2*x**2)**p*a*c**3*d 
**3*p**4*x - 2646*(c**2 - d**2*x**2)**p*a*c**3*d**3*p**3*x - 1244*(c**2 - 
d**2*x**2)**p*a*c**3*d**3*p**2*x + 1494*(c**2 - d**2*x**2)**p*a*c**3*d**3* 
p*x + 1260*(c**2 - d**2*x**2)**p*a*c**3*d**3*x + 32*(c**2 - d**2*x**2)**p* 
a*c**2*d**4*p**6*x**2 + 448*(c**2 - d**2*x**2)**p*a*c**2*d**4*p**5*x**2 + 
2512*(c**2 - d**2*x**2)**p*a*c**2*d**4*p**4*x**2 + 7136*(c**2 - d**2*x**2) 
**p*a*c**2*d**4*p**3*x**2 + 10626*(c**2 - d**2*x**2)**p*a*c**2*d**4*p**2*x 
**2 + 7596*(c**2 - d**2*x**2)**p*a*c**2*d**4*p*x**2 + 1890*(c**2 - d**2*x* 
*2)**p*a*c**2*d**4*x**2 + 48*(c**2 - d**2*x**2)**p*a*c*d**5*p**6*x**3 + 60 
0*(c**2 - d**2*x**2)**p*a*c*d**5*p**5*x**3 + 2964*(c**2 - d**2*x**2)**p*a* 
c*d**5*p**4*x**3 + 7314*(c**2 - d**2*x**2)**p*a*c*d**5*p**3*x**3 + 9336*(c 
**2 - d**2*x**2)**p*a*c*d**5*p**2*x**3 + 5694*(c**2 - d**2*x**2)**p*a*c*d* 
*5*p*x**3 + 1260*(c**2 - d**2*x**2)**p*a*c*d**5*x**3 + 16*(c**2 - d**2*x** 
2)**p*a*d**6*p**6*x**4 + 192*(c**2 - d**2*x**2)**p*a*d**6*p**5*x**4 + 904* 
(c**2 - d**2*x**2)**p*a*d**6*p**4*x**4 + 2112*(c**2 - d**2*x**2)**p*a*d...