\(\int (c+d x)^3 (a+b x^2)^{3/2} (A+B x+C x^2+D x^3) \, dx\) [71]

Optimal result

Integrand size = 34, antiderivative size = 591 \[ \int (c+d x)^3 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {a \left (96 A b^3 c^3-a \left (16 b^2 c \left (c^2 C+3 B c d+3 A d^2\right )+3 a^2 d^3 D-6 a b d \left (3 c C d+B d^2+3 c^2 D\right )\right )\right ) x \sqrt {a+b x^2}}{256 b^3}+\frac {\left (96 A b^3 c^3-a \left (16 b^2 c \left (c^2 C+3 B c d+3 A d^2\right )+3 a^2 d^3 D-6 a b d \left (3 c C d+B d^2+3 c^2 D\right )\right )\right ) x \left (a+b x^2\right )^{3/2}}{384 b^3}+\frac {\left (b^2 c^2 (B c+3 A d)+a^2 d^2 (C d+3 c D)-a b \left (3 c^2 C d+3 B c d^2+A d^3+c^3 D\right )\right ) \left (a+b x^2\right )^{5/2}}{5 b^3}+\frac {\left (16 b^2 c \left (c^2 C+3 B c d+3 A d^2\right )+3 a^2 d^3 D-6 a b d \left (3 c C d+B d^2+3 c^2 D\right )\right ) x \left (a+b x^2\right )^{5/2}}{96 b^3}-\frac {d \left (a d^2 D-2 b \left (3 c C d+B d^2+3 c^2 D\right )\right ) x^3 \left (a+b x^2\right )^{5/2}}{16 b^2}+\frac {d^3 D x^5 \left (a+b x^2\right )^{5/2}}{10 b}-\frac {\left (2 a d^2 (C d+3 c D)-b \left (3 c^2 C d+3 B c d^2+A d^3+c^3 D\right )\right ) \left (a+b x^2\right )^{7/2}}{7 b^3}+\frac {d^2 (C d+3 c D) \left (a+b x^2\right )^{9/2}}{9 b^3}+\frac {a^2 \left (96 A b^3 c^3-a \left (16 b^2 c \left (c^2 C+3 B c d+3 A d^2\right )+3 a^2 d^3 D-6 a b d \left (3 c C d+B d^2+3 c^2 D\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{256 b^{7/2}} \] Output:

1/256*a*(96*A*b^3*c^3-a*(16*b^2*c*(3*A*d^2+3*B*c*d+C*c^2)+3*a^2*d^3*D-6*a* 
b*d*(B*d^2+3*C*c*d+3*D*c^2)))*x*(b*x^2+a)^(1/2)/b^3+1/384*(96*A*b^3*c^3-a* 
(16*b^2*c*(3*A*d^2+3*B*c*d+C*c^2)+3*a^2*d^3*D-6*a*b*d*(B*d^2+3*C*c*d+3*D*c 
^2)))*x*(b*x^2+a)^(3/2)/b^3+1/5*(b^2*c^2*(3*A*d+B*c)+a^2*d^2*(C*d+3*D*c)-a 
*b*(A*d^3+3*B*c*d^2+3*C*c^2*d+D*c^3))*(b*x^2+a)^(5/2)/b^3+1/96*(16*b^2*c*( 
3*A*d^2+3*B*c*d+C*c^2)+3*a^2*d^3*D-6*a*b*d*(B*d^2+3*C*c*d+3*D*c^2))*x*(b*x 
^2+a)^(5/2)/b^3-1/16*d*(a*d^2*D-2*b*(B*d^2+3*C*c*d+3*D*c^2))*x^3*(b*x^2+a) 
^(5/2)/b^2+1/10*d^3*D*x^5*(b*x^2+a)^(5/2)/b-1/7*(2*a*d^2*(C*d+3*D*c)-b*(A* 
d^3+3*B*c*d^2+3*C*c^2*d+D*c^3))*(b*x^2+a)^(7/2)/b^3+1/9*d^2*(C*d+3*D*c)*(b 
*x^2+a)^(9/2)/b^3+1/256*a^2*(96*A*b^3*c^3-a*(16*b^2*c*(3*A*d^2+3*B*c*d+C*c 
^2)+3*a^2*d^3*D-6*a*b*d*(B*d^2+3*C*c*d+3*D*c^2)))*arctanh(b^(1/2)*x/(b*x^2 
+a)^(1/2))/b^(7/2)
 

Mathematica [A] (verified)

Time = 4.41 (sec) , antiderivative size = 627, normalized size of antiderivative = 1.06 \[ \int (c+d x)^3 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {\sqrt {b} \sqrt {a+b x^2} \left (a^4 d^2 (2048 C d+6144 c D+945 d D x)+12 a^2 b^2 \left (12 A d \left (336 c^2+105 c d x+16 d^2 x^2\right )+3 B \left (448 c^3+420 c^2 d x+192 c d^2 x^2+35 d^3 x^3\right )+x \left (12 c^3 (35 C+16 D x)+2 d^3 x^3 (32 C+21 D x)+9 c^2 d x (64 C+35 D x)+3 c d^2 x^2 (105 C+64 D x)\right )\right )-2 a^3 b \left (2304 c^3 D+27 c^2 d (256 C+105 D x)+3 c d^2 (2304 B+x (945 C+512 D x))+d^3 \left (2304 A+x \left (945 B+512 C x+315 D x^2\right )\right )\right )+32 b^4 x^3 \left (18 A \left (35 c^3+84 c^2 d x+70 c d^2 x^2+20 d^3 x^3\right )+x \left (9 B \left (56 c^3+140 c^2 d x+120 c d^2 x^2+35 d^3 x^3\right )+x \left (60 c^3 (7 C+6 D x)+135 c^2 d x (8 C+7 D x)+105 c d^2 x^2 (9 C+8 D x)+28 d^3 x^3 (10 C+9 D x)\right )\right )\right )+16 a b^3 x \left (18 A \left (175 c^3+336 c^2 d x+245 c d^2 x^2+64 d^3 x^3\right )+x \left (9 B \left (224 c^3+490 c^2 d x+384 c d^2 x^2+105 d^3 x^3\right )+x \left (27 c^2 d x (128 C+105 D x)+15 c d^2 x^2 (189 C+160 D x)+6 c^3 (245 C+192 D x)+d^3 x^3 (800 C+693 D x)\right )\right )\right )\right )+315 a^2 \left (48 A b^2 c \left (-2 b c^2+a d^2\right )+a \left (16 b^2 c^2 (c C+3 B d)+3 a^2 d^3 D-6 a b d \left (3 c C d+B d^2+3 c^2 D\right )\right )\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{80640 b^{7/2}} \] Input:

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

Output:

(Sqrt[b]*Sqrt[a + b*x^2]*(a^4*d^2*(2048*C*d + 6144*c*D + 945*d*D*x) + 12*a 
^2*b^2*(12*A*d*(336*c^2 + 105*c*d*x + 16*d^2*x^2) + 3*B*(448*c^3 + 420*c^2 
*d*x + 192*c*d^2*x^2 + 35*d^3*x^3) + x*(12*c^3*(35*C + 16*D*x) + 2*d^3*x^3 
*(32*C + 21*D*x) + 9*c^2*d*x*(64*C + 35*D*x) + 3*c*d^2*x^2*(105*C + 64*D*x 
))) - 2*a^3*b*(2304*c^3*D + 27*c^2*d*(256*C + 105*D*x) + 3*c*d^2*(2304*B + 
 x*(945*C + 512*D*x)) + d^3*(2304*A + x*(945*B + 512*C*x + 315*D*x^2))) + 
32*b^4*x^3*(18*A*(35*c^3 + 84*c^2*d*x + 70*c*d^2*x^2 + 20*d^3*x^3) + x*(9* 
B*(56*c^3 + 140*c^2*d*x + 120*c*d^2*x^2 + 35*d^3*x^3) + x*(60*c^3*(7*C + 6 
*D*x) + 135*c^2*d*x*(8*C + 7*D*x) + 105*c*d^2*x^2*(9*C + 8*D*x) + 28*d^3*x 
^3*(10*C + 9*D*x)))) + 16*a*b^3*x*(18*A*(175*c^3 + 336*c^2*d*x + 245*c*d^2 
*x^2 + 64*d^3*x^3) + x*(9*B*(224*c^3 + 490*c^2*d*x + 384*c*d^2*x^2 + 105*d 
^3*x^3) + x*(27*c^2*d*x*(128*C + 105*D*x) + 15*c*d^2*x^2*(189*C + 160*D*x) 
 + 6*c^3*(245*C + 192*D*x) + d^3*x^3*(800*C + 693*D*x))))) + 315*a^2*(48*A 
*b^2*c*(-2*b*c^2 + a*d^2) + a*(16*b^2*c^2*(c*C + 3*B*d) + 3*a^2*d^3*D - 6* 
a*b*d*(3*c*C*d + B*d^2 + 3*c^2*D)))*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/( 
80640*b^(7/2))
 

Rubi [A] (verified)

Time = 1.19 (sec) , antiderivative size = 554, normalized size of antiderivative = 0.94, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {2185, 27, 2185, 27, 687, 687, 27, 676, 211, 211, 224, 219}

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.

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

\(\Big \downarrow \) 2185

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2185

\(\displaystyle \frac {\frac {\int b d^3 (c+d x)^3 \left (d (18 A b d-8 a C d+3 a c D)-\left (9 a D d^2+2 b \left (-3 D c^2+5 C d c-9 B d^2\right )\right ) x\right ) \left (b x^2+a\right )^{3/2}dx}{9 b d^2}+\frac {1}{9} d \left (a+b x^2\right )^{5/2} (c+d x)^4 (2 C d-3 c D)}{2 b d^3}+\frac {D \left (a+b x^2\right )^{5/2} (c+d x)^5}{10 b d^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{9} d \int (c+d x)^3 \left (d (18 A b d-8 a C d+3 a c D)-\left (9 a D d^2+2 b \left (-3 D c^2+5 C d c-9 B d^2\right )\right ) x\right ) \left (b x^2+a\right )^{3/2}dx+\frac {1}{9} d \left (a+b x^2\right )^{5/2} (c+d x)^4 (2 C d-3 c D)}{2 b d^3}+\frac {D \left (a+b x^2\right )^{5/2} (c+d x)^5}{10 b d^2}\)

\(\Big \downarrow \) 687

\(\displaystyle \frac {\frac {1}{9} d \left (\frac {\int (c+d x)^2 \left (d \left (144 A c d b^2+a \left (27 a d^2 D-b \left (-6 D c^2+34 C d c+54 B d^2\right )\right )\right )-b \left (a (64 C d+3 c D) d^2+6 b \left (-3 D c^3+5 C d c^2-9 B d^2 c-24 A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{3/2}dx}{8 b}-\frac {\left (a+b x^2\right )^{5/2} (c+d x)^3 \left (9 a d^2 D-18 b B d^2-6 b c^2 D+10 b c C d\right )}{8 b}\right )+\frac {1}{9} d \left (a+b x^2\right )^{5/2} (c+d x)^4 (2 C d-3 c D)}{2 b d^3}+\frac {D \left (a+b x^2\right )^{5/2} (c+d x)^5}{10 b d^2}\)

\(\Big \downarrow \) 687

\(\displaystyle \frac {\frac {1}{9} d \left (\frac {\frac {\int b (c+d x) \left (d \left (144 A b d \left (7 b c^2-2 a d^2\right )+a \left (a d^2 (128 C d+195 c D)-2 b c \left (-3 D c^2+89 C d c+243 B d^2\right )\right )\right )+3 \left (63 a^2 D d^4-2 a b \left (-6 D c^2+61 C d c+63 B d^2\right ) d^2-4 b^2 c \left (-3 D c^3+5 C d c^2-9 B d^2 c-108 A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{3/2}dx}{7 b}-\frac {1}{7} \left (a+b x^2\right )^{5/2} (c+d x)^2 \left (a d^2 (3 c D+64 C d)+6 b \left (-24 A d^3-9 B c d^2-3 c^3 D+5 c^2 C d\right )\right )}{8 b}-\frac {\left (a+b x^2\right )^{5/2} (c+d x)^3 \left (9 a d^2 D-18 b B d^2-6 b c^2 D+10 b c C d\right )}{8 b}\right )+\frac {1}{9} d \left (a+b x^2\right )^{5/2} (c+d x)^4 (2 C d-3 c D)}{2 b d^3}+\frac {D \left (a+b x^2\right )^{5/2} (c+d x)^5}{10 b d^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{9} d \left (\frac {\frac {1}{7} \int (c+d x) \left (d \left (144 A b d \left (7 b c^2-2 a d^2\right )+a \left (a d^2 (128 C d+195 c D)-2 b c \left (-3 D c^2+89 C d c+243 B d^2\right )\right )\right )+3 \left (63 a^2 D d^4-2 a b \left (-6 D c^2+61 C d c+63 B d^2\right ) d^2-4 b^2 c \left (-3 D c^3+5 C d c^2-9 B d^2 c-108 A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{3/2}dx-\frac {1}{7} \left (a+b x^2\right )^{5/2} (c+d x)^2 \left (a d^2 (3 c D+64 C d)+6 b \left (-24 A d^3-9 B c d^2-3 c^3 D+5 c^2 C d\right )\right )}{8 b}-\frac {\left (a+b x^2\right )^{5/2} (c+d x)^3 \left (9 a d^2 D-18 b B d^2-6 b c^2 D+10 b c C d\right )}{8 b}\right )+\frac {1}{9} d \left (a+b x^2\right )^{5/2} (c+d x)^4 (2 C d-3 c D)}{2 b d^3}+\frac {D \left (a+b x^2\right )^{5/2} (c+d x)^5}{10 b d^2}\)

\(\Big \downarrow \) 676

\(\displaystyle \frac {\frac {1}{9} d \left (\frac {\frac {1}{7} \left (\frac {21 d^2 \left (48 A b^2 c \left (2 b c^2-a d^2\right )-a \left (3 a^2 d^3 D-6 a b d \left (B d^2+3 c^2 D+3 c C d\right )+16 b^2 c^2 (3 B d+c C)\right )\right ) \int \left (b x^2+a\right )^{3/2}dx}{2 b}+\frac {2 \left (a+b x^2\right )^{5/2} \left (64 a^2 d^4 (3 c D+C d)-a b d^2 \left (144 A d^3+432 B c d^2-21 c^3 D+272 c^2 C d\right )-6 b^2 c^2 \left (-192 A d^3-9 B c d^2-3 c^3 D+5 c^2 C d\right )\right )}{5 b}+\frac {d x \left (a+b x^2\right )^{5/2} \left (63 a^2 d^4 D-2 a b d^2 \left (63 B d^2-6 c^2 D+61 c C d\right )-4 b^2 c \left (-108 A d^3-9 B c d^2-3 c^3 D+5 c^2 C d\right )\right )}{2 b}\right )-\frac {1}{7} \left (a+b x^2\right )^{5/2} (c+d x)^2 \left (a d^2 (3 c D+64 C d)+6 b \left (-24 A d^3-9 B c d^2-3 c^3 D+5 c^2 C d\right )\right )}{8 b}-\frac {\left (a+b x^2\right )^{5/2} (c+d x)^3 \left (9 a d^2 D-18 b B d^2-6 b c^2 D+10 b c C d\right )}{8 b}\right )+\frac {1}{9} d \left (a+b x^2\right )^{5/2} (c+d x)^4 (2 C d-3 c D)}{2 b d^3}+\frac {D \left (a+b x^2\right )^{5/2} (c+d x)^5}{10 b d^2}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\frac {1}{9} d \left (\frac {\frac {1}{7} \left (\frac {21 d^2 \left (48 A b^2 c \left (2 b c^2-a d^2\right )-a \left (3 a^2 d^3 D-6 a b d \left (B d^2+3 c^2 D+3 c C d\right )+16 b^2 c^2 (3 B d+c C)\right )\right ) \left (\frac {3}{4} a \int \sqrt {b x^2+a}dx+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )}{2 b}+\frac {2 \left (a+b x^2\right )^{5/2} \left (64 a^2 d^4 (3 c D+C d)-a b d^2 \left (144 A d^3+432 B c d^2-21 c^3 D+272 c^2 C d\right )-6 b^2 c^2 \left (-192 A d^3-9 B c d^2-3 c^3 D+5 c^2 C d\right )\right )}{5 b}+\frac {d x \left (a+b x^2\right )^{5/2} \left (63 a^2 d^4 D-2 a b d^2 \left (63 B d^2-6 c^2 D+61 c C d\right )-4 b^2 c \left (-108 A d^3-9 B c d^2-3 c^3 D+5 c^2 C d\right )\right )}{2 b}\right )-\frac {1}{7} \left (a+b x^2\right )^{5/2} (c+d x)^2 \left (a d^2 (3 c D+64 C d)+6 b \left (-24 A d^3-9 B c d^2-3 c^3 D+5 c^2 C d\right )\right )}{8 b}-\frac {\left (a+b x^2\right )^{5/2} (c+d x)^3 \left (9 a d^2 D-18 b B d^2-6 b c^2 D+10 b c C d\right )}{8 b}\right )+\frac {1}{9} d \left (a+b x^2\right )^{5/2} (c+d x)^4 (2 C d-3 c D)}{2 b d^3}+\frac {D \left (a+b x^2\right )^{5/2} (c+d x)^5}{10 b d^2}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\frac {1}{9} d \left (\frac {\frac {1}{7} \left (\frac {21 d^2 \left (48 A b^2 c \left (2 b c^2-a d^2\right )-a \left (3 a^2 d^3 D-6 a b d \left (B d^2+3 c^2 D+3 c C d\right )+16 b^2 c^2 (3 B d+c C)\right )\right ) \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{\sqrt {b x^2+a}}dx+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )}{2 b}+\frac {2 \left (a+b x^2\right )^{5/2} \left (64 a^2 d^4 (3 c D+C d)-a b d^2 \left (144 A d^3+432 B c d^2-21 c^3 D+272 c^2 C d\right )-6 b^2 c^2 \left (-192 A d^3-9 B c d^2-3 c^3 D+5 c^2 C d\right )\right )}{5 b}+\frac {d x \left (a+b x^2\right )^{5/2} \left (63 a^2 d^4 D-2 a b d^2 \left (63 B d^2-6 c^2 D+61 c C d\right )-4 b^2 c \left (-108 A d^3-9 B c d^2-3 c^3 D+5 c^2 C d\right )\right )}{2 b}\right )-\frac {1}{7} \left (a+b x^2\right )^{5/2} (c+d x)^2 \left (a d^2 (3 c D+64 C d)+6 b \left (-24 A d^3-9 B c d^2-3 c^3 D+5 c^2 C d\right )\right )}{8 b}-\frac {\left (a+b x^2\right )^{5/2} (c+d x)^3 \left (9 a d^2 D-18 b B d^2-6 b c^2 D+10 b c C d\right )}{8 b}\right )+\frac {1}{9} d \left (a+b x^2\right )^{5/2} (c+d x)^4 (2 C d-3 c D)}{2 b d^3}+\frac {D \left (a+b x^2\right )^{5/2} (c+d x)^5}{10 b d^2}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\frac {1}{9} d \left (\frac {\frac {1}{7} \left (\frac {21 d^2 \left (48 A b^2 c \left (2 b c^2-a d^2\right )-a \left (3 a^2 d^3 D-6 a b d \left (B d^2+3 c^2 D+3 c C d\right )+16 b^2 c^2 (3 B d+c C)\right )\right ) \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )}{2 b}+\frac {2 \left (a+b x^2\right )^{5/2} \left (64 a^2 d^4 (3 c D+C d)-a b d^2 \left (144 A d^3+432 B c d^2-21 c^3 D+272 c^2 C d\right )-6 b^2 c^2 \left (-192 A d^3-9 B c d^2-3 c^3 D+5 c^2 C d\right )\right )}{5 b}+\frac {d x \left (a+b x^2\right )^{5/2} \left (63 a^2 d^4 D-2 a b d^2 \left (63 B d^2-6 c^2 D+61 c C d\right )-4 b^2 c \left (-108 A d^3-9 B c d^2-3 c^3 D+5 c^2 C d\right )\right )}{2 b}\right )-\frac {1}{7} \left (a+b x^2\right )^{5/2} (c+d x)^2 \left (a d^2 (3 c D+64 C d)+6 b \left (-24 A d^3-9 B c d^2-3 c^3 D+5 c^2 C d\right )\right )}{8 b}-\frac {\left (a+b x^2\right )^{5/2} (c+d x)^3 \left (9 a d^2 D-18 b B d^2-6 b c^2 D+10 b c C d\right )}{8 b}\right )+\frac {1}{9} d \left (a+b x^2\right )^{5/2} (c+d x)^4 (2 C d-3 c D)}{2 b d^3}+\frac {D \left (a+b x^2\right )^{5/2} (c+d x)^5}{10 b d^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {1}{9} d \left (\frac {\frac {1}{7} \left (\frac {21 d^2 \left (\frac {3}{4} a \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right ) \left (48 A b^2 c \left (2 b c^2-a d^2\right )-a \left (3 a^2 d^3 D-6 a b d \left (B d^2+3 c^2 D+3 c C d\right )+16 b^2 c^2 (3 B d+c C)\right )\right )}{2 b}+\frac {2 \left (a+b x^2\right )^{5/2} \left (64 a^2 d^4 (3 c D+C d)-a b d^2 \left (144 A d^3+432 B c d^2-21 c^3 D+272 c^2 C d\right )-6 b^2 c^2 \left (-192 A d^3-9 B c d^2-3 c^3 D+5 c^2 C d\right )\right )}{5 b}+\frac {d x \left (a+b x^2\right )^{5/2} \left (63 a^2 d^4 D-2 a b d^2 \left (63 B d^2-6 c^2 D+61 c C d\right )-4 b^2 c \left (-108 A d^3-9 B c d^2-3 c^3 D+5 c^2 C d\right )\right )}{2 b}\right )-\frac {1}{7} \left (a+b x^2\right )^{5/2} (c+d x)^2 \left (a d^2 (3 c D+64 C d)+6 b \left (-24 A d^3-9 B c d^2-3 c^3 D+5 c^2 C d\right )\right )}{8 b}-\frac {\left (a+b x^2\right )^{5/2} (c+d x)^3 \left (9 a d^2 D-18 b B d^2-6 b c^2 D+10 b c C d\right )}{8 b}\right )+\frac {1}{9} d \left (a+b x^2\right )^{5/2} (c+d x)^4 (2 C d-3 c D)}{2 b d^3}+\frac {D \left (a+b x^2\right )^{5/2} (c+d x)^5}{10 b d^2}\)

Input:

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

Output:

(D*(c + d*x)^5*(a + b*x^2)^(5/2))/(10*b*d^2) + ((d*(2*C*d - 3*c*D)*(c + d* 
x)^4*(a + b*x^2)^(5/2))/9 + (d*(-1/8*((10*b*c*C*d - 18*b*B*d^2 - 6*b*c^2*D 
 + 9*a*d^2*D)*(c + d*x)^3*(a + b*x^2)^(5/2))/b + (-1/7*((a*d^2*(64*C*d + 3 
*c*D) + 6*b*(5*c^2*C*d - 9*B*c*d^2 - 24*A*d^3 - 3*c^3*D))*(c + d*x)^2*(a + 
 b*x^2)^(5/2)) + ((2*(64*a^2*d^4*(C*d + 3*c*D) - a*b*d^2*(272*c^2*C*d + 43 
2*B*c*d^2 + 144*A*d^3 - 21*c^3*D) - 6*b^2*c^2*(5*c^2*C*d - 9*B*c*d^2 - 192 
*A*d^3 - 3*c^3*D))*(a + b*x^2)^(5/2))/(5*b) + (d*(63*a^2*d^4*D - 2*a*b*d^2 
*(61*c*C*d + 63*B*d^2 - 6*c^2*D) - 4*b^2*c*(5*c^2*C*d - 9*B*c*d^2 - 108*A* 
d^3 - 3*c^3*D))*x*(a + b*x^2)^(5/2))/(2*b) + (21*d^2*(48*A*b^2*c*(2*b*c^2 
- a*d^2) - a*(16*b^2*c^2*(c*C + 3*B*d) + 3*a^2*d^3*D - 6*a*b*d*(3*c*C*d + 
B*d^2 + 3*c^2*D)))*((x*(a + b*x^2)^(3/2))/4 + (3*a*((x*Sqrt[a + b*x^2])/2 
+ (a*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*Sqrt[b])))/4))/(2*b))/7)/(8* 
b)))/9)/(2*b*d^3)
 

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 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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[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 676

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x 
_Symbol] :> Simp[(e*f + d*g)*((a + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + (Sim 
p[e*g*x*((a + c*x^2)^(p + 1)/(c*(2*p + 3))), x] - Simp[(a*e*g - c*d*f*(2*p 
+ 3))/(c*(2*p + 3))   Int[(a + c*x^2)^p, x], x]) /; FreeQ[{a, c, d, e, f, g 
, p}, x] &&  !LeQ[p, -1]
 

rule 687

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[g*(d + e*x)^m*((a + c*x^2)^(p + 1)/(c*(m + 2*p + 2)) 
), x] + Simp[1/(c*(m + 2*p + 2))   Int[(d + e*x)^(m - 1)*(a + c*x^2)^p*Simp 
[c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x 
] /; FreeQ[{a, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && 
 (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) &&  !(IGtQ[m, 0] && Eq 
Q[f, 0])
 

rule 2185

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : 
> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x) 
^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*e^(q - 1)*(m + q + 2*p + 1))), x] + Si 
mp[1/(b*e^q*(m + q + 2*p + 1))   Int[(d + e*x)^m*(a + b*x^2)^p*ExpandToSum[ 
b*e^q*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x 
)^(q - 2)*(a*e^2*(m + q - 1) - b*d^2*(m + q + 2*p + 1) - 2*b*d*e*(m + q + p 
)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, d 
, e, m, p}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] &&  !(EqQ[d, 0] && 
True) &&  !(IGtQ[m, 0] && RationalQ[a, b, d, e] && (IntegerQ[p] || ILtQ[p + 
 1/2, 0]))
 

Maple [A] (verified)

Time = 1.34 (sec) , antiderivative size = 545, normalized size of antiderivative = 0.92

method	result	size
default	\(A \,c^{3} \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )+\frac {c^{2} \left (3 A d +B c \right ) \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{5 b}+d^{2} \left (C d +3 D c \right ) \left (\frac {x^{4} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{9 b}-\frac {4 a \left (\frac {x^{2} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{7 b}-\frac {2 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{35 b^{2}}\right )}{9 b}\right )+c \left (3 A \,d^{2}+3 B c d +C \,c^{2}\right ) \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6 b}\right )+d \left (B \,d^{2}+3 C c d +3 D c^{2}\right ) \left (\frac {x^{3} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{8 b}-\frac {3 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6 b}\right )}{8 b}\right )+\left (A \,d^{3}+3 B c \,d^{2}+3 C \,c^{2} d +D c^{3}\right ) \left (\frac {x^{2} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{7 b}-\frac {2 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{35 b^{2}}\right )+D d^{3} \left (\frac {x^{5} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{10 b}-\frac {a \left (\frac {x^{3} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{8 b}-\frac {3 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6 b}\right )}{8 b}\right )}{2 b}\right )\)	\(545\)

Input:

int((d*x+c)^3*(b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 1643, normalized size of antiderivative = 2.78 \[ \int (c+d x)^3 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)^3*(b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A),x, algorithm="fric 
as")
 

Output:

[-1/161280*(315*(16*(C*a^3*b^2 - 6*A*a^2*b^3)*c^3 - 6*(3*D*a^4*b - 8*B*a^3 
*b^2)*c^2*d - 6*(3*C*a^4*b - 8*A*a^3*b^2)*c*d^2 + 3*(D*a^5 - 2*B*a^4*b)*d^ 
3)*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) - 2*(8064*D*b^5 
*d^3*x^9 + 8960*(3*D*b^5*c*d^2 + C*b^5*d^3)*x^8 + 1008*(30*D*b^5*c^2*d + 3 
0*C*b^5*c*d^2 + (11*D*a*b^4 + 10*B*b^5)*d^3)*x^7 + 1280*(9*D*b^5*c^3 + 27* 
C*b^5*c^2*d + 3*(10*D*a*b^4 + 9*B*b^5)*c*d^2 + (10*C*a*b^4 + 9*A*b^5)*d^3) 
*x^6 + 168*(80*C*b^5*c^3 + 30*(9*D*a*b^4 + 8*B*b^5)*c^2*d + 30*(9*C*a*b^4 
+ 8*A*b^5)*c*d^2 + 3*(D*a^2*b^3 + 30*B*a*b^4)*d^3)*x^5 + 768*(3*(8*D*a*b^4 
 + 7*B*b^5)*c^3 + 9*(8*C*a*b^4 + 7*A*b^5)*c^2*d + 3*(D*a^2*b^3 + 24*B*a*b^ 
4)*c*d^2 + (C*a^2*b^3 + 24*A*a*b^4)*d^3)*x^4 - 2304*(2*D*a^3*b^2 - 7*B*a^2 
*b^3)*c^3 - 6912*(2*C*a^3*b^2 - 7*A*a^2*b^3)*c^2*d + 1536*(4*D*a^4*b - 9*B 
*a^3*b^2)*c*d^2 + 512*(4*C*a^4*b - 9*A*a^3*b^2)*d^3 + 210*(16*(7*C*a*b^4 + 
 6*A*b^5)*c^3 + 6*(3*D*a^2*b^3 + 56*B*a*b^4)*c^2*d + 6*(3*C*a^2*b^3 + 56*A 
*a*b^4)*c*d^2 - 3*(D*a^3*b^2 - 2*B*a^2*b^3)*d^3)*x^3 + 256*(9*(D*a^2*b^3 + 
 14*B*a*b^4)*c^3 + 27*(C*a^2*b^3 + 14*A*a*b^4)*c^2*d - 3*(4*D*a^3*b^2 - 9* 
B*a^2*b^3)*c*d^2 - (4*C*a^3*b^2 - 9*A*a^2*b^3)*d^3)*x^2 + 315*(16*(C*a^2*b 
^3 + 10*A*a*b^4)*c^3 - 6*(3*D*a^3*b^2 - 8*B*a^2*b^3)*c^2*d - 6*(3*C*a^3*b^ 
2 - 8*A*a^2*b^3)*c*d^2 + 3*(D*a^4*b - 2*B*a^3*b^2)*d^3)*x)*sqrt(b*x^2 + a) 
)/b^4, 1/80640*(315*(16*(C*a^3*b^2 - 6*A*a^2*b^3)*c^3 - 6*(3*D*a^4*b - 8*B 
*a^3*b^2)*c^2*d - 6*(3*C*a^4*b - 8*A*a^3*b^2)*c*d^2 + 3*(D*a^5 - 2*B*a^...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1991 vs. \(2 (598) = 1196\).

Time = 0.88 (sec) , antiderivative size = 1991, normalized size of antiderivative = 3.37 \[ \int (c+d x)^3 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 727, normalized size of antiderivative = 1.23 \[ \int (c+d x)^3 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx =\text {Too large to display} \] Input:

integrate((d*x+c)^3*(b*x^2+a)^(3/2)*(D*x^3+C*x^2+B*x+A),x, algorithm="maxi 
ma")
 

Output:

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

Giac [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 878, normalized size of antiderivative = 1.49 \[ \int (c+d x)^3 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

1/80640*sqrt(b*x^2 + a)*((2*((4*((2*(7*(8*(9*D*b*d^3*x + 10*(3*D*b^9*c*d^2 
 + C*b^9*d^3)/b^8)*x + 9*(30*D*b^9*c^2*d + 30*C*b^9*c*d^2 + 11*D*a*b^8*d^3 
 + 10*B*b^9*d^3)/b^8)*x + 80*(9*D*b^9*c^3 + 27*C*b^9*c^2*d + 30*D*a*b^8*c* 
d^2 + 27*B*b^9*c*d^2 + 10*C*a*b^8*d^3 + 9*A*b^9*d^3)/b^8)*x + 21*(80*C*b^9 
*c^3 + 270*D*a*b^8*c^2*d + 240*B*b^9*c^2*d + 270*C*a*b^8*c*d^2 + 240*A*b^9 
*c*d^2 + 3*D*a^2*b^7*d^3 + 90*B*a*b^8*d^3)/b^8)*x + 96*(24*D*a*b^8*c^3 + 2 
1*B*b^9*c^3 + 72*C*a*b^8*c^2*d + 63*A*b^9*c^2*d + 3*D*a^2*b^7*c*d^2 + 72*B 
*a*b^8*c*d^2 + C*a^2*b^7*d^3 + 24*A*a*b^8*d^3)/b^8)*x + 105*(112*C*a*b^8*c 
^3 + 96*A*b^9*c^3 + 18*D*a^2*b^7*c^2*d + 336*B*a*b^8*c^2*d + 18*C*a^2*b^7* 
c*d^2 + 336*A*a*b^8*c*d^2 - 3*D*a^3*b^6*d^3 + 6*B*a^2*b^7*d^3)/b^8)*x + 12 
8*(9*D*a^2*b^7*c^3 + 126*B*a*b^8*c^3 + 27*C*a^2*b^7*c^2*d + 378*A*a*b^8*c^ 
2*d - 12*D*a^3*b^6*c*d^2 + 27*B*a^2*b^7*c*d^2 - 4*C*a^3*b^6*d^3 + 9*A*a^2* 
b^7*d^3)/b^8)*x + 315*(16*C*a^2*b^7*c^3 + 160*A*a*b^8*c^3 - 18*D*a^3*b^6*c 
^2*d + 48*B*a^2*b^7*c^2*d - 18*C*a^3*b^6*c*d^2 + 48*A*a^2*b^7*c*d^2 + 3*D* 
a^4*b^5*d^3 - 6*B*a^3*b^6*d^3)/b^8)*x - 256*(18*D*a^3*b^6*c^3 - 63*B*a^2*b 
^7*c^3 + 54*C*a^3*b^6*c^2*d - 189*A*a^2*b^7*c^2*d - 24*D*a^4*b^5*c*d^2 + 5 
4*B*a^3*b^6*c*d^2 - 8*C*a^4*b^5*d^3 + 18*A*a^3*b^6*d^3)/b^8) + 1/256*(16*C 
*a^3*b^2*c^3 - 96*A*a^2*b^3*c^3 - 18*D*a^4*b*c^2*d + 48*B*a^3*b^2*c^2*d - 
18*C*a^4*b*c*d^2 + 48*A*a^3*b^2*c*d^2 + 3*D*a^5*d^3 - 6*B*a^4*b*d^3)*log(a 
bs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(7/2)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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