\(\int (c+d x)^3 (a+b x^2)^{5/2} (A+B x+C x^2+D x^3) \, dx\) [85]

Optimal result

Integrand size = 34, antiderivative size = 682 \[ \int (c+d x)^3 \left (a+b x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {a^2 \left (320 A b^3 c^3-a \left (40 b^2 c \left (c^2 C+3 B c d+3 A d^2\right )+5 a^2 d^3 D-12 a b d \left (3 c C d+B d^2+3 c^2 D\right )\right )\right ) x \sqrt {a+b x^2}}{1024 b^3}+\frac {a \left (320 A b^3 c^3-a \left (40 b^2 c \left (c^2 C+3 B c d+3 A d^2\right )+5 a^2 d^3 D-12 a b d \left (3 c C d+B d^2+3 c^2 D\right )\right )\right ) x \left (a+b x^2\right )^{3/2}}{1536 b^3}+\frac {\left (320 A b^3 c^3-a \left (40 b^2 c \left (c^2 C+3 B c d+3 A d^2\right )+5 a^2 d^3 D-12 a b d \left (3 c C d+B d^2+3 c^2 D\right )\right )\right ) x \left (a+b x^2\right )^{5/2}}{1920 b^3}+\frac {\left (b^2 c^2 (B c+3 A d)+a^2 d^2 (C d+3 c D)-a b \left (3 c^2 C d+3 B c d^2+A d^3+c^3 D\right )\right ) \left (a+b x^2\right )^{7/2}}{7 b^3}+\frac {\left (40 b^2 c \left (c^2 C+3 B c d+3 A d^2\right )+5 a^2 d^3 D-12 a b d \left (3 c C d+B d^2+3 c^2 D\right )\right ) x \left (a+b x^2\right )^{7/2}}{320 b^3}-\frac {d \left (5 a d^2 D-12 b \left (3 c C d+B d^2+3 c^2 D\right )\right ) x^3 \left (a+b x^2\right )^{7/2}}{120 b^2}+\frac {d^3 D x^5 \left (a+b x^2\right )^{7/2}}{12 b}-\frac {\left (2 a d^2 (C d+3 c D)-b \left (3 c^2 C d+3 B c d^2+A d^3+c^3 D\right )\right ) \left (a+b x^2\right )^{9/2}}{9 b^3}+\frac {d^2 (C d+3 c D) \left (a+b x^2\right )^{11/2}}{11 b^3}+\frac {a^3 \left (320 A b^3 c^3-a \left (40 b^2 c \left (c^2 C+3 B c d+3 A d^2\right )+5 a^2 d^3 D-12 a b d \left (3 c C d+B d^2+3 c^2 D\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{1024 b^{7/2}} \] Output:

1/1024*a^2*(320*A*b^3*c^3-a*(40*b^2*c*(3*A*d^2+3*B*c*d+C*c^2)+5*a^2*d^3*D- 
12*a*b*d*(B*d^2+3*C*c*d+3*D*c^2)))*x*(b*x^2+a)^(1/2)/b^3+1/1536*a*(320*A*b 
^3*c^3-a*(40*b^2*c*(3*A*d^2+3*B*c*d+C*c^2)+5*a^2*d^3*D-12*a*b*d*(B*d^2+3*C 
*c*d+3*D*c^2)))*x*(b*x^2+a)^(3/2)/b^3+1/1920*(320*A*b^3*c^3-a*(40*b^2*c*(3 
*A*d^2+3*B*c*d+C*c^2)+5*a^2*d^3*D-12*a*b*d*(B*d^2+3*C*c*d+3*D*c^2)))*x*(b* 
x^2+a)^(5/2)/b^3+1/7*(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)^(7/2)/b^3+1/320*(40*b^2*c*(3*A*d^2+3* 
B*c*d+C*c^2)+5*a^2*d^3*D-12*a*b*d*(B*d^2+3*C*c*d+3*D*c^2))*x*(b*x^2+a)^(7/ 
2)/b^3-1/120*d*(5*a*d^2*D-12*b*(B*d^2+3*C*c*d+3*D*c^2))*x^3*(b*x^2+a)^(7/2 
)/b^2+1/12*d^3*D*x^5*(b*x^2+a)^(7/2)/b-1/9*(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)^(9/2)/b^3+1/11*d^2*(C*d+3*D*c)*(b*x^2 
+a)^(11/2)/b^3+1/1024*a^3*(320*A*b^3*c^3-a*(40*b^2*c*(3*A*d^2+3*B*c*d+C*c^ 
2)+5*a^2*d^3*D-12*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 = 6.44 (sec) , antiderivative size = 776, normalized size of antiderivative = 1.14 \[ \int (c+d x)^3 \left (a+b x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {\sqrt {b} \sqrt {a+b x^2} \left (5 a^5 d^2 (8192 C d+24576 c D+3465 d D x)+40 a^3 b^2 \left (11 A d \left (3456 c^2+945 c d x+128 d^2 x^2\right )+33 B \left (384 c^3+315 c^2 d x+128 c d^2 x^2+21 d^3 x^3\right )+x \left (33 c^2 d x (128 C+63 D x)+3 d^3 x^3 (128 C+77 D x)+9 c d^2 x^2 (231 C+128 D x)+11 c^3 (315 C+128 D x)\right )\right )-10 a^4 b \left (11264 c^3 D+66 c^2 d (512 C+189 D x)+6 c d^2 (5632 B+x (2079 C+1024 D x))+d^3 \left (11264 A+x \left (4158 B+2048 C x+1155 D x^2\right )\right )\right )+128 b^5 x^5 \left (55 A \left (84 c^3+216 c^2 d x+189 c d^2 x^2+56 d^3 x^3\right )+x \left (33 B \left (120 c^3+315 c^2 d x+280 c d^2 x^2+84 d^3 x^3\right )+7 x \left (55 c^3 (9 C+8 D x)+132 c^2 d x (10 C+9 D x)+108 c d^2 x^2 (11 C+10 D x)+30 d^3 x^3 (12 C+11 D x)\right )\right )\right )+64 a b^4 x^3 \left (55 A \left (546 c^3+1296 c^2 d x+1071 c d^2 x^2+304 d^3 x^3\right )+x \left (33 B \left (720 c^3+1785 c^2 d x+1520 c d^2 x^2+441 d^3 x^3\right )+x \left (70 d^3 x^3 (184 C+165 D x)+55 c^3 (357 C+304 D x)+33 c^2 d x (1520 C+1323 D x)+21 c d^2 x^2 (2079 C+1840 D x)\right )\right )\right )+16 a^2 b^3 x \left (165 A \left (924 c^3+1728 c^2 d x+1239 c d^2 x^2+320 d^3 x^3\right )+x \left (99 B \left (960 c^3+2065 c^2 d x+1600 c d^2 x^2+434 d^3 x^3\right )+x \left (165 c^3 (413 C+320 D x)+198 c^2 d x (800 C+651 D x)+5 d^3 x^3 (7232 C+6237 D x)+6 c d^2 x^2 (21483 C+18080 D x)\right )\right )\right )\right )+3465 a^3 \left (-40 A b^2 c \left (8 b c^2-3 a d^2\right )+a \left (40 b^2 c^2 (c C+3 B d)+5 a^2 d^3 D-12 a b d \left (3 c C d+B d^2+3 c^2 D\right )\right )\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{3548160 b^{7/2}} \] Input:

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

Output:

(Sqrt[b]*Sqrt[a + b*x^2]*(5*a^5*d^2*(8192*C*d + 24576*c*D + 3465*d*D*x) + 
40*a^3*b^2*(11*A*d*(3456*c^2 + 945*c*d*x + 128*d^2*x^2) + 33*B*(384*c^3 + 
315*c^2*d*x + 128*c*d^2*x^2 + 21*d^3*x^3) + x*(33*c^2*d*x*(128*C + 63*D*x) 
 + 3*d^3*x^3*(128*C + 77*D*x) + 9*c*d^2*x^2*(231*C + 128*D*x) + 11*c^3*(31 
5*C + 128*D*x))) - 10*a^4*b*(11264*c^3*D + 66*c^2*d*(512*C + 189*D*x) + 6* 
c*d^2*(5632*B + x*(2079*C + 1024*D*x)) + d^3*(11264*A + x*(4158*B + 2048*C 
*x + 1155*D*x^2))) + 128*b^5*x^5*(55*A*(84*c^3 + 216*c^2*d*x + 189*c*d^2*x 
^2 + 56*d^3*x^3) + x*(33*B*(120*c^3 + 315*c^2*d*x + 280*c*d^2*x^2 + 84*d^3 
*x^3) + 7*x*(55*c^3*(9*C + 8*D*x) + 132*c^2*d*x*(10*C + 9*D*x) + 108*c*d^2 
*x^2*(11*C + 10*D*x) + 30*d^3*x^3*(12*C + 11*D*x)))) + 64*a*b^4*x^3*(55*A* 
(546*c^3 + 1296*c^2*d*x + 1071*c*d^2*x^2 + 304*d^3*x^3) + x*(33*B*(720*c^3 
 + 1785*c^2*d*x + 1520*c*d^2*x^2 + 441*d^3*x^3) + x*(70*d^3*x^3*(184*C + 1 
65*D*x) + 55*c^3*(357*C + 304*D*x) + 33*c^2*d*x*(1520*C + 1323*D*x) + 21*c 
*d^2*x^2*(2079*C + 1840*D*x)))) + 16*a^2*b^3*x*(165*A*(924*c^3 + 1728*c^2* 
d*x + 1239*c*d^2*x^2 + 320*d^3*x^3) + x*(99*B*(960*c^3 + 2065*c^2*d*x + 16 
00*c*d^2*x^2 + 434*d^3*x^3) + x*(165*c^3*(413*C + 320*D*x) + 198*c^2*d*x*( 
800*C + 651*D*x) + 5*d^3*x^3*(7232*C + 6237*D*x) + 6*c*d^2*x^2*(21483*C + 
18080*D*x))))) + 3465*a^3*(-40*A*b^2*c*(8*b*c^2 - 3*a*d^2) + a*(40*b^2*c^2 
*(c*C + 3*B*d) + 5*a^2*d^3*D - 12*a*b*d*(3*c*C*d + B*d^2 + 3*c^2*D)))*Log[ 
-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/(3548160*b^(7/2))
 

Rubi [A] (verified)

Time = 1.21 (sec) , antiderivative size = 577, normalized size of antiderivative = 0.85, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.382, Rules used = {2185, 2185, 27, 687, 27, 687, 27, 676, 211, 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.

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

\(\Big \downarrow \) 2185

\(\displaystyle \frac {\int (c+d x)^3 \left (b x^2+a\right )^{5/2} \left (b (12 C d-19 c D) x^2 d^2+(12 A b d-5 a c D) d^2+\left (-7 b D c^2+12 b B d^2-5 a d^2 D\right ) x d\right )dx}{12 b d^3}+\frac {D \left (a+b x^2\right )^{7/2} (c+d x)^5}{12 b d^2}\)

\(\Big \downarrow \) 2185

\(\displaystyle \frac {\frac {\int b d^3 (c+d x)^3 \left (3 d (44 A b d-16 a C d+7 a c D)-\left (55 a D d^2+4 b \left (-14 D c^2+21 C d c-33 B d^2\right )\right ) x\right ) \left (b x^2+a\right )^{5/2}dx}{11 b d^2}+\frac {1}{11} d \left (a+b x^2\right )^{7/2} (c+d x)^4 (12 C d-19 c D)}{12 b d^3}+\frac {D \left (a+b x^2\right )^{7/2} (c+d x)^5}{12 b d^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{11} d \int (c+d x)^3 \left (3 d (44 A b d-16 a C d+7 a c D)-\left (55 a D d^2+4 b \left (-14 D c^2+21 C d c-33 B d^2\right )\right ) x\right ) \left (b x^2+a\right )^{5/2}dx+\frac {1}{11} d \left (a+b x^2\right )^{7/2} (c+d x)^4 (12 C d-19 c D)}{12 b d^3}+\frac {D \left (a+b x^2\right )^{7/2} (c+d x)^5}{12 b d^2}\)

\(\Big \downarrow \) 687

\(\displaystyle \frac {\frac {1}{11} d \left (\frac {\int 3 (c+d x)^2 \left (d \left (440 A c d b^2+a \left (55 a d^2 D-2 b \left (-7 D c^2+38 C d c+66 B d^2\right )\right )\right )-b \left (5 a (32 C d-3 c D) d^2+4 b \left (-14 D c^3+21 C d c^2-33 B d^2 c-110 A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{5/2}dx}{10 b}-\frac {\left (a+b x^2\right )^{7/2} (c+d x)^3 \left (55 a d^2 D-132 b B d^2-56 b c^2 D+84 b c C d\right )}{10 b}\right )+\frac {1}{11} d \left (a+b x^2\right )^{7/2} (c+d x)^4 (12 C d-19 c D)}{12 b d^3}+\frac {D \left (a+b x^2\right )^{7/2} (c+d x)^5}{12 b d^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{11} d \left (\frac {3 \int (c+d x)^2 \left (d \left (440 A c d b^2+a \left (55 a d^2 D-2 b \left (-7 D c^2+38 C d c+66 B d^2\right )\right )\right )-b \left (5 a (32 C d-3 c D) d^2+4 b \left (-14 D c^3+21 C d c^2-33 B d^2 c-110 A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{5/2}dx}{10 b}-\frac {\left (a+b x^2\right )^{7/2} (c+d x)^3 \left (55 a d^2 D-132 b B d^2-56 b c^2 D+84 b c C d\right )}{10 b}\right )+\frac {1}{11} d \left (a+b x^2\right )^{7/2} (c+d x)^4 (12 C d-19 c D)}{12 b d^3}+\frac {D \left (a+b x^2\right )^{7/2} (c+d x)^5}{12 b d^2}\)

\(\Big \downarrow \) 687

\(\displaystyle \frac {\frac {1}{11} d \left (\frac {3 \left (\frac {\int b (c+d x) \left (d \left (440 A b d \left (9 b c^2-2 a d^2\right )+a \left (5 a d^2 (64 C d+93 c D)-2 b c \left (-7 D c^2+258 C d c+726 B d^2\right )\right )\right )+\left (495 a^2 D d^4-4 a b \left (-39 D c^2+251 C d c+297 B d^2\right ) d^2-8 b^2 c \left (-14 D c^3+21 C d c^2-33 B d^2 c-605 A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{5/2}dx}{9 b}-\frac {1}{9} \left (a+b x^2\right )^{7/2} (c+d x)^2 \left (5 a d^2 (32 C d-3 c D)+4 b \left (-110 A d^3-33 B c d^2-14 c^3 D+21 c^2 C d\right )\right )\right )}{10 b}-\frac {\left (a+b x^2\right )^{7/2} (c+d x)^3 \left (55 a d^2 D-132 b B d^2-56 b c^2 D+84 b c C d\right )}{10 b}\right )+\frac {1}{11} d \left (a+b x^2\right )^{7/2} (c+d x)^4 (12 C d-19 c D)}{12 b d^3}+\frac {D \left (a+b x^2\right )^{7/2} (c+d x)^5}{12 b d^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{11} d \left (\frac {3 \left (\frac {1}{9} \int (c+d x) \left (d \left (440 A b d \left (9 b c^2-2 a d^2\right )+a \left (5 a d^2 (64 C d+93 c D)-2 b c \left (-7 D c^2+258 C d c+726 B d^2\right )\right )\right )+\left (495 a^2 D d^4-4 a b \left (-39 D c^2+251 C d c+297 B d^2\right ) d^2-8 b^2 c \left (-14 D c^3+21 C d c^2-33 B d^2 c-605 A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{5/2}dx-\frac {1}{9} \left (a+b x^2\right )^{7/2} (c+d x)^2 \left (5 a d^2 (32 C d-3 c D)+4 b \left (-110 A d^3-33 B c d^2-14 c^3 D+21 c^2 C d\right )\right )\right )}{10 b}-\frac {\left (a+b x^2\right )^{7/2} (c+d x)^3 \left (55 a d^2 D-132 b B d^2-56 b c^2 D+84 b c C d\right )}{10 b}\right )+\frac {1}{11} d \left (a+b x^2\right )^{7/2} (c+d x)^4 (12 C d-19 c D)}{12 b d^3}+\frac {D \left (a+b x^2\right )^{7/2} (c+d x)^5}{12 b d^2}\)

\(\Big \downarrow \) 676

\(\displaystyle \frac {\frac {1}{11} d \left (\frac {3 \left (\frac {1}{9} \left (\frac {99 d^2 \left (40 A b^2 c \left (8 b c^2-3 a d^2\right )-a \left (5 a^2 d^3 D-12 a b d \left (B d^2+3 c^2 D+3 c C d\right )+40 b^2 c^2 (3 B d+c C)\right )\right ) \int \left (b x^2+a\right )^{5/2}dx}{8 b}+\frac {2 \left (a+b x^2\right )^{7/2} \left (160 a^2 d^4 (3 c D+C d)-5 a b d^2 \left (88 A d^3+264 B c d^2-17 c^3 D+152 c^2 C d\right )-4 b^2 c^2 \left (-1100 A d^3-33 B c d^2-14 c^3 D+21 c^2 C d\right )\right )}{7 b}+\frac {d x \left (a+b x^2\right )^{7/2} \left (495 a^2 d^4 D-4 a b d^2 \left (297 B d^2-39 c^2 D+251 c C d\right )-8 b^2 c \left (-605 A d^3-33 B c d^2-14 c^3 D+21 c^2 C d\right )\right )}{8 b}\right )-\frac {1}{9} \left (a+b x^2\right )^{7/2} (c+d x)^2 \left (5 a d^2 (32 C d-3 c D)+4 b \left (-110 A d^3-33 B c d^2-14 c^3 D+21 c^2 C d\right )\right )\right )}{10 b}-\frac {\left (a+b x^2\right )^{7/2} (c+d x)^3 \left (55 a d^2 D-132 b B d^2-56 b c^2 D+84 b c C d\right )}{10 b}\right )+\frac {1}{11} d \left (a+b x^2\right )^{7/2} (c+d x)^4 (12 C d-19 c D)}{12 b d^3}+\frac {D \left (a+b x^2\right )^{7/2} (c+d x)^5}{12 b d^2}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\frac {1}{11} d \left (\frac {3 \left (\frac {1}{9} \left (\frac {99 d^2 \left (40 A b^2 c \left (8 b c^2-3 a d^2\right )-a \left (5 a^2 d^3 D-12 a b d \left (B d^2+3 c^2 D+3 c C d\right )+40 b^2 c^2 (3 B d+c C)\right )\right ) \left (\frac {5}{6} a \int \left (b x^2+a\right )^{3/2}dx+\frac {1}{6} x \left (a+b x^2\right )^{5/2}\right )}{8 b}+\frac {2 \left (a+b x^2\right )^{7/2} \left (160 a^2 d^4 (3 c D+C d)-5 a b d^2 \left (88 A d^3+264 B c d^2-17 c^3 D+152 c^2 C d\right )-4 b^2 c^2 \left (-1100 A d^3-33 B c d^2-14 c^3 D+21 c^2 C d\right )\right )}{7 b}+\frac {d x \left (a+b x^2\right )^{7/2} \left (495 a^2 d^4 D-4 a b d^2 \left (297 B d^2-39 c^2 D+251 c C d\right )-8 b^2 c \left (-605 A d^3-33 B c d^2-14 c^3 D+21 c^2 C d\right )\right )}{8 b}\right )-\frac {1}{9} \left (a+b x^2\right )^{7/2} (c+d x)^2 \left (5 a d^2 (32 C d-3 c D)+4 b \left (-110 A d^3-33 B c d^2-14 c^3 D+21 c^2 C d\right )\right )\right )}{10 b}-\frac {\left (a+b x^2\right )^{7/2} (c+d x)^3 \left (55 a d^2 D-132 b B d^2-56 b c^2 D+84 b c C d\right )}{10 b}\right )+\frac {1}{11} d \left (a+b x^2\right )^{7/2} (c+d x)^4 (12 C d-19 c D)}{12 b d^3}+\frac {D \left (a+b x^2\right )^{7/2} (c+d x)^5}{12 b d^2}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\frac {1}{11} d \left (\frac {3 \left (\frac {1}{9} \left (\frac {99 d^2 \left (40 A b^2 c \left (8 b c^2-3 a d^2\right )-a \left (5 a^2 d^3 D-12 a b d \left (B d^2+3 c^2 D+3 c C d\right )+40 b^2 c^2 (3 B d+c C)\right )\right ) \left (\frac {5}{6} a \left (\frac {3}{4} a \int \sqrt {b x^2+a}dx+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )+\frac {1}{6} x \left (a+b x^2\right )^{5/2}\right )}{8 b}+\frac {2 \left (a+b x^2\right )^{7/2} \left (160 a^2 d^4 (3 c D+C d)-5 a b d^2 \left (88 A d^3+264 B c d^2-17 c^3 D+152 c^2 C d\right )-4 b^2 c^2 \left (-1100 A d^3-33 B c d^2-14 c^3 D+21 c^2 C d\right )\right )}{7 b}+\frac {d x \left (a+b x^2\right )^{7/2} \left (495 a^2 d^4 D-4 a b d^2 \left (297 B d^2-39 c^2 D+251 c C d\right )-8 b^2 c \left (-605 A d^3-33 B c d^2-14 c^3 D+21 c^2 C d\right )\right )}{8 b}\right )-\frac {1}{9} \left (a+b x^2\right )^{7/2} (c+d x)^2 \left (5 a d^2 (32 C d-3 c D)+4 b \left (-110 A d^3-33 B c d^2-14 c^3 D+21 c^2 C d\right )\right )\right )}{10 b}-\frac {\left (a+b x^2\right )^{7/2} (c+d x)^3 \left (55 a d^2 D-132 b B d^2-56 b c^2 D+84 b c C d\right )}{10 b}\right )+\frac {1}{11} d \left (a+b x^2\right )^{7/2} (c+d x)^4 (12 C d-19 c D)}{12 b d^3}+\frac {D \left (a+b x^2\right )^{7/2} (c+d x)^5}{12 b d^2}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\frac {1}{11} d \left (\frac {3 \left (\frac {1}{9} \left (\frac {99 d^2 \left (40 A b^2 c \left (8 b c^2-3 a d^2\right )-a \left (5 a^2 d^3 D-12 a b d \left (B d^2+3 c^2 D+3 c C d\right )+40 b^2 c^2 (3 B d+c C)\right )\right ) \left (\frac {5}{6} a \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{\sqrt {b x^2+a}}dx+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )+\frac {1}{6} x \left (a+b x^2\right )^{5/2}\right )}{8 b}+\frac {2 \left (a+b x^2\right )^{7/2} \left (160 a^2 d^4 (3 c D+C d)-5 a b d^2 \left (88 A d^3+264 B c d^2-17 c^3 D+152 c^2 C d\right )-4 b^2 c^2 \left (-1100 A d^3-33 B c d^2-14 c^3 D+21 c^2 C d\right )\right )}{7 b}+\frac {d x \left (a+b x^2\right )^{7/2} \left (495 a^2 d^4 D-4 a b d^2 \left (297 B d^2-39 c^2 D+251 c C d\right )-8 b^2 c \left (-605 A d^3-33 B c d^2-14 c^3 D+21 c^2 C d\right )\right )}{8 b}\right )-\frac {1}{9} \left (a+b x^2\right )^{7/2} (c+d x)^2 \left (5 a d^2 (32 C d-3 c D)+4 b \left (-110 A d^3-33 B c d^2-14 c^3 D+21 c^2 C d\right )\right )\right )}{10 b}-\frac {\left (a+b x^2\right )^{7/2} (c+d x)^3 \left (55 a d^2 D-132 b B d^2-56 b c^2 D+84 b c C d\right )}{10 b}\right )+\frac {1}{11} d \left (a+b x^2\right )^{7/2} (c+d x)^4 (12 C d-19 c D)}{12 b d^3}+\frac {D \left (a+b x^2\right )^{7/2} (c+d x)^5}{12 b d^2}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\frac {1}{11} d \left (\frac {3 \left (\frac {1}{9} \left (\frac {99 d^2 \left (40 A b^2 c \left (8 b c^2-3 a d^2\right )-a \left (5 a^2 d^3 D-12 a b d \left (B d^2+3 c^2 D+3 c C d\right )+40 b^2 c^2 (3 B d+c C)\right )\right ) \left (\frac {5}{6} a \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )+\frac {1}{6} x \left (a+b x^2\right )^{5/2}\right )}{8 b}+\frac {2 \left (a+b x^2\right )^{7/2} \left (160 a^2 d^4 (3 c D+C d)-5 a b d^2 \left (88 A d^3+264 B c d^2-17 c^3 D+152 c^2 C d\right )-4 b^2 c^2 \left (-1100 A d^3-33 B c d^2-14 c^3 D+21 c^2 C d\right )\right )}{7 b}+\frac {d x \left (a+b x^2\right )^{7/2} \left (495 a^2 d^4 D-4 a b d^2 \left (297 B d^2-39 c^2 D+251 c C d\right )-8 b^2 c \left (-605 A d^3-33 B c d^2-14 c^3 D+21 c^2 C d\right )\right )}{8 b}\right )-\frac {1}{9} \left (a+b x^2\right )^{7/2} (c+d x)^2 \left (5 a d^2 (32 C d-3 c D)+4 b \left (-110 A d^3-33 B c d^2-14 c^3 D+21 c^2 C d\right )\right )\right )}{10 b}-\frac {\left (a+b x^2\right )^{7/2} (c+d x)^3 \left (55 a d^2 D-132 b B d^2-56 b c^2 D+84 b c C d\right )}{10 b}\right )+\frac {1}{11} d \left (a+b x^2\right )^{7/2} (c+d x)^4 (12 C d-19 c D)}{12 b d^3}+\frac {D \left (a+b x^2\right )^{7/2} (c+d x)^5}{12 b d^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {1}{11} d \left (\frac {3 \left (\frac {1}{9} \left (\frac {99 d^2 \left (\frac {5}{6} a \left (\frac {3}{4} a \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )+\frac {1}{6} x \left (a+b x^2\right )^{5/2}\right ) \left (40 A b^2 c \left (8 b c^2-3 a d^2\right )-a \left (5 a^2 d^3 D-12 a b d \left (B d^2+3 c^2 D+3 c C d\right )+40 b^2 c^2 (3 B d+c C)\right )\right )}{8 b}+\frac {2 \left (a+b x^2\right )^{7/2} \left (160 a^2 d^4 (3 c D+C d)-5 a b d^2 \left (88 A d^3+264 B c d^2-17 c^3 D+152 c^2 C d\right )-4 b^2 c^2 \left (-1100 A d^3-33 B c d^2-14 c^3 D+21 c^2 C d\right )\right )}{7 b}+\frac {d x \left (a+b x^2\right )^{7/2} \left (495 a^2 d^4 D-4 a b d^2 \left (297 B d^2-39 c^2 D+251 c C d\right )-8 b^2 c \left (-605 A d^3-33 B c d^2-14 c^3 D+21 c^2 C d\right )\right )}{8 b}\right )-\frac {1}{9} \left (a+b x^2\right )^{7/2} (c+d x)^2 \left (5 a d^2 (32 C d-3 c D)+4 b \left (-110 A d^3-33 B c d^2-14 c^3 D+21 c^2 C d\right )\right )\right )}{10 b}-\frac {\left (a+b x^2\right )^{7/2} (c+d x)^3 \left (55 a d^2 D-132 b B d^2-56 b c^2 D+84 b c C d\right )}{10 b}\right )+\frac {1}{11} d \left (a+b x^2\right )^{7/2} (c+d x)^4 (12 C d-19 c D)}{12 b d^3}+\frac {D \left (a+b x^2\right )^{7/2} (c+d x)^5}{12 b d^2}\)

Input:

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

Output:

(D*(c + d*x)^5*(a + b*x^2)^(7/2))/(12*b*d^2) + ((d*(12*C*d - 19*c*D)*(c + 
d*x)^4*(a + b*x^2)^(7/2))/11 + (d*(-1/10*((84*b*c*C*d - 132*b*B*d^2 - 56*b 
*c^2*D + 55*a*d^2*D)*(c + d*x)^3*(a + b*x^2)^(7/2))/b + (3*(-1/9*((5*a*d^2 
*(32*C*d - 3*c*D) + 4*b*(21*c^2*C*d - 33*B*c*d^2 - 110*A*d^3 - 14*c^3*D))* 
(c + d*x)^2*(a + b*x^2)^(7/2)) + ((2*(160*a^2*d^4*(C*d + 3*c*D) - 5*a*b*d^ 
2*(152*c^2*C*d + 264*B*c*d^2 + 88*A*d^3 - 17*c^3*D) - 4*b^2*c^2*(21*c^2*C* 
d - 33*B*c*d^2 - 1100*A*d^3 - 14*c^3*D))*(a + b*x^2)^(7/2))/(7*b) + (d*(49 
5*a^2*d^4*D - 4*a*b*d^2*(251*c*C*d + 297*B*d^2 - 39*c^2*D) - 8*b^2*c*(21*c 
^2*C*d - 33*B*c*d^2 - 605*A*d^3 - 14*c^3*D))*x*(a + b*x^2)^(7/2))/(8*b) + 
(99*d^2*(40*A*b^2*c*(8*b*c^2 - 3*a*d^2) - a*(40*b^2*c^2*(c*C + 3*B*d) + 5* 
a^2*d^3*D - 12*a*b*d*(3*c*C*d + B*d^2 + 3*c^2*D)))*((x*(a + b*x^2)^(5/2))/ 
6 + (5*a*((x*(a + b*x^2)^(3/2))/4 + (3*a*((x*Sqrt[a + b*x^2])/2 + (a*ArcTa 
nh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*Sqrt[b])))/4))/6))/(8*b))/9))/(10*b))) 
/11)/(12*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 = 1.41 (sec) , antiderivative size = 609, normalized size of antiderivative = 0.89

method	result	size
default	\(A \,c^{3} \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 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}\right )+\frac {c^{2} \left (3 A d +B c \right ) \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{7 b}+d^{2} \left (C d +3 D c \right ) \left (\frac {x^{4} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{11 b}-\frac {4 a \left (\frac {x^{2} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{9 b}-\frac {2 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{63 b^{2}}\right )}{11 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 {7}{2}}}{8 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 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}\right )}{8 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 {7}{2}}}{10 b}-\frac {3 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{8 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 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}\right )}{8 b}\right )}{10 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 {7}{2}}}{9 b}-\frac {2 a \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{63 b^{2}}\right )+D d^{3} \left (\frac {x^{5} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{12 b}-\frac {5 a \left (\frac {x^{3} \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{10 b}-\frac {3 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {7}{2}}}{8 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6}+\frac {5 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}\right )}{8 b}\right )}{10 b}\right )}{12 b}\right )\)	\(609\)

Input:

int((d*x+c)^3*(b*x^2+a)^(5/2)*(D*x^3+C*x^2+B*x+A),x,method=_RETURNVERBOSE)
 

Output:

A*c^3*(1/6*x*(b*x^2+a)^(5/2)+5/6*a*(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)))))+1/7*c^2*(3*A*d 
+B*c)*(b*x^2+a)^(7/2)/b+d^2*(C*d+3*D*c)*(1/11*x^4*(b*x^2+a)^(7/2)/b-4/11*a 
/b*(1/9*x^2*(b*x^2+a)^(7/2)/b-2/63*a/b^2*(b*x^2+a)^(7/2)))+c*(3*A*d^2+3*B* 
c*d+C*c^2)*(1/8*x*(b*x^2+a)^(7/2)/b-1/8*a/b*(1/6*x*(b*x^2+a)^(5/2)+5/6*a*( 
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)*(1/10*x^3*(b*x^2+a)^(7/ 
2)/b-3/10*a/b*(1/8*x*(b*x^2+a)^(7/2)/b-1/8*a/b*(1/6*x*(b*x^2+a)^(5/2)+5/6* 
a*(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)*(1/9*x^2*(b 
*x^2+a)^(7/2)/b-2/63*a/b^2*(b*x^2+a)^(7/2))+D*d^3*(1/12*x^5*(b*x^2+a)^(7/2 
)/b-5/12*a/b*(1/10*x^3*(b*x^2+a)^(7/2)/b-3/10*a/b*(1/8*x*(b*x^2+a)^(7/2)/b 
-1/8*a/b*(1/6*x*(b*x^2+a)^(5/2)+5/6*a*(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 = 0.23 (sec) , antiderivative size = 2061, normalized size of antiderivative = 3.02 \[ \int (c+d x)^3 \left (a+b x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)^3*(b*x^2+a)^(5/2)*(D*x^3+C*x^2+B*x+A),x, algorithm="fric 
as")
 

Output:

[-1/7096320*(3465*(40*(C*a^4*b^2 - 8*A*a^3*b^3)*c^3 - 12*(3*D*a^5*b - 10*B 
*a^4*b^2)*c^2*d - 12*(3*C*a^5*b - 10*A*a^4*b^2)*c*d^2 + (5*D*a^6 - 12*B*a^ 
5*b)*d^3)*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) - 2*(295 
680*D*b^6*d^3*x^11 + 322560*(3*D*b^6*c*d^2 + C*b^6*d^3)*x^10 + 29568*(36*D 
*b^6*c^2*d + 36*C*b^6*c*d^2 + (25*D*a*b^5 + 12*B*b^6)*d^3)*x^9 + 35840*(11 
*D*b^6*c^3 + 33*C*b^6*c^2*d + 3*(23*D*a*b^5 + 11*B*b^6)*c*d^2 + (23*C*a*b^ 
5 + 11*A*b^6)*d^3)*x^8 + 11088*(40*C*b^6*c^3 + 12*(21*D*a*b^5 + 10*B*b^6)* 
c^2*d + 12*(21*C*a*b^5 + 10*A*b^6)*c*d^2 + 3*(15*D*a^2*b^4 + 28*B*a*b^5)*d 
^3)*x^7 + 5120*(11*(19*D*a*b^5 + 9*B*b^6)*c^3 + 33*(19*C*a*b^5 + 9*A*b^6)* 
c^2*d + 3*(113*D*a^2*b^4 + 209*B*a*b^5)*c*d^2 + (113*C*a^2*b^4 + 209*A*a*b 
^5)*d^3)*x^6 + 1848*(40*(17*C*a*b^5 + 8*A*b^6)*c^3 + 12*(93*D*a^2*b^4 + 17 
0*B*a*b^5)*c^2*d + 12*(93*C*a^2*b^4 + 170*A*a*b^5)*c*d^2 + (5*D*a^3*b^3 + 
372*B*a^2*b^4)*d^3)*x^5 + 15360*(11*(5*D*a^2*b^4 + 9*B*a*b^5)*c^3 + 33*(5* 
C*a^2*b^4 + 9*A*a*b^5)*c^2*d + 3*(D*a^3*b^3 + 55*B*a^2*b^4)*c*d^2 + (C*a^3 
*b^3 + 55*A*a^2*b^4)*d^3)*x^4 - 56320*(2*D*a^4*b^2 - 9*B*a^3*b^3)*c^3 - 16 
8960*(2*C*a^4*b^2 - 9*A*a^3*b^3)*c^2*d + 30720*(4*D*a^5*b - 11*B*a^4*b^2)* 
c*d^2 + 10240*(4*C*a^5*b - 11*A*a^4*b^2)*d^3 + 2310*(8*(59*C*a^2*b^4 + 104 
*A*a*b^5)*c^3 + 12*(3*D*a^3*b^3 + 118*B*a^2*b^4)*c^2*d + 12*(3*C*a^3*b^3 + 
 118*A*a^2*b^4)*c*d^2 - (5*D*a^4*b^2 - 12*B*a^3*b^3)*d^3)*x^3 + 5120*(11*( 
D*a^3*b^3 + 27*B*a^2*b^4)*c^3 + 33*(C*a^3*b^3 + 27*A*a^2*b^4)*c^2*d - 3...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3186 vs. \(2 (695) = 1390\).

Time = 1.00 (sec) , antiderivative size = 3186, normalized size of antiderivative = 4.67 \[ \int (c+d x)^3 \left (a+b x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

Piecewise((sqrt(a + b*x**2)*(D*b**2*d**3*x**11/12 + x**10*(C*b**3*d**3 + 3 
*D*b**3*c*d**2)/(11*b) + x**9*(B*b**3*d**3 + 3*C*b**3*c*d**2 + 25*D*a*b**2 
*d**3/12 + 3*D*b**3*c**2*d)/(10*b) + x**8*(A*b**3*d**3 + 3*B*b**3*c*d**2 + 
 3*C*a*b**2*d**3 + 3*C*b**3*c**2*d + 9*D*a*b**2*c*d**2 + D*b**3*c**3 - 10* 
a*(C*b**3*d**3 + 3*D*b**3*c*d**2)/(11*b))/(9*b) + x**7*(3*A*b**3*c*d**2 + 
3*B*a*b**2*d**3 + 3*B*b**3*c**2*d + 9*C*a*b**2*c*d**2 + C*b**3*c**3 + 3*D* 
a**2*b*d**3 + 9*D*a*b**2*c**2*d - 9*a*(B*b**3*d**3 + 3*C*b**3*c*d**2 + 25* 
D*a*b**2*d**3/12 + 3*D*b**3*c**2*d)/(10*b))/(8*b) + x**6*(3*A*a*b**2*d**3 
+ 3*A*b**3*c**2*d + 9*B*a*b**2*c*d**2 + B*b**3*c**3 + 3*C*a**2*b*d**3 + 9* 
C*a*b**2*c**2*d + 9*D*a**2*b*c*d**2 + 3*D*a*b**2*c**3 - 8*a*(A*b**3*d**3 + 
 3*B*b**3*c*d**2 + 3*C*a*b**2*d**3 + 3*C*b**3*c**2*d + 9*D*a*b**2*c*d**2 + 
 D*b**3*c**3 - 10*a*(C*b**3*d**3 + 3*D*b**3*c*d**2)/(11*b))/(9*b))/(7*b) + 
 x**5*(9*A*a*b**2*c*d**2 + A*b**3*c**3 + 3*B*a**2*b*d**3 + 9*B*a*b**2*c**2 
*d + 9*C*a**2*b*c*d**2 + 3*C*a*b**2*c**3 + D*a**3*d**3 + 9*D*a**2*b*c**2*d 
 - 7*a*(3*A*b**3*c*d**2 + 3*B*a*b**2*d**3 + 3*B*b**3*c**2*d + 9*C*a*b**2*c 
*d**2 + C*b**3*c**3 + 3*D*a**2*b*d**3 + 9*D*a*b**2*c**2*d - 9*a*(B*b**3*d* 
*3 + 3*C*b**3*c*d**2 + 25*D*a*b**2*d**3/12 + 3*D*b**3*c**2*d)/(10*b))/(8*b 
))/(6*b) + x**4*(3*A*a**2*b*d**3 + 9*A*a*b**2*c**2*d + 9*B*a**2*b*c*d**2 + 
 3*B*a*b**2*c**3 + C*a**3*d**3 + 9*C*a**2*b*c**2*d + 3*D*a**3*c*d**2 + 3*D 
*a**2*b*c**3 - 6*a*(3*A*a*b**2*d**3 + 3*A*b**3*c**2*d + 9*B*a*b**2*c*d*...
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 844, normalized size of antiderivative = 1.24 \[ \int (c+d x)^3 \left (a+b x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)^3*(b*x^2+a)^(5/2)*(D*x^3+C*x^2+B*x+A),x, algorithm="maxi 
ma")
 

Output:

1/12*(b*x^2 + a)^(7/2)*D*d^3*x^5/b - 1/24*(b*x^2 + a)^(7/2)*D*a*d^3*x^3/b^ 
2 + 1/6*(b*x^2 + a)^(5/2)*A*c^3*x + 5/24*(b*x^2 + a)^(3/2)*A*a*c^3*x + 5/1 
6*sqrt(b*x^2 + a)*A*a^2*c^3*x + 1/64*(b*x^2 + a)^(7/2)*D*a^2*d^3*x/b^3 - 1 
/384*(b*x^2 + a)^(5/2)*D*a^3*d^3*x/b^3 - 5/1536*(b*x^2 + a)^(3/2)*D*a^4*d^ 
3*x/b^3 - 5/1024*sqrt(b*x^2 + a)*D*a^5*d^3*x/b^3 + 1/11*(3*D*c*d^2 + C*d^3 
)*(b*x^2 + a)^(7/2)*x^4/b + 5/16*A*a^3*c^3*arcsinh(b*x/sqrt(a*b))/sqrt(b) 
- 5/1024*D*a^6*d^3*arcsinh(b*x/sqrt(a*b))/b^(7/2) + 1/7*(b*x^2 + a)^(7/2)* 
B*c^3/b + 3/7*(b*x^2 + a)^(7/2)*A*c^2*d/b + 1/10*(3*D*c^2*d + 3*C*c*d^2 + 
B*d^3)*(b*x^2 + a)^(7/2)*x^3/b - 4/99*(3*D*c*d^2 + C*d^3)*(b*x^2 + a)^(7/2 
)*a*x^2/b^2 + 1/9*(D*c^3 + 3*C*c^2*d + 3*B*c*d^2 + A*d^3)*(b*x^2 + a)^(7/2 
)*x^2/b - 3/80*(3*D*c^2*d + 3*C*c*d^2 + B*d^3)*(b*x^2 + a)^(7/2)*a*x/b^2 + 
 1/160*(3*D*c^2*d + 3*C*c*d^2 + B*d^3)*(b*x^2 + a)^(5/2)*a^2*x/b^2 + 1/128 
*(3*D*c^2*d + 3*C*c*d^2 + B*d^3)*(b*x^2 + a)^(3/2)*a^3*x/b^2 + 3/256*(3*D* 
c^2*d + 3*C*c*d^2 + B*d^3)*sqrt(b*x^2 + a)*a^4*x/b^2 + 1/8*(C*c^3 + 3*B*c^ 
2*d + 3*A*c*d^2)*(b*x^2 + a)^(7/2)*x/b - 1/48*(C*c^3 + 3*B*c^2*d + 3*A*c*d 
^2)*(b*x^2 + a)^(5/2)*a*x/b - 5/192*(C*c^3 + 3*B*c^2*d + 3*A*c*d^2)*(b*x^2 
 + a)^(3/2)*a^2*x/b - 5/128*(C*c^3 + 3*B*c^2*d + 3*A*c*d^2)*sqrt(b*x^2 + a 
)*a^3*x/b + 3/256*(3*D*c^2*d + 3*C*c*d^2 + B*d^3)*a^5*arcsinh(b*x/sqrt(a*b 
))/b^(5/2) - 5/128*(C*c^3 + 3*B*c^2*d + 3*A*c*d^2)*a^4*arcsinh(b*x/sqrt(a* 
b))/b^(3/2) + 8/693*(3*D*c*d^2 + C*d^3)*(b*x^2 + a)^(7/2)*a^2/b^3 - 2/6...
 

Giac [A] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 1100, normalized size of antiderivative = 1.61 \[ \int (c+d x)^3 \left (a+b x^2\right )^{5/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

1/3548160*sqrt(b*x^2 + a)*((2*((4*((2*(7*(8*(3*(10*(11*D*b^2*d^3*x + 12*(3 
*D*b^12*c*d^2 + C*b^12*d^3)/b^10)*x + 11*(36*D*b^12*c^2*d + 36*C*b^12*c*d^ 
2 + 25*D*a*b^11*d^3 + 12*B*b^12*d^3)/b^10)*x + 40*(11*D*b^12*c^3 + 33*C*b^ 
12*c^2*d + 69*D*a*b^11*c*d^2 + 33*B*b^12*c*d^2 + 23*C*a*b^11*d^3 + 11*A*b^ 
12*d^3)/b^10)*x + 99*(40*C*b^12*c^3 + 252*D*a*b^11*c^2*d + 120*B*b^12*c^2* 
d + 252*C*a*b^11*c*d^2 + 120*A*b^12*c*d^2 + 45*D*a^2*b^10*d^3 + 84*B*a*b^1 
1*d^3)/b^10)*x + 320*(209*D*a*b^11*c^3 + 99*B*b^12*c^3 + 627*C*a*b^11*c^2* 
d + 297*A*b^12*c^2*d + 339*D*a^2*b^10*c*d^2 + 627*B*a*b^11*c*d^2 + 113*C*a 
^2*b^10*d^3 + 209*A*a*b^11*d^3)/b^10)*x + 231*(680*C*a*b^11*c^3 + 320*A*b^ 
12*c^3 + 1116*D*a^2*b^10*c^2*d + 2040*B*a*b^11*c^2*d + 1116*C*a^2*b^10*c*d 
^2 + 2040*A*a*b^11*c*d^2 + 5*D*a^3*b^9*d^3 + 372*B*a^2*b^10*d^3)/b^10)*x + 
 1920*(55*D*a^2*b^10*c^3 + 99*B*a*b^11*c^3 + 165*C*a^2*b^10*c^2*d + 297*A* 
a*b^11*c^2*d + 3*D*a^3*b^9*c*d^2 + 165*B*a^2*b^10*c*d^2 + C*a^3*b^9*d^3 + 
55*A*a^2*b^10*d^3)/b^10)*x + 1155*(472*C*a^2*b^10*c^3 + 832*A*a*b^11*c^3 + 
 36*D*a^3*b^9*c^2*d + 1416*B*a^2*b^10*c^2*d + 36*C*a^3*b^9*c*d^2 + 1416*A* 
a^2*b^10*c*d^2 - 5*D*a^4*b^8*d^3 + 12*B*a^3*b^9*d^3)/b^10)*x + 2560*(11*D* 
a^3*b^9*c^3 + 297*B*a^2*b^10*c^3 + 33*C*a^3*b^9*c^2*d + 891*A*a^2*b^10*c^2 
*d - 12*D*a^4*b^8*c*d^2 + 33*B*a^3*b^9*c*d^2 - 4*C*a^4*b^8*d^3 + 11*A*a^3* 
b^9*d^3)/b^10)*x + 3465*(40*C*a^3*b^9*c^3 + 704*A*a^2*b^10*c^3 - 36*D*a^4* 
b^8*c^2*d + 120*B*a^3*b^9*c^2*d - 36*C*a^4*b^8*c*d^2 + 120*A*a^3*b^9*c*...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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