\(\int x^3 (c+d x) (a+b x^2)^{3/2} (A+B x+C x^2) \, dx\) [47]

Optimal result

Integrand size = 30, antiderivative size = 311 \[ \int x^3 (c+d x) \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right ) \, dx=\frac {3 a^3 (a C d-2 b (B c+A d)) x \sqrt {a+b x^2}}{256 b^3}-\frac {a^2 (a C d-2 b (B c+A d)) x^3 \sqrt {a+b x^2}}{128 b^2}-\frac {a (a C d-2 b (B c+A d)) x^5 \sqrt {a+b x^2}}{32 b}-\frac {(a C d-2 b (B c+A d)) x^5 \left (a+b x^2\right )^{3/2}}{16 b}-\frac {a (A b c-a c C-a B d) \left (a+b x^2\right )^{5/2}}{5 b^3}+\frac {C d x^5 \left (a+b x^2\right )^{5/2}}{10 b}+\frac {(A b c-2 a (c C+B d)) \left (a+b x^2\right )^{7/2}}{7 b^3}+\frac {(c C+B d) \left (a+b x^2\right )^{9/2}}{9 b^3}-\frac {3 a^4 (a C d-2 b (B c+A d)) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{256 b^{7/2}} \] Output:

3/256*a^3*(a*C*d-2*b*(A*d+B*c))*x*(b*x^2+a)^(1/2)/b^3-1/128*a^2*(a*C*d-2*b 
*(A*d+B*c))*x^3*(b*x^2+a)^(1/2)/b^2-1/32*a*(a*C*d-2*b*(A*d+B*c))*x^5*(b*x^ 
2+a)^(1/2)/b-1/16*(a*C*d-2*b*(A*d+B*c))*x^5*(b*x^2+a)^(3/2)/b-1/5*a*(A*b*c 
-B*a*d-C*a*c)*(b*x^2+a)^(5/2)/b^3+1/10*C*d*x^5*(b*x^2+a)^(5/2)/b+1/7*(A*b* 
c-2*a*(B*d+C*c))*(b*x^2+a)^(7/2)/b^3+1/9*(B*d+C*c)*(b*x^2+a)^(9/2)/b^3-3/2 
56*a^4*(a*C*d-2*b*(A*d+B*c))*arctanh(b^(1/2)*x/(b*x^2+a)^(1/2))/b^(7/2)
 

Mathematica [A] (verified)

Time = 1.41 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.86 \[ \int x^3 (c+d x) \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right ) \, dx=\frac {\sqrt {b} \sqrt {a+b x^2} \left (a^4 (2048 c C+2048 B d+945 C d x)+12 a^2 b^2 x^2 \left (3 A (64 c+35 d x)+x \left (105 B c+64 c C x+64 B d x+42 C d x^2\right )\right )-2 a^3 b \left (9 A (256 c+105 d x)+x \left (945 B c+512 c C x+512 B d x+315 C d x^2\right )\right )+16 a b^3 x^4 \left (9 A (128 c+105 d x)+x \left (945 B c+800 c C x+800 B d x+693 C d x^2\right )\right )+32 b^4 x^6 (45 A (8 c+7 d x)+7 x (5 B (9 c+8 d x)+4 C x (10 c+9 d x)))\right )+945 a^4 (a C d-2 b (B c+A d)) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{80640 b^{7/2}} \] Input:

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

Output:

(Sqrt[b]*Sqrt[a + b*x^2]*(a^4*(2048*c*C + 2048*B*d + 945*C*d*x) + 12*a^2*b 
^2*x^2*(3*A*(64*c + 35*d*x) + x*(105*B*c + 64*c*C*x + 64*B*d*x + 42*C*d*x^ 
2)) - 2*a^3*b*(9*A*(256*c + 105*d*x) + x*(945*B*c + 512*c*C*x + 512*B*d*x 
+ 315*C*d*x^2)) + 16*a*b^3*x^4*(9*A*(128*c + 105*d*x) + x*(945*B*c + 800*c 
*C*x + 800*B*d*x + 693*C*d*x^2)) + 32*b^4*x^6*(45*A*(8*c + 7*d*x) + 7*x*(5 
*B*(9*c + 8*d*x) + 4*C*x*(10*c + 9*d*x)))) + 945*a^4*(a*C*d - 2*b*(B*c + A 
*d))*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/(80640*b^(7/2))
 

Rubi [A] (verified)

Time = 2.73 (sec) , antiderivative size = 483, normalized size of antiderivative = 1.55, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.433, Rules used = {2185, 27, 2185, 25, 2185, 2185, 25, 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.

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

\(\Big \downarrow \) 2185

\(\displaystyle \frac {\int -5 (c+d x) \left (b x^2+a\right )^{3/2} \left (b d^4 (7 c C-2 B d) x^4+d^3 \left (9 b C c^2-2 A b d^2+a C d^2\right ) x^3+c C d^2 \left (5 b c^2+3 a d^2\right ) x^2+c^2 C d \left (b c^2+3 a d^2\right ) x+a c^3 C d^2\right )dx}{10 b d^5}+\frac {C \left (a+b x^2\right )^{5/2} (c+d x)^5}{10 b d^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {C \left (a+b x^2\right )^{5/2} (c+d x)^5}{10 b d^4}-\frac {\int (c+d x) \left (b x^2+a\right )^{3/2} \left (b d^4 (7 c C-2 B d) x^4+d^3 \left (9 b C c^2-2 A b d^2+a C d^2\right ) x^3+c C d^2 \left (5 b c^2+3 a d^2\right ) x^2+c^2 C d \left (b c^2+3 a d^2\right ) x+a c^3 C d^2\right )dx}{2 b d^5}\)

\(\Big \downarrow \) 2185

\(\displaystyle \frac {C \left (a+b x^2\right )^{5/2} (c+d x)^5}{10 b d^4}-\frac {\frac {\int -\left ((c+d x) \left (b x^2+a\right )^{3/2} \left (-b \left (9 a C d^2-2 b \left (40 C c^2-23 B d c+9 A d^2\right )\right ) x^3 d^7+b \left (2 b (44 c C-19 B d) c^2+a d^2 (c C-8 B d)\right ) x^2 d^6+a b c^2 (19 c C-8 B d) d^6+b c \left (2 b (13 c C-5 B d) c^2+a d^2 (29 c C-16 B d)\right ) x d^5\right )\right )dx}{9 b d^4}+\frac {1}{9} d \left (a+b x^2\right )^{5/2} (c+d x)^4 (7 c C-2 B d)}{2 b d^5}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {C \left (a+b x^2\right )^{5/2} (c+d x)^5}{10 b d^4}-\frac {\frac {1}{9} d \left (a+b x^2\right )^{5/2} (c+d x)^4 (7 c C-2 B d)-\frac {\int (c+d x) \left (b x^2+a\right )^{3/2} \left (-b \left (9 a C d^2-2 b \left (40 C c^2-23 B d c+9 A d^2\right )\right ) x^3 d^7+b \left (2 b (44 c C-19 B d) c^2+a d^2 (c C-8 B d)\right ) x^2 d^6+a b c^2 (19 c C-8 B d) d^6+b c \left (2 b (13 c C-5 B d) c^2+a d^2 (29 c C-16 B d)\right ) x d^5\right )dx}{9 b d^4}}{2 b d^5}\)

\(\Big \downarrow \) 2185

\(\displaystyle \frac {C \left (a+b x^2\right )^{5/2} (c+d x)^5}{10 b d^4}-\frac {\frac {1}{9} d \left (a+b x^2\right )^{5/2} (c+d x)^4 (7 c C-2 B d)-\frac {\frac {\int (c+d x) \left (b x^2+a\right )^{3/2} \left (b^2 \left (a d^2 (125 c C-64 B d)-6 b c \left (56 C c^2-49 B d c+39 A d^2\right )\right ) x^2 d^9+a b c \left (27 a C d^2-b \left (88 C c^2-74 B d c+54 A d^2\right )\right ) d^9+b \left (27 a^2 C d^4+a b \left (37 C c^2+10 B d c-54 A d^2\right ) d^2-6 b^2 c^2 \left (32 C c^2-25 B d c+15 A d^2\right )\right ) x d^8\right )dx}{8 b d^3}+\frac {1}{8} d^5 \left (a+b x^2\right )^{5/2} (c+d x)^3 \left (-9 a C d^2+18 A b d^2-46 b B c d+80 b c^2 C\right )}{9 b d^4}}{2 b d^5}\)

\(\Big \downarrow \) 2185

\(\displaystyle \frac {C \left (a+b x^2\right )^{5/2} (c+d x)^5}{10 b d^4}-\frac {\frac {1}{9} d \left (a+b x^2\right )^{5/2} (c+d x)^4 (7 c C-2 B d)-\frac {\frac {\frac {\int -b^2 d^{10} (c+d x) \left (a d \left (a d^2 (61 c C-128 B d)-2 b c \left (28 C c^2-35 B d c+45 A d^2\right )\right )-3 \left (63 a^2 C d^4-2 a b \left (61 C c^2-65 B d c+63 A d^2\right ) d^2+4 b^2 c^2 \left (28 C c^2-35 B d c+45 A d^2\right )\right ) x\right ) \left (b x^2+a\right )^{3/2}dx}{7 b d^2}+\frac {1}{7} b d^8 \left (a+b x^2\right )^{5/2} (c+d x)^2 \left (a d^2 (125 c C-64 B d)-6 b c \left (39 A d^2-49 B c d+56 c^2 C\right )\right )}{8 b d^3}+\frac {1}{8} d^5 \left (a+b x^2\right )^{5/2} (c+d x)^3 \left (-9 a C d^2+18 A b d^2-46 b B c d+80 b c^2 C\right )}{9 b d^4}}{2 b d^5}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {C \left (a+b x^2\right )^{5/2} (c+d x)^5}{10 b d^4}-\frac {\frac {1}{9} d \left (a+b x^2\right )^{5/2} (c+d x)^4 (7 c C-2 B d)-\frac {\frac {\frac {1}{7} b d^8 \left (a+b x^2\right )^{5/2} (c+d x)^2 \left (a d^2 (125 c C-64 B d)-6 b c \left (39 A d^2-49 B c d+56 c^2 C\right )\right )-\frac {\int b^2 d^{10} (c+d x) \left (a d \left (a d^2 (61 c C-128 B d)-2 b c \left (28 C c^2-35 B d c+45 A d^2\right )\right )-3 \left (63 a^2 C d^4-2 a b \left (61 C c^2-65 B d c+63 A d^2\right ) d^2+4 b^2 c^2 \left (28 C c^2-35 B d c+45 A d^2\right )\right ) x\right ) \left (b x^2+a\right )^{3/2}dx}{7 b d^2}}{8 b d^3}+\frac {1}{8} d^5 \left (a+b x^2\right )^{5/2} (c+d x)^3 \left (-9 a C d^2+18 A b d^2-46 b B c d+80 b c^2 C\right )}{9 b d^4}}{2 b d^5}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {C \left (a+b x^2\right )^{5/2} (c+d x)^5}{10 b d^4}-\frac {\frac {1}{9} d \left (a+b x^2\right )^{5/2} (c+d x)^4 (7 c C-2 B d)-\frac {\frac {\frac {1}{7} b d^8 \left (a+b x^2\right )^{5/2} (c+d x)^2 \left (a d^2 (125 c C-64 B d)-6 b c \left (39 A d^2-49 B c d+56 c^2 C\right )\right )-\frac {1}{7} b d^8 \int (c+d x) \left (a d \left (a d^2 (61 c C-128 B d)-2 b c \left (28 C c^2-35 B d c+45 A d^2\right )\right )-3 \left (63 a^2 C d^4-2 a b \left (61 C c^2-65 B d c+63 A d^2\right ) d^2+4 b^2 c^2 \left (28 C c^2-35 B d c+45 A d^2\right )\right ) x\right ) \left (b x^2+a\right )^{3/2}dx}{8 b d^3}+\frac {1}{8} d^5 \left (a+b x^2\right )^{5/2} (c+d x)^3 \left (-9 a C d^2+18 A b d^2-46 b B c d+80 b c^2 C\right )}{9 b d^4}}{2 b d^5}\)

\(\Big \downarrow \) 676

\(\displaystyle \frac {C \left (a+b x^2\right )^{5/2} (c+d x)^5}{10 b d^4}-\frac {\frac {1}{9} d \left (a+b x^2\right )^{5/2} (c+d x)^4 (7 c C-2 B d)-\frac {\frac {\frac {1}{7} b d^8 \left (a+b x^2\right )^{5/2} (c+d x)^2 \left (a d^2 (125 c C-64 B d)-6 b c \left (39 A d^2-49 B c d+56 c^2 C\right )\right )-\frac {1}{7} b d^8 \left (\frac {63 a^2 d^4 (a C d-2 b (A d+B c)) \int \left (b x^2+a\right )^{3/2}dx}{2 b}-\frac {d x \left (a+b x^2\right )^{5/2} \left (63 a^2 C d^4-2 a b d^2 \left (63 A d^2-65 B c d+61 c^2 C\right )+4 b^2 c^2 \left (45 A d^2-35 B c d+28 c^2 C\right )\right )}{2 b}-\frac {2 \left (a+b x^2\right )^{5/2} \left (64 a^2 d^4 (B d+c C)-a b c d^2 \left (144 A d^2-160 B c d+155 c^2 C\right )+6 b^2 c^3 \left (45 A d^2-35 B c d+28 c^2 C\right )\right )}{5 b}\right )}{8 b d^3}+\frac {1}{8} d^5 \left (a+b x^2\right )^{5/2} (c+d x)^3 \left (-9 a C d^2+18 A b d^2-46 b B c d+80 b c^2 C\right )}{9 b d^4}}{2 b d^5}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {C \left (a+b x^2\right )^{5/2} (c+d x)^5}{10 b d^4}-\frac {\frac {1}{9} d \left (a+b x^2\right )^{5/2} (c+d x)^4 (7 c C-2 B d)-\frac {\frac {\frac {1}{7} b d^8 \left (a+b x^2\right )^{5/2} (c+d x)^2 \left (a d^2 (125 c C-64 B d)-6 b c \left (39 A d^2-49 B c d+56 c^2 C\right )\right )-\frac {1}{7} b d^8 \left (\frac {63 a^2 d^4 (a C d-2 b (A d+B c)) \left (\frac {3}{4} a \int \sqrt {b x^2+a}dx+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )}{2 b}-\frac {d x \left (a+b x^2\right )^{5/2} \left (63 a^2 C d^4-2 a b d^2 \left (63 A d^2-65 B c d+61 c^2 C\right )+4 b^2 c^2 \left (45 A d^2-35 B c d+28 c^2 C\right )\right )}{2 b}-\frac {2 \left (a+b x^2\right )^{5/2} \left (64 a^2 d^4 (B d+c C)-a b c d^2 \left (144 A d^2-160 B c d+155 c^2 C\right )+6 b^2 c^3 \left (45 A d^2-35 B c d+28 c^2 C\right )\right )}{5 b}\right )}{8 b d^3}+\frac {1}{8} d^5 \left (a+b x^2\right )^{5/2} (c+d x)^3 \left (-9 a C d^2+18 A b d^2-46 b B c d+80 b c^2 C\right )}{9 b d^4}}{2 b d^5}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {C \left (a+b x^2\right )^{5/2} (c+d x)^5}{10 b d^4}-\frac {\frac {1}{9} d \left (a+b x^2\right )^{5/2} (c+d x)^4 (7 c C-2 B d)-\frac {\frac {\frac {1}{7} b d^8 \left (a+b x^2\right )^{5/2} (c+d x)^2 \left (a d^2 (125 c C-64 B d)-6 b c \left (39 A d^2-49 B c d+56 c^2 C\right )\right )-\frac {1}{7} b d^8 \left (\frac {63 a^2 d^4 (a C d-2 b (A d+B c)) \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{\sqrt {b x^2+a}}dx+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )}{2 b}-\frac {d x \left (a+b x^2\right )^{5/2} \left (63 a^2 C d^4-2 a b d^2 \left (63 A d^2-65 B c d+61 c^2 C\right )+4 b^2 c^2 \left (45 A d^2-35 B c d+28 c^2 C\right )\right )}{2 b}-\frac {2 \left (a+b x^2\right )^{5/2} \left (64 a^2 d^4 (B d+c C)-a b c d^2 \left (144 A d^2-160 B c d+155 c^2 C\right )+6 b^2 c^3 \left (45 A d^2-35 B c d+28 c^2 C\right )\right )}{5 b}\right )}{8 b d^3}+\frac {1}{8} d^5 \left (a+b x^2\right )^{5/2} (c+d x)^3 \left (-9 a C d^2+18 A b d^2-46 b B c d+80 b c^2 C\right )}{9 b d^4}}{2 b d^5}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {C \left (a+b x^2\right )^{5/2} (c+d x)^5}{10 b d^4}-\frac {\frac {1}{9} d \left (a+b x^2\right )^{5/2} (c+d x)^4 (7 c C-2 B d)-\frac {\frac {\frac {1}{7} b d^8 \left (a+b x^2\right )^{5/2} (c+d x)^2 \left (a d^2 (125 c C-64 B d)-6 b c \left (39 A d^2-49 B c d+56 c^2 C\right )\right )-\frac {1}{7} b d^8 \left (\frac {63 a^2 d^4 (a C d-2 b (A d+B c)) \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )}{2 b}-\frac {d x \left (a+b x^2\right )^{5/2} \left (63 a^2 C d^4-2 a b d^2 \left (63 A d^2-65 B c d+61 c^2 C\right )+4 b^2 c^2 \left (45 A d^2-35 B c d+28 c^2 C\right )\right )}{2 b}-\frac {2 \left (a+b x^2\right )^{5/2} \left (64 a^2 d^4 (B d+c C)-a b c d^2 \left (144 A d^2-160 B c d+155 c^2 C\right )+6 b^2 c^3 \left (45 A d^2-35 B c d+28 c^2 C\right )\right )}{5 b}\right )}{8 b d^3}+\frac {1}{8} d^5 \left (a+b x^2\right )^{5/2} (c+d x)^3 \left (-9 a C d^2+18 A b d^2-46 b B c d+80 b c^2 C\right )}{9 b d^4}}{2 b d^5}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {C \left (a+b x^2\right )^{5/2} (c+d x)^5}{10 b d^4}-\frac {\frac {1}{9} d \left (a+b x^2\right )^{5/2} (c+d x)^4 (7 c C-2 B d)-\frac {\frac {\frac {1}{7} b d^8 \left (a+b x^2\right )^{5/2} (c+d x)^2 \left (a d^2 (125 c C-64 B d)-6 b c \left (39 A d^2-49 B c d+56 c^2 C\right )\right )-\frac {1}{7} b d^8 \left (\frac {63 a^2 d^4 \left (\frac {3}{4} a \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right ) (a C d-2 b (A d+B c))}{2 b}-\frac {d x \left (a+b x^2\right )^{5/2} \left (63 a^2 C d^4-2 a b d^2 \left (63 A d^2-65 B c d+61 c^2 C\right )+4 b^2 c^2 \left (45 A d^2-35 B c d+28 c^2 C\right )\right )}{2 b}-\frac {2 \left (a+b x^2\right )^{5/2} \left (64 a^2 d^4 (B d+c C)-a b c d^2 \left (144 A d^2-160 B c d+155 c^2 C\right )+6 b^2 c^3 \left (45 A d^2-35 B c d+28 c^2 C\right )\right )}{5 b}\right )}{8 b d^3}+\frac {1}{8} d^5 \left (a+b x^2\right )^{5/2} (c+d x)^3 \left (-9 a C d^2+18 A b d^2-46 b B c d+80 b c^2 C\right )}{9 b d^4}}{2 b d^5}\)

Input:

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

Output:

(C*(c + d*x)^5*(a + b*x^2)^(5/2))/(10*b*d^4) - ((d*(7*c*C - 2*B*d)*(c + d* 
x)^4*(a + b*x^2)^(5/2))/9 - ((d^5*(80*b*c^2*C - 46*b*B*c*d + 18*A*b*d^2 - 
9*a*C*d^2)*(c + d*x)^3*(a + b*x^2)^(5/2))/8 + ((b*d^8*(a*d^2*(125*c*C - 64 
*B*d) - 6*b*c*(56*c^2*C - 49*B*c*d + 39*A*d^2))*(c + d*x)^2*(a + b*x^2)^(5 
/2))/7 - (b*d^8*((-2*(64*a^2*d^4*(c*C + B*d) + 6*b^2*c^3*(28*c^2*C - 35*B* 
c*d + 45*A*d^2) - a*b*c*d^2*(155*c^2*C - 160*B*c*d + 144*A*d^2))*(a + b*x^ 
2)^(5/2))/(5*b) - (d*(63*a^2*C*d^4 + 4*b^2*c^2*(28*c^2*C - 35*B*c*d + 45*A 
*d^2) - 2*a*b*d^2*(61*c^2*C - 65*B*c*d + 63*A*d^2))*x*(a + b*x^2)^(5/2))/( 
2*b) + (63*a^2*d^4*(a*C*d - 2*b*(B*c + A*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*Sq 
rt[b])))/4))/(2*b)))/7)/(8*b*d^3))/(9*b*d^4))/(2*b*d^5)
 

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 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 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 = 0.26 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.07

method	result	size
default	\(\left (A d +B c \right ) \left (\frac {x^{3} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{8 b}-\frac {3 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6 b}\right )}{8 b}\right )+\left (B d +C c \right ) \left (\frac {x^{4} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{9 b}-\frac {4 a \left (\frac {x^{2} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{7 b}-\frac {2 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{35 b^{2}}\right )}{9 b}\right )+A c \left (\frac {x^{2} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{7 b}-\frac {2 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{35 b^{2}}\right )+d C \left (\frac {x^{5} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{10 b}-\frac {a \left (\frac {x^{3} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{8 b}-\frac {3 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6 b}\right )}{8 b}\right )}{2 b}\right )\)	\(332\)
risch	\(-\frac {\left (-8064 C \,b^{4} d \,x^{9}-8960 B \,b^{4} d \,x^{8}-8960 C \,b^{4} c \,x^{8}-10080 A \,b^{4} d \,x^{7}-10080 B \,b^{4} c \,x^{7}-11088 C a \,b^{3} d \,x^{7}-11520 A \,b^{4} c \,x^{6}-12800 B a \,b^{3} d \,x^{6}-12800 C a \,b^{3} c \,x^{6}-15120 A a \,b^{3} d \,x^{5}-15120 B a \,b^{3} c \,x^{5}-504 C \,a^{2} b^{2} d \,x^{5}-18432 A a \,b^{3} c \,x^{4}-768 B \,a^{2} b^{2} d \,x^{4}-768 C \,a^{2} b^{2} c \,x^{4}-1260 A \,a^{2} b^{2} d \,x^{3}-1260 B \,a^{2} b^{2} c \,x^{3}+630 C \,a^{3} b d \,x^{3}-2304 A \,a^{2} b^{2} c \,x^{2}+1024 B \,a^{3} b d \,x^{2}+1024 C \,a^{3} b c \,x^{2}+1890 A \,a^{3} b d x +1890 B \,a^{3} b c x -945 C \,a^{4} d x +4608 A \,a^{3} b c -2048 B \,a^{4} d -2048 C \,a^{4} c \right ) \sqrt {b \,x^{2}+a}}{80640 b^{3}}+\frac {3 a^{4} \left (2 A b d +2 B b c -a C d \right ) \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{256 b^{\frac {7}{2}}}\)	\(342\)

Input:

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

Output:

(A*d+B*c)*(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))))))+(B*d+C*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)))+A*c*(1/7*x^2*(b*x 
^2+a)^(5/2)/b-2/35*a/b^2*(b*x^2+a)^(5/2))+d*C*(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 = 0.12 (sec) , antiderivative size = 716, normalized size of antiderivative = 2.30 \[ \int x^3 (c+d x) \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

[1/161280*(945*(2*B*a^4*b*c - (C*a^5 - 2*A*a^4*b)*d)*sqrt(b)*log(-2*b*x^2 
- 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 2*(8064*C*b^5*d*x^9 + 8960*(C*b^5*c + 
 B*b^5*d)*x^8 + 1008*(10*B*b^5*c + (11*C*a*b^4 + 10*A*b^5)*d)*x^7 + 2048*B 
*a^4*b*d + 1280*(10*B*a*b^4*d + (10*C*a*b^4 + 9*A*b^5)*c)*x^6 + 504*(30*B* 
a*b^4*c + (C*a^2*b^3 + 30*A*a*b^4)*d)*x^5 + 768*(B*a^2*b^3*d + (C*a^2*b^3 
+ 24*A*a*b^4)*c)*x^4 + 630*(2*B*a^2*b^3*c - (C*a^3*b^2 - 2*A*a^2*b^3)*d)*x 
^3 - 256*(4*B*a^3*b^2*d + (4*C*a^3*b^2 - 9*A*a^2*b^3)*c)*x^2 + 512*(4*C*a^ 
4*b - 9*A*a^3*b^2)*c - 945*(2*B*a^3*b^2*c - (C*a^4*b - 2*A*a^3*b^2)*d)*x)* 
sqrt(b*x^2 + a))/b^4, -1/80640*(945*(2*B*a^4*b*c - (C*a^5 - 2*A*a^4*b)*d)* 
sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) - (8064*C*b^5*d*x^9 + 8960*(C* 
b^5*c + B*b^5*d)*x^8 + 1008*(10*B*b^5*c + (11*C*a*b^4 + 10*A*b^5)*d)*x^7 + 
 2048*B*a^4*b*d + 1280*(10*B*a*b^4*d + (10*C*a*b^4 + 9*A*b^5)*c)*x^6 + 504 
*(30*B*a*b^4*c + (C*a^2*b^3 + 30*A*a*b^4)*d)*x^5 + 768*(B*a^2*b^3*d + (C*a 
^2*b^3 + 24*A*a*b^4)*c)*x^4 + 630*(2*B*a^2*b^3*c - (C*a^3*b^2 - 2*A*a^2*b^ 
3)*d)*x^3 - 256*(4*B*a^3*b^2*d + (4*C*a^3*b^2 - 9*A*a^2*b^3)*c)*x^2 + 512* 
(4*C*a^4*b - 9*A*a^3*b^2)*c - 945*(2*B*a^3*b^2*c - (C*a^4*b - 2*A*a^3*b^2) 
*d)*x)*sqrt(b*x^2 + a))/b^4]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 790 vs. \(2 (294) = 588\).

Time = 0.67 (sec) , antiderivative size = 790, normalized size of antiderivative = 2.54 \[ \int x^3 (c+d x) \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

Piecewise((3*a**2*(A*a**2*d + B*a**2*c - 5*a*(2*A*a*b*d + 2*B*a*b*c + C*a* 
*2*d - 7*a*(A*b**2*d + B*b**2*c + 11*C*a*b*d/10)/(8*b))/(6*b))*Piecewise(( 
log(2*sqrt(b)*sqrt(a + b*x**2) + 2*b*x)/sqrt(b), Ne(a, 0)), (x*log(x)/sqrt 
(b*x**2), True))/(8*b**2) + sqrt(a + b*x**2)*(C*b*d*x**9/10 - 3*a*x*(A*a** 
2*d + B*a**2*c - 5*a*(2*A*a*b*d + 2*B*a*b*c + C*a**2*d - 7*a*(A*b**2*d + B 
*b**2*c + 11*C*a*b*d/10)/(8*b))/(6*b))/(8*b**2) - 2*a*(A*a**2*c - 4*a*(2*A 
*a*b*c + B*a**2*d + C*a**2*c - 6*a*(A*b**2*c + 2*B*a*b*d + 2*C*a*b*c - 8*a 
*(B*b**2*d + C*b**2*c)/(9*b))/(7*b))/(5*b))/(3*b**2) + x**8*(B*b**2*d + C* 
b**2*c)/(9*b) + x**7*(A*b**2*d + B*b**2*c + 11*C*a*b*d/10)/(8*b) + x**6*(A 
*b**2*c + 2*B*a*b*d + 2*C*a*b*c - 8*a*(B*b**2*d + C*b**2*c)/(9*b))/(7*b) + 
 x**5*(2*A*a*b*d + 2*B*a*b*c + C*a**2*d - 7*a*(A*b**2*d + B*b**2*c + 11*C* 
a*b*d/10)/(8*b))/(6*b) + x**4*(2*A*a*b*c + B*a**2*d + C*a**2*c - 6*a*(A*b* 
*2*c + 2*B*a*b*d + 2*C*a*b*c - 8*a*(B*b**2*d + C*b**2*c)/(9*b))/(7*b))/(5* 
b) + x**3*(A*a**2*d + B*a**2*c - 5*a*(2*A*a*b*d + 2*B*a*b*c + C*a**2*d - 7 
*a*(A*b**2*d + B*b**2*c + 11*C*a*b*d/10)/(8*b))/(6*b))/(4*b) + x**2*(A*a** 
2*c - 4*a*(2*A*a*b*c + B*a**2*d + C*a**2*c - 6*a*(A*b**2*c + 2*B*a*b*d + 2 
*C*a*b*c - 8*a*(B*b**2*d + C*b**2*c)/(9*b))/(7*b))/(5*b))/(3*b)), Ne(b, 0) 
), (a**(3/2)*(A*c*x**4/4 + C*d*x**7/7 + x**6*(B*d + C*c)/6 + x**5*(A*d + B 
*c)/5), True))
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.12 \[ \int x^3 (c+d x) \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right ) \, dx=\frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} C d x^{5}}{10 \, b} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} C a d x^{3}}{16 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (C c + B d\right )} x^{4}}{9 \, b} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A c x^{2}}{7 \, b} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (B c + A d\right )} x^{3}}{8 \, b} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} C a^{2} d x}{32 \, b^{3}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} C a^{3} d x}{128 \, b^{3}} - \frac {3 \, \sqrt {b x^{2} + a} C a^{4} d x}{256 \, b^{3}} - \frac {4 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (C c + B d\right )} a x^{2}}{63 \, b^{2}} - \frac {3 \, C a^{5} d \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{256 \, b^{\frac {7}{2}}} - \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} A a c}{35 \, b^{2}} - \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (B c + A d\right )} a x}{16 \, b^{2}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (B c + A d\right )} a^{2} x}{64 \, b^{2}} + \frac {3 \, \sqrt {b x^{2} + a} {\left (B c + A d\right )} a^{3} x}{128 \, b^{2}} + \frac {3 \, {\left (B c + A d\right )} a^{4} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{128 \, b^{\frac {5}{2}}} + \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (C c + B d\right )} a^{2}}{315 \, b^{3}} \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.21 \[ \int x^3 (c+d x) \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right ) \, dx=\frac {1}{80640} \, \sqrt {b x^{2} + a} {\left ({\left (2 \, {\left ({\left (4 \, {\left ({\left (2 \, {\left (7 \, {\left (8 \, {\left (9 \, C b d x + \frac {10 \, {\left (C b^{9} c + B b^{9} d\right )}}{b^{8}}\right )} x + \frac {9 \, {\left (10 \, B b^{9} c + 11 \, C a b^{8} d + 10 \, A b^{9} d\right )}}{b^{8}}\right )} x + \frac {80 \, {\left (10 \, C a b^{8} c + 9 \, A b^{9} c + 10 \, B a b^{8} d\right )}}{b^{8}}\right )} x + \frac {63 \, {\left (30 \, B a b^{8} c + C a^{2} b^{7} d + 30 \, A a b^{8} d\right )}}{b^{8}}\right )} x + \frac {96 \, {\left (C a^{2} b^{7} c + 24 \, A a b^{8} c + B a^{2} b^{7} d\right )}}{b^{8}}\right )} x + \frac {315 \, {\left (2 \, B a^{2} b^{7} c - C a^{3} b^{6} d + 2 \, A a^{2} b^{7} d\right )}}{b^{8}}\right )} x - \frac {128 \, {\left (4 \, C a^{3} b^{6} c - 9 \, A a^{2} b^{7} c + 4 \, B a^{3} b^{6} d\right )}}{b^{8}}\right )} x - \frac {945 \, {\left (2 \, B a^{3} b^{6} c - C a^{4} b^{5} d + 2 \, A a^{3} b^{6} d\right )}}{b^{8}}\right )} x + \frac {512 \, {\left (4 \, C a^{4} b^{5} c - 9 \, A a^{3} b^{6} c + 4 \, B a^{4} b^{5} d\right )}}{b^{8}}\right )} - \frac {3 \, {\left (2 \, B a^{4} b c - C a^{5} d + 2 \, A a^{4} b d\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{256 \, b^{\frac {7}{2}}} \] Input:

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

Output:

1/80640*sqrt(b*x^2 + a)*((2*((4*((2*(7*(8*(9*C*b*d*x + 10*(C*b^9*c + B*b^9 
*d)/b^8)*x + 9*(10*B*b^9*c + 11*C*a*b^8*d + 10*A*b^9*d)/b^8)*x + 80*(10*C* 
a*b^8*c + 9*A*b^9*c + 10*B*a*b^8*d)/b^8)*x + 63*(30*B*a*b^8*c + C*a^2*b^7* 
d + 30*A*a*b^8*d)/b^8)*x + 96*(C*a^2*b^7*c + 24*A*a*b^8*c + B*a^2*b^7*d)/b 
^8)*x + 315*(2*B*a^2*b^7*c - C*a^3*b^6*d + 2*A*a^2*b^7*d)/b^8)*x - 128*(4* 
C*a^3*b^6*c - 9*A*a^2*b^7*c + 4*B*a^3*b^6*d)/b^8)*x - 945*(2*B*a^3*b^6*c - 
 C*a^4*b^5*d + 2*A*a^3*b^6*d)/b^8)*x + 512*(4*C*a^4*b^5*c - 9*A*a^3*b^6*c 
+ 4*B*a^4*b^5*d)/b^8) - 3/256*(2*B*a^4*b*c - C*a^5*d + 2*A*a^4*b*d)*log(ab 
s(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(7/2)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 604, normalized size of antiderivative = 1.94 \[ \int x^3 (c+d x) \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right ) \, dx=\frac {-1890 \sqrt {b \,x^{2}+a}\, a^{3} b^{3} c x -1024 \sqrt {b \,x^{2}+a}\, a^{3} b^{3} d \,x^{2}+1260 \sqrt {b \,x^{2}+a}\, a^{2} b^{4} c \,x^{3}+768 \sqrt {b \,x^{2}+a}\, a^{2} b^{4} d \,x^{4}+15120 \sqrt {b \,x^{2}+a}\, a \,b^{5} c \,x^{5}+945 \sqrt {b \,x^{2}+a}\, a^{4} b c d x -630 \sqrt {b \,x^{2}+a}\, a^{3} b^{2} c d \,x^{3}+504 \sqrt {b \,x^{2}+a}\, a^{2} b^{3} c d \,x^{5}+11088 \sqrt {b \,x^{2}+a}\, a \,b^{4} c d \,x^{7}+2048 \sqrt {b \,x^{2}+a}\, a^{4} b^{2} d +12800 \sqrt {b \,x^{2}+a}\, a \,b^{5} d \,x^{6}+1890 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{4} b^{2} c -1890 \sqrt {b \,x^{2}+a}\, a^{4} b^{2} d x +2304 \sqrt {b \,x^{2}+a}\, a^{3} b^{3} c \,x^{2}+1260 \sqrt {b \,x^{2}+a}\, a^{3} b^{3} d \,x^{3}-1024 \sqrt {b \,x^{2}+a}\, a^{3} b^{2} c^{2} x^{2}+18432 \sqrt {b \,x^{2}+a}\, a^{2} b^{4} c \,x^{4}+15120 \sqrt {b \,x^{2}+a}\, a^{2} b^{4} d \,x^{5}+768 \sqrt {b \,x^{2}+a}\, a^{2} b^{3} c^{2} x^{4}+11520 \sqrt {b \,x^{2}+a}\, a \,b^{5} c \,x^{6}+10080 \sqrt {b \,x^{2}+a}\, a \,b^{5} d \,x^{7}+12800 \sqrt {b \,x^{2}+a}\, a \,b^{4} c^{2} x^{6}+8064 \sqrt {b \,x^{2}+a}\, b^{5} c d \,x^{9}+1890 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{5} b d -945 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a^{5} c d -4608 \sqrt {b \,x^{2}+a}\, a^{4} b^{2} c +2048 \sqrt {b \,x^{2}+a}\, a^{4} b \,c^{2}+10080 \sqrt {b \,x^{2}+a}\, b^{6} c \,x^{7}+8960 \sqrt {b \,x^{2}+a}\, b^{6} d \,x^{8}+8960 \sqrt {b \,x^{2}+a}\, b^{5} c^{2} x^{8}}{80640 b^{4}} \] Input:

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

Output:

( - 4608*sqrt(a + b*x**2)*a**4*b**2*c - 1890*sqrt(a + b*x**2)*a**4*b**2*d* 
x + 2048*sqrt(a + b*x**2)*a**4*b**2*d + 2048*sqrt(a + b*x**2)*a**4*b*c**2 
+ 945*sqrt(a + b*x**2)*a**4*b*c*d*x + 2304*sqrt(a + b*x**2)*a**3*b**3*c*x* 
*2 - 1890*sqrt(a + b*x**2)*a**3*b**3*c*x + 1260*sqrt(a + b*x**2)*a**3*b**3 
*d*x**3 - 1024*sqrt(a + b*x**2)*a**3*b**3*d*x**2 - 1024*sqrt(a + b*x**2)*a 
**3*b**2*c**2*x**2 - 630*sqrt(a + b*x**2)*a**3*b**2*c*d*x**3 + 18432*sqrt( 
a + b*x**2)*a**2*b**4*c*x**4 + 1260*sqrt(a + b*x**2)*a**2*b**4*c*x**3 + 15 
120*sqrt(a + b*x**2)*a**2*b**4*d*x**5 + 768*sqrt(a + b*x**2)*a**2*b**4*d*x 
**4 + 768*sqrt(a + b*x**2)*a**2*b**3*c**2*x**4 + 504*sqrt(a + b*x**2)*a**2 
*b**3*c*d*x**5 + 11520*sqrt(a + b*x**2)*a*b**5*c*x**6 + 15120*sqrt(a + b*x 
**2)*a*b**5*c*x**5 + 10080*sqrt(a + b*x**2)*a*b**5*d*x**7 + 12800*sqrt(a + 
 b*x**2)*a*b**5*d*x**6 + 12800*sqrt(a + b*x**2)*a*b**4*c**2*x**6 + 11088*s 
qrt(a + b*x**2)*a*b**4*c*d*x**7 + 10080*sqrt(a + b*x**2)*b**6*c*x**7 + 896 
0*sqrt(a + b*x**2)*b**6*d*x**8 + 8960*sqrt(a + b*x**2)*b**5*c**2*x**8 + 80 
64*sqrt(a + b*x**2)*b**5*c*d*x**9 + 1890*sqrt(b)*log((sqrt(a + b*x**2) + s 
qrt(b)*x)/sqrt(a))*a**5*b*d - 945*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)* 
x)/sqrt(a))*a**5*c*d + 1890*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqr 
t(a))*a**4*b**2*c)/(80640*b**4)