\(\int (c+d x)^2 (a+b x^2)^{3/2} (A+B x+C x^2+D x^3) \, dx\) [72]

Optimal result

Integrand size = 34, antiderivative size = 417 \[ \int (c+d x)^2 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {a \left (8 A b \left (6 b c^2-a d^2\right )-a (8 b c (c C+2 B d)-3 a d (C d+2 c D))\right ) x \sqrt {a+b x^2}}{128 b^2}+\frac {\left (8 A b \left (6 b c^2-a d^2\right )-a (8 b c (c C+2 B d)-3 a d (C d+2 c D))\right ) x \left (a+b x^2\right )^{3/2}}{192 b^2}+\frac {\left (b^2 c (B c+2 A d)+a^2 d^2 D-a b \left (2 c C d+B d^2+c^2 D\right )\right ) \left (a+b x^2\right )^{5/2}}{5 b^3}+\frac {\left (8 b \left (c^2 C+2 B c d+A d^2\right )-3 a d (C d+2 c D)\right ) x \left (a+b x^2\right )^{5/2}}{48 b^2}+\frac {d (C d+2 c D) x^3 \left (a+b x^2\right )^{5/2}}{8 b}-\frac {\left (2 a d^2 D-b \left (2 c C d+B d^2+c^2 D\right )\right ) \left (a+b x^2\right )^{7/2}}{7 b^3}+\frac {d^2 D \left (a+b x^2\right )^{9/2}}{9 b^3}+\frac {a^2 \left (8 A b \left (6 b c^2-a d^2\right )-a (8 b c (c C+2 B d)-3 a d (C d+2 c D))\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{128 b^{5/2}} \] Output:

1/128*a*(8*A*b*(-a*d^2+6*b*c^2)-a*(8*b*c*(2*B*d+C*c)-3*a*d*(C*d+2*D*c)))*x 
*(b*x^2+a)^(1/2)/b^2+1/192*(8*A*b*(-a*d^2+6*b*c^2)-a*(8*b*c*(2*B*d+C*c)-3* 
a*d*(C*d+2*D*c)))*x*(b*x^2+a)^(3/2)/b^2+1/5*(b^2*c*(2*A*d+B*c)+a^2*d^2*D-a 
*b*(B*d^2+2*C*c*d+D*c^2))*(b*x^2+a)^(5/2)/b^3+1/48*(8*b*(A*d^2+2*B*c*d+C*c 
^2)-3*a*d*(C*d+2*D*c))*x*(b*x^2+a)^(5/2)/b^2+1/8*d*(C*d+2*D*c)*x^3*(b*x^2+ 
a)^(5/2)/b-1/7*(2*a*d^2*D-b*(B*d^2+2*C*c*d+D*c^2))*(b*x^2+a)^(7/2)/b^3+1/9 
*d^2*D*(b*x^2+a)^(9/2)/b^3+1/128*a^2*(8*A*b*(-a*d^2+6*b*c^2)-a*(8*b*c*(2*B 
*d+C*c)-3*a*d*(C*d+2*D*c)))*arctanh(b^(1/2)*x/(b*x^2+a)^(1/2))/b^(5/2)
 

Mathematica [A] (verified)

Time = 2.59 (sec) , antiderivative size = 438, normalized size of antiderivative = 1.05 \[ \int (c+d x)^2 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {\sqrt {a+b x^2} \left (1024 a^4 d^2 D+6 a^2 b^2 \left (84 A d (32 c+5 d x)+24 B \left (56 c^2+35 c d x+8 d^2 x^2\right )+x \left (12 c^2 (35 C+16 D x)+6 c d x (64 C+35 D x)+d^2 x^2 (105 C+64 D x)\right )\right )-a^3 b \left (2304 c^2 D+18 c d (256 C+105 D x)+d^2 (2304 B+x (945 C+512 D x))\right )+16 b^4 x^3 \left (42 A \left (15 c^2+24 c d x+10 d^2 x^2\right )+x \left (24 B \left (21 c^2+35 c d x+15 d^2 x^2\right )+5 x \left (12 c^2 (7 C+6 D x)+18 c d x (8 C+7 D x)+7 d^2 x^2 (9 C+8 D x)\right )\right )\right )+8 a b^3 x \left (42 A \left (75 c^2+96 c d x+35 d^2 x^2\right )+x \left (12 B \left (168 c^2+245 c d x+96 d^2 x^2\right )+x \left (18 c d x (128 C+105 D x)+5 d^2 x^2 (189 C+160 D x)+6 c^2 (245 C+192 D x)\right )\right )\right )\right )-315 a^2 \sqrt {b} \left (8 A b \left (6 b c^2-a d^2\right )+a (-8 b c (c C+2 B d)+3 a d (C d+2 c D))\right ) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{40320 b^3} \] Input:

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

Output:

(Sqrt[a + b*x^2]*(1024*a^4*d^2*D + 6*a^2*b^2*(84*A*d*(32*c + 5*d*x) + 24*B 
*(56*c^2 + 35*c*d*x + 8*d^2*x^2) + x*(12*c^2*(35*C + 16*D*x) + 6*c*d*x*(64 
*C + 35*D*x) + d^2*x^2*(105*C + 64*D*x))) - a^3*b*(2304*c^2*D + 18*c*d*(25 
6*C + 105*D*x) + d^2*(2304*B + x*(945*C + 512*D*x))) + 16*b^4*x^3*(42*A*(1 
5*c^2 + 24*c*d*x + 10*d^2*x^2) + x*(24*B*(21*c^2 + 35*c*d*x + 15*d^2*x^2) 
+ 5*x*(12*c^2*(7*C + 6*D*x) + 18*c*d*x*(8*C + 7*D*x) + 7*d^2*x^2*(9*C + 8* 
D*x)))) + 8*a*b^3*x*(42*A*(75*c^2 + 96*c*d*x + 35*d^2*x^2) + x*(12*B*(168* 
c^2 + 245*c*d*x + 96*d^2*x^2) + x*(18*c*d*x*(128*C + 105*D*x) + 5*d^2*x^2* 
(189*C + 160*D*x) + 6*c^2*(245*C + 192*D*x))))) - 315*a^2*Sqrt[b]*(8*A*b*( 
6*b*c^2 - a*d^2) + a*(-8*b*c*(c*C + 2*B*d) + 3*a*d*(C*d + 2*c*D)))*Log[-(S 
qrt[b]*x) + Sqrt[a + b*x^2]])/(40320*b^3)
 

Rubi [A] (verified)

Time = 0.96 (sec) , antiderivative size = 409, normalized size of antiderivative = 0.98, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.265, Rules used = {2185, 2185, 27, 687, 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.

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

\(\Big \downarrow \) 2185

\(\displaystyle \frac {\int (c+d x)^2 \left (b x^2+a\right )^{3/2} \left (b (9 C d-14 c D) x^2 d^2+(9 A b d-4 a c D) d^2+\left (-5 b D c^2+9 b B d^2-4 a d^2 D\right ) x d\right )dx}{9 b d^3}+\frac {D \left (a+b x^2\right )^{5/2} (c+d x)^4}{9 b d^2}\)

\(\Big \downarrow \) 2185

\(\displaystyle \frac {\frac {\int b d^3 (c+d x)^2 \left (d (72 A b d-27 a C d+10 a c D)-\left (32 a D d^2+b \left (-30 D c^2+45 C d c-72 B d^2\right )\right ) x\right ) \left (b x^2+a\right )^{3/2}dx}{8 b d^2}+\frac {1}{8} d \left (a+b x^2\right )^{5/2} (c+d x)^3 (9 C d-14 c D)}{9 b d^3}+\frac {D \left (a+b x^2\right )^{5/2} (c+d x)^4}{9 b d^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{8} d \int (c+d x)^2 \left (d (72 A b d-27 a C d+10 a c D)-\left (32 a D d^2+b \left (-30 D c^2+45 C d c-72 B d^2\right )\right ) x\right ) \left (b x^2+a\right )^{3/2}dx+\frac {1}{8} d \left (a+b x^2\right )^{5/2} (c+d x)^3 (9 C d-14 c D)}{9 b d^3}+\frac {D \left (a+b x^2\right )^{5/2} (c+d x)^4}{9 b d^2}\)

\(\Big \downarrow \) 687

\(\displaystyle \frac {\frac {1}{8} d \left (\frac {\int (c+d x) \left (d \left (504 A c d b^2+a \left (64 a d^2 D-b \left (-10 D c^2+99 C d c+144 B d^2\right )\right )\right )-3 b \left (a (63 C d-2 c D) d^2+2 b \left (-10 D c^3+15 C d c^2-24 B d^2 c-84 A d^3\right )\right ) x\right ) \left (b x^2+a\right )^{3/2}dx}{7 b}-\frac {\left (a+b x^2\right )^{5/2} (c+d x)^2 \left (32 a d^2 D+b \left (-72 B d^2-30 c^2 D+45 c C d\right )\right )}{7 b}\right )+\frac {1}{8} d \left (a+b x^2\right )^{5/2} (c+d x)^3 (9 C d-14 c D)}{9 b d^3}+\frac {D \left (a+b x^2\right )^{5/2} (c+d x)^4}{9 b d^2}\)

\(\Big \downarrow \) 676

\(\displaystyle \frac {\frac {1}{8} d \left (\frac {\frac {21}{2} d^2 \left (8 A b \left (6 b c^2-a d^2\right )-a (8 b c (2 B d+c C)-3 a d (2 c D+C d))\right ) \int \left (b x^2+a\right )^{3/2}dx+\frac {2 \left (a+b x^2\right )^{5/2} \left (32 a^2 d^4 D-8 a b d^2 \left (9 B d^2+c^2 (-D)+18 c C d\right )-3 b^2 c \left (-168 A d^3-24 B c d^2-10 c^3 D+15 c^2 C d\right )\right )}{5 b}-\frac {1}{2} d x \left (a+b x^2\right )^{5/2} \left (a d^2 (63 C d-2 c D)+2 b \left (-84 A d^3-24 B c d^2-10 c^3 D+15 c^2 C d\right )\right )}{7 b}-\frac {\left (a+b x^2\right )^{5/2} (c+d x)^2 \left (32 a d^2 D+b \left (-72 B d^2-30 c^2 D+45 c C d\right )\right )}{7 b}\right )+\frac {1}{8} d \left (a+b x^2\right )^{5/2} (c+d x)^3 (9 C d-14 c D)}{9 b d^3}+\frac {D \left (a+b x^2\right )^{5/2} (c+d x)^4}{9 b d^2}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\frac {1}{8} d \left (\frac {\frac {21}{2} d^2 \left (8 A b \left (6 b c^2-a d^2\right )-a (8 b c (2 B d+c C)-3 a d (2 c D+C d))\right ) \left (\frac {3}{4} a \int \sqrt {b x^2+a}dx+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )+\frac {2 \left (a+b x^2\right )^{5/2} \left (32 a^2 d^4 D-8 a b d^2 \left (9 B d^2+c^2 (-D)+18 c C d\right )-3 b^2 c \left (-168 A d^3-24 B c d^2-10 c^3 D+15 c^2 C d\right )\right )}{5 b}-\frac {1}{2} d x \left (a+b x^2\right )^{5/2} \left (a d^2 (63 C d-2 c D)+2 b \left (-84 A d^3-24 B c d^2-10 c^3 D+15 c^2 C d\right )\right )}{7 b}-\frac {\left (a+b x^2\right )^{5/2} (c+d x)^2 \left (32 a d^2 D+b \left (-72 B d^2-30 c^2 D+45 c C d\right )\right )}{7 b}\right )+\frac {1}{8} d \left (a+b x^2\right )^{5/2} (c+d x)^3 (9 C d-14 c D)}{9 b d^3}+\frac {D \left (a+b x^2\right )^{5/2} (c+d x)^4}{9 b d^2}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\frac {1}{8} d \left (\frac {\frac {21}{2} d^2 \left (8 A b \left (6 b c^2-a d^2\right )-a (8 b c (2 B d+c C)-3 a d (2 c D+C d))\right ) \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{\sqrt {b x^2+a}}dx+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )+\frac {2 \left (a+b x^2\right )^{5/2} \left (32 a^2 d^4 D-8 a b d^2 \left (9 B d^2+c^2 (-D)+18 c C d\right )-3 b^2 c \left (-168 A d^3-24 B c d^2-10 c^3 D+15 c^2 C d\right )\right )}{5 b}-\frac {1}{2} d x \left (a+b x^2\right )^{5/2} \left (a d^2 (63 C d-2 c D)+2 b \left (-84 A d^3-24 B c d^2-10 c^3 D+15 c^2 C d\right )\right )}{7 b}-\frac {\left (a+b x^2\right )^{5/2} (c+d x)^2 \left (32 a d^2 D+b \left (-72 B d^2-30 c^2 D+45 c C d\right )\right )}{7 b}\right )+\frac {1}{8} d \left (a+b x^2\right )^{5/2} (c+d x)^3 (9 C d-14 c D)}{9 b d^3}+\frac {D \left (a+b x^2\right )^{5/2} (c+d x)^4}{9 b d^2}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\frac {1}{8} d \left (\frac {\frac {21}{2} d^2 \left (8 A b \left (6 b c^2-a d^2\right )-a (8 b c (2 B d+c C)-3 a d (2 c D+C d))\right ) \left (\frac {3}{4} a \left (\frac {1}{2} a \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right )+\frac {2 \left (a+b x^2\right )^{5/2} \left (32 a^2 d^4 D-8 a b d^2 \left (9 B d^2+c^2 (-D)+18 c C d\right )-3 b^2 c \left (-168 A d^3-24 B c d^2-10 c^3 D+15 c^2 C d\right )\right )}{5 b}-\frac {1}{2} d x \left (a+b x^2\right )^{5/2} \left (a d^2 (63 C d-2 c D)+2 b \left (-84 A d^3-24 B c d^2-10 c^3 D+15 c^2 C d\right )\right )}{7 b}-\frac {\left (a+b x^2\right )^{5/2} (c+d x)^2 \left (32 a d^2 D+b \left (-72 B d^2-30 c^2 D+45 c C d\right )\right )}{7 b}\right )+\frac {1}{8} d \left (a+b x^2\right )^{5/2} (c+d x)^3 (9 C d-14 c D)}{9 b d^3}+\frac {D \left (a+b x^2\right )^{5/2} (c+d x)^4}{9 b d^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {1}{8} d \left (\frac {\frac {2 \left (a+b x^2\right )^{5/2} \left (32 a^2 d^4 D-8 a b d^2 \left (9 B d^2+c^2 (-D)+18 c C d\right )-3 b^2 c \left (-168 A d^3-24 B c d^2-10 c^3 D+15 c^2 C d\right )\right )}{5 b}+\frac {21}{2} d^2 \left (\frac {3}{4} a \left (\frac {a \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}}+\frac {1}{2} x \sqrt {a+b x^2}\right )+\frac {1}{4} x \left (a+b x^2\right )^{3/2}\right ) \left (8 A b \left (6 b c^2-a d^2\right )-a (8 b c (2 B d+c C)-3 a d (2 c D+C d))\right )-\frac {1}{2} d x \left (a+b x^2\right )^{5/2} \left (a d^2 (63 C d-2 c D)+2 b \left (-84 A d^3-24 B c d^2-10 c^3 D+15 c^2 C d\right )\right )}{7 b}-\frac {\left (a+b x^2\right )^{5/2} (c+d x)^2 \left (32 a d^2 D+b \left (-72 B d^2-30 c^2 D+45 c C d\right )\right )}{7 b}\right )+\frac {1}{8} d \left (a+b x^2\right )^{5/2} (c+d x)^3 (9 C d-14 c D)}{9 b d^3}+\frac {D \left (a+b x^2\right )^{5/2} (c+d x)^4}{9 b d^2}\)

Input:

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

Output:

(D*(c + d*x)^4*(a + b*x^2)^(5/2))/(9*b*d^2) + ((d*(9*C*d - 14*c*D)*(c + d* 
x)^3*(a + b*x^2)^(5/2))/8 + (d*(-1/7*((32*a*d^2*D + b*(45*c*C*d - 72*B*d^2 
 - 30*c^2*D))*(c + d*x)^2*(a + b*x^2)^(5/2))/b + ((2*(32*a^2*d^4*D - 8*a*b 
*d^2*(18*c*C*d + 9*B*d^2 - c^2*D) - 3*b^2*c*(15*c^2*C*d - 24*B*c*d^2 - 168 
*A*d^3 - 10*c^3*D))*(a + b*x^2)^(5/2))/(5*b) - (d*(a*d^2*(63*C*d - 2*c*D) 
+ 2*b*(15*c^2*C*d - 24*B*c*d^2 - 84*A*d^3 - 10*c^3*D))*x*(a + b*x^2)^(5/2) 
)/2 + (21*d^2*(8*A*b*(6*b*c^2 - a*d^2) - a*(8*b*c*(c*C + 2*B*d) - 3*a*d*(C 
*d + 2*c*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)/(7*b)))/8)/(9*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 = 1.23 (sec) , antiderivative size = 390, normalized size of antiderivative = 0.94

method	result	size
default	\(A \,c^{2} \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )+\frac {c \left (2 A d +B c \right ) \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{5 b}+d \left (C d +2 D c \right ) \left (\frac {x^{3} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{8 b}-\frac {3 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6 b}\right )}{8 b}\right )+\left (A \,d^{2}+2 B c d +C \,c^{2}\right ) \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6 b}\right )+\left (B \,d^{2}+2 C c d +D c^{2}\right ) \left (\frac {x^{2} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{7 b}-\frac {2 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{35 b^{2}}\right )+D d^{2} \left (\frac {x^{4} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{9 b}-\frac {4 a \left (\frac {x^{2} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{7 b}-\frac {2 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{35 b^{2}}\right )}{9 b}\right )\)	\(390\)

Input:

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

Output:

A*c^2*(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*A*d+B*c)/b*(b*x^2+a)^(5/2)+d*(C*d+2 
*D*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))))))+(A*d^2+2*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)*l 
n(b^(1/2)*x+(b*x^2+a)^(1/2)))))+(B*d^2+2*C*c*d+D*c^2)*(1/7*x^2*(b*x^2+a)^( 
5/2)/b-2/35*a/b^2*(b*x^2+a)^(5/2))+D*d^2*(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)))
 

Fricas [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 1119, normalized size of antiderivative = 2.68 \[ \int (c+d x)^2 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

[1/80640*(315*(8*(C*a^3*b - 6*A*a^2*b^2)*c^2 - 2*(3*D*a^4 - 8*B*a^3*b)*c*d 
 - (3*C*a^4 - 8*A*a^3*b)*d^2)*sqrt(b)*log(-2*b*x^2 + 2*sqrt(b*x^2 + a)*sqr 
t(b)*x - a) + 2*(4480*D*b^4*d^2*x^8 + 5040*(2*D*b^4*c*d + C*b^4*d^2)*x^7 + 
 640*(9*D*b^4*c^2 + 18*C*b^4*c*d + (10*D*a*b^3 + 9*B*b^4)*d^2)*x^6 + 840*( 
8*C*b^4*c^2 + 2*(9*D*a*b^3 + 8*B*b^4)*c*d + (9*C*a*b^3 + 8*A*b^4)*d^2)*x^5 
 + 384*(3*(8*D*a*b^3 + 7*B*b^4)*c^2 + 6*(8*C*a*b^3 + 7*A*b^4)*c*d + (D*a^2 
*b^2 + 24*B*a*b^3)*d^2)*x^4 + 210*(8*(7*C*a*b^3 + 6*A*b^4)*c^2 + 2*(3*D*a^ 
2*b^2 + 56*B*a*b^3)*c*d + (3*C*a^2*b^2 + 56*A*a*b^3)*d^2)*x^3 - 1152*(2*D* 
a^3*b - 7*B*a^2*b^2)*c^2 - 2304*(2*C*a^3*b - 7*A*a^2*b^2)*c*d + 256*(4*D*a 
^4 - 9*B*a^3*b)*d^2 + 128*(9*(D*a^2*b^2 + 14*B*a*b^3)*c^2 + 18*(C*a^2*b^2 
+ 14*A*a*b^3)*c*d - (4*D*a^3*b - 9*B*a^2*b^2)*d^2)*x^2 + 315*(8*(C*a^2*b^2 
 + 10*A*a*b^3)*c^2 - 2*(3*D*a^3*b - 8*B*a^2*b^2)*c*d - (3*C*a^3*b - 8*A*a^ 
2*b^2)*d^2)*x)*sqrt(b*x^2 + a))/b^3, 1/40320*(315*(8*(C*a^3*b - 6*A*a^2*b^ 
2)*c^2 - 2*(3*D*a^4 - 8*B*a^3*b)*c*d - (3*C*a^4 - 8*A*a^3*b)*d^2)*sqrt(-b) 
*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)) + (4480*D*b^4*d^2*x^8 + 5040*(2*D*b^4* 
c*d + C*b^4*d^2)*x^7 + 640*(9*D*b^4*c^2 + 18*C*b^4*c*d + (10*D*a*b^3 + 9*B 
*b^4)*d^2)*x^6 + 840*(8*C*b^4*c^2 + 2*(9*D*a*b^3 + 8*B*b^4)*c*d + (9*C*a*b 
^3 + 8*A*b^4)*d^2)*x^5 + 384*(3*(8*D*a*b^3 + 7*B*b^4)*c^2 + 6*(8*C*a*b^3 + 
 7*A*b^4)*c*d + (D*a^2*b^2 + 24*B*a*b^3)*d^2)*x^4 + 210*(8*(7*C*a*b^3 + 6* 
A*b^4)*c^2 + 2*(3*D*a^2*b^2 + 56*B*a*b^3)*c*d + (3*C*a^2*b^2 + 56*A*a*b...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1302 vs. \(2 (403) = 806\).

Time = 0.87 (sec) , antiderivative size = 1302, normalized size of antiderivative = 3.12 \[ \int (c+d x)^2 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\text {Too large to display} \] Input:

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

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 489, normalized size of antiderivative = 1.17 \[ \int (c+d x)^2 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} D d^{2} x^{4}}{9 \, b} - \frac {4 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} D a d^{2} x^{2}}{63 \, b^{2}} + \frac {1}{4} \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} A c^{2} x + \frac {3}{8} \, \sqrt {b x^{2} + a} A a c^{2} x + \frac {{\left (2 \, D c d + C d^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {5}{2}} x^{3}}{8 \, b} + \frac {3 \, A a^{2} c^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {b}} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} B c^{2}}{5 \, b} + \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} A c d}{5 \, b} + \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} D a^{2} d^{2}}{315 \, b^{3}} + \frac {{\left (D c^{2} + 2 \, C c d + B d^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {5}{2}} x^{2}}{7 \, b} - \frac {{\left (2 \, D c d + C d^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {5}{2}} a x}{16 \, b^{2}} + \frac {{\left (2 \, D c d + C d^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} x}{64 \, b^{2}} + \frac {3 \, {\left (2 \, D c d + C d^{2}\right )} \sqrt {b x^{2} + a} a^{3} x}{128 \, b^{2}} + \frac {{\left (C c^{2} + 2 \, B c d + A d^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {5}{2}} x}{6 \, b} - \frac {{\left (C c^{2} + 2 \, B c d + A d^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {3}{2}} a x}{24 \, b} - \frac {{\left (C c^{2} + 2 \, B c d + A d^{2}\right )} \sqrt {b x^{2} + a} a^{2} x}{16 \, b} + \frac {3 \, {\left (2 \, D c d + C d^{2}\right )} a^{4} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{128 \, b^{\frac {5}{2}}} - \frac {{\left (C c^{2} + 2 \, B c d + A d^{2}\right )} a^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{16 \, b^{\frac {3}{2}}} - \frac {2 \, {\left (D c^{2} + 2 \, C c d + B d^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {5}{2}} a}{35 \, b^{2}} \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 596, normalized size of antiderivative = 1.43 \[ \int (c+d x)^2 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2+D x^3\right ) \, dx=\frac {1}{40320} \, \sqrt {b x^{2} + a} {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, {\left (2 \, {\left (7 \, {\left (8 \, D b d^{2} x + \frac {9 \, {\left (2 \, D b^{8} c d + C b^{8} d^{2}\right )}}{b^{7}}\right )} x + \frac {8 \, {\left (9 \, D b^{8} c^{2} + 18 \, C b^{8} c d + 10 \, D a b^{7} d^{2} + 9 \, B b^{8} d^{2}\right )}}{b^{7}}\right )} x + \frac {21 \, {\left (8 \, C b^{8} c^{2} + 18 \, D a b^{7} c d + 16 \, B b^{8} c d + 9 \, C a b^{7} d^{2} + 8 \, A b^{8} d^{2}\right )}}{b^{7}}\right )} x + \frac {48 \, {\left (24 \, D a b^{7} c^{2} + 21 \, B b^{8} c^{2} + 48 \, C a b^{7} c d + 42 \, A b^{8} c d + D a^{2} b^{6} d^{2} + 24 \, B a b^{7} d^{2}\right )}}{b^{7}}\right )} x + \frac {105 \, {\left (56 \, C a b^{7} c^{2} + 48 \, A b^{8} c^{2} + 6 \, D a^{2} b^{6} c d + 112 \, B a b^{7} c d + 3 \, C a^{2} b^{6} d^{2} + 56 \, A a b^{7} d^{2}\right )}}{b^{7}}\right )} x + \frac {64 \, {\left (9 \, D a^{2} b^{6} c^{2} + 126 \, B a b^{7} c^{2} + 18 \, C a^{2} b^{6} c d + 252 \, A a b^{7} c d - 4 \, D a^{3} b^{5} d^{2} + 9 \, B a^{2} b^{6} d^{2}\right )}}{b^{7}}\right )} x + \frac {315 \, {\left (8 \, C a^{2} b^{6} c^{2} + 80 \, A a b^{7} c^{2} - 6 \, D a^{3} b^{5} c d + 16 \, B a^{2} b^{6} c d - 3 \, C a^{3} b^{5} d^{2} + 8 \, A a^{2} b^{6} d^{2}\right )}}{b^{7}}\right )} x - \frac {128 \, {\left (18 \, D a^{3} b^{5} c^{2} - 63 \, B a^{2} b^{6} c^{2} + 36 \, C a^{3} b^{5} c d - 126 \, A a^{2} b^{6} c d - 8 \, D a^{4} b^{4} d^{2} + 18 \, B a^{3} b^{5} d^{2}\right )}}{b^{7}}\right )} + \frac {{\left (8 \, C a^{3} b c^{2} - 48 \, A a^{2} b^{2} c^{2} - 6 \, D a^{4} c d + 16 \, B a^{3} b c d - 3 \, C a^{4} d^{2} + 8 \, A a^{3} b d^{2}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{128 \, b^{\frac {5}{2}}} \] Input:

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

Output:

1/40320*sqrt(b*x^2 + a)*((2*((4*(5*(2*(7*(8*D*b*d^2*x + 9*(2*D*b^8*c*d + C 
*b^8*d^2)/b^7)*x + 8*(9*D*b^8*c^2 + 18*C*b^8*c*d + 10*D*a*b^7*d^2 + 9*B*b^ 
8*d^2)/b^7)*x + 21*(8*C*b^8*c^2 + 18*D*a*b^7*c*d + 16*B*b^8*c*d + 9*C*a*b^ 
7*d^2 + 8*A*b^8*d^2)/b^7)*x + 48*(24*D*a*b^7*c^2 + 21*B*b^8*c^2 + 48*C*a*b 
^7*c*d + 42*A*b^8*c*d + D*a^2*b^6*d^2 + 24*B*a*b^7*d^2)/b^7)*x + 105*(56*C 
*a*b^7*c^2 + 48*A*b^8*c^2 + 6*D*a^2*b^6*c*d + 112*B*a*b^7*c*d + 3*C*a^2*b^ 
6*d^2 + 56*A*a*b^7*d^2)/b^7)*x + 64*(9*D*a^2*b^6*c^2 + 126*B*a*b^7*c^2 + 1 
8*C*a^2*b^6*c*d + 252*A*a*b^7*c*d - 4*D*a^3*b^5*d^2 + 9*B*a^2*b^6*d^2)/b^7 
)*x + 315*(8*C*a^2*b^6*c^2 + 80*A*a*b^7*c^2 - 6*D*a^3*b^5*c*d + 16*B*a^2*b 
^6*c*d - 3*C*a^3*b^5*d^2 + 8*A*a^2*b^6*d^2)/b^7)*x - 128*(18*D*a^3*b^5*c^2 
 - 63*B*a^2*b^6*c^2 + 36*C*a^3*b^5*c*d - 126*A*a^2*b^6*c*d - 8*D*a^4*b^4*d 
^2 + 18*B*a^3*b^5*d^2)/b^7) + 1/128*(8*C*a^3*b*c^2 - 48*A*a^2*b^2*c^2 - 6* 
D*a^4*c*d + 16*B*a^3*b*c*d - 3*C*a^4*d^2 + 8*A*a^3*b*d^2)*log(abs(-sqrt(b) 
*x + sqrt(b*x^2 + a)))/b^(5/2)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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