\(\int x (c+d x)^2 (a+b x^2)^{3/2} (A+B x+C x^2) \, dx\) [63]

Optimal result

Integrand size = 30, antiderivative size = 349 \[ \int x (c+d x)^2 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right ) \, dx=\frac {a^2 (8 b c (B c+2 A d)-3 a d (2 c C+B d)) x \sqrt {a+b x^2}}{128 b^2}+\frac {a (8 b c (B c+2 A d)-3 a d (2 c C+B d)) x^3 \sqrt {a+b x^2}}{64 b}+\frac {(8 b c (B c+2 A d)-3 a d (2 c C+B d)) x^3 \left (a+b x^2\right )^{3/2}}{48 b}+\frac {\left (A b \left (b c^2-a d^2\right )+a \left (a C d^2-b c (c C+2 B d)\right )\right ) \left (a+b x^2\right )^{5/2}}{5 b^3}+\frac {d (2 c C+B d) x^3 \left (a+b x^2\right )^{5/2}}{8 b}-\frac {\left (2 a C d^2-b \left (c^2 C+2 B c d+A d^2\right )\right ) \left (a+b x^2\right )^{7/2}}{7 b^3}+\frac {C d^2 \left (a+b x^2\right )^{9/2}}{9 b^3}-\frac {a^3 (8 b c (B c+2 A d)-3 a d (2 c C+B d)) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{128 b^{5/2}} \] Output:

1/128*a^2*(8*b*c*(2*A*d+B*c)-3*a*d*(B*d+2*C*c))*x*(b*x^2+a)^(1/2)/b^2+1/64 
*a*(8*b*c*(2*A*d+B*c)-3*a*d*(B*d+2*C*c))*x^3*(b*x^2+a)^(1/2)/b+1/48*(8*b*c 
*(2*A*d+B*c)-3*a*d*(B*d+2*C*c))*x^3*(b*x^2+a)^(3/2)/b+1/5*(A*b*(-a*d^2+b*c 
^2)+a*(a*C*d^2-b*c*(2*B*d+C*c)))*(b*x^2+a)^(5/2)/b^3+1/8*d*(B*d+2*C*c)*x^3 
*(b*x^2+a)^(5/2)/b-1/7*(2*a*C*d^2-b*(A*d^2+2*B*c*d+C*c^2))*(b*x^2+a)^(7/2) 
/b^3+1/9*C*d^2*(b*x^2+a)^(9/2)/b^3-1/128*a^3*(8*b*c*(2*A*d+B*c)-3*a*d*(B*d 
+2*C*c))*arctanh(b^(1/2)*x/(b*x^2+a)^(1/2))/b^(5/2)
 

Mathematica [A] (verified)

Time = 1.72 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.05 \[ \int x (c+d x)^2 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right ) \, dx=\frac {\sqrt {a+b x^2} \left (1024 a^4 C d^2-a^3 b \left (2304 c^2 C+18 c d (256 B+105 C x)+d^2 \left (2304 A+945 B x+512 C x^2\right )\right )+16 b^4 x^4 \left (24 A \left (21 c^2+35 c d x+15 d^2 x^2\right )+5 x \left (3 B \left (28 c^2+48 c d x+21 d^2 x^2\right )+2 C x \left (36 c^2+63 c d x+28 d^2 x^2\right )\right )\right )+6 a^2 b^2 \left (24 A \left (56 c^2+35 c d x+8 d^2 x^2\right )+x \left (2 C x \left (96 c^2+105 c d x+32 d^2 x^2\right )+3 B \left (140 c^2+128 c d x+35 d^2 x^2\right )\right )\right )+8 a b^3 x^2 \left (12 A \left (168 c^2+245 c d x+96 d^2 x^2\right )+x \left (3 B \left (490 c^2+768 c d x+315 d^2 x^2\right )+2 C x \left (576 c^2+945 c d x+400 d^2 x^2\right )\right )\right )\right )-315 a^3 \sqrt {b} (-8 b c (B c+2 A d)+3 a d (2 c C+B d)) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{40320 b^3} \] Input:

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

Output:

(Sqrt[a + b*x^2]*(1024*a^4*C*d^2 - a^3*b*(2304*c^2*C + 18*c*d*(256*B + 105 
*C*x) + d^2*(2304*A + 945*B*x + 512*C*x^2)) + 16*b^4*x^4*(24*A*(21*c^2 + 3 
5*c*d*x + 15*d^2*x^2) + 5*x*(3*B*(28*c^2 + 48*c*d*x + 21*d^2*x^2) + 2*C*x* 
(36*c^2 + 63*c*d*x + 28*d^2*x^2))) + 6*a^2*b^2*(24*A*(56*c^2 + 35*c*d*x + 
8*d^2*x^2) + x*(2*C*x*(96*c^2 + 105*c*d*x + 32*d^2*x^2) + 3*B*(140*c^2 + 1 
28*c*d*x + 35*d^2*x^2))) + 8*a*b^3*x^2*(12*A*(168*c^2 + 245*c*d*x + 96*d^2 
*x^2) + x*(3*B*(490*c^2 + 768*c*d*x + 315*d^2*x^2) + 2*C*x*(576*c^2 + 945* 
c*d*x + 400*d^2*x^2)))) - 315*a^3*Sqrt[b]*(-8*b*c*(B*c + 2*A*d) + 3*a*d*(2 
*c*C + B*d))*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/(40320*b^3)
 

Rubi [A] (verified)

Time = 1.52 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.07, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.367, Rules used = {2185, 25, 2185, 25, 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.

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

\(\Big \downarrow \) 2185

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2185

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 687

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

\(\Big \downarrow \) 676

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

\(\Big \downarrow \) 211

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

\(\Big \downarrow \) 211

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

(C*(c + d*x)^4*(a + b*x^2)^(5/2))/(9*b*d^2) - ((d*(14*c*C - 9*B*d)*(c + d* 
x)^3*(a + b*x^2)^(5/2))/8 - (d*(-1/7*((32*a*C*d^2 - b*(30*c^2*C - 45*B*c*d 
 + 72*A*d^2))*(c + d*x)^2*(a + b*x^2)^(5/2))/b + ((2*(32*a^2*C*d^4 + 8*a*b 
*d^2*(c^2*C - 18*B*c*d - 9*A*d^2) + 3*b^2*c^2*(10*c^2*C - 15*B*c*d + 24*A* 
d^2))*(a + b*x^2)^(5/2))/(5*b) + (d*(a*d^2*(2*c*C - 63*B*d) + 2*b*c*(10*c^ 
2*C - 15*B*c*d + 24*A*d^2))*x*(a + b*x^2)^(5/2))/2 - (21*a*d^2*(8*b*c*(B*c 
 + 2*A*d) - 3*a*d*(2*c*C + B*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 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 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 = 0.26 (sec) , antiderivative size = 322, normalized size of antiderivative = 0.92

method	result	size
default	\(c \left (2 A d +B c \right ) \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6 b}\right )+d \left (B d +2 C c \right ) \left (\frac {x^{3} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{8 b}-\frac {3 a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{6 b}-\frac {a \left (\frac {x \left (b \,x^{2}+a \right )^{\frac {3}{2}}}{4}+\frac {3 a \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{4}\right )}{6 b}\right )}{8 b}\right )+\left (A \,d^{2}+2 B c d +C \,c^{2}\right ) \left (\frac {x^{2} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{7 b}-\frac {2 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{35 b^{2}}\right )+\frac {A \,c^{2} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{5 b}+C \,d^{2} \left (\frac {x^{4} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{9 b}-\frac {4 a \left (\frac {x^{2} \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{7 b}-\frac {2 a \left (b \,x^{2}+a \right )^{\frac {5}{2}}}{35 b^{2}}\right )}{9 b}\right )\)	\(322\)
risch	\(-\frac {\left (-4480 C \,b^{4} d^{2} x^{8}-5040 B \,b^{4} d^{2} x^{7}-10080 C \,b^{4} c d \,x^{7}-5760 A \,b^{4} d^{2} x^{6}-11520 B \,b^{4} c d \,x^{6}-6400 C a \,b^{3} d^{2} x^{6}-5760 C \,b^{4} c^{2} x^{6}-13440 A \,b^{4} c d \,x^{5}-7560 B a \,b^{3} d^{2} x^{5}-6720 B \,b^{4} c^{2} x^{5}-15120 C a \,b^{3} c d \,x^{5}-9216 A a \,b^{3} d^{2} x^{4}-8064 A \,b^{4} c^{2} x^{4}-18432 B a \,b^{3} c d \,x^{4}-384 C \,a^{2} b^{2} d^{2} x^{4}-9216 C a \,b^{3} c^{2} x^{4}-23520 A a \,b^{3} c d \,x^{3}-630 B \,a^{2} b^{2} d^{2} x^{3}-11760 B a \,b^{3} c^{2} x^{3}-1260 C \,a^{2} b^{2} c d \,x^{3}-1152 A \,a^{2} b^{2} d^{2} x^{2}-16128 A a \,b^{3} c^{2} x^{2}-2304 B \,a^{2} b^{2} c d \,x^{2}+512 C \,a^{3} b \,d^{2} x^{2}-1152 C \,a^{2} b^{2} c^{2} x^{2}-5040 A \,a^{2} b^{2} c d x +945 B \,a^{3} b \,d^{2} x -2520 B \,a^{2} b^{2} c^{2} x +1890 C \,a^{3} b c d x +2304 A \,a^{3} b \,d^{2}-8064 A \,a^{2} b^{2} c^{2}+4608 B \,a^{3} b c d -1024 C \,a^{4} d^{2}+2304 C \,a^{3} b \,c^{2}\right ) \sqrt {b \,x^{2}+a}}{40320 b^{3}}-\frac {a^{3} \left (16 A b c d -3 a B \,d^{2}+8 b B \,c^{2}-6 C a c d \right ) \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{128 b^{\frac {5}{2}}}\)	\(484\)

Input:

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

Output:

c*(2*A*d+B*c)*(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*C*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)*(1/7*x^2*(b*x^2+a)^( 
5/2)/b-2/35*a/b^2*(b*x^2+a)^(5/2))+1/5*A*c^2*(b*x^2+a)^(5/2)/b+C*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 = 0.16 (sec) , antiderivative size = 923, normalized size of antiderivative = 2.64 \[ \int x (c+d x)^2 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

[-1/80640*(315*(8*B*a^3*b*c^2 - 3*B*a^4*d^2 - 2*(3*C*a^4 - 8*A*a^3*b)*c*d) 
*sqrt(b)*log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) - 2*(4480*C*b^4*d 
^2*x^8 + 5040*(2*C*b^4*c*d + B*b^4*d^2)*x^7 - 4608*B*a^3*b*c*d + 640*(9*C* 
b^4*c^2 + 18*B*b^4*c*d + (10*C*a*b^3 + 9*A*b^4)*d^2)*x^6 + 840*(8*B*b^4*c^ 
2 + 9*B*a*b^3*d^2 + 2*(9*C*a*b^3 + 8*A*b^4)*c*d)*x^5 + 384*(48*B*a*b^3*c*d 
 + 3*(8*C*a*b^3 + 7*A*b^4)*c^2 + (C*a^2*b^2 + 24*A*a*b^3)*d^2)*x^4 + 210*( 
56*B*a*b^3*c^2 + 3*B*a^2*b^2*d^2 + 2*(3*C*a^2*b^2 + 56*A*a*b^3)*c*d)*x^3 - 
 1152*(2*C*a^3*b - 7*A*a^2*b^2)*c^2 + 256*(4*C*a^4 - 9*A*a^3*b)*d^2 + 128* 
(18*B*a^2*b^2*c*d + 9*(C*a^2*b^2 + 14*A*a*b^3)*c^2 - (4*C*a^3*b - 9*A*a^2* 
b^2)*d^2)*x^2 + 315*(8*B*a^2*b^2*c^2 - 3*B*a^3*b*d^2 - 2*(3*C*a^3*b - 8*A* 
a^2*b^2)*c*d)*x)*sqrt(b*x^2 + a))/b^3, 1/40320*(315*(8*B*a^3*b*c^2 - 3*B*a 
^4*d^2 - 2*(3*C*a^4 - 8*A*a^3*b)*c*d)*sqrt(-b)*arctan(sqrt(-b)*x/sqrt(b*x^ 
2 + a)) + (4480*C*b^4*d^2*x^8 + 5040*(2*C*b^4*c*d + B*b^4*d^2)*x^7 - 4608* 
B*a^3*b*c*d + 640*(9*C*b^4*c^2 + 18*B*b^4*c*d + (10*C*a*b^3 + 9*A*b^4)*d^2 
)*x^6 + 840*(8*B*b^4*c^2 + 9*B*a*b^3*d^2 + 2*(9*C*a*b^3 + 8*A*b^4)*c*d)*x^ 
5 + 384*(48*B*a*b^3*c*d + 3*(8*C*a*b^3 + 7*A*b^4)*c^2 + (C*a^2*b^2 + 24*A* 
a*b^3)*d^2)*x^4 + 210*(56*B*a*b^3*c^2 + 3*B*a^2*b^2*d^2 + 2*(3*C*a^2*b^2 + 
 56*A*a*b^3)*c*d)*x^3 - 1152*(2*C*a^3*b - 7*A*a^2*b^2)*c^2 + 256*(4*C*a^4 
- 9*A*a^3*b)*d^2 + 128*(18*B*a^2*b^2*c*d + 9*(C*a^2*b^2 + 14*A*a*b^3)*c^2 
- (4*C*a^3*b - 9*A*a^2*b^2)*d^2)*x^2 + 315*(8*B*a^2*b^2*c^2 - 3*B*a^3*b...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1086 vs. \(2 (333) = 666\).

Time = 0.65 (sec) , antiderivative size = 1086, normalized size of antiderivative = 3.11 \[ \int x (c+d x)^2 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.14 \[ \int x (c+d x)^2 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right ) \, dx=\frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} C d^{2} x^{4}}{9 \, b} - \frac {4 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} C a d^{2} x^{2}}{63 \, b^{2}} + \frac {{\left (2 \, C c d + B d^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {5}{2}} x^{3}}{8 \, b} + \frac {{\left (b x^{2} + a\right )}^{\frac {5}{2}} A c^{2}}{5 \, b} + \frac {8 \, {\left (b x^{2} + a\right )}^{\frac {5}{2}} C a^{2} d^{2}}{315 \, b^{3}} + \frac {{\left (C c^{2} + 2 \, B c d + A d^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {5}{2}} x^{2}}{7 \, b} - \frac {{\left (2 \, C c d + B d^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {5}{2}} a x}{16 \, b^{2}} + \frac {{\left (2 \, C c d + B d^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {3}{2}} a^{2} x}{64 \, b^{2}} + \frac {3 \, {\left (2 \, C c d + B d^{2}\right )} \sqrt {b x^{2} + a} a^{3} x}{128 \, b^{2}} + \frac {{\left (B c^{2} + 2 \, A c d\right )} {\left (b x^{2} + a\right )}^{\frac {5}{2}} x}{6 \, b} - \frac {{\left (B c^{2} + 2 \, A c d\right )} {\left (b x^{2} + a\right )}^{\frac {3}{2}} a x}{24 \, b} - \frac {{\left (B c^{2} + 2 \, A c d\right )} \sqrt {b x^{2} + a} a^{2} x}{16 \, b} + \frac {3 \, {\left (2 \, C c d + B d^{2}\right )} a^{4} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{128 \, b^{\frac {5}{2}}} - \frac {{\left (B c^{2} + 2 \, A c d\right )} a^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{16 \, b^{\frac {3}{2}}} - \frac {2 \, {\left (C c^{2} + 2 \, B c d + A d^{2}\right )} {\left (b x^{2} + a\right )}^{\frac {5}{2}} a}{35 \, b^{2}} \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 496, normalized size of antiderivative = 1.42 \[ \int x (c+d x)^2 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right ) \, dx=\frac {1}{40320} \, \sqrt {b x^{2} + a} {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, {\left (2 \, {\left (7 \, {\left (8 \, C b d^{2} x + \frac {9 \, {\left (2 \, C b^{8} c d + B b^{8} d^{2}\right )}}{b^{7}}\right )} x + \frac {8 \, {\left (9 \, C b^{8} c^{2} + 18 \, B b^{8} c d + 10 \, C a b^{7} d^{2} + 9 \, A b^{8} d^{2}\right )}}{b^{7}}\right )} x + \frac {21 \, {\left (8 \, B b^{8} c^{2} + 18 \, C a b^{7} c d + 16 \, A b^{8} c d + 9 \, B a b^{7} d^{2}\right )}}{b^{7}}\right )} x + \frac {48 \, {\left (24 \, C a b^{7} c^{2} + 21 \, A b^{8} c^{2} + 48 \, B a b^{7} c d + C a^{2} b^{6} d^{2} + 24 \, A a b^{7} d^{2}\right )}}{b^{7}}\right )} x + \frac {105 \, {\left (56 \, B a b^{7} c^{2} + 6 \, C a^{2} b^{6} c d + 112 \, A a b^{7} c d + 3 \, B a^{2} b^{6} d^{2}\right )}}{b^{7}}\right )} x + \frac {64 \, {\left (9 \, C a^{2} b^{6} c^{2} + 126 \, A a b^{7} c^{2} + 18 \, B a^{2} b^{6} c d - 4 \, C a^{3} b^{5} d^{2} + 9 \, A a^{2} b^{6} d^{2}\right )}}{b^{7}}\right )} x + \frac {315 \, {\left (8 \, B a^{2} b^{6} c^{2} - 6 \, C a^{3} b^{5} c d + 16 \, A a^{2} b^{6} c d - 3 \, B a^{3} b^{5} d^{2}\right )}}{b^{7}}\right )} x - \frac {128 \, {\left (18 \, C a^{3} b^{5} c^{2} - 63 \, A a^{2} b^{6} c^{2} + 36 \, B a^{3} b^{5} c d - 8 \, C a^{4} b^{4} d^{2} + 18 \, A a^{3} b^{5} d^{2}\right )}}{b^{7}}\right )} + \frac {{\left (8 \, B a^{3} b c^{2} - 6 \, C a^{4} c d + 16 \, A a^{3} b c d - 3 \, B a^{4} d^{2}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{128 \, b^{\frac {5}{2}}} \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 9.75 (sec) , antiderivative size = 807, normalized size of antiderivative = 2.31 \[ \int x (c+d x)^2 \left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

( - 2304*sqrt(a + b*x**2)*a**4*b*d**2 + 1024*sqrt(a + b*x**2)*a**4*c*d**2 
+ 8064*sqrt(a + b*x**2)*a**3*b**2*c**2 + 5040*sqrt(a + b*x**2)*a**3*b**2*c 
*d*x - 4608*sqrt(a + b*x**2)*a**3*b**2*c*d + 1152*sqrt(a + b*x**2)*a**3*b* 
*2*d**2*x**2 - 945*sqrt(a + b*x**2)*a**3*b**2*d**2*x - 2304*sqrt(a + b*x** 
2)*a**3*b*c**3 - 1890*sqrt(a + b*x**2)*a**3*b*c**2*d*x - 512*sqrt(a + b*x* 
*2)*a**3*b*c*d**2*x**2 + 16128*sqrt(a + b*x**2)*a**2*b**3*c**2*x**2 + 2520 
*sqrt(a + b*x**2)*a**2*b**3*c**2*x + 23520*sqrt(a + b*x**2)*a**2*b**3*c*d* 
x**3 + 2304*sqrt(a + b*x**2)*a**2*b**3*c*d*x**2 + 9216*sqrt(a + b*x**2)*a* 
*2*b**3*d**2*x**4 + 630*sqrt(a + b*x**2)*a**2*b**3*d**2*x**3 + 1152*sqrt(a 
 + b*x**2)*a**2*b**2*c**3*x**2 + 1260*sqrt(a + b*x**2)*a**2*b**2*c**2*d*x* 
*3 + 384*sqrt(a + b*x**2)*a**2*b**2*c*d**2*x**4 + 8064*sqrt(a + b*x**2)*a* 
b**4*c**2*x**4 + 11760*sqrt(a + b*x**2)*a*b**4*c**2*x**3 + 13440*sqrt(a + 
b*x**2)*a*b**4*c*d*x**5 + 18432*sqrt(a + b*x**2)*a*b**4*c*d*x**4 + 5760*sq 
rt(a + b*x**2)*a*b**4*d**2*x**6 + 7560*sqrt(a + b*x**2)*a*b**4*d**2*x**5 + 
 9216*sqrt(a + b*x**2)*a*b**3*c**3*x**4 + 15120*sqrt(a + b*x**2)*a*b**3*c* 
*2*d*x**5 + 6400*sqrt(a + b*x**2)*a*b**3*c*d**2*x**6 + 6720*sqrt(a + b*x** 
2)*b**5*c**2*x**5 + 11520*sqrt(a + b*x**2)*b**5*c*d*x**6 + 5040*sqrt(a + b 
*x**2)*b**5*d**2*x**7 + 5760*sqrt(a + b*x**2)*b**4*c**3*x**6 + 10080*sqrt( 
a + b*x**2)*b**4*c**2*d*x**7 + 4480*sqrt(a + b*x**2)*b**4*c*d**2*x**8 - 50 
40*sqrt(b)*log((sqrt(a + b*x**2) + sqrt(b)*x)/sqrt(a))*a**4*b*c*d + 945...