\(\int x (c+d x)^3 (a x+b x^2)^{3/2} \, dx\) [80]

Optimal result

Integrand size = 22, antiderivative size = 476 \[ \int x (c+d x)^3 \left (a x+b x^2\right )^{3/2} \, dx=\frac {3 a^4 \left (128 b^3 c^3-a d \left (224 b^2 c^2-144 a b c d+33 a^2 d^2\right )\right ) \sqrt {a x+b x^2}}{16384 b^6}-\frac {a^3 \left (128 b^3 c^3-a d \left (224 b^2 c^2-144 a b c d+33 a^2 d^2\right )\right ) x \sqrt {a x+b x^2}}{8192 b^5}+\frac {a^2 \left (128 b^3 c^3-a d \left (224 b^2 c^2-144 a b c d+33 a^2 d^2\right )\right ) x^2 \sqrt {a x+b x^2}}{10240 b^4}+\frac {11 a \left (128 b^3 c^3-a d \left (224 b^2 c^2-144 a b c d+33 a^2 d^2\right )\right ) x^3 \sqrt {a x+b x^2}}{5120 b^3}+\frac {\left (128 b^3 c^3-a d \left (224 b^2 c^2-144 a b c d+33 a^2 d^2\right )\right ) x^4 \sqrt {a x+b x^2}}{640 b^2}+\frac {d \left (224 b^2 c^2-144 a b c d+33 a^2 d^2\right ) x \left (a x+b x^2\right )^{5/2}}{448 b^3}+\frac {d^2 (48 b c-11 a d) x^2 \left (a x+b x^2\right )^{5/2}}{112 b^2}+\frac {d^3 x^3 \left (a x+b x^2\right )^{5/2}}{8 b}-\frac {3 a^5 \left (128 b^3 c^3-a d \left (224 b^2 c^2-144 a b c d+33 a^2 d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{16384 b^{13/2}} \] Output:

3/16384*a^4*(128*b^3*c^3-a*d*(33*a^2*d^2-144*a*b*c*d+224*b^2*c^2))*(b*x^2+ 
a*x)^(1/2)/b^6-1/8192*a^3*(128*b^3*c^3-a*d*(33*a^2*d^2-144*a*b*c*d+224*b^2 
*c^2))*x*(b*x^2+a*x)^(1/2)/b^5+1/10240*a^2*(128*b^3*c^3-a*d*(33*a^2*d^2-14 
4*a*b*c*d+224*b^2*c^2))*x^2*(b*x^2+a*x)^(1/2)/b^4+11/5120*a*(128*b^3*c^3-a 
*d*(33*a^2*d^2-144*a*b*c*d+224*b^2*c^2))*x^3*(b*x^2+a*x)^(1/2)/b^3+1/640*( 
128*b^3*c^3-a*d*(33*a^2*d^2-144*a*b*c*d+224*b^2*c^2))*x^4*(b*x^2+a*x)^(1/2 
)/b^2+1/448*d*(33*a^2*d^2-144*a*b*c*d+224*b^2*c^2)*x*(b*x^2+a*x)^(5/2)/b^3 
+1/112*d^2*(-11*a*d+48*b*c)*x^2*(b*x^2+a*x)^(5/2)/b^2+1/8*d^3*x^3*(b*x^2+a 
*x)^(5/2)/b-3/16384*a^5*(128*b^3*c^3-a*d*(33*a^2*d^2-144*a*b*c*d+224*b^2*c 
^2))*arctanh(b^(1/2)*x/(b*x^2+a*x)^(1/2))/b^(13/2)
 

Mathematica [A] (verified)

Time = 2.20 (sec) , antiderivative size = 413, normalized size of antiderivative = 0.87 \[ \int x (c+d x)^3 \left (a x+b x^2\right )^{3/2} \, dx=\frac {\sqrt {x} \sqrt {a+b x} \left (\sqrt {b} \sqrt {x} \sqrt {a+b x} \left (-3465 a^7 d^3+210 a^6 b d^2 (72 c+11 d x)-168 a^5 b^2 d \left (140 c^2+60 c d x+11 d^2 x^2\right )+256 a^2 b^5 x^2 \left (28 c^3+42 c^2 d x+24 c d^2 x^2+5 d^3 x^3\right )-128 a^3 b^4 x \left (70 c^3+98 c^2 d x+54 c d^2 x^2+11 d^3 x^3\right )+2048 b^7 x^4 \left (56 c^3+140 c^2 d x+120 c d^2 x^2+35 d^3 x^3\right )+1024 a b^6 x^3 \left (154 c^3+364 c^2 d x+300 c d^2 x^2+85 d^3 x^3\right )+16 a^4 b^3 \left (840 c^3+980 c^2 d x+504 c d^2 x^2+99 d^3 x^3\right )\right )+3360 a^5 b c \left (8 b^2 c^2+9 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}-\sqrt {a+b x}}\right )+210 a^6 d \left (224 b^2 c^2+33 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{-\sqrt {a}+\sqrt {a+b x}}\right )\right )}{573440 b^{13/2} \sqrt {x (a+b x)}} \] Input:

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

Output:

(Sqrt[x]*Sqrt[a + b*x]*(Sqrt[b]*Sqrt[x]*Sqrt[a + b*x]*(-3465*a^7*d^3 + 210 
*a^6*b*d^2*(72*c + 11*d*x) - 168*a^5*b^2*d*(140*c^2 + 60*c*d*x + 11*d^2*x^ 
2) + 256*a^2*b^5*x^2*(28*c^3 + 42*c^2*d*x + 24*c*d^2*x^2 + 5*d^3*x^3) - 12 
8*a^3*b^4*x*(70*c^3 + 98*c^2*d*x + 54*c*d^2*x^2 + 11*d^3*x^3) + 2048*b^7*x 
^4*(56*c^3 + 140*c^2*d*x + 120*c*d^2*x^2 + 35*d^3*x^3) + 1024*a*b^6*x^3*(1 
54*c^3 + 364*c^2*d*x + 300*c*d^2*x^2 + 85*d^3*x^3) + 16*a^4*b^3*(840*c^3 + 
 980*c^2*d*x + 504*c*d^2*x^2 + 99*d^3*x^3)) + 3360*a^5*b*c*(8*b^2*c^2 + 9* 
a^2*d^2)*ArcTanh[(Sqrt[b]*Sqrt[x])/(Sqrt[a] - Sqrt[a + b*x])] + 210*a^6*d* 
(224*b^2*c^2 + 33*a^2*d^2)*ArcTanh[(Sqrt[b]*Sqrt[x])/(-Sqrt[a] + Sqrt[a + 
b*x])]))/(573440*b^(13/2)*Sqrt[x*(a + b*x)])
 

Rubi [A] (verified)

Time = 1.02 (sec) , antiderivative size = 306, normalized size of antiderivative = 0.64, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {1262, 27, 2169, 27, 1225, 1087, 1087, 1091, 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.

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

\(\Big \downarrow \) 1262

\(\displaystyle \frac {\int \frac {1}{2} x \left (b x^2+a x\right )^{3/2} \left (16 b c^3+48 b d x c^2+d^2 (48 b c-11 a d) x^2\right )dx}{8 b}+\frac {d^3 x^3 \left (a x+b x^2\right )^{5/2}}{8 b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int x \left (b x^2+a x\right )^{3/2} \left (16 b c^3+48 b d x c^2+d^2 (48 b c-11 a d) x^2\right )dx}{16 b}+\frac {d^3 x^3 \left (a x+b x^2\right )^{5/2}}{8 b}\)

\(\Big \downarrow \) 2169

\(\displaystyle \frac {\frac {\int \frac {1}{2} x \left (224 b^2 c^3+3 d \left (224 b^2 c^2-144 a b d c+33 a^2 d^2\right ) x\right ) \left (b x^2+a x\right )^{3/2}dx}{7 b}+\frac {d^2 x^2 \left (a x+b x^2\right )^{5/2} (48 b c-11 a d)}{7 b}}{16 b}+\frac {d^3 x^3 \left (a x+b x^2\right )^{5/2}}{8 b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int x \left (224 b^2 c^3+3 d \left (224 b^2 c^2-144 a b d c+33 a^2 d^2\right ) x\right ) \left (b x^2+a x\right )^{3/2}dx}{14 b}+\frac {d^2 x^2 \left (a x+b x^2\right )^{5/2} (48 b c-11 a d)}{7 b}}{16 b}+\frac {d^3 x^3 \left (a x+b x^2\right )^{5/2}}{8 b}\)

\(\Big \downarrow \) 1225

\(\displaystyle \frac {\frac {\frac {\left (a x+b x^2\right )^{5/2} \left (10 b d x \left (33 a^2 d^2-144 a b c d+224 b^2 c^2\right )+7 \left (128 b^3 c^3-a d \left (33 a^2 d^2-144 a b c d+224 b^2 c^2\right )\right )\right )}{20 b^2}-\frac {7 a \left (128 b^3 c^3-a d \left (33 a^2 d^2-144 a b c d+224 b^2 c^2\right )\right ) \int \left (b x^2+a x\right )^{3/2}dx}{8 b^2}}{14 b}+\frac {d^2 x^2 \left (a x+b x^2\right )^{5/2} (48 b c-11 a d)}{7 b}}{16 b}+\frac {d^3 x^3 \left (a x+b x^2\right )^{5/2}}{8 b}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {\frac {\frac {\left (a x+b x^2\right )^{5/2} \left (10 b d x \left (33 a^2 d^2-144 a b c d+224 b^2 c^2\right )+7 \left (128 b^3 c^3-a d \left (33 a^2 d^2-144 a b c d+224 b^2 c^2\right )\right )\right )}{20 b^2}-\frac {7 a \left (128 b^3 c^3-a d \left (33 a^2 d^2-144 a b c d+224 b^2 c^2\right )\right ) \left (\frac {(a+2 b x) \left (a x+b x^2\right )^{3/2}}{8 b}-\frac {3 a^2 \int \sqrt {b x^2+a x}dx}{16 b}\right )}{8 b^2}}{14 b}+\frac {d^2 x^2 \left (a x+b x^2\right )^{5/2} (48 b c-11 a d)}{7 b}}{16 b}+\frac {d^3 x^3 \left (a x+b x^2\right )^{5/2}}{8 b}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {\frac {\frac {\left (a x+b x^2\right )^{5/2} \left (10 b d x \left (33 a^2 d^2-144 a b c d+224 b^2 c^2\right )+7 \left (128 b^3 c^3-a d \left (33 a^2 d^2-144 a b c d+224 b^2 c^2\right )\right )\right )}{20 b^2}-\frac {7 a \left (128 b^3 c^3-a d \left (33 a^2 d^2-144 a b c d+224 b^2 c^2\right )\right ) \left (\frac {(a+2 b x) \left (a x+b x^2\right )^{3/2}}{8 b}-\frac {3 a^2 \left (\frac {(a+2 b x) \sqrt {a x+b x^2}}{4 b}-\frac {a^2 \int \frac {1}{\sqrt {b x^2+a x}}dx}{8 b}\right )}{16 b}\right )}{8 b^2}}{14 b}+\frac {d^2 x^2 \left (a x+b x^2\right )^{5/2} (48 b c-11 a d)}{7 b}}{16 b}+\frac {d^3 x^3 \left (a x+b x^2\right )^{5/2}}{8 b}\)

\(\Big \downarrow \) 1091

\(\displaystyle \frac {\frac {\frac {\left (a x+b x^2\right )^{5/2} \left (10 b d x \left (33 a^2 d^2-144 a b c d+224 b^2 c^2\right )+7 \left (128 b^3 c^3-a d \left (33 a^2 d^2-144 a b c d+224 b^2 c^2\right )\right )\right )}{20 b^2}-\frac {7 a \left (128 b^3 c^3-a d \left (33 a^2 d^2-144 a b c d+224 b^2 c^2\right )\right ) \left (\frac {(a+2 b x) \left (a x+b x^2\right )^{3/2}}{8 b}-\frac {3 a^2 \left (\frac {(a+2 b x) \sqrt {a x+b x^2}}{4 b}-\frac {a^2 \int \frac {1}{1-\frac {b x^2}{b x^2+a x}}d\frac {x}{\sqrt {b x^2+a x}}}{4 b}\right )}{16 b}\right )}{8 b^2}}{14 b}+\frac {d^2 x^2 \left (a x+b x^2\right )^{5/2} (48 b c-11 a d)}{7 b}}{16 b}+\frac {d^3 x^3 \left (a x+b x^2\right )^{5/2}}{8 b}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {\left (a x+b x^2\right )^{5/2} \left (10 b d x \left (33 a^2 d^2-144 a b c d+224 b^2 c^2\right )+7 \left (128 b^3 c^3-a d \left (33 a^2 d^2-144 a b c d+224 b^2 c^2\right )\right )\right )}{20 b^2}-\frac {7 a \left (\frac {(a+2 b x) \left (a x+b x^2\right )^{3/2}}{8 b}-\frac {3 a^2 \left (\frac {(a+2 b x) \sqrt {a x+b x^2}}{4 b}-\frac {a^2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{4 b^{3/2}}\right )}{16 b}\right ) \left (128 b^3 c^3-a d \left (33 a^2 d^2-144 a b c d+224 b^2 c^2\right )\right )}{8 b^2}}{14 b}+\frac {d^2 x^2 \left (a x+b x^2\right )^{5/2} (48 b c-11 a d)}{7 b}}{16 b}+\frac {d^3 x^3 \left (a x+b x^2\right )^{5/2}}{8 b}\)

Input:

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

Output:

(d^3*x^3*(a*x + b*x^2)^(5/2))/(8*b) + ((d^2*(48*b*c - 11*a*d)*x^2*(a*x + b 
*x^2)^(5/2))/(7*b) + (((7*(128*b^3*c^3 - a*d*(224*b^2*c^2 - 144*a*b*c*d + 
33*a^2*d^2)) + 10*b*d*(224*b^2*c^2 - 144*a*b*c*d + 33*a^2*d^2)*x)*(a*x + b 
*x^2)^(5/2))/(20*b^2) - (7*a*(128*b^3*c^3 - a*d*(224*b^2*c^2 - 144*a*b*c*d 
 + 33*a^2*d^2))*(((a + 2*b*x)*(a*x + b*x^2)^(3/2))/(8*b) - (3*a^2*(((a + 2 
*b*x)*Sqrt[a*x + b*x^2])/(4*b) - (a^2*ArcTanh[(Sqrt[b]*x)/Sqrt[a*x + b*x^2 
]])/(4*b^(3/2))))/(16*b)))/(8*b^2))/(14*b))/(16*b)
 

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 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 1087

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) 
*((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* 
p + 1)))   Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && 
GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
 

rule 1091

Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[Int[1/(1 
 - c*x^2), x], x, x/Sqrt[b*x + c*x^2]], x] /; FreeQ[{b, c}, x]
 

rule 1225

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

rule 1262

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x_ 
) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[g^n*(d + e*x)^(m + n - 1)*((a + b 
*x + c*x^2)^(p + 1)/(c*e^(n - 1)*(m + n + 2*p + 1))), x] + Simp[1/(c*e^n*(m 
 + n + 2*p + 1))   Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^n*(m 
 + n + 2*p + 1)*(f + g*x)^n - c*g^n*(m + n + 2*p + 1)*(d + e*x)^n + e*g^n*( 
m + p + n)*(d + e*x)^(n - 2)*(b*d - 2*a*e + (2*c*d - b*e)*x), x], x], x] /; 
 FreeQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && 
IGtQ[n, 0] && NeQ[m + n + 2*p + 1, 0]
 

rule 2169

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p 
_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, S 
imp[f*(d + e*x)^(m + q - 1)*((a + b*x + c*x^2)^(p + 1)/(c*e^(q - 1)*(m + q 
+ 2*p + 1))), x] + Simp[1/(c*e^q*(m + q + 2*p + 1))   Int[(d + e*x)^m*(a + 
b*x + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 
1)*(d + e*x)^q + e*f*(m + p + q)*(d + e*x)^(q - 2)*(b*d - 2*a*e + (2*c*d - 
b*e)*x), x], x], x] /; NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, d, e, m, 
 p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2 
, 0]
 

Maple [A] (verified)

Time = 0.64 (sec) , antiderivative size = 306, normalized size of antiderivative = 0.64

method	result	size
pseudoelliptic	\(\frac {\frac {99 a^{5} \left (-\frac {48}{11} a^{2} b c \,d^{2}+\frac {224}{33} a \,b^{2} c^{2} d -\frac {128}{33} b^{3} c^{3}+a^{3} d^{3}\right ) \operatorname {arctanh}\left (\frac {\sqrt {x \left (b x +a \right )}}{x \sqrt {b}}\right )}{16384}-\frac {99 \left (-\frac {1024 x^{2} \left (\frac {d x}{2}+c \right ) \left (\frac {5}{14} d^{2} x^{2}+c d x +c^{2}\right ) a^{2} b^{\frac {11}{2}}}{495}-\frac {2048 \left (\frac {85}{154} d^{3} x^{3}+\frac {150}{77} c \,d^{2} x^{2}+\frac {26}{11} c^{2} d x +c^{3}\right ) x^{3} a \,b^{\frac {13}{2}}}{45}-\frac {16384 \left (\frac {5}{8} d^{3} x^{3}+\frac {15}{7} c \,d^{2} x^{2}+\frac {5}{2} c^{2} d x +c^{3}\right ) x^{4} b^{\frac {15}{2}}}{495}+a^{3} \left (-\frac {128 \left (\frac {33}{280} d^{3} x^{3}+\frac {3}{5} c \,d^{2} x^{2}+\frac {7}{6} c^{2} d x +c^{3}\right ) a \,b^{\frac {7}{2}}}{33}+\frac {256 \left (\frac {11}{70} d^{3} x^{3}+\frac {27}{35} c \,d^{2} x^{2}+\frac {7}{5} c^{2} d x +c^{3}\right ) x \,b^{\frac {9}{2}}}{99}+d \,a^{2} \left (\left (\frac {32}{11} c d x +\frac {224}{33} c^{2}+\frac {8}{15} d^{2} x^{2}\right ) b^{\frac {5}{2}}+d \left (\left (-\frac {48 c}{11}-\frac {2 d x}{3}\right ) b^{\frac {3}{2}}+\sqrt {b}\, a d \right ) a \right )\right )\right ) \sqrt {x \left (b x +a \right )}}{16384}}{b^{\frac {13}{2}}}\)	\(306\)
risch	\(-\frac {\left (-71680 b^{7} d^{3} x^{7}-87040 a \,b^{6} d^{3} x^{6}-245760 b^{7} c \,d^{2} x^{6}-1280 a^{2} b^{5} d^{3} x^{5}-307200 a \,b^{6} c \,d^{2} x^{5}-286720 b^{7} c^{2} d \,x^{5}+1408 a^{3} b^{4} d^{3} x^{4}-6144 a^{2} b^{5} c \,d^{2} x^{4}-372736 a \,b^{6} c^{2} d \,x^{4}-114688 b^{7} c^{3} x^{4}-1584 a^{4} b^{3} d^{3} x^{3}+6912 a^{3} b^{4} c \,d^{2} x^{3}-10752 a^{2} b^{5} c^{2} d \,x^{3}-157696 a \,b^{6} c^{3} x^{3}+1848 a^{5} b^{2} d^{3} x^{2}-8064 a^{4} b^{3} c \,d^{2} x^{2}+12544 a^{3} b^{4} c^{2} d \,x^{2}-7168 a^{2} b^{5} c^{3} x^{2}-2310 a^{6} b \,d^{3} x +10080 a^{5} b^{2} c \,d^{2} x -15680 a^{4} b^{3} c^{2} d x +8960 a^{3} b^{4} c^{3} x +3465 a^{7} d^{3}-15120 a^{6} b c \,d^{2}+23520 a^{5} b^{2} c^{2} d -13440 a^{4} b^{3} c^{3}\right ) x \left (b x +a \right )}{573440 b^{6} \sqrt {x \left (b x +a \right )}}+\frac {3 a^{5} \left (33 a^{3} d^{3}-144 a^{2} b c \,d^{2}+224 a \,b^{2} c^{2} d -128 b^{3} c^{3}\right ) \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{32768 b^{\frac {13}{2}}}\)	\(422\)
default	\(c^{3} \left (\frac {\left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{5 b}-\frac {a \left (\frac {\left (2 b x +a \right ) \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{8 b}-\frac {3 a^{2} \left (\frac {\left (2 b x +a \right ) \sqrt {b \,x^{2}+a x}}{4 b}-\frac {a^{2} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{8 b^{\frac {3}{2}}}\right )}{16 b}\right )}{2 b}\right )+d^{3} \left (\frac {x^{3} \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{8 b}-\frac {11 a \left (\frac {x^{2} \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{7 b}-\frac {9 a \left (\frac {x \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{6 b}-\frac {7 a \left (\frac {\left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{5 b}-\frac {a \left (\frac {\left (2 b x +a \right ) \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{8 b}-\frac {3 a^{2} \left (\frac {\left (2 b x +a \right ) \sqrt {b \,x^{2}+a x}}{4 b}-\frac {a^{2} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{8 b^{\frac {3}{2}}}\right )}{16 b}\right )}{2 b}\right )}{12 b}\right )}{14 b}\right )}{16 b}\right )+3 c \,d^{2} \left (\frac {x^{2} \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{7 b}-\frac {9 a \left (\frac {x \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{6 b}-\frac {7 a \left (\frac {\left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{5 b}-\frac {a \left (\frac {\left (2 b x +a \right ) \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{8 b}-\frac {3 a^{2} \left (\frac {\left (2 b x +a \right ) \sqrt {b \,x^{2}+a x}}{4 b}-\frac {a^{2} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{8 b^{\frac {3}{2}}}\right )}{16 b}\right )}{2 b}\right )}{12 b}\right )}{14 b}\right )+3 c^{2} d \left (\frac {x \left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{6 b}-\frac {7 a \left (\frac {\left (b \,x^{2}+a x \right )^{\frac {5}{2}}}{5 b}-\frac {a \left (\frac {\left (2 b x +a \right ) \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{8 b}-\frac {3 a^{2} \left (\frac {\left (2 b x +a \right ) \sqrt {b \,x^{2}+a x}}{4 b}-\frac {a^{2} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{8 b^{\frac {3}{2}}}\right )}{16 b}\right )}{2 b}\right )}{12 b}\right )\)	\(608\)

Input:

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

Output:

99/16384/b^(13/2)*(a^5*(-48/11*a^2*b*c*d^2+224/33*a*b^2*c^2*d-128/33*b^3*c 
^3+a^3*d^3)*arctanh((x*(b*x+a))^(1/2)/x/b^(1/2))-(-1024/495*x^2*(1/2*d*x+c 
)*(5/14*d^2*x^2+c*d*x+c^2)*a^2*b^(11/2)-2048/45*(85/154*d^3*x^3+150/77*c*d 
^2*x^2+26/11*c^2*d*x+c^3)*x^3*a*b^(13/2)-16384/495*(5/8*d^3*x^3+15/7*c*d^2 
*x^2+5/2*c^2*d*x+c^3)*x^4*b^(15/2)+a^3*(-128/33*(33/280*d^3*x^3+3/5*c*d^2* 
x^2+7/6*c^2*d*x+c^3)*a*b^(7/2)+256/99*(11/70*d^3*x^3+27/35*c*d^2*x^2+7/5*c 
^2*d*x+c^3)*x*b^(9/2)+d*a^2*((32/11*c*d*x+224/33*c^2+8/15*d^2*x^2)*b^(5/2) 
+d*((-48/11*c-2/3*d*x)*b^(3/2)+b^(1/2)*a*d)*a)))*(x*(b*x+a))^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 810, normalized size of antiderivative = 1.70 \[ \int x (c+d x)^3 \left (a x+b x^2\right )^{3/2} \, dx =\text {Too large to display} \] Input:

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

Output:

[-1/1146880*(105*(128*a^5*b^3*c^3 - 224*a^6*b^2*c^2*d + 144*a^7*b*c*d^2 - 
33*a^8*d^3)*sqrt(b)*log(2*b*x + a + 2*sqrt(b*x^2 + a*x)*sqrt(b)) - 2*(7168 
0*b^8*d^3*x^7 + 13440*a^4*b^4*c^3 - 23520*a^5*b^3*c^2*d + 15120*a^6*b^2*c* 
d^2 - 3465*a^7*b*d^3 + 5120*(48*b^8*c*d^2 + 17*a*b^7*d^3)*x^6 + 1280*(224* 
b^8*c^2*d + 240*a*b^7*c*d^2 + a^2*b^6*d^3)*x^5 + 128*(896*b^8*c^3 + 2912*a 
*b^7*c^2*d + 48*a^2*b^6*c*d^2 - 11*a^3*b^5*d^3)*x^4 + 16*(9856*a*b^7*c^3 + 
 672*a^2*b^6*c^2*d - 432*a^3*b^5*c*d^2 + 99*a^4*b^4*d^3)*x^3 + 56*(128*a^2 
*b^6*c^3 - 224*a^3*b^5*c^2*d + 144*a^4*b^4*c*d^2 - 33*a^5*b^3*d^3)*x^2 - 7 
0*(128*a^3*b^5*c^3 - 224*a^4*b^4*c^2*d + 144*a^5*b^3*c*d^2 - 33*a^6*b^2*d^ 
3)*x)*sqrt(b*x^2 + a*x))/b^7, 1/573440*(105*(128*a^5*b^3*c^3 - 224*a^6*b^2 
*c^2*d + 144*a^7*b*c*d^2 - 33*a^8*d^3)*sqrt(-b)*arctan(sqrt(b*x^2 + a*x)*s 
qrt(-b)/(b*x + a)) + (71680*b^8*d^3*x^7 + 13440*a^4*b^4*c^3 - 23520*a^5*b^ 
3*c^2*d + 15120*a^6*b^2*c*d^2 - 3465*a^7*b*d^3 + 5120*(48*b^8*c*d^2 + 17*a 
*b^7*d^3)*x^6 + 1280*(224*b^8*c^2*d + 240*a*b^7*c*d^2 + a^2*b^6*d^3)*x^5 + 
 128*(896*b^8*c^3 + 2912*a*b^7*c^2*d + 48*a^2*b^6*c*d^2 - 11*a^3*b^5*d^3)* 
x^4 + 16*(9856*a*b^7*c^3 + 672*a^2*b^6*c^2*d - 432*a^3*b^5*c*d^2 + 99*a^4* 
b^4*d^3)*x^3 + 56*(128*a^2*b^6*c^3 - 224*a^3*b^5*c^2*d + 144*a^4*b^4*c*d^2 
 - 33*a^5*b^3*d^3)*x^2 - 70*(128*a^3*b^5*c^3 - 224*a^4*b^4*c^2*d + 144*a^5 
*b^3*c*d^2 - 33*a^6*b^2*d^3)*x)*sqrt(b*x^2 + a*x))/b^7]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1006 vs. \(2 (469) = 938\).

Time = 0.64 (sec) , antiderivative size = 1006, normalized size of antiderivative = 2.11 \[ \int x (c+d x)^3 \left (a x+b x^2\right )^{3/2} \, dx =\text {Too large to display} \] Input:

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

Output:

Piecewise((-5*a**3*(a**2*c**3 - 7*a*(3*a**2*c**2*d + 2*a*b*c**3 - 9*a*(3*a 
**2*c*d**2 + 6*a*b*c**2*d - 11*a*(a**2*d**3 + 6*a*b*c*d**2 - 13*a*(17*a*b* 
d**3/16 + 3*b**2*c*d**2)/(14*b) + 3*b**2*c**2*d)/(12*b) + b**2*c**3)/(10*b 
))/(8*b))*Piecewise((log(a + 2*sqrt(b)*sqrt(a*x + b*x**2) + 2*b*x)/sqrt(b) 
, Ne(a**2/b, 0)), ((a/(2*b) + x)*log(a/(2*b) + x)/sqrt(b*(a/(2*b) + x)**2) 
, True))/(16*b**3) + sqrt(a*x + b*x**2)*(5*a**2*(a**2*c**3 - 7*a*(3*a**2*c 
**2*d + 2*a*b*c**3 - 9*a*(3*a**2*c*d**2 + 6*a*b*c**2*d - 11*a*(a**2*d**3 + 
 6*a*b*c*d**2 - 13*a*(17*a*b*d**3/16 + 3*b**2*c*d**2)/(14*b) + 3*b**2*c**2 
*d)/(12*b) + b**2*c**3)/(10*b))/(8*b))/(8*b**3) - 5*a*x*(a**2*c**3 - 7*a*( 
3*a**2*c**2*d + 2*a*b*c**3 - 9*a*(3*a**2*c*d**2 + 6*a*b*c**2*d - 11*a*(a** 
2*d**3 + 6*a*b*c*d**2 - 13*a*(17*a*b*d**3/16 + 3*b**2*c*d**2)/(14*b) + 3*b 
**2*c**2*d)/(12*b) + b**2*c**3)/(10*b))/(8*b))/(12*b**2) + b*d**3*x**7/8 + 
 x**6*(17*a*b*d**3/16 + 3*b**2*c*d**2)/(7*b) + x**5*(a**2*d**3 + 6*a*b*c*d 
**2 - 13*a*(17*a*b*d**3/16 + 3*b**2*c*d**2)/(14*b) + 3*b**2*c**2*d)/(6*b) 
+ x**4*(3*a**2*c*d**2 + 6*a*b*c**2*d - 11*a*(a**2*d**3 + 6*a*b*c*d**2 - 13 
*a*(17*a*b*d**3/16 + 3*b**2*c*d**2)/(14*b) + 3*b**2*c**2*d)/(12*b) + b**2* 
c**3)/(5*b) + x**3*(3*a**2*c**2*d + 2*a*b*c**3 - 9*a*(3*a**2*c*d**2 + 6*a* 
b*c**2*d - 11*a*(a**2*d**3 + 6*a*b*c*d**2 - 13*a*(17*a*b*d**3/16 + 3*b**2* 
c*d**2)/(14*b) + 3*b**2*c**2*d)/(12*b) + b**2*c**3)/(10*b))/(4*b) + x**2*( 
a**2*c**3 - 7*a*(3*a**2*c**2*d + 2*a*b*c**3 - 9*a*(3*a**2*c*d**2 + 6*a*...
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 724, normalized size of antiderivative = 1.52 \[ \int x (c+d x)^3 \left (a x+b x^2\right )^{3/2} \, dx =\text {Too large to display} \] Input:

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

Output:

1/8*(b*x^2 + a*x)^(5/2)*d^3*x^3/b + 3/7*(b*x^2 + a*x)^(5/2)*c*d^2*x^2/b - 
11/112*(b*x^2 + a*x)^(5/2)*a*d^3*x^2/b^2 + 3/64*sqrt(b*x^2 + a*x)*a^3*c^3* 
x/b^2 - 1/8*(b*x^2 + a*x)^(3/2)*a*c^3*x/b - 21/256*sqrt(b*x^2 + a*x)*a^4*c 
^2*d*x/b^3 + 7/32*(b*x^2 + a*x)^(3/2)*a^2*c^2*d*x/b^2 + 1/2*(b*x^2 + a*x)^ 
(5/2)*c^2*d*x/b + 27/512*sqrt(b*x^2 + a*x)*a^5*c*d^2*x/b^4 - 9/64*(b*x^2 + 
 a*x)^(3/2)*a^3*c*d^2*x/b^3 - 9/28*(b*x^2 + a*x)^(5/2)*a*c*d^2*x/b^2 - 99/ 
8192*sqrt(b*x^2 + a*x)*a^6*d^3*x/b^5 + 33/1024*(b*x^2 + a*x)^(3/2)*a^4*d^3 
*x/b^4 + 33/448*(b*x^2 + a*x)^(5/2)*a^2*d^3*x/b^3 - 3/256*a^5*c^3*log(2*b* 
x + a + 2*sqrt(b*x^2 + a*x)*sqrt(b))/b^(7/2) + 21/1024*a^6*c^2*d*log(2*b*x 
 + a + 2*sqrt(b*x^2 + a*x)*sqrt(b))/b^(9/2) - 27/2048*a^7*c*d^2*log(2*b*x 
+ a + 2*sqrt(b*x^2 + a*x)*sqrt(b))/b^(11/2) + 99/32768*a^8*d^3*log(2*b*x + 
 a + 2*sqrt(b*x^2 + a*x)*sqrt(b))/b^(13/2) + 3/128*sqrt(b*x^2 + a*x)*a^4*c 
^3/b^3 - 1/16*(b*x^2 + a*x)^(3/2)*a^2*c^3/b^2 + 1/5*(b*x^2 + a*x)^(5/2)*c^ 
3/b - 21/512*sqrt(b*x^2 + a*x)*a^5*c^2*d/b^4 + 7/64*(b*x^2 + a*x)^(3/2)*a^ 
3*c^2*d/b^3 - 7/20*(b*x^2 + a*x)^(5/2)*a*c^2*d/b^2 + 27/1024*sqrt(b*x^2 + 
a*x)*a^6*c*d^2/b^5 - 9/128*(b*x^2 + a*x)^(3/2)*a^4*c*d^2/b^4 + 9/40*(b*x^2 
 + a*x)^(5/2)*a^2*c*d^2/b^3 - 99/16384*sqrt(b*x^2 + a*x)*a^7*d^3/b^6 + 33/ 
2048*(b*x^2 + a*x)^(3/2)*a^5*d^3/b^5 - 33/640*(b*x^2 + a*x)^(5/2)*a^3*d^3/ 
b^4
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 427, normalized size of antiderivative = 0.90 \[ \int x (c+d x)^3 \left (a x+b x^2\right )^{3/2} \, dx=\frac {1}{573440} \, \sqrt {b x^{2} + a x} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, {\left (4 \, {\left (14 \, b d^{3} x + \frac {48 \, b^{8} c d^{2} + 17 \, a b^{7} d^{3}}{b^{7}}\right )} x + \frac {224 \, b^{8} c^{2} d + 240 \, a b^{7} c d^{2} + a^{2} b^{6} d^{3}}{b^{7}}\right )} x + \frac {896 \, b^{8} c^{3} + 2912 \, a b^{7} c^{2} d + 48 \, a^{2} b^{6} c d^{2} - 11 \, a^{3} b^{5} d^{3}}{b^{7}}\right )} x + \frac {9856 \, a b^{7} c^{3} + 672 \, a^{2} b^{6} c^{2} d - 432 \, a^{3} b^{5} c d^{2} + 99 \, a^{4} b^{4} d^{3}}{b^{7}}\right )} x + \frac {7 \, {\left (128 \, a^{2} b^{6} c^{3} - 224 \, a^{3} b^{5} c^{2} d + 144 \, a^{4} b^{4} c d^{2} - 33 \, a^{5} b^{3} d^{3}\right )}}{b^{7}}\right )} x - \frac {35 \, {\left (128 \, a^{3} b^{5} c^{3} - 224 \, a^{4} b^{4} c^{2} d + 144 \, a^{5} b^{3} c d^{2} - 33 \, a^{6} b^{2} d^{3}\right )}}{b^{7}}\right )} x + \frac {105 \, {\left (128 \, a^{4} b^{4} c^{3} - 224 \, a^{5} b^{3} c^{2} d + 144 \, a^{6} b^{2} c d^{2} - 33 \, a^{7} b d^{3}\right )}}{b^{7}}\right )} + \frac {3 \, {\left (128 \, a^{5} b^{3} c^{3} - 224 \, a^{6} b^{2} c^{2} d + 144 \, a^{7} b c d^{2} - 33 \, a^{8} d^{3}\right )} \log \left ({\left | 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} \sqrt {b} + a \right |}\right )}{32768 \, b^{\frac {13}{2}}} \] Input:

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

Output:

1/573440*sqrt(b*x^2 + a*x)*(2*(4*(2*(8*(10*(4*(14*b*d^3*x + (48*b^8*c*d^2 
+ 17*a*b^7*d^3)/b^7)*x + (224*b^8*c^2*d + 240*a*b^7*c*d^2 + a^2*b^6*d^3)/b 
^7)*x + (896*b^8*c^3 + 2912*a*b^7*c^2*d + 48*a^2*b^6*c*d^2 - 11*a^3*b^5*d^ 
3)/b^7)*x + (9856*a*b^7*c^3 + 672*a^2*b^6*c^2*d - 432*a^3*b^5*c*d^2 + 99*a 
^4*b^4*d^3)/b^7)*x + 7*(128*a^2*b^6*c^3 - 224*a^3*b^5*c^2*d + 144*a^4*b^4* 
c*d^2 - 33*a^5*b^3*d^3)/b^7)*x - 35*(128*a^3*b^5*c^3 - 224*a^4*b^4*c^2*d + 
 144*a^5*b^3*c*d^2 - 33*a^6*b^2*d^3)/b^7)*x + 105*(128*a^4*b^4*c^3 - 224*a 
^5*b^3*c^2*d + 144*a^6*b^2*c*d^2 - 33*a^7*b*d^3)/b^7) + 3/32768*(128*a^5*b 
^3*c^3 - 224*a^6*b^2*c^2*d + 144*a^7*b*c*d^2 - 33*a^8*d^3)*log(abs(2*(sqrt 
(b)*x - sqrt(b*x^2 + a*x))*sqrt(b) + a))/b^(13/2)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 155.06 (sec) , antiderivative size = 670, normalized size of antiderivative = 1.41 \[ \int x (c+d x)^3 \left (a x+b x^2\right )^{3/2} \, dx=\frac {3465 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a^{8} d^{3}+15120 \sqrt {x}\, \sqrt {b x +a}\, a^{6} b^{2} c \,d^{2}+2310 \sqrt {x}\, \sqrt {b x +a}\, a^{6} b^{2} d^{3} x -23520 \sqrt {x}\, \sqrt {b x +a}\, a^{5} b^{3} c^{2} d -1848 \sqrt {x}\, \sqrt {b x +a}\, a^{5} b^{3} d^{3} x^{2}+1584 \sqrt {x}\, \sqrt {b x +a}\, a^{4} b^{4} d^{3} x^{3}-8960 \sqrt {x}\, \sqrt {b x +a}\, a^{3} b^{5} c^{3} x -1408 \sqrt {x}\, \sqrt {b x +a}\, a^{3} b^{5} d^{3} x^{4}+7168 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b^{6} c^{3} x^{2}+1280 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b^{6} d^{3} x^{5}+157696 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{7} c^{3} x^{3}+87040 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{7} d^{3} x^{6}+286720 \sqrt {x}\, \sqrt {b x +a}\, b^{8} c^{2} d \,x^{5}+245760 \sqrt {x}\, \sqrt {b x +a}\, b^{8} c \,d^{2} x^{6}-15120 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a^{7} b c \,d^{2}+23520 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a^{6} b^{2} c^{2} d -3465 \sqrt {x}\, \sqrt {b x +a}\, a^{7} b \,d^{3}+13440 \sqrt {x}\, \sqrt {b x +a}\, a^{4} b^{4} c^{3}+114688 \sqrt {x}\, \sqrt {b x +a}\, b^{8} c^{3} x^{4}+71680 \sqrt {x}\, \sqrt {b x +a}\, b^{8} d^{3} x^{7}-13440 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a^{5} b^{3} c^{3}-10080 \sqrt {x}\, \sqrt {b x +a}\, a^{5} b^{3} c \,d^{2} x +15680 \sqrt {x}\, \sqrt {b x +a}\, a^{4} b^{4} c^{2} d x +8064 \sqrt {x}\, \sqrt {b x +a}\, a^{4} b^{4} c \,d^{2} x^{2}-12544 \sqrt {x}\, \sqrt {b x +a}\, a^{3} b^{5} c^{2} d \,x^{2}-6912 \sqrt {x}\, \sqrt {b x +a}\, a^{3} b^{5} c \,d^{2} x^{3}+10752 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b^{6} c^{2} d \,x^{3}+6144 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b^{6} c \,d^{2} x^{4}+372736 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{7} c^{2} d \,x^{4}+307200 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{7} c \,d^{2} x^{5}}{573440 b^{7}} \] Input:

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

Output:

( - 3465*sqrt(x)*sqrt(a + b*x)*a**7*b*d**3 + 15120*sqrt(x)*sqrt(a + b*x)*a 
**6*b**2*c*d**2 + 2310*sqrt(x)*sqrt(a + b*x)*a**6*b**2*d**3*x - 23520*sqrt 
(x)*sqrt(a + b*x)*a**5*b**3*c**2*d - 10080*sqrt(x)*sqrt(a + b*x)*a**5*b**3 
*c*d**2*x - 1848*sqrt(x)*sqrt(a + b*x)*a**5*b**3*d**3*x**2 + 13440*sqrt(x) 
*sqrt(a + b*x)*a**4*b**4*c**3 + 15680*sqrt(x)*sqrt(a + b*x)*a**4*b**4*c**2 
*d*x + 8064*sqrt(x)*sqrt(a + b*x)*a**4*b**4*c*d**2*x**2 + 1584*sqrt(x)*sqr 
t(a + b*x)*a**4*b**4*d**3*x**3 - 8960*sqrt(x)*sqrt(a + b*x)*a**3*b**5*c**3 
*x - 12544*sqrt(x)*sqrt(a + b*x)*a**3*b**5*c**2*d*x**2 - 6912*sqrt(x)*sqrt 
(a + b*x)*a**3*b**5*c*d**2*x**3 - 1408*sqrt(x)*sqrt(a + b*x)*a**3*b**5*d** 
3*x**4 + 7168*sqrt(x)*sqrt(a + b*x)*a**2*b**6*c**3*x**2 + 10752*sqrt(x)*sq 
rt(a + b*x)*a**2*b**6*c**2*d*x**3 + 6144*sqrt(x)*sqrt(a + b*x)*a**2*b**6*c 
*d**2*x**4 + 1280*sqrt(x)*sqrt(a + b*x)*a**2*b**6*d**3*x**5 + 157696*sqrt( 
x)*sqrt(a + b*x)*a*b**7*c**3*x**3 + 372736*sqrt(x)*sqrt(a + b*x)*a*b**7*c* 
*2*d*x**4 + 307200*sqrt(x)*sqrt(a + b*x)*a*b**7*c*d**2*x**5 + 87040*sqrt(x 
)*sqrt(a + b*x)*a*b**7*d**3*x**6 + 114688*sqrt(x)*sqrt(a + b*x)*b**8*c**3* 
x**4 + 286720*sqrt(x)*sqrt(a + b*x)*b**8*c**2*d*x**5 + 245760*sqrt(x)*sqrt 
(a + b*x)*b**8*c*d**2*x**6 + 71680*sqrt(x)*sqrt(a + b*x)*b**8*d**3*x**7 + 
3465*sqrt(b)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a))*a**8*d**3 - 15 
120*sqrt(b)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a))*a**7*b*c*d**2 + 
 23520*sqrt(b)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a))*a**6*b**2...