\(\int x (c+d x)^3 \sqrt {a x+b x^2} \, dx\) [20]

Optimal result

Integrand size = 22, antiderivative size = 349 \[ \int x (c+d x)^3 \sqrt {a x+b x^2} \, dx=-\frac {a^2 \left (64 b^3 c^3-3 a d \left (40 b^2 c^2-28 a b c d+7 a^2 d^2\right )\right ) \sqrt {a x+b x^2}}{512 b^5}+\frac {a \left (64 b^3 c^3-3 a d \left (40 b^2 c^2-28 a b c d+7 a^2 d^2\right )\right ) x \sqrt {a x+b x^2}}{768 b^4}+\frac {1}{192} \left (64 c^3-\frac {3 a d \left (40 b^2 c^2-28 a b c d+7 a^2 d^2\right )}{b^3}\right ) x^2 \sqrt {a x+b x^2}+\frac {3 d \left (40 b^2 c^2-28 a b c d+7 a^2 d^2\right ) x \left (a x+b x^2\right )^{3/2}}{160 b^3}+\frac {3 d^2 (4 b c-a d) x^2 \left (a x+b x^2\right )^{3/2}}{20 b^2}+\frac {d^3 x^3 \left (a x+b x^2\right )^{3/2}}{6 b}+\frac {a^3 \left (64 b^3 c^3-3 a d \left (40 b^2 c^2-28 a b c d+7 a^2 d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{512 b^{11/2}} \] Output:

-1/512*a^2*(64*b^3*c^3-3*a*d*(7*a^2*d^2-28*a*b*c*d+40*b^2*c^2))*(b*x^2+a*x 
)^(1/2)/b^5+1/768*a*(64*b^3*c^3-3*a*d*(7*a^2*d^2-28*a*b*c*d+40*b^2*c^2))*x 
*(b*x^2+a*x)^(1/2)/b^4+1/192*(64*c^3-3*a*d*(7*a^2*d^2-28*a*b*c*d+40*b^2*c^ 
2)/b^3)*x^2*(b*x^2+a*x)^(1/2)+3/160*d*(7*a^2*d^2-28*a*b*c*d+40*b^2*c^2)*x* 
(b*x^2+a*x)^(3/2)/b^3+3/20*d^2*(-a*d+4*b*c)*x^2*(b*x^2+a*x)^(3/2)/b^2+1/6* 
d^3*x^3*(b*x^2+a*x)^(3/2)/b+1/512*a^3*(64*b^3*c^3-3*a*d*(7*a^2*d^2-28*a*b* 
c*d+40*b^2*c^2))*arctanh(b^(1/2)*x/(b*x^2+a*x)^(1/2))/b^(11/2)
 

Mathematica [A] (verified)

Time = 1.60 (sec) , antiderivative size = 330, normalized size of antiderivative = 0.95 \[ \int x (c+d x)^3 \sqrt {a x+b x^2} \, dx=\frac {\sqrt {x} \sqrt {a+b x} \left (\sqrt {b} \sqrt {x} \sqrt {a+b x} \left (315 a^5 d^3-210 a^4 b d^2 (6 c+d x)+24 a^3 b^2 d \left (75 c^2+35 c d x+7 d^2 x^2\right )+64 a b^4 x \left (10 c^3+15 c^2 d x+9 c d^2 x^2+2 d^3 x^3\right )-48 a^2 b^3 \left (20 c^3+25 c^2 d x+14 c d^2 x^2+3 d^3 x^3\right )+128 b^5 x^2 \left (20 c^3+45 c^2 d x+36 c d^2 x^2+10 d^3 x^3\right )\right )+90 a^4 d \left (40 b^2 c^2+7 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}-\sqrt {a+b x}}\right )+120 a^3 b c \left (16 b^2 c^2+21 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{-\sqrt {a}+\sqrt {a+b x}}\right )\right )}{7680 b^{11/2} \sqrt {x (a+b x)}} \] Input:

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

Output:

(Sqrt[x]*Sqrt[a + b*x]*(Sqrt[b]*Sqrt[x]*Sqrt[a + b*x]*(315*a^5*d^3 - 210*a 
^4*b*d^2*(6*c + d*x) + 24*a^3*b^2*d*(75*c^2 + 35*c*d*x + 7*d^2*x^2) + 64*a 
*b^4*x*(10*c^3 + 15*c^2*d*x + 9*c*d^2*x^2 + 2*d^3*x^3) - 48*a^2*b^3*(20*c^ 
3 + 25*c^2*d*x + 14*c*d^2*x^2 + 3*d^3*x^3) + 128*b^5*x^2*(20*c^3 + 45*c^2* 
d*x + 36*c*d^2*x^2 + 10*d^3*x^3)) + 90*a^4*d*(40*b^2*c^2 + 7*a^2*d^2)*ArcT 
anh[(Sqrt[b]*Sqrt[x])/(Sqrt[a] - Sqrt[a + b*x])] + 120*a^3*b*c*(16*b^2*c^2 
 + 21*a^2*d^2)*ArcTanh[(Sqrt[b]*Sqrt[x])/(-Sqrt[a] + Sqrt[a + b*x])]))/(76 
80*b^(11/2)*Sqrt[x*(a + b*x)])
 

Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 269, normalized size of antiderivative = 0.77, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {1262, 27, 2169, 27, 1225, 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.

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

\(\Big \downarrow \) 1262

\(\displaystyle \frac {\int \frac {3}{2} x \sqrt {b x^2+a x} \left (4 b c^3+12 b d x c^2+3 d^2 (4 b c-a d) x^2\right )dx}{6 b}+\frac {d^3 x^3 \left (a x+b x^2\right )^{3/2}}{6 b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int x \sqrt {b x^2+a x} \left (4 b c^3+12 b d x c^2+3 d^2 (4 b c-a d) x^2\right )dx}{4 b}+\frac {d^3 x^3 \left (a x+b x^2\right )^{3/2}}{6 b}\)

\(\Big \downarrow \) 2169

\(\displaystyle \frac {\frac {\int \frac {1}{2} x \left (40 b^2 c^3+3 d \left (40 b^2 c^2-28 a b d c+7 a^2 d^2\right ) x\right ) \sqrt {b x^2+a x}dx}{5 b}+\frac {3 d^2 x^2 \left (a x+b x^2\right )^{3/2} (4 b c-a d)}{5 b}}{4 b}+\frac {d^3 x^3 \left (a x+b x^2\right )^{3/2}}{6 b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int x \left (40 b^2 c^3+3 d \left (40 b^2 c^2-28 a b d c+7 a^2 d^2\right ) x\right ) \sqrt {b x^2+a x}dx}{10 b}+\frac {3 d^2 x^2 \left (a x+b x^2\right )^{3/2} (4 b c-a d)}{5 b}}{4 b}+\frac {d^3 x^3 \left (a x+b x^2\right )^{3/2}}{6 b}\)

\(\Big \downarrow \) 1225

\(\displaystyle \frac {\frac {\frac {\left (a x+b x^2\right )^{3/2} \left (18 b d x \left (7 a^2 d^2-28 a b c d+40 b^2 c^2\right )+5 \left (64 b^3 c^3-3 a d \left (7 a^2 d^2-28 a b c d+40 b^2 c^2\right )\right )\right )}{24 b^2}-\frac {5 a \left (64 b^3 c^3-3 a d \left (7 a^2 d^2-28 a b c d+40 b^2 c^2\right )\right ) \int \sqrt {b x^2+a x}dx}{16 b^2}}{10 b}+\frac {3 d^2 x^2 \left (a x+b x^2\right )^{3/2} (4 b c-a d)}{5 b}}{4 b}+\frac {d^3 x^3 \left (a x+b x^2\right )^{3/2}}{6 b}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {\frac {\frac {\left (a x+b x^2\right )^{3/2} \left (18 b d x \left (7 a^2 d^2-28 a b c d+40 b^2 c^2\right )+5 \left (64 b^3 c^3-3 a d \left (7 a^2 d^2-28 a b c d+40 b^2 c^2\right )\right )\right )}{24 b^2}-\frac {5 a \left (64 b^3 c^3-3 a d \left (7 a^2 d^2-28 a b c d+40 b^2 c^2\right )\right ) \left (\frac {(a+2 b x) \sqrt {a x+b x^2}}{4 b}-\frac {a^2 \int \frac {1}{\sqrt {b x^2+a x}}dx}{8 b}\right )}{16 b^2}}{10 b}+\frac {3 d^2 x^2 \left (a x+b x^2\right )^{3/2} (4 b c-a d)}{5 b}}{4 b}+\frac {d^3 x^3 \left (a x+b x^2\right )^{3/2}}{6 b}\)

\(\Big \downarrow \) 1091

\(\displaystyle \frac {\frac {\frac {\left (a x+b x^2\right )^{3/2} \left (18 b d x \left (7 a^2 d^2-28 a b c d+40 b^2 c^2\right )+5 \left (64 b^3 c^3-3 a d \left (7 a^2 d^2-28 a b c d+40 b^2 c^2\right )\right )\right )}{24 b^2}-\frac {5 a \left (64 b^3 c^3-3 a d \left (7 a^2 d^2-28 a b c d+40 b^2 c^2\right )\right ) \left (\frac {(a+2 b x) \sqrt {a x+b x^2}}{4 b}-\frac {a^2 \int \frac {1}{1-\frac {b x^2}{b x^2+a x}}d\frac {x}{\sqrt {b x^2+a x}}}{4 b}\right )}{16 b^2}}{10 b}+\frac {3 d^2 x^2 \left (a x+b x^2\right )^{3/2} (4 b c-a d)}{5 b}}{4 b}+\frac {d^3 x^3 \left (a x+b x^2\right )^{3/2}}{6 b}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {\left (a x+b x^2\right )^{3/2} \left (18 b d x \left (7 a^2 d^2-28 a b c d+40 b^2 c^2\right )+5 \left (64 b^3 c^3-3 a d \left (7 a^2 d^2-28 a b c d+40 b^2 c^2\right )\right )\right )}{24 b^2}-\frac {5 a \left (\frac {(a+2 b x) \sqrt {a x+b x^2}}{4 b}-\frac {a^2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{4 b^{3/2}}\right ) \left (64 b^3 c^3-3 a d \left (7 a^2 d^2-28 a b c d+40 b^2 c^2\right )\right )}{16 b^2}}{10 b}+\frac {3 d^2 x^2 \left (a x+b x^2\right )^{3/2} (4 b c-a d)}{5 b}}{4 b}+\frac {d^3 x^3 \left (a x+b x^2\right )^{3/2}}{6 b}\)

Input:

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

Output:

(d^3*x^3*(a*x + b*x^2)^(3/2))/(6*b) + ((3*d^2*(4*b*c - a*d)*x^2*(a*x + b*x 
^2)^(3/2))/(5*b) + (((5*(64*b^3*c^3 - 3*a*d*(40*b^2*c^2 - 28*a*b*c*d + 7*a 
^2*d^2)) + 18*b*d*(40*b^2*c^2 - 28*a*b*c*d + 7*a^2*d^2)*x)*(a*x + b*x^2)^( 
3/2))/(24*b^2) - (5*a*(64*b^3*c^3 - 3*a*d*(40*b^2*c^2 - 28*a*b*c*d + 7*a^2 
*d^2))*(((a + 2*b*x)*Sqrt[a*x + b*x^2])/(4*b) - (a^2*ArcTanh[(Sqrt[b]*x)/S 
qrt[a*x + b*x^2]])/(4*b^(3/2))))/(16*b^2))/(10*b))/(4*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 = 0.60 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.65

method	result	size
pseudoelliptic	\(-\frac {21 \left (a^{3} \left (a^{3} d^{3}-4 a^{2} b c \,d^{2}+\frac {40}{7} a \,b^{2} c^{2} d -\frac {64}{21} b^{3} c^{3}\right ) \operatorname {arctanh}\left (\frac {\sqrt {x \left (b x +a \right )}}{x \sqrt {b}}\right )-\left (\frac {512 \left (\frac {1}{2} d^{3} x^{3}+\frac {9}{5} c \,d^{2} x^{2}+\frac {9}{4} c^{2} d x +c^{3}\right ) x^{2} b^{\frac {11}{2}}}{63}+\left (-\frac {64 \left (\frac {3}{20} d^{3} x^{3}+\frac {7}{10} c \,d^{2} x^{2}+\frac {5}{4} c^{2} d x +c^{3}\right ) a \,b^{\frac {7}{2}}}{21}+\frac {128 \left (\frac {2}{5} d^{2} x^{2}+c d x +c^{2}\right ) x \left (\frac {d x}{2}+c \right ) b^{\frac {9}{2}}}{63}+d \,a^{2} \left (\left (\frac {8}{15} d^{2} x^{2}+\frac {8}{3} c d x +\frac {40}{7} c^{2}\right ) b^{\frac {5}{2}}+d a \left (\left (-\frac {2 d x}{3}-4 c \right ) b^{\frac {3}{2}}+\sqrt {b}\, a d \right )\right )\right ) a \right ) \sqrt {x \left (b x +a \right )}\right )}{512 b^{\frac {11}{2}}}\)	\(228\)
risch	\(\frac {\left (1280 b^{5} d^{3} x^{5}+128 a \,b^{4} d^{3} x^{4}+4608 b^{5} c \,d^{2} x^{4}-144 a^{2} b^{3} d^{3} x^{3}+576 a \,b^{4} c \,d^{2} x^{3}+5760 b^{5} c^{2} d \,x^{3}+168 a^{3} b^{2} d^{3} x^{2}-672 a^{2} b^{3} c \,d^{2} x^{2}+960 a \,b^{4} c^{2} d \,x^{2}+2560 b^{5} c^{3} x^{2}-210 a^{4} b \,d^{3} x +840 a^{3} b^{2} c \,d^{2} x -1200 a^{2} b^{3} c^{2} d x +640 a \,b^{4} c^{3} x +315 a^{5} d^{3}-1260 a^{4} b c \,d^{2}+1800 a^{3} b^{2} c^{2} d -960 a^{2} b^{3} c^{3}\right ) x \left (b x +a \right )}{7680 b^{5} \sqrt {x \left (b x +a \right )}}-\frac {a^{3} \left (21 a^{3} d^{3}-84 a^{2} b c \,d^{2}+120 a \,b^{2} c^{2} d -64 b^{3} c^{3}\right ) \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{1024 b^{\frac {11}{2}}}\)	\(306\)
default	\(c^{3} \left (\frac {\left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{3 b}-\frac {a \left (\frac {\left (2 b x +a \right ) \sqrt {b \,x^{2}+a x}}{4 b}-\frac {a^{2} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{8 b^{\frac {3}{2}}}\right )}{2 b}\right )+d^{3} \left (\frac {x^{3} \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{6 b}-\frac {3 a \left (\frac {x^{2} \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{5 b}-\frac {7 a \left (\frac {x \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{4 b}-\frac {5 a \left (\frac {\left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{3 b}-\frac {a \left (\frac {\left (2 b x +a \right ) \sqrt {b \,x^{2}+a x}}{4 b}-\frac {a^{2} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{8 b^{\frac {3}{2}}}\right )}{2 b}\right )}{8 b}\right )}{10 b}\right )}{4 b}\right )+3 c \,d^{2} \left (\frac {x^{2} \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{5 b}-\frac {7 a \left (\frac {x \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{4 b}-\frac {5 a \left (\frac {\left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{3 b}-\frac {a \left (\frac {\left (2 b x +a \right ) \sqrt {b \,x^{2}+a x}}{4 b}-\frac {a^{2} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{8 b^{\frac {3}{2}}}\right )}{2 b}\right )}{8 b}\right )}{10 b}\right )+3 c^{2} d \left (\frac {x \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{4 b}-\frac {5 a \left (\frac {\left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{3 b}-\frac {a \left (\frac {\left (2 b x +a \right ) \sqrt {b \,x^{2}+a x}}{4 b}-\frac {a^{2} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{8 b^{\frac {3}{2}}}\right )}{2 b}\right )}{8 b}\right )\)	\(484\)

Input:

int(x*(d*x+c)^3*(b*x^2+a*x)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-21/512/b^(11/2)*(a^3*(a^3*d^3-4*a^2*b*c*d^2+40/7*a*b^2*c^2*d-64/21*b^3*c^ 
3)*arctanh((x*(b*x+a))^(1/2)/x/b^(1/2))-(512/63*(1/2*d^3*x^3+9/5*c*d^2*x^2 
+9/4*c^2*d*x+c^3)*x^2*b^(11/2)+(-64/21*(3/20*d^3*x^3+7/10*c*d^2*x^2+5/4*c^ 
2*d*x+c^3)*a*b^(7/2)+128/63*(2/5*d^2*x^2+c*d*x+c^2)*x*(1/2*d*x+c)*b^(9/2)+ 
d*a^2*((8/15*d^2*x^2+8/3*c*d*x+40/7*c^2)*b^(5/2)+d*a*((-2/3*d*x-4*c)*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 = 603, normalized size of antiderivative = 1.73 \[ \int x (c+d x)^3 \sqrt {a x+b x^2} \, dx=\left [-\frac {15 \, {\left (64 \, a^{3} b^{3} c^{3} - 120 \, a^{4} b^{2} c^{2} d + 84 \, a^{5} b c d^{2} - 21 \, a^{6} d^{3}\right )} \sqrt {b} \log \left (2 \, b x + a - 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right ) - 2 \, {\left (1280 \, b^{6} d^{3} x^{5} - 960 \, a^{2} b^{4} c^{3} + 1800 \, a^{3} b^{3} c^{2} d - 1260 \, a^{4} b^{2} c d^{2} + 315 \, a^{5} b d^{3} + 128 \, {\left (36 \, b^{6} c d^{2} + a b^{5} d^{3}\right )} x^{4} + 144 \, {\left (40 \, b^{6} c^{2} d + 4 \, a b^{5} c d^{2} - a^{2} b^{4} d^{3}\right )} x^{3} + 8 \, {\left (320 \, b^{6} c^{3} + 120 \, a b^{5} c^{2} d - 84 \, a^{2} b^{4} c d^{2} + 21 \, a^{3} b^{3} d^{3}\right )} x^{2} + 10 \, {\left (64 \, a b^{5} c^{3} - 120 \, a^{2} b^{4} c^{2} d + 84 \, a^{3} b^{3} c d^{2} - 21 \, a^{4} b^{2} d^{3}\right )} x\right )} \sqrt {b x^{2} + a x}}{15360 \, b^{6}}, -\frac {15 \, {\left (64 \, a^{3} b^{3} c^{3} - 120 \, a^{4} b^{2} c^{2} d + 84 \, a^{5} b c d^{2} - 21 \, a^{6} d^{3}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x^{2} + a x} \sqrt {-b}}{b x + a}\right ) - {\left (1280 \, b^{6} d^{3} x^{5} - 960 \, a^{2} b^{4} c^{3} + 1800 \, a^{3} b^{3} c^{2} d - 1260 \, a^{4} b^{2} c d^{2} + 315 \, a^{5} b d^{3} + 128 \, {\left (36 \, b^{6} c d^{2} + a b^{5} d^{3}\right )} x^{4} + 144 \, {\left (40 \, b^{6} c^{2} d + 4 \, a b^{5} c d^{2} - a^{2} b^{4} d^{3}\right )} x^{3} + 8 \, {\left (320 \, b^{6} c^{3} + 120 \, a b^{5} c^{2} d - 84 \, a^{2} b^{4} c d^{2} + 21 \, a^{3} b^{3} d^{3}\right )} x^{2} + 10 \, {\left (64 \, a b^{5} c^{3} - 120 \, a^{2} b^{4} c^{2} d + 84 \, a^{3} b^{3} c d^{2} - 21 \, a^{4} b^{2} d^{3}\right )} x\right )} \sqrt {b x^{2} + a x}}{7680 \, b^{6}}\right ] \] Input:

integrate(x*(d*x+c)^3*(b*x^2+a*x)^(1/2),x, algorithm="fricas")
 

Output:

[-1/15360*(15*(64*a^3*b^3*c^3 - 120*a^4*b^2*c^2*d + 84*a^5*b*c*d^2 - 21*a^ 
6*d^3)*sqrt(b)*log(2*b*x + a - 2*sqrt(b*x^2 + a*x)*sqrt(b)) - 2*(1280*b^6* 
d^3*x^5 - 960*a^2*b^4*c^3 + 1800*a^3*b^3*c^2*d - 1260*a^4*b^2*c*d^2 + 315* 
a^5*b*d^3 + 128*(36*b^6*c*d^2 + a*b^5*d^3)*x^4 + 144*(40*b^6*c^2*d + 4*a*b 
^5*c*d^2 - a^2*b^4*d^3)*x^3 + 8*(320*b^6*c^3 + 120*a*b^5*c^2*d - 84*a^2*b^ 
4*c*d^2 + 21*a^3*b^3*d^3)*x^2 + 10*(64*a*b^5*c^3 - 120*a^2*b^4*c^2*d + 84* 
a^3*b^3*c*d^2 - 21*a^4*b^2*d^3)*x)*sqrt(b*x^2 + a*x))/b^6, -1/7680*(15*(64 
*a^3*b^3*c^3 - 120*a^4*b^2*c^2*d + 84*a^5*b*c*d^2 - 21*a^6*d^3)*sqrt(-b)*a 
rctan(sqrt(b*x^2 + a*x)*sqrt(-b)/(b*x + a)) - (1280*b^6*d^3*x^5 - 960*a^2* 
b^4*c^3 + 1800*a^3*b^3*c^2*d - 1260*a^4*b^2*c*d^2 + 315*a^5*b*d^3 + 128*(3 
6*b^6*c*d^2 + a*b^5*d^3)*x^4 + 144*(40*b^6*c^2*d + 4*a*b^5*c*d^2 - a^2*b^4 
*d^3)*x^3 + 8*(320*b^6*c^3 + 120*a*b^5*c^2*d - 84*a^2*b^4*c*d^2 + 21*a^3*b 
^3*d^3)*x^2 + 10*(64*a*b^5*c^3 - 120*a^2*b^4*c^2*d + 84*a^3*b^3*c*d^2 - 21 
*a^4*b^2*d^3)*x)*sqrt(b*x^2 + a*x))/b^6]
 

Sympy [A] (verification not implemented)

Time = 0.60 (sec) , antiderivative size = 525, normalized size of antiderivative = 1.50 \[ \int x (c+d x)^3 \sqrt {a x+b x^2} \, dx=\begin {cases} \frac {3 a^{2} \left (a c^{3} - \frac {5 a \left (3 a c^{2} d - \frac {7 a \left (3 a c d^{2} - \frac {9 a \left (\frac {a d^{3}}{12} + 3 b c d^{2}\right )}{10 b} + 3 b c^{2} d\right )}{8 b} + b c^{3}\right )}{6 b}\right ) \left (\begin {cases} \frac {\log {\left (a + 2 \sqrt {b} \sqrt {a x + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: \frac {a^{2}}{b} \neq 0 \\\frac {\left (\frac {a}{2 b} + x\right ) \log {\left (\frac {a}{2 b} + x \right )}}{\sqrt {b \left (\frac {a}{2 b} + x\right )^{2}}} & \text {otherwise} \end {cases}\right )}{8 b^{2}} + \sqrt {a x + b x^{2}} \left (- \frac {3 a \left (a c^{3} - \frac {5 a \left (3 a c^{2} d - \frac {7 a \left (3 a c d^{2} - \frac {9 a \left (\frac {a d^{3}}{12} + 3 b c d^{2}\right )}{10 b} + 3 b c^{2} d\right )}{8 b} + b c^{3}\right )}{6 b}\right )}{4 b^{2}} + \frac {d^{3} x^{5}}{6} + \frac {x^{4} \left (\frac {a d^{3}}{12} + 3 b c d^{2}\right )}{5 b} + \frac {x^{3} \cdot \left (3 a c d^{2} - \frac {9 a \left (\frac {a d^{3}}{12} + 3 b c d^{2}\right )}{10 b} + 3 b c^{2} d\right )}{4 b} + \frac {x^{2} \cdot \left (3 a c^{2} d - \frac {7 a \left (3 a c d^{2} - \frac {9 a \left (\frac {a d^{3}}{12} + 3 b c d^{2}\right )}{10 b} + 3 b c^{2} d\right )}{8 b} + b c^{3}\right )}{3 b} + \frac {x \left (a c^{3} - \frac {5 a \left (3 a c^{2} d - \frac {7 a \left (3 a c d^{2} - \frac {9 a \left (\frac {a d^{3}}{12} + 3 b c d^{2}\right )}{10 b} + 3 b c^{2} d\right )}{8 b} + b c^{3}\right )}{6 b}\right )}{2 b}\right ) & \text {for}\: b \neq 0 \\\frac {2 \left (\frac {c^{3} \left (a x\right )^{\frac {5}{2}}}{5} + \frac {3 c^{2} d \left (a x\right )^{\frac {7}{2}}}{7 a} + \frac {c d^{2} \left (a x\right )^{\frac {9}{2}}}{3 a^{2}} + \frac {d^{3} \left (a x\right )^{\frac {11}{2}}}{11 a^{3}}\right )}{a^{2}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \] Input:

integrate(x*(d*x+c)**3*(b*x**2+a*x)**(1/2),x)
 

Output:

Piecewise((3*a**2*(a*c**3 - 5*a*(3*a*c**2*d - 7*a*(3*a*c*d**2 - 9*a*(a*d** 
3/12 + 3*b*c*d**2)/(10*b) + 3*b*c**2*d)/(8*b) + b*c**3)/(6*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))/(8*b**2) + s 
qrt(a*x + b*x**2)*(-3*a*(a*c**3 - 5*a*(3*a*c**2*d - 7*a*(3*a*c*d**2 - 9*a* 
(a*d**3/12 + 3*b*c*d**2)/(10*b) + 3*b*c**2*d)/(8*b) + b*c**3)/(6*b))/(4*b* 
*2) + d**3*x**5/6 + x**4*(a*d**3/12 + 3*b*c*d**2)/(5*b) + x**3*(3*a*c*d**2 
 - 9*a*(a*d**3/12 + 3*b*c*d**2)/(10*b) + 3*b*c**2*d)/(4*b) + x**2*(3*a*c** 
2*d - 7*a*(3*a*c*d**2 - 9*a*(a*d**3/12 + 3*b*c*d**2)/(10*b) + 3*b*c**2*d)/ 
(8*b) + b*c**3)/(3*b) + x*(a*c**3 - 5*a*(3*a*c**2*d - 7*a*(3*a*c*d**2 - 9* 
a*(a*d**3/12 + 3*b*c*d**2)/(10*b) + 3*b*c**2*d)/(8*b) + b*c**3)/(6*b))/(2* 
b)), Ne(b, 0)), (2*(c**3*(a*x)**(5/2)/5 + 3*c**2*d*(a*x)**(7/2)/(7*a) + c* 
d**2*(a*x)**(9/2)/(3*a**2) + d**3*(a*x)**(11/2)/(11*a**3))/a**2, Ne(a, 0)) 
, (0, True))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 540, normalized size of antiderivative = 1.55 \[ \int x (c+d x)^3 \sqrt {a x+b x^2} \, dx=\frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} d^{3} x^{3}}{6 \, b} + \frac {3 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} c d^{2} x^{2}}{5 \, b} - \frac {3 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a d^{3} x^{2}}{20 \, b^{2}} - \frac {\sqrt {b x^{2} + a x} a c^{3} x}{4 \, b} + \frac {15 \, \sqrt {b x^{2} + a x} a^{2} c^{2} d x}{32 \, b^{2}} + \frac {3 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} c^{2} d x}{4 \, b} - \frac {21 \, \sqrt {b x^{2} + a x} a^{3} c d^{2} x}{64 \, b^{3}} - \frac {21 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a c d^{2} x}{40 \, b^{2}} + \frac {21 \, \sqrt {b x^{2} + a x} a^{4} d^{3} x}{256 \, b^{4}} + \frac {21 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{2} d^{3} x}{160 \, b^{3}} + \frac {a^{3} c^{3} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{16 \, b^{\frac {5}{2}}} - \frac {15 \, a^{4} c^{2} d \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{128 \, b^{\frac {7}{2}}} + \frac {21 \, a^{5} c d^{2} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{256 \, b^{\frac {9}{2}}} - \frac {21 \, a^{6} d^{3} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{1024 \, b^{\frac {11}{2}}} - \frac {\sqrt {b x^{2} + a x} a^{2} c^{3}}{8 \, b^{2}} + \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} c^{3}}{3 \, b} + \frac {15 \, \sqrt {b x^{2} + a x} a^{3} c^{2} d}{64 \, b^{3}} - \frac {5 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a c^{2} d}{8 \, b^{2}} - \frac {21 \, \sqrt {b x^{2} + a x} a^{4} c d^{2}}{128 \, b^{4}} + \frac {7 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{2} c d^{2}}{16 \, b^{3}} + \frac {21 \, \sqrt {b x^{2} + a x} a^{5} d^{3}}{512 \, b^{5}} - \frac {7 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{3} d^{3}}{64 \, b^{4}} \] Input:

integrate(x*(d*x+c)^3*(b*x^2+a*x)^(1/2),x, algorithm="maxima")
 

Output:

1/6*(b*x^2 + a*x)^(3/2)*d^3*x^3/b + 3/5*(b*x^2 + a*x)^(3/2)*c*d^2*x^2/b - 
3/20*(b*x^2 + a*x)^(3/2)*a*d^3*x^2/b^2 - 1/4*sqrt(b*x^2 + a*x)*a*c^3*x/b + 
 15/32*sqrt(b*x^2 + a*x)*a^2*c^2*d*x/b^2 + 3/4*(b*x^2 + a*x)^(3/2)*c^2*d*x 
/b - 21/64*sqrt(b*x^2 + a*x)*a^3*c*d^2*x/b^3 - 21/40*(b*x^2 + a*x)^(3/2)*a 
*c*d^2*x/b^2 + 21/256*sqrt(b*x^2 + a*x)*a^4*d^3*x/b^4 + 21/160*(b*x^2 + a* 
x)^(3/2)*a^2*d^3*x/b^3 + 1/16*a^3*c^3*log(2*b*x + a + 2*sqrt(b*x^2 + a*x)* 
sqrt(b))/b^(5/2) - 15/128*a^4*c^2*d*log(2*b*x + a + 2*sqrt(b*x^2 + a*x)*sq 
rt(b))/b^(7/2) + 21/256*a^5*c*d^2*log(2*b*x + a + 2*sqrt(b*x^2 + a*x)*sqrt 
(b))/b^(9/2) - 21/1024*a^6*d^3*log(2*b*x + a + 2*sqrt(b*x^2 + a*x)*sqrt(b) 
)/b^(11/2) - 1/8*sqrt(b*x^2 + a*x)*a^2*c^3/b^2 + 1/3*(b*x^2 + a*x)^(3/2)*c 
^3/b + 15/64*sqrt(b*x^2 + a*x)*a^3*c^2*d/b^3 - 5/8*(b*x^2 + a*x)^(3/2)*a*c 
^2*d/b^2 - 21/128*sqrt(b*x^2 + a*x)*a^4*c*d^2/b^4 + 7/16*(b*x^2 + a*x)^(3/ 
2)*a^2*c*d^2/b^3 + 21/512*sqrt(b*x^2 + a*x)*a^5*d^3/b^5 - 7/64*(b*x^2 + a* 
x)^(3/2)*a^3*d^3/b^4
 

Giac [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 311, normalized size of antiderivative = 0.89 \[ \int x (c+d x)^3 \sqrt {a x+b x^2} \, dx=\frac {1}{7680} \, \sqrt {b x^{2} + a x} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, d^{3} x + \frac {36 \, b^{5} c d^{2} + a b^{4} d^{3}}{b^{5}}\right )} x + \frac {9 \, {\left (40 \, b^{5} c^{2} d + 4 \, a b^{4} c d^{2} - a^{2} b^{3} d^{3}\right )}}{b^{5}}\right )} x + \frac {320 \, b^{5} c^{3} + 120 \, a b^{4} c^{2} d - 84 \, a^{2} b^{3} c d^{2} + 21 \, a^{3} b^{2} d^{3}}{b^{5}}\right )} x + \frac {5 \, {\left (64 \, a b^{4} c^{3} - 120 \, a^{2} b^{3} c^{2} d + 84 \, a^{3} b^{2} c d^{2} - 21 \, a^{4} b d^{3}\right )}}{b^{5}}\right )} x - \frac {15 \, {\left (64 \, a^{2} b^{3} c^{3} - 120 \, a^{3} b^{2} c^{2} d + 84 \, a^{4} b c d^{2} - 21 \, a^{5} d^{3}\right )}}{b^{5}}\right )} - \frac {{\left (64 \, a^{3} b^{3} c^{3} - 120 \, a^{4} b^{2} c^{2} d + 84 \, a^{5} b c d^{2} - 21 \, a^{6} d^{3}\right )} \log \left ({\left | 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} \sqrt {b} + a \right |}\right )}{1024 \, b^{\frac {11}{2}}} \] Input:

integrate(x*(d*x+c)^3*(b*x^2+a*x)^(1/2),x, algorithm="giac")
 

Output:

1/7680*sqrt(b*x^2 + a*x)*(2*(4*(2*(8*(10*d^3*x + (36*b^5*c*d^2 + a*b^4*d^3 
)/b^5)*x + 9*(40*b^5*c^2*d + 4*a*b^4*c*d^2 - a^2*b^3*d^3)/b^5)*x + (320*b^ 
5*c^3 + 120*a*b^4*c^2*d - 84*a^2*b^3*c*d^2 + 21*a^3*b^2*d^3)/b^5)*x + 5*(6 
4*a*b^4*c^3 - 120*a^2*b^3*c^2*d + 84*a^3*b^2*c*d^2 - 21*a^4*b*d^3)/b^5)*x 
- 15*(64*a^2*b^3*c^3 - 120*a^3*b^2*c^2*d + 84*a^4*b*c*d^2 - 21*a^5*d^3)/b^ 
5) - 1/1024*(64*a^3*b^3*c^3 - 120*a^4*b^2*c^2*d + 84*a^5*b*c*d^2 - 21*a^6* 
d^3)*log(abs(2*(sqrt(b)*x - sqrt(b*x^2 + a*x))*sqrt(b) + a))/b^(11/2)
 

Mupad [B] (verification not implemented)

Time = 9.43 (sec) , antiderivative size = 451, normalized size of antiderivative = 1.29 \[ \int x (c+d x)^3 \sqrt {a x+b x^2} \, dx=\frac {d^3\,x^3\,{\left (b\,x^2+a\,x\right )}^{3/2}}{6\,b}+\frac {a^3\,c^3\,\ln \left (\frac {a+2\,b\,x}{\sqrt {b}}+2\,\sqrt {b\,x^2+a\,x}\right )}{16\,b^{5/2}}+\frac {c^3\,\sqrt {b\,x^2+a\,x}\,\left (-3\,a^2+2\,a\,b\,x+8\,b^2\,x^2\right )}{24\,b^2}+\frac {3\,a\,d^3\,\left (\frac {7\,a\,\left (\frac {x\,{\left (b\,x^2+a\,x\right )}^{3/2}}{4\,b}-\frac {5\,a\,\left (\frac {a^3\,\ln \left (\frac {a+2\,b\,x}{\sqrt {b}}+2\,\sqrt {b\,x^2+a\,x}\right )}{16\,b^{5/2}}+\frac {\sqrt {b\,x^2+a\,x}\,\left (-3\,a^2+2\,a\,b\,x+8\,b^2\,x^2\right )}{24\,b^2}\right )}{8\,b}\right )}{10\,b}-\frac {x^2\,{\left (b\,x^2+a\,x\right )}^{3/2}}{5\,b}\right )}{4\,b}+\frac {3\,c^2\,d\,x\,{\left (b\,x^2+a\,x\right )}^{3/2}}{4\,b}-\frac {15\,a\,c^2\,d\,\left (\frac {a^3\,\ln \left (\frac {a+2\,b\,x}{\sqrt {b}}+2\,\sqrt {b\,x^2+a\,x}\right )}{16\,b^{5/2}}+\frac {\sqrt {b\,x^2+a\,x}\,\left (-3\,a^2+2\,a\,b\,x+8\,b^2\,x^2\right )}{24\,b^2}\right )}{8\,b}-\frac {21\,a\,c\,d^2\,\left (\frac {x\,{\left (b\,x^2+a\,x\right )}^{3/2}}{4\,b}-\frac {5\,a\,\left (\frac {a^3\,\ln \left (\frac {a+2\,b\,x}{\sqrt {b}}+2\,\sqrt {b\,x^2+a\,x}\right )}{16\,b^{5/2}}+\frac {\sqrt {b\,x^2+a\,x}\,\left (-3\,a^2+2\,a\,b\,x+8\,b^2\,x^2\right )}{24\,b^2}\right )}{8\,b}\right )}{10\,b}+\frac {3\,c\,d^2\,x^2\,{\left (b\,x^2+a\,x\right )}^{3/2}}{5\,b} \] Input:

int(x*(a*x + b*x^2)^(1/2)*(c + d*x)^3,x)
                                                                                    
                                                                                    
 

Output:

(d^3*x^3*(a*x + b*x^2)^(3/2))/(6*b) + (a^3*c^3*log((a + 2*b*x)/b^(1/2) + 2 
*(a*x + b*x^2)^(1/2)))/(16*b^(5/2)) + (c^3*(a*x + b*x^2)^(1/2)*(8*b^2*x^2 
- 3*a^2 + 2*a*b*x))/(24*b^2) + (3*a*d^3*((7*a*((x*(a*x + b*x^2)^(3/2))/(4* 
b) - (5*a*((a^3*log((a + 2*b*x)/b^(1/2) + 2*(a*x + b*x^2)^(1/2)))/(16*b^(5 
/2)) + ((a*x + b*x^2)^(1/2)*(8*b^2*x^2 - 3*a^2 + 2*a*b*x))/(24*b^2)))/(8*b 
)))/(10*b) - (x^2*(a*x + b*x^2)^(3/2))/(5*b)))/(4*b) + (3*c^2*d*x*(a*x + b 
*x^2)^(3/2))/(4*b) - (15*a*c^2*d*((a^3*log((a + 2*b*x)/b^(1/2) + 2*(a*x + 
b*x^2)^(1/2)))/(16*b^(5/2)) + ((a*x + b*x^2)^(1/2)*(8*b^2*x^2 - 3*a^2 + 2* 
a*b*x))/(24*b^2)))/(8*b) - (21*a*c*d^2*((x*(a*x + b*x^2)^(3/2))/(4*b) - (5 
*a*((a^3*log((a + 2*b*x)/b^(1/2) + 2*(a*x + b*x^2)^(1/2)))/(16*b^(5/2)) + 
((a*x + b*x^2)^(1/2)*(8*b^2*x^2 - 3*a^2 + 2*a*b*x))/(24*b^2)))/(8*b)))/(10 
*b) + (3*c*d^2*x^2*(a*x + b*x^2)^(3/2))/(5*b)
 

Reduce [B] (verification not implemented)

Time = 12.00 (sec) , antiderivative size = 490, normalized size of antiderivative = 1.40 \[ \int x (c+d x)^3 \sqrt {a x+b x^2} \, dx=\frac {315 \sqrt {x}\, \sqrt {b x +a}\, a^{5} b \,d^{3}-1260 \sqrt {x}\, \sqrt {b x +a}\, a^{4} b^{2} c \,d^{2}-210 \sqrt {x}\, \sqrt {b x +a}\, a^{4} b^{2} d^{3} x +1800 \sqrt {x}\, \sqrt {b x +a}\, a^{3} b^{3} c^{2} d +840 \sqrt {x}\, \sqrt {b x +a}\, a^{3} b^{3} c \,d^{2} x +168 \sqrt {x}\, \sqrt {b x +a}\, a^{3} b^{3} d^{3} x^{2}-960 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b^{4} c^{3}-1200 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b^{4} c^{2} d x -672 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b^{4} c \,d^{2} x^{2}-144 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b^{4} d^{3} x^{3}+640 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{5} c^{3} x +960 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{5} c^{2} d \,x^{2}+576 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{5} c \,d^{2} x^{3}+128 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{5} d^{3} x^{4}+2560 \sqrt {x}\, \sqrt {b x +a}\, b^{6} c^{3} x^{2}+5760 \sqrt {x}\, \sqrt {b x +a}\, b^{6} c^{2} d \,x^{3}+4608 \sqrt {x}\, \sqrt {b x +a}\, b^{6} c \,d^{2} x^{4}+1280 \sqrt {x}\, \sqrt {b x +a}\, b^{6} d^{3} x^{5}-315 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a^{6} d^{3}+1260 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a^{5} b c \,d^{2}-1800 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a^{4} b^{2} c^{2} d +960 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a^{3} b^{3} c^{3}}{7680 b^{6}} \] Input:

int(x*(d*x+c)^3*(b*x^2+a*x)^(1/2),x)
 

Output:

(315*sqrt(x)*sqrt(a + b*x)*a**5*b*d**3 - 1260*sqrt(x)*sqrt(a + b*x)*a**4*b 
**2*c*d**2 - 210*sqrt(x)*sqrt(a + b*x)*a**4*b**2*d**3*x + 1800*sqrt(x)*sqr 
t(a + b*x)*a**3*b**3*c**2*d + 840*sqrt(x)*sqrt(a + b*x)*a**3*b**3*c*d**2*x 
 + 168*sqrt(x)*sqrt(a + b*x)*a**3*b**3*d**3*x**2 - 960*sqrt(x)*sqrt(a + b* 
x)*a**2*b**4*c**3 - 1200*sqrt(x)*sqrt(a + b*x)*a**2*b**4*c**2*d*x - 672*sq 
rt(x)*sqrt(a + b*x)*a**2*b**4*c*d**2*x**2 - 144*sqrt(x)*sqrt(a + b*x)*a**2 
*b**4*d**3*x**3 + 640*sqrt(x)*sqrt(a + b*x)*a*b**5*c**3*x + 960*sqrt(x)*sq 
rt(a + b*x)*a*b**5*c**2*d*x**2 + 576*sqrt(x)*sqrt(a + b*x)*a*b**5*c*d**2*x 
**3 + 128*sqrt(x)*sqrt(a + b*x)*a*b**5*d**3*x**4 + 2560*sqrt(x)*sqrt(a + b 
*x)*b**6*c**3*x**2 + 5760*sqrt(x)*sqrt(a + b*x)*b**6*c**2*d*x**3 + 4608*sq 
rt(x)*sqrt(a + b*x)*b**6*c*d**2*x**4 + 1280*sqrt(x)*sqrt(a + b*x)*b**6*d** 
3*x**5 - 315*sqrt(b)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a))*a**6*d 
**3 + 1260*sqrt(b)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a))*a**5*b*c 
*d**2 - 1800*sqrt(b)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a))*a**4*b 
**2*c**2*d + 960*sqrt(b)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a))*a* 
*3*b**3*c**3)/(7680*b**6)