\(\int x^2 (c+d x)^2 \sqrt {a x+b x^2} \, dx\) [10]

Optimal result

Integrand size = 24, antiderivative size = 294 \[ \int x^2 (c+d x)^2 \sqrt {a x+b x^2} \, dx=\frac {a^3 \left (40 b^2 c^2-7 a d (8 b c-3 a d)\right ) \sqrt {a x+b x^2}}{512 b^5}-\frac {a^2 \left (40 b^2 c^2-7 a d (8 b c-3 a d)\right ) x \sqrt {a x+b x^2}}{768 b^4}+\frac {a \left (40 b^2 c^2-7 a d (8 b c-3 a d)\right ) x^2 \sqrt {a x+b x^2}}{960 b^3}+\frac {1}{160} \left (40 c^2-\frac {7 a d (8 b c-3 a d)}{b^2}\right ) x^3 \sqrt {a x+b x^2}+\frac {d (8 b c-3 a d) x^2 \left (a x+b x^2\right )^{3/2}}{20 b^2}+\frac {d^2 x^3 \left (a x+b x^2\right )^{3/2}}{6 b}-\frac {a^4 \left (40 b^2 c^2-7 a d (8 b c-3 a d)\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{512 b^{11/2}} \] Output:

1/512*a^3*(40*b^2*c^2-7*a*d*(-3*a*d+8*b*c))*(b*x^2+a*x)^(1/2)/b^5-1/768*a^ 
2*(40*b^2*c^2-7*a*d*(-3*a*d+8*b*c))*x*(b*x^2+a*x)^(1/2)/b^4+1/960*a*(40*b^ 
2*c^2-7*a*d*(-3*a*d+8*b*c))*x^2*(b*x^2+a*x)^(1/2)/b^3+1/160*(40*c^2-7*a*d* 
(-3*a*d+8*b*c)/b^2)*x^3*(b*x^2+a*x)^(1/2)+1/20*d*(-3*a*d+8*b*c)*x^2*(b*x^2 
+a*x)^(3/2)/b^2+1/6*d^2*x^3*(b*x^2+a*x)^(3/2)/b-1/512*a^4*(40*b^2*c^2-7*a* 
d*(-3*a*d+8*b*c))*arctanh(b^(1/2)*x/(b*x^2+a*x)^(1/2))/b^(11/2)
 

Mathematica [A] (verified)

Time = 1.42 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.95 \[ \int x^2 (c+d x)^2 \sqrt {a x+b x^2} \, dx=\frac {\sqrt {x (a+b x)} \left (600 a^3 b^2 c^2-840 a^4 b c d+315 a^5 d^2-400 a^2 b^3 c^2 x+560 a^3 b^2 c d x-210 a^4 b d^2 x+320 a b^4 c^2 x^2-448 a^2 b^3 c d x^2+168 a^3 b^2 d^2 x^2+1920 b^5 c^2 x^3+384 a b^4 c d x^3-144 a^2 b^3 d^2 x^3+3072 b^5 c d x^4+128 a b^4 d^2 x^4+1280 b^5 d^2 x^5\right )}{7680 b^5}-\frac {a^4 \left (40 b^2 c^2-56 a b c d+21 a^2 d^2\right ) \sqrt {x (a+b x)} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{-\sqrt {a}+\sqrt {a+b x}}\right )}{256 b^{11/2} \sqrt {x} \sqrt {a+b x}} \] Input:

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

Output:

(Sqrt[x*(a + b*x)]*(600*a^3*b^2*c^2 - 840*a^4*b*c*d + 315*a^5*d^2 - 400*a^ 
2*b^3*c^2*x + 560*a^3*b^2*c*d*x - 210*a^4*b*d^2*x + 320*a*b^4*c^2*x^2 - 44 
8*a^2*b^3*c*d*x^2 + 168*a^3*b^2*d^2*x^2 + 1920*b^5*c^2*x^3 + 384*a*b^4*c*d 
*x^3 - 144*a^2*b^3*d^2*x^3 + 3072*b^5*c*d*x^4 + 128*a*b^4*d^2*x^4 + 1280*b 
^5*d^2*x^5))/(7680*b^5) - (a^4*(40*b^2*c^2 - 56*a*b*c*d + 21*a^2*d^2)*Sqrt 
[x*(a + b*x)]*ArcTanh[(Sqrt[b]*Sqrt[x])/(-Sqrt[a] + Sqrt[a + b*x])])/(256* 
b^(11/2)*Sqrt[x]*Sqrt[a + b*x])
 

Rubi [A] (verified)

Time = 0.73 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.74, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1262, 27, 1221, 1134, 1160, 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.

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

\(\Big \downarrow \) 1262

\(\displaystyle \frac {\int \frac {3}{2} x^2 \left (4 b c^2+d (8 b c-3 a d) x\right ) \sqrt {b x^2+a x}dx}{6 b}+\frac {d^2 x^3 \left (a x+b x^2\right )^{3/2}}{6 b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int x^2 \left (4 b c^2+d (8 b c-3 a d) x\right ) \sqrt {b x^2+a x}dx}{4 b}+\frac {d^2 x^3 \left (a x+b x^2\right )^{3/2}}{6 b}\)

\(\Big \downarrow \) 1221

\(\displaystyle \frac {\frac {\left (21 a^2 d^2-56 a b c d+40 b^2 c^2\right ) \int x^2 \sqrt {b x^2+a x}dx}{10 b}+\frac {d x^2 \left (a x+b x^2\right )^{3/2} (8 b c-3 a d)}{5 b}}{4 b}+\frac {d^2 x^3 \left (a x+b x^2\right )^{3/2}}{6 b}\)

\(\Big \downarrow \) 1134

\(\displaystyle \frac {\frac {\left (21 a^2 d^2-56 a b c d+40 b^2 c^2\right ) \left (\frac {x \left (a x+b x^2\right )^{3/2}}{4 b}-\frac {5 a \int x \sqrt {b x^2+a x}dx}{8 b}\right )}{10 b}+\frac {d x^2 \left (a x+b x^2\right )^{3/2} (8 b c-3 a d)}{5 b}}{4 b}+\frac {d^2 x^3 \left (a x+b x^2\right )^{3/2}}{6 b}\)

\(\Big \downarrow \) 1160

\(\displaystyle \frac {\frac {\left (21 a^2 d^2-56 a b c d+40 b^2 c^2\right ) \left (\frac {x \left (a x+b x^2\right )^{3/2}}{4 b}-\frac {5 a \left (\frac {\left (a x+b x^2\right )^{3/2}}{3 b}-\frac {a \int \sqrt {b x^2+a x}dx}{2 b}\right )}{8 b}\right )}{10 b}+\frac {d x^2 \left (a x+b x^2\right )^{3/2} (8 b c-3 a d)}{5 b}}{4 b}+\frac {d^2 x^3 \left (a x+b x^2\right )^{3/2}}{6 b}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {\frac {\left (21 a^2 d^2-56 a b c d+40 b^2 c^2\right ) \left (\frac {x \left (a x+b x^2\right )^{3/2}}{4 b}-\frac {5 a \left (\frac {\left (a x+b x^2\right )^{3/2}}{3 b}-\frac {a \left (\frac {(a+2 b x) \sqrt {a x+b x^2}}{4 b}-\frac {a^2 \int \frac {1}{\sqrt {b x^2+a x}}dx}{8 b}\right )}{2 b}\right )}{8 b}\right )}{10 b}+\frac {d x^2 \left (a x+b x^2\right )^{3/2} (8 b c-3 a d)}{5 b}}{4 b}+\frac {d^2 x^3 \left (a x+b x^2\right )^{3/2}}{6 b}\)

\(\Big \downarrow \) 1091

\(\displaystyle \frac {\frac {\left (21 a^2 d^2-56 a b c d+40 b^2 c^2\right ) \left (\frac {x \left (a x+b x^2\right )^{3/2}}{4 b}-\frac {5 a \left (\frac {\left (a x+b x^2\right )^{3/2}}{3 b}-\frac {a \left (\frac {(a+2 b x) \sqrt {a x+b x^2}}{4 b}-\frac {a^2 \int \frac {1}{1-\frac {b x^2}{b x^2+a x}}d\frac {x}{\sqrt {b x^2+a x}}}{4 b}\right )}{2 b}\right )}{8 b}\right )}{10 b}+\frac {d x^2 \left (a x+b x^2\right )^{3/2} (8 b c-3 a d)}{5 b}}{4 b}+\frac {d^2 x^3 \left (a x+b x^2\right )^{3/2}}{6 b}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\left (\frac {x \left (a x+b x^2\right )^{3/2}}{4 b}-\frac {5 a \left (\frac {\left (a x+b x^2\right )^{3/2}}{3 b}-\frac {a \left (\frac {(a+2 b x) \sqrt {a x+b x^2}}{4 b}-\frac {a^2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{4 b^{3/2}}\right )}{2 b}\right )}{8 b}\right ) \left (21 a^2 d^2-56 a b c d+40 b^2 c^2\right )}{10 b}+\frac {d x^2 \left (a x+b x^2\right )^{3/2} (8 b c-3 a d)}{5 b}}{4 b}+\frac {d^2 x^3 \left (a x+b x^2\right )^{3/2}}{6 b}\)

Input:

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

Output:

(d^2*x^3*(a*x + b*x^2)^(3/2))/(6*b) + ((d*(8*b*c - 3*a*d)*x^2*(a*x + b*x^2 
)^(3/2))/(5*b) + ((40*b^2*c^2 - 56*a*b*c*d + 21*a^2*d^2)*((x*(a*x + b*x^2) 
^(3/2))/(4*b) - (5*a*((a*x + b*x^2)^(3/2)/(3*b) - (a*(((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 
))))/(2*b)))/(8*b)))/(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 1134

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 
 1))), x] + Simp[(m + p)*((2*c*d - b*e)/(c*(m + 2*p + 1)))   Int[(d + e*x)^ 
(m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[ 
c*d^2 - b*d*e + a*e^2, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && IntegerQ[2 
*p]
 

rule 1160

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

rule 1221

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

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]
 

Maple [A] (verified)

Time = 0.59 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.64

method	result	size
pseudoelliptic	\(-\frac {21 \left (a^{4} \left (a^{2} d^{2}-\frac {8}{3} a b c d +\frac {40}{21} b^{2} c^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {x \left (b x +a \right )}}{x \sqrt {b}}\right )-\sqrt {x \left (b x +a \right )}\, \left (\frac {128 x^{3} \left (\frac {2}{3} d^{2} x^{2}+\frac {8}{5} c d x +c^{2}\right ) b^{\frac {11}{2}}}{21}+\left (\frac {40 \left (\frac {7}{25} d^{2} x^{2}+\frac {14}{15} c d x +c^{2}\right ) a^{2} b^{\frac {5}{2}}}{21}-\frac {80 x a \left (\frac {9}{25} d^{2} x^{2}+\frac {28}{25} c d x +c^{2}\right ) b^{\frac {7}{2}}}{63}+\frac {64 x^{2} \left (\frac {2}{5} d^{2} x^{2}+\frac {6}{5} c d x +c^{2}\right ) b^{\frac {9}{2}}}{63}+d \left (\frac {2 \left (-d x -4 c \right ) b^{\frac {3}{2}}}{3}+\sqrt {b}\, a d \right ) a^{3}\right ) a \right )\right )}{512 b^{\frac {11}{2}}}\)	\(189\)
risch	\(\frac {\left (1280 b^{5} d^{2} x^{5}+128 a \,b^{4} d^{2} x^{4}+3072 b^{5} c d \,x^{4}-144 a^{2} b^{3} d^{2} x^{3}+384 a \,b^{4} c d \,x^{3}+1920 b^{5} c^{2} x^{3}+168 a^{3} b^{2} d^{2} x^{2}-448 a^{2} b^{3} c d \,x^{2}+320 a \,b^{4} c^{2} x^{2}-210 a^{4} b \,d^{2} x +560 a^{3} b^{2} c d x -400 a^{2} b^{3} c^{2} x +315 a^{5} d^{2}-840 a^{4} b c d +600 a^{3} b^{2} c^{2}\right ) x \left (b x +a \right )}{7680 b^{5} \sqrt {x \left (b x +a \right )}}-\frac {a^{4} \left (21 a^{2} d^{2}-56 a b c d +40 b^{2} c^{2}\right ) \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{1024 b^{\frac {11}{2}}}\)	\(246\)
default	\(c^{2} \left (\frac {x \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{4 b}-\frac {5 a \left (\frac {\left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{3 b}-\frac {a \left (\frac {\left (2 b x +a \right ) \sqrt {b \,x^{2}+a x}}{4 b}-\frac {a^{2} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{8 b^{\frac {3}{2}}}\right )}{2 b}\right )}{8 b}\right )+d^{2} \left (\frac {x^{3} \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{6 b}-\frac {3 a \left (\frac {x^{2} \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{5 b}-\frac {7 a \left (\frac {x \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{4 b}-\frac {5 a \left (\frac {\left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{3 b}-\frac {a \left (\frac {\left (2 b x +a \right ) \sqrt {b \,x^{2}+a x}}{4 b}-\frac {a^{2} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{8 b^{\frac {3}{2}}}\right )}{2 b}\right )}{8 b}\right )}{10 b}\right )}{4 b}\right )+2 c d \left (\frac {x^{2} \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{5 b}-\frac {7 a \left (\frac {x \left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{4 b}-\frac {5 a \left (\frac {\left (b \,x^{2}+a x \right )^{\frac {3}{2}}}{3 b}-\frac {a \left (\frac {\left (2 b x +a \right ) \sqrt {b \,x^{2}+a x}}{4 b}-\frac {a^{2} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{8 b^{\frac {3}{2}}}\right )}{2 b}\right )}{8 b}\right )}{10 b}\right )\)	\(398\)

Input:

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

Output:

-21/512/b^(11/2)*(a^4*(a^2*d^2-8/3*a*b*c*d+40/21*b^2*c^2)*arctanh((x*(b*x+ 
a))^(1/2)/x/b^(1/2))-(x*(b*x+a))^(1/2)*(128/21*x^3*(2/3*d^2*x^2+8/5*c*d*x+ 
c^2)*b^(11/2)+(40/21*(7/25*d^2*x^2+14/15*c*d*x+c^2)*a^2*b^(5/2)-80/63*x*a* 
(9/25*d^2*x^2+28/25*c*d*x+c^2)*b^(7/2)+64/63*x^2*(2/5*d^2*x^2+6/5*c*d*x+c^ 
2)*b^(9/2)+d*(2/3*(-d*x-4*c)*b^(3/2)+b^(1/2)*a*d)*a^3)*a))
 

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 490, normalized size of antiderivative = 1.67 \[ \int x^2 (c+d x)^2 \sqrt {a x+b x^2} \, dx=\left [\frac {15 \, {\left (40 \, a^{4} b^{2} c^{2} - 56 \, a^{5} b c d + 21 \, a^{6} d^{2}\right )} \sqrt {b} \log \left (2 \, b x + a - 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right ) + 2 \, {\left (1280 \, b^{6} d^{2} x^{5} + 600 \, a^{3} b^{3} c^{2} - 840 \, a^{4} b^{2} c d + 315 \, a^{5} b d^{2} + 128 \, {\left (24 \, b^{6} c d + a b^{5} d^{2}\right )} x^{4} + 48 \, {\left (40 \, b^{6} c^{2} + 8 \, a b^{5} c d - 3 \, a^{2} b^{4} d^{2}\right )} x^{3} + 8 \, {\left (40 \, a b^{5} c^{2} - 56 \, a^{2} b^{4} c d + 21 \, a^{3} b^{3} d^{2}\right )} x^{2} - 10 \, {\left (40 \, a^{2} b^{4} c^{2} - 56 \, a^{3} b^{3} c d + 21 \, a^{4} b^{2} d^{2}\right )} x\right )} \sqrt {b x^{2} + a x}}{15360 \, b^{6}}, \frac {15 \, {\left (40 \, a^{4} b^{2} c^{2} - 56 \, a^{5} b c d + 21 \, a^{6} d^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x^{2} + a x} \sqrt {-b}}{b x + a}\right ) + {\left (1280 \, b^{6} d^{2} x^{5} + 600 \, a^{3} b^{3} c^{2} - 840 \, a^{4} b^{2} c d + 315 \, a^{5} b d^{2} + 128 \, {\left (24 \, b^{6} c d + a b^{5} d^{2}\right )} x^{4} + 48 \, {\left (40 \, b^{6} c^{2} + 8 \, a b^{5} c d - 3 \, a^{2} b^{4} d^{2}\right )} x^{3} + 8 \, {\left (40 \, a b^{5} c^{2} - 56 \, a^{2} b^{4} c d + 21 \, a^{3} b^{3} d^{2}\right )} x^{2} - 10 \, {\left (40 \, a^{2} b^{4} c^{2} - 56 \, a^{3} b^{3} c d + 21 \, a^{4} b^{2} d^{2}\right )} x\right )} \sqrt {b x^{2} + a x}}{7680 \, b^{6}}\right ] \] Input:

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

Output:

[1/15360*(15*(40*a^4*b^2*c^2 - 56*a^5*b*c*d + 21*a^6*d^2)*sqrt(b)*log(2*b* 
x + a - 2*sqrt(b*x^2 + a*x)*sqrt(b)) + 2*(1280*b^6*d^2*x^5 + 600*a^3*b^3*c 
^2 - 840*a^4*b^2*c*d + 315*a^5*b*d^2 + 128*(24*b^6*c*d + a*b^5*d^2)*x^4 + 
48*(40*b^6*c^2 + 8*a*b^5*c*d - 3*a^2*b^4*d^2)*x^3 + 8*(40*a*b^5*c^2 - 56*a 
^2*b^4*c*d + 21*a^3*b^3*d^2)*x^2 - 10*(40*a^2*b^4*c^2 - 56*a^3*b^3*c*d + 2 
1*a^4*b^2*d^2)*x)*sqrt(b*x^2 + a*x))/b^6, 1/7680*(15*(40*a^4*b^2*c^2 - 56* 
a^5*b*c*d + 21*a^6*d^2)*sqrt(-b)*arctan(sqrt(b*x^2 + a*x)*sqrt(-b)/(b*x + 
a)) + (1280*b^6*d^2*x^5 + 600*a^3*b^3*c^2 - 840*a^4*b^2*c*d + 315*a^5*b*d^ 
2 + 128*(24*b^6*c*d + a*b^5*d^2)*x^4 + 48*(40*b^6*c^2 + 8*a*b^5*c*d - 3*a^ 
2*b^4*d^2)*x^3 + 8*(40*a*b^5*c^2 - 56*a^2*b^4*c*d + 21*a^3*b^3*d^2)*x^2 - 
10*(40*a^2*b^4*c^2 - 56*a^3*b^3*c*d + 21*a^4*b^2*d^2)*x)*sqrt(b*x^2 + a*x) 
)/b^6]
 

Sympy [A] (verification not implemented)

Time = 0.48 (sec) , antiderivative size = 406, normalized size of antiderivative = 1.38 \[ \int x^2 (c+d x)^2 \sqrt {a x+b x^2} \, dx=\begin {cases} - \frac {5 a^{3} \left (a c^{2} - \frac {7 a \left (2 a c d - \frac {9 a \left (\frac {a d^{2}}{12} + 2 b c d\right )}{10 b} + b c^{2}\right )}{8 b}\right ) \left (\begin {cases} \frac {\log {\left (a + 2 \sqrt {b} \sqrt {a x + b x^{2}} + 2 b x \right )}}{\sqrt {b}} & \text {for}\: \frac {a^{2}}{b} \neq 0 \\\frac {\left (\frac {a}{2 b} + x\right ) \log {\left (\frac {a}{2 b} + x \right )}}{\sqrt {b \left (\frac {a}{2 b} + x\right )^{2}}} & \text {otherwise} \end {cases}\right )}{16 b^{3}} + \sqrt {a x + b x^{2}} \cdot \left (\frac {5 a^{2} \left (a c^{2} - \frac {7 a \left (2 a c d - \frac {9 a \left (\frac {a d^{2}}{12} + 2 b c d\right )}{10 b} + b c^{2}\right )}{8 b}\right )}{8 b^{3}} - \frac {5 a x \left (a c^{2} - \frac {7 a \left (2 a c d - \frac {9 a \left (\frac {a d^{2}}{12} + 2 b c d\right )}{10 b} + b c^{2}\right )}{8 b}\right )}{12 b^{2}} + \frac {d^{2} x^{5}}{6} + \frac {x^{4} \left (\frac {a d^{2}}{12} + 2 b c d\right )}{5 b} + \frac {x^{3} \cdot \left (2 a c d - \frac {9 a \left (\frac {a d^{2}}{12} + 2 b c d\right )}{10 b} + b c^{2}\right )}{4 b} + \frac {x^{2} \left (a c^{2} - \frac {7 a \left (2 a c d - \frac {9 a \left (\frac {a d^{2}}{12} + 2 b c d\right )}{10 b} + b c^{2}\right )}{8 b}\right )}{3 b}\right ) & \text {for}\: b \neq 0 \\\frac {2 \left (\frac {c^{2} \left (a x\right )^{\frac {7}{2}}}{7} + \frac {2 c d \left (a x\right )^{\frac {9}{2}}}{9 a} + \frac {d^{2} \left (a x\right )^{\frac {11}{2}}}{11 a^{2}}\right )}{a^{3}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \] Input:

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 427, normalized size of antiderivative = 1.45 \[ \int x^2 (c+d x)^2 \sqrt {a x+b x^2} \, dx=\frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} d^{2} x^{3}}{6 \, b} + \frac {2 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} c d x^{2}}{5 \, b} - \frac {3 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a d^{2} x^{2}}{20 \, b^{2}} + \frac {5 \, \sqrt {b x^{2} + a x} a^{2} c^{2} x}{32 \, b^{2}} + \frac {{\left (b x^{2} + a x\right )}^{\frac {3}{2}} c^{2} x}{4 \, b} - \frac {7 \, \sqrt {b x^{2} + a x} a^{3} c d x}{32 \, b^{3}} - \frac {7 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a c d x}{20 \, b^{2}} + \frac {21 \, \sqrt {b x^{2} + a x} a^{4} d^{2} x}{256 \, b^{4}} + \frac {21 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{2} d^{2} x}{160 \, b^{3}} - \frac {5 \, a^{4} c^{2} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{128 \, b^{\frac {7}{2}}} + \frac {7 \, a^{5} c d \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{128 \, b^{\frac {9}{2}}} - \frac {21 \, a^{6} d^{2} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right )}{1024 \, b^{\frac {11}{2}}} + \frac {5 \, \sqrt {b x^{2} + a x} a^{3} c^{2}}{64 \, b^{3}} - \frac {5 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a c^{2}}{24 \, b^{2}} - \frac {7 \, \sqrt {b x^{2} + a x} a^{4} c d}{64 \, b^{4}} + \frac {7 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{2} c d}{24 \, b^{3}} + \frac {21 \, \sqrt {b x^{2} + a x} a^{5} d^{2}}{512 \, b^{5}} - \frac {7 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} a^{3} d^{2}}{64 \, b^{4}} \] Input:

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

Output:

1/6*(b*x^2 + a*x)^(3/2)*d^2*x^3/b + 2/5*(b*x^2 + a*x)^(3/2)*c*d*x^2/b - 3/ 
20*(b*x^2 + a*x)^(3/2)*a*d^2*x^2/b^2 + 5/32*sqrt(b*x^2 + a*x)*a^2*c^2*x/b^ 
2 + 1/4*(b*x^2 + a*x)^(3/2)*c^2*x/b - 7/32*sqrt(b*x^2 + a*x)*a^3*c*d*x/b^3 
 - 7/20*(b*x^2 + a*x)^(3/2)*a*c*d*x/b^2 + 21/256*sqrt(b*x^2 + a*x)*a^4*d^2 
*x/b^4 + 21/160*(b*x^2 + a*x)^(3/2)*a^2*d^2*x/b^3 - 5/128*a^4*c^2*log(2*b* 
x + a + 2*sqrt(b*x^2 + a*x)*sqrt(b))/b^(7/2) + 7/128*a^5*c*d*log(2*b*x + a 
 + 2*sqrt(b*x^2 + a*x)*sqrt(b))/b^(9/2) - 21/1024*a^6*d^2*log(2*b*x + a + 
2*sqrt(b*x^2 + a*x)*sqrt(b))/b^(11/2) + 5/64*sqrt(b*x^2 + a*x)*a^3*c^2/b^3 
 - 5/24*(b*x^2 + a*x)^(3/2)*a*c^2/b^2 - 7/64*sqrt(b*x^2 + a*x)*a^4*c*d/b^4 
 + 7/24*(b*x^2 + a*x)^(3/2)*a^2*c*d/b^3 + 21/512*sqrt(b*x^2 + a*x)*a^5*d^2 
/b^5 - 7/64*(b*x^2 + a*x)^(3/2)*a^3*d^2/b^4
 

Giac [A] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.87 \[ \int x^2 (c+d x)^2 \sqrt {a x+b x^2} \, dx=\frac {1}{7680} \, \sqrt {b x^{2} + a x} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, d^{2} x + \frac {24 \, b^{5} c d + a b^{4} d^{2}}{b^{5}}\right )} x + \frac {3 \, {\left (40 \, b^{5} c^{2} + 8 \, a b^{4} c d - 3 \, a^{2} b^{3} d^{2}\right )}}{b^{5}}\right )} x + \frac {40 \, a b^{4} c^{2} - 56 \, a^{2} b^{3} c d + 21 \, a^{3} b^{2} d^{2}}{b^{5}}\right )} x - \frac {5 \, {\left (40 \, a^{2} b^{3} c^{2} - 56 \, a^{3} b^{2} c d + 21 \, a^{4} b d^{2}\right )}}{b^{5}}\right )} x + \frac {15 \, {\left (40 \, a^{3} b^{2} c^{2} - 56 \, a^{4} b c d + 21 \, a^{5} d^{2}\right )}}{b^{5}}\right )} + \frac {{\left (40 \, a^{4} b^{2} c^{2} - 56 \, a^{5} b c d + 21 \, a^{6} d^{2}\right )} \log \left ({\left | 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} \sqrt {b} + a \right |}\right )}{1024 \, b^{\frac {11}{2}}} \] Input:

integrate(x^2*(d*x+c)^2*(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^2*x + (24*b^5*c*d + a*b^4*d^2)/ 
b^5)*x + 3*(40*b^5*c^2 + 8*a*b^4*c*d - 3*a^2*b^3*d^2)/b^5)*x + (40*a*b^4*c 
^2 - 56*a^2*b^3*c*d + 21*a^3*b^2*d^2)/b^5)*x - 5*(40*a^2*b^3*c^2 - 56*a^3* 
b^2*c*d + 21*a^4*b*d^2)/b^5)*x + 15*(40*a^3*b^2*c^2 - 56*a^4*b*c*d + 21*a^ 
5*d^2)/b^5) + 1/1024*(40*a^4*b^2*c^2 - 56*a^5*b*c*d + 21*a^6*d^2)*log(abs( 
2*(sqrt(b)*x - sqrt(b*x^2 + a*x))*sqrt(b) + a))/b^(11/2)
 

Mupad [B] (verification not implemented)

Time = 9.27 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.26 \[ \int x^2 (c+d x)^2 \sqrt {a x+b x^2} \, dx=\frac {d^2\,x^3\,{\left (b\,x^2+a\,x\right )}^{3/2}}{6\,b}-\frac {5\,a\,c^2\,\left (\frac {a^3\,\ln \left (\frac {a+2\,b\,x}{\sqrt {b}}+2\,\sqrt {b\,x^2+a\,x}\right )}{16\,b^{5/2}}+\frac {\sqrt {b\,x^2+a\,x}\,\left (-3\,a^2+2\,a\,b\,x+8\,b^2\,x^2\right )}{24\,b^2}\right )}{8\,b}+\frac {3\,a\,d^2\,\left (\frac {7\,a\,\left (\frac {x\,{\left (b\,x^2+a\,x\right )}^{3/2}}{4\,b}-\frac {5\,a\,\left (\frac {a^3\,\ln \left (\frac {a+2\,b\,x}{\sqrt {b}}+2\,\sqrt {b\,x^2+a\,x}\right )}{16\,b^{5/2}}+\frac {\sqrt {b\,x^2+a\,x}\,\left (-3\,a^2+2\,a\,b\,x+8\,b^2\,x^2\right )}{24\,b^2}\right )}{8\,b}\right )}{10\,b}-\frac {x^2\,{\left (b\,x^2+a\,x\right )}^{3/2}}{5\,b}\right )}{4\,b}+\frac {c^2\,x\,{\left (b\,x^2+a\,x\right )}^{3/2}}{4\,b}-\frac {7\,a\,c\,d\,\left (\frac {x\,{\left (b\,x^2+a\,x\right )}^{3/2}}{4\,b}-\frac {5\,a\,\left (\frac {a^3\,\ln \left (\frac {a+2\,b\,x}{\sqrt {b}}+2\,\sqrt {b\,x^2+a\,x}\right )}{16\,b^{5/2}}+\frac {\sqrt {b\,x^2+a\,x}\,\left (-3\,a^2+2\,a\,b\,x+8\,b^2\,x^2\right )}{24\,b^2}\right )}{8\,b}\right )}{5\,b}+\frac {2\,c\,d\,x^2\,{\left (b\,x^2+a\,x\right )}^{3/2}}{5\,b} \] Input:

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

Output:

(d^2*x^3*(a*x + b*x^2)^(3/2))/(6*b) - (5*a*c^2*((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) + (3*a*d^2*((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) + (c^2*x*(a 
*x + b*x^2)^(3/2))/(4*b) - (7*a*c*d*((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)))/(5*b) 
+ (2*c*d*x^2*(a*x + b*x^2)^(3/2))/(5*b)
 

Reduce [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 386, normalized size of antiderivative = 1.31 \[ \int x^2 (c+d x)^2 \sqrt {a x+b x^2} \, dx=\frac {315 \sqrt {x}\, \sqrt {b x +a}\, a^{5} b \,d^{2}-840 \sqrt {x}\, \sqrt {b x +a}\, a^{4} b^{2} c d -210 \sqrt {x}\, \sqrt {b x +a}\, a^{4} b^{2} d^{2} x +600 \sqrt {x}\, \sqrt {b x +a}\, a^{3} b^{3} c^{2}+560 \sqrt {x}\, \sqrt {b x +a}\, a^{3} b^{3} c d x +168 \sqrt {x}\, \sqrt {b x +a}\, a^{3} b^{3} d^{2} x^{2}-400 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b^{4} c^{2} x -448 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b^{4} c d \,x^{2}-144 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b^{4} d^{2} x^{3}+320 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{5} c^{2} x^{2}+384 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{5} c d \,x^{3}+128 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{5} d^{2} x^{4}+1920 \sqrt {x}\, \sqrt {b x +a}\, b^{6} c^{2} x^{3}+3072 \sqrt {x}\, \sqrt {b x +a}\, b^{6} c d \,x^{4}+1280 \sqrt {x}\, \sqrt {b x +a}\, b^{6} d^{2} x^{5}-315 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a^{6} d^{2}+840 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a^{5} b c d -600 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a^{4} b^{2} c^{2}}{7680 b^{6}} \] Input:

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

Output:

(315*sqrt(x)*sqrt(a + b*x)*a**5*b*d**2 - 840*sqrt(x)*sqrt(a + b*x)*a**4*b* 
*2*c*d - 210*sqrt(x)*sqrt(a + b*x)*a**4*b**2*d**2*x + 600*sqrt(x)*sqrt(a + 
 b*x)*a**3*b**3*c**2 + 560*sqrt(x)*sqrt(a + b*x)*a**3*b**3*c*d*x + 168*sqr 
t(x)*sqrt(a + b*x)*a**3*b**3*d**2*x**2 - 400*sqrt(x)*sqrt(a + b*x)*a**2*b* 
*4*c**2*x - 448*sqrt(x)*sqrt(a + b*x)*a**2*b**4*c*d*x**2 - 144*sqrt(x)*sqr 
t(a + b*x)*a**2*b**4*d**2*x**3 + 320*sqrt(x)*sqrt(a + b*x)*a*b**5*c**2*x** 
2 + 384*sqrt(x)*sqrt(a + b*x)*a*b**5*c*d*x**3 + 128*sqrt(x)*sqrt(a + b*x)* 
a*b**5*d**2*x**4 + 1920*sqrt(x)*sqrt(a + b*x)*b**6*c**2*x**3 + 3072*sqrt(x 
)*sqrt(a + b*x)*b**6*c*d*x**4 + 1280*sqrt(x)*sqrt(a + b*x)*b**6*d**2*x**5 
- 315*sqrt(b)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a))*a**6*d**2 + 8 
40*sqrt(b)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a))*a**5*b*c*d - 600 
*sqrt(b)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a))*a**4*b**2*c**2)/(7 
680*b**6)