\(\int (b d+2 c d x)^2 (a+b x+c x^2)^{5/2} \, dx\) [180]

Optimal result

Integrand size = 26, antiderivative size = 207 \[ \int (b d+2 c d x)^2 \left (a+b x+c x^2\right )^{5/2} \, dx=-\frac {5 \left (b^2-4 a c\right )^3 d^2 (b+2 c x) \sqrt {a+b x+c x^2}}{4096 c^3}+\frac {5 \left (b^2-4 a c\right )^2 d^2 (b+2 c x)^3 \sqrt {a+b x+c x^2}}{2048 c^3}-\frac {5 \left (b^2-4 a c\right ) d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}}{384 c^2}+\frac {d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{5/2}}{16 c}-\frac {5 \left (b^2-4 a c\right )^4 d^2 \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8192 c^{7/2}} \] Output:

-5/4096*(-4*a*c+b^2)^3*d^2*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c^3+5/2048*(-4*a* 
c+b^2)^2*d^2*(2*c*x+b)^3*(c*x^2+b*x+a)^(1/2)/c^3-5/384*(-4*a*c+b^2)*d^2*(2 
*c*x+b)^3*(c*x^2+b*x+a)^(3/2)/c^2+1/16*d^2*(2*c*x+b)^3*(c*x^2+b*x+a)^(5/2) 
/c-5/8192*(-4*a*c+b^2)^4*d^2*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^( 
1/2))/c^(7/2)
 

Mathematica [A] (verified)

Time = 2.53 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.08 \[ \int (b d+2 c d x)^2 \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {d^2 \left (\sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)} \left (15 b^6-40 b^5 c x+44 b^4 c \left (-5 a+2 c x^2\right )+64 b^3 c^2 x \left (9 a+52 c x^2\right )+128 b c^3 x \left (59 a^2+136 a c x^2+72 c^2 x^4\right )+16 b^2 c^2 \left (73 a^2+580 a c x^2+584 c^2 x^4\right )+64 c^3 \left (15 a^3+118 a^2 c x^2+136 a c^2 x^4+48 c^3 x^6\right )\right )-15 \left (b^2-4 a c\right )^4 \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {a}+\sqrt {a+x (b+c x)}}\right )\right )}{12288 c^{7/2}} \] Input:

Integrate[(b*d + 2*c*d*x)^2*(a + b*x + c*x^2)^(5/2),x]
 

Output:

(d^2*(Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)]*(15*b^6 - 40*b^5*c*x + 44* 
b^4*c*(-5*a + 2*c*x^2) + 64*b^3*c^2*x*(9*a + 52*c*x^2) + 128*b*c^3*x*(59*a 
^2 + 136*a*c*x^2 + 72*c^2*x^4) + 16*b^2*c^2*(73*a^2 + 580*a*c*x^2 + 584*c^ 
2*x^4) + 64*c^3*(15*a^3 + 118*a^2*c*x^2 + 136*a*c^2*x^4 + 48*c^3*x^6)) - 1 
5*(b^2 - 4*a*c)^4*ArcTanh[(Sqrt[c]*x)/(-Sqrt[a] + Sqrt[a + x*(b + c*x)])]) 
)/(12288*c^(7/2))
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {1109, 27, 1109, 1109, 1116, 1092, 219}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \left (a+b x+c x^2\right )^{5/2} (b d+2 c d x)^2 \, dx\)

\(\Big \downarrow \) 1109

\(\displaystyle \frac {d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{5/2}}{16 c}-\frac {5 \left (b^2-4 a c\right ) \int d^2 (b+2 c x)^2 \left (c x^2+b x+a\right )^{3/2}dx}{32 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{5/2}}{16 c}-\frac {5 d^2 \left (b^2-4 a c\right ) \int (b+2 c x)^2 \left (c x^2+b x+a\right )^{3/2}dx}{32 c}\)

\(\Big \downarrow \) 1109

\(\displaystyle \frac {d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{5/2}}{16 c}-\frac {5 d^2 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}}{12 c}-\frac {\left (b^2-4 a c\right ) \int (b+2 c x)^2 \sqrt {c x^2+b x+a}dx}{8 c}\right )}{32 c}\)

\(\Big \downarrow \) 1109

\(\displaystyle \frac {d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{5/2}}{16 c}-\frac {5 d^2 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}}{12 c}-\frac {\left (b^2-4 a c\right ) \left (\frac {(b+2 c x)^3 \sqrt {a+b x+c x^2}}{8 c}-\frac {\left (b^2-4 a c\right ) \int \frac {(b+2 c x)^2}{\sqrt {c x^2+b x+a}}dx}{16 c}\right )}{8 c}\right )}{32 c}\)

\(\Big \downarrow \) 1116

\(\displaystyle \frac {d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{5/2}}{16 c}-\frac {5 d^2 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}}{12 c}-\frac {\left (b^2-4 a c\right ) \left (\frac {(b+2 c x)^3 \sqrt {a+b x+c x^2}}{8 c}-\frac {\left (b^2-4 a c\right ) \left (\frac {1}{2} \left (b^2-4 a c\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx+(b+2 c x) \sqrt {a+b x+c x^2}\right )}{16 c}\right )}{8 c}\right )}{32 c}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{5/2}}{16 c}-\frac {5 d^2 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}}{12 c}-\frac {\left (b^2-4 a c\right ) \left (\frac {(b+2 c x)^3 \sqrt {a+b x+c x^2}}{8 c}-\frac {\left (b^2-4 a c\right ) \left (\left (b^2-4 a c\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}+(b+2 c x) \sqrt {a+b x+c x^2}\right )}{16 c}\right )}{8 c}\right )}{32 c}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {d^2 (b+2 c x)^3 \left (a+b x+c x^2\right )^{5/2}}{16 c}-\frac {5 d^2 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x)^3 \left (a+b x+c x^2\right )^{3/2}}{12 c}-\frac {\left (b^2-4 a c\right ) \left (\frac {(b+2 c x)^3 \sqrt {a+b x+c x^2}}{8 c}-\frac {\left (b^2-4 a c\right ) \left (\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {c}}+(b+2 c x) \sqrt {a+b x+c x^2}\right )}{16 c}\right )}{8 c}\right )}{32 c}\)

Input:

Int[(b*d + 2*c*d*x)^2*(a + b*x + c*x^2)^(5/2),x]
 

Output:

(d^2*(b + 2*c*x)^3*(a + b*x + c*x^2)^(5/2))/(16*c) - (5*(b^2 - 4*a*c)*d^2* 
(((b + 2*c*x)^3*(a + b*x + c*x^2)^(3/2))/(12*c) - ((b^2 - 4*a*c)*(((b + 2* 
c*x)^3*Sqrt[a + b*x + c*x^2])/(8*c) - ((b^2 - 4*a*c)*((b + 2*c*x)*Sqrt[a + 
 b*x + c*x^2] + ((b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x 
 + c*x^2])])/(2*Sqrt[c])))/(16*c)))/(8*c)))/(32*c)
 

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 1092

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[I 
nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a 
, b, c}, x]
 

rule 1109

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_S 
ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*x + c*x^2)^p/(e*(m + 2*p + 1))), x 
] - Simp[d*p*((b^2 - 4*a*c)/(b*e*(m + 2*p + 1)))   Int[(d + e*x)^m*(a + b*x 
 + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[2*c*d - b* 
e, 0] && NeQ[m + 2*p + 3, 0] && GtQ[p, 0] &&  !LtQ[m, -1] &&  !(IGtQ[(m - 1 
)/2, 0] && ( !IntegerQ[p] || LtQ[m, 2*p])) && RationalQ[m] && IntegerQ[2*p]
 

rule 1116

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_S 
ymbol] :> Simp[2*d*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(b*(m + 2*p 
 + 1))), x] + Simp[d^2*(m - 1)*((b^2 - 4*a*c)/(b^2*(m + 2*p + 1)))   Int[(d 
 + e*x)^(m - 2)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] 
 && EqQ[2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && GtQ[m, 1] && NeQ[m + 2*p 
+ 1, 0] && (IntegerQ[2*p] || (IntegerQ[m] && RationalQ[p]) || OddQ[m])
 

Maple [A] (verified)

Time = 0.99 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.42

method	result	size
risch	\(\frac {\left (6144 c^{7} x^{7}+21504 b \,c^{6} x^{6}+17408 a \,c^{6} x^{5}+27904 b^{2} c^{5} x^{5}+43520 a b \,c^{5} x^{4}+16000 b^{3} c^{4} x^{4}+15104 a^{2} c^{5} x^{3}+35968 a \,b^{2} c^{4} x^{3}+3504 b^{4} c^{3} x^{3}+22656 a^{2} b \,c^{4} x^{2}+10432 a \,b^{3} c^{3} x^{2}+8 b^{5} c^{2} x^{2}+1920 a^{3} c^{4} x +9888 a^{2} b^{2} c^{3} x +136 a \,b^{4} c^{2} x -10 b^{6} c x +960 a^{3} b \,c^{3}+1168 a^{2} b^{3} c^{2}-220 a \,b^{5} c +15 b^{7}\right ) \sqrt {c \,x^{2}+b x +a}\, d^{2}}{12288 c^{3}}-\frac {5 \left (256 a^{4} c^{4}-256 a^{3} b^{2} c^{3}+96 a^{2} b^{4} c^{2}-16 a \,b^{6} c +b^{8}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) d^{2}}{8192 c^{\frac {7}{2}}}\)	\(294\)
default	\(d^{2} \left (b^{2} \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{12 c}+\frac {5 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )+4 c^{2} \left (\frac {x \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}{8 c}-\frac {9 b \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}{7 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{12 c}+\frac {5 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )}{2 c}\right )}{16 c}-\frac {a \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{12 c}+\frac {5 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )}{8 c}\right )+4 b c \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {7}{2}}}{7 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}{12 c}+\frac {5 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{8 c}+\frac {3 \left (4 a c -b^{2}\right ) \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )}{2 c}\right )\right )\)	\(666\)

Input:

int((2*c*d*x+b*d)^2*(c*x^2+b*x+a)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

1/12288/c^3*(6144*c^7*x^7+21504*b*c^6*x^6+17408*a*c^6*x^5+27904*b^2*c^5*x^ 
5+43520*a*b*c^5*x^4+16000*b^3*c^4*x^4+15104*a^2*c^5*x^3+35968*a*b^2*c^4*x^ 
3+3504*b^4*c^3*x^3+22656*a^2*b*c^4*x^2+10432*a*b^3*c^3*x^2+8*b^5*c^2*x^2+1 
920*a^3*c^4*x+9888*a^2*b^2*c^3*x+136*a*b^4*c^2*x-10*b^6*c*x+960*a^3*b*c^3+ 
1168*a^2*b^3*c^2-220*a*b^5*c+15*b^7)*(c*x^2+b*x+a)^(1/2)*d^2-5/8192*(256*a 
^4*c^4-256*a^3*b^2*c^3+96*a^2*b^4*c^2-16*a*b^6*c+b^8)/c^(7/2)*ln((1/2*b+c* 
x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d^2
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 675, normalized size of antiderivative = 3.26 \[ \int (b d+2 c d x)^2 \left (a+b x+c x^2\right )^{5/2} \, dx=\left [\frac {15 \, {\left (b^{8} - 16 \, a b^{6} c + 96 \, a^{2} b^{4} c^{2} - 256 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}\right )} \sqrt {c} d^{2} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} + 4 \, \sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {c} - 4 \, a c\right ) + 4 \, {\left (6144 \, c^{8} d^{2} x^{7} + 21504 \, b c^{7} d^{2} x^{6} + 256 \, {\left (109 \, b^{2} c^{6} + 68 \, a c^{7}\right )} d^{2} x^{5} + 640 \, {\left (25 \, b^{3} c^{5} + 68 \, a b c^{6}\right )} d^{2} x^{4} + 16 \, {\left (219 \, b^{4} c^{4} + 2248 \, a b^{2} c^{5} + 944 \, a^{2} c^{6}\right )} d^{2} x^{3} + 8 \, {\left (b^{5} c^{3} + 1304 \, a b^{3} c^{4} + 2832 \, a^{2} b c^{5}\right )} d^{2} x^{2} - 2 \, {\left (5 \, b^{6} c^{2} - 68 \, a b^{4} c^{3} - 4944 \, a^{2} b^{2} c^{4} - 960 \, a^{3} c^{5}\right )} d^{2} x + {\left (15 \, b^{7} c - 220 \, a b^{5} c^{2} + 1168 \, a^{2} b^{3} c^{3} + 960 \, a^{3} b c^{4}\right )} d^{2}\right )} \sqrt {c x^{2} + b x + a}}{49152 \, c^{4}}, \frac {15 \, {\left (b^{8} - 16 \, a b^{6} c + 96 \, a^{2} b^{4} c^{2} - 256 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}\right )} \sqrt {-c} d^{2} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + 2 \, {\left (6144 \, c^{8} d^{2} x^{7} + 21504 \, b c^{7} d^{2} x^{6} + 256 \, {\left (109 \, b^{2} c^{6} + 68 \, a c^{7}\right )} d^{2} x^{5} + 640 \, {\left (25 \, b^{3} c^{5} + 68 \, a b c^{6}\right )} d^{2} x^{4} + 16 \, {\left (219 \, b^{4} c^{4} + 2248 \, a b^{2} c^{5} + 944 \, a^{2} c^{6}\right )} d^{2} x^{3} + 8 \, {\left (b^{5} c^{3} + 1304 \, a b^{3} c^{4} + 2832 \, a^{2} b c^{5}\right )} d^{2} x^{2} - 2 \, {\left (5 \, b^{6} c^{2} - 68 \, a b^{4} c^{3} - 4944 \, a^{2} b^{2} c^{4} - 960 \, a^{3} c^{5}\right )} d^{2} x + {\left (15 \, b^{7} c - 220 \, a b^{5} c^{2} + 1168 \, a^{2} b^{3} c^{3} + 960 \, a^{3} b c^{4}\right )} d^{2}\right )} \sqrt {c x^{2} + b x + a}}{24576 \, c^{4}}\right ] \] Input:

integrate((2*c*d*x+b*d)^2*(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")
 

Output:

[1/49152*(15*(b^8 - 16*a*b^6*c + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 + 256*a^ 
4*c^4)*sqrt(c)*d^2*log(-8*c^2*x^2 - 8*b*c*x - b^2 + 4*sqrt(c*x^2 + b*x + a 
)*(2*c*x + b)*sqrt(c) - 4*a*c) + 4*(6144*c^8*d^2*x^7 + 21504*b*c^7*d^2*x^6 
 + 256*(109*b^2*c^6 + 68*a*c^7)*d^2*x^5 + 640*(25*b^3*c^5 + 68*a*b*c^6)*d^ 
2*x^4 + 16*(219*b^4*c^4 + 2248*a*b^2*c^5 + 944*a^2*c^6)*d^2*x^3 + 8*(b^5*c 
^3 + 1304*a*b^3*c^4 + 2832*a^2*b*c^5)*d^2*x^2 - 2*(5*b^6*c^2 - 68*a*b^4*c^ 
3 - 4944*a^2*b^2*c^4 - 960*a^3*c^5)*d^2*x + (15*b^7*c - 220*a*b^5*c^2 + 11 
68*a^2*b^3*c^3 + 960*a^3*b*c^4)*d^2)*sqrt(c*x^2 + b*x + a))/c^4, 1/24576*( 
15*(b^8 - 16*a*b^6*c + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 + 256*a^4*c^4)*sqr 
t(-c)*d^2*arctan(1/2*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(-c)/(c^2*x^2 + 
 b*c*x + a*c)) + 2*(6144*c^8*d^2*x^7 + 21504*b*c^7*d^2*x^6 + 256*(109*b^2* 
c^6 + 68*a*c^7)*d^2*x^5 + 640*(25*b^3*c^5 + 68*a*b*c^6)*d^2*x^4 + 16*(219* 
b^4*c^4 + 2248*a*b^2*c^5 + 944*a^2*c^6)*d^2*x^3 + 8*(b^5*c^3 + 1304*a*b^3* 
c^4 + 2832*a^2*b*c^5)*d^2*x^2 - 2*(5*b^6*c^2 - 68*a*b^4*c^3 - 4944*a^2*b^2 
*c^4 - 960*a^3*c^5)*d^2*x + (15*b^7*c - 220*a*b^5*c^2 + 1168*a^2*b^3*c^3 + 
 960*a^3*b*c^4)*d^2)*sqrt(c*x^2 + b*x + a))/c^4]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2958 vs. \(2 (202) = 404\).

Time = 0.71 (sec) , antiderivative size = 2958, normalized size of antiderivative = 14.29 \[ \int (b d+2 c d x)^2 \left (a+b x+c x^2\right )^{5/2} \, dx=\text {Too large to display} \] Input:

integrate((2*c*d*x+b*d)**2*(c*x**2+b*x+a)**(5/2),x)
 

Output:

Piecewise((sqrt(a + b*x + c*x**2)*(7*b*c**3*d**2*x**6/4 + c**4*d**2*x**7/2 
 + x**5*(17*a*c**4*d**2/2 + 109*b**2*c**3*d**2/8)/(6*c) + x**4*(51*a*b*c** 
3*d**2/2 + 19*b**3*c**2*d**2 - 11*b*(17*a*c**4*d**2/2 + 109*b**2*c**3*d**2 
/8)/(12*c))/(5*c) + x**3*(12*a**2*c**3*d**2 + 39*a*b**2*c**2*d**2 - 5*a*(1 
7*a*c**4*d**2/2 + 109*b**2*c**3*d**2/8)/(6*c) + 7*b**4*c*d**2 - 9*b*(51*a* 
b*c**3*d**2/2 + 19*b**3*c**2*d**2 - 11*b*(17*a*c**4*d**2/2 + 109*b**2*c**3 
*d**2/8)/(12*c))/(10*c))/(4*c) + x**2*(24*a**2*b*c**2*d**2 + 18*a*b**3*c*d 
**2 - 4*a*(51*a*b*c**3*d**2/2 + 19*b**3*c**2*d**2 - 11*b*(17*a*c**4*d**2/2 
 + 109*b**2*c**3*d**2/8)/(12*c))/(5*c) + b**5*d**2 - 7*b*(12*a**2*c**3*d** 
2 + 39*a*b**2*c**2*d**2 - 5*a*(17*a*c**4*d**2/2 + 109*b**2*c**3*d**2/8)/(6 
*c) + 7*b**4*c*d**2 - 9*b*(51*a*b*c**3*d**2/2 + 19*b**3*c**2*d**2 - 11*b*( 
17*a*c**4*d**2/2 + 109*b**2*c**3*d**2/8)/(12*c))/(10*c))/(8*c))/(3*c) + x* 
(4*a**3*c**2*d**2 + 15*a**2*b**2*c*d**2 + 3*a*b**4*d**2 - 3*a*(12*a**2*c** 
3*d**2 + 39*a*b**2*c**2*d**2 - 5*a*(17*a*c**4*d**2/2 + 109*b**2*c**3*d**2/ 
8)/(6*c) + 7*b**4*c*d**2 - 9*b*(51*a*b*c**3*d**2/2 + 19*b**3*c**2*d**2 - 1 
1*b*(17*a*c**4*d**2/2 + 109*b**2*c**3*d**2/8)/(12*c))/(10*c))/(4*c) - 5*b* 
(24*a**2*b*c**2*d**2 + 18*a*b**3*c*d**2 - 4*a*(51*a*b*c**3*d**2/2 + 19*b** 
3*c**2*d**2 - 11*b*(17*a*c**4*d**2/2 + 109*b**2*c**3*d**2/8)/(12*c))/(5*c) 
 + b**5*d**2 - 7*b*(12*a**2*c**3*d**2 + 39*a*b**2*c**2*d**2 - 5*a*(17*a*c* 
*4*d**2/2 + 109*b**2*c**3*d**2/8)/(6*c) + 7*b**4*c*d**2 - 9*b*(51*a*b*c...
 

Maxima [F(-2)]

Exception generated. \[ \int (b d+2 c d x)^2 \left (a+b x+c x^2\right )^{5/2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((2*c*d*x+b*d)^2*(c*x^2+b*x+a)^(5/2),x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for 
 more deta
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 387 vs. \(2 (181) = 362\).

Time = 0.35 (sec) , antiderivative size = 387, normalized size of antiderivative = 1.87 \[ \int (b d+2 c d x)^2 \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {1}{12288} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, {\left (12 \, {\left (2 \, c^{4} d^{2} x + 7 \, b c^{3} d^{2}\right )} x + \frac {109 \, b^{2} c^{9} d^{2} + 68 \, a c^{10} d^{2}}{c^{7}}\right )} x + \frac {5 \, {\left (25 \, b^{3} c^{8} d^{2} + 68 \, a b c^{9} d^{2}\right )}}{c^{7}}\right )} x + \frac {219 \, b^{4} c^{7} d^{2} + 2248 \, a b^{2} c^{8} d^{2} + 944 \, a^{2} c^{9} d^{2}}{c^{7}}\right )} x + \frac {b^{5} c^{6} d^{2} + 1304 \, a b^{3} c^{7} d^{2} + 2832 \, a^{2} b c^{8} d^{2}}{c^{7}}\right )} x - \frac {5 \, b^{6} c^{5} d^{2} - 68 \, a b^{4} c^{6} d^{2} - 4944 \, a^{2} b^{2} c^{7} d^{2} - 960 \, a^{3} c^{8} d^{2}}{c^{7}}\right )} x + \frac {15 \, b^{7} c^{4} d^{2} - 220 \, a b^{5} c^{5} d^{2} + 1168 \, a^{2} b^{3} c^{6} d^{2} + 960 \, a^{3} b c^{7} d^{2}}{c^{7}}\right )} + \frac {5 \, {\left (b^{8} d^{2} - 16 \, a b^{6} c d^{2} + 96 \, a^{2} b^{4} c^{2} d^{2} - 256 \, a^{3} b^{2} c^{3} d^{2} + 256 \, a^{4} c^{4} d^{2}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{8192 \, c^{\frac {7}{2}}} \] Input:

integrate((2*c*d*x+b*d)^2*(c*x^2+b*x+a)^(5/2),x, algorithm="giac")
 

Output:

1/12288*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(2*(12*(2*c^4*d^2*x + 7*b*c^3*d^ 
2)*x + (109*b^2*c^9*d^2 + 68*a*c^10*d^2)/c^7)*x + 5*(25*b^3*c^8*d^2 + 68*a 
*b*c^9*d^2)/c^7)*x + (219*b^4*c^7*d^2 + 2248*a*b^2*c^8*d^2 + 944*a^2*c^9*d 
^2)/c^7)*x + (b^5*c^6*d^2 + 1304*a*b^3*c^7*d^2 + 2832*a^2*b*c^8*d^2)/c^7)* 
x - (5*b^6*c^5*d^2 - 68*a*b^4*c^6*d^2 - 4944*a^2*b^2*c^7*d^2 - 960*a^3*c^8 
*d^2)/c^7)*x + (15*b^7*c^4*d^2 - 220*a*b^5*c^5*d^2 + 1168*a^2*b^3*c^6*d^2 
+ 960*a^3*b*c^7*d^2)/c^7) + 5/8192*(b^8*d^2 - 16*a*b^6*c*d^2 + 96*a^2*b^4* 
c^2*d^2 - 256*a^3*b^2*c^3*d^2 + 256*a^4*c^4*d^2)*log(abs(2*(sqrt(c)*x - sq 
rt(c*x^2 + b*x + a))*sqrt(c) + b))/c^(7/2)
 

Mupad [F(-1)]

Timed out. \[ \int (b d+2 c d x)^2 \left (a+b x+c x^2\right )^{5/2} \, dx=\int {\left (b\,d+2\,c\,d\,x\right )}^2\,{\left (c\,x^2+b\,x+a\right )}^{5/2} \,d x \] Input:

int((b*d + 2*c*d*x)^2*(a + b*x + c*x^2)^(5/2),x)
 

Output:

int((b*d + 2*c*d*x)^2*(a + b*x + c*x^2)^(5/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 663, normalized size of antiderivative = 3.20 \[ \int (b d+2 c d x)^2 \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {d^{2} \left (2336 \sqrt {c \,x^{2}+b x +a}\, a^{2} b^{3} c^{3}+30208 \sqrt {c \,x^{2}+b x +a}\, a^{2} c^{6} x^{3}-440 \sqrt {c \,x^{2}+b x +a}\, a \,b^{5} c^{2}+34816 \sqrt {c \,x^{2}+b x +a}\, a \,c^{7} x^{5}-20 \sqrt {c \,x^{2}+b x +a}\, b^{6} c^{2} x +16 \sqrt {c \,x^{2}+b x +a}\, b^{5} c^{3} x^{2}+7008 \sqrt {c \,x^{2}+b x +a}\, b^{4} c^{4} x^{3}+32000 \sqrt {c \,x^{2}+b x +a}\, b^{3} c^{5} x^{4}+55808 \sqrt {c \,x^{2}+b x +a}\, b^{2} c^{6} x^{5}+43008 \sqrt {c \,x^{2}+b x +a}\, b \,c^{7} x^{6}-3840 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) a^{4} c^{4}+1920 \sqrt {c \,x^{2}+b x +a}\, a^{3} b \,c^{4}+3840 \sqrt {c \,x^{2}+b x +a}\, a^{3} c^{5} x +19776 \sqrt {c \,x^{2}+b x +a}\, a^{2} b^{2} c^{4} x +45312 \sqrt {c \,x^{2}+b x +a}\, a^{2} b \,c^{5} x^{2}+272 \sqrt {c \,x^{2}+b x +a}\, a \,b^{4} c^{3} x +20864 \sqrt {c \,x^{2}+b x +a}\, a \,b^{3} c^{4} x^{2}+71936 \sqrt {c \,x^{2}+b x +a}\, a \,b^{2} c^{5} x^{3}+87040 \sqrt {c \,x^{2}+b x +a}\, a b \,c^{6} x^{4}+3840 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) a^{3} b^{2} c^{3}-1440 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) a^{2} b^{4} c^{2}+240 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) a \,b^{6} c +30 \sqrt {c \,x^{2}+b x +a}\, b^{7} c +12288 \sqrt {c \,x^{2}+b x +a}\, c^{8} x^{7}-15 \sqrt {c}\, \mathrm {log}\left (\frac {2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) b^{8}\right )}{24576 c^{4}} \] Input:

int((2*c*d*x+b*d)^2*(c*x^2+b*x+a)^(5/2),x)
 

Output:

(d**2*(1920*sqrt(a + b*x + c*x**2)*a**3*b*c**4 + 3840*sqrt(a + b*x + c*x** 
2)*a**3*c**5*x + 2336*sqrt(a + b*x + c*x**2)*a**2*b**3*c**3 + 19776*sqrt(a 
 + b*x + c*x**2)*a**2*b**2*c**4*x + 45312*sqrt(a + b*x + c*x**2)*a**2*b*c* 
*5*x**2 + 30208*sqrt(a + b*x + c*x**2)*a**2*c**6*x**3 - 440*sqrt(a + b*x + 
 c*x**2)*a*b**5*c**2 + 272*sqrt(a + b*x + c*x**2)*a*b**4*c**3*x + 20864*sq 
rt(a + b*x + c*x**2)*a*b**3*c**4*x**2 + 71936*sqrt(a + b*x + c*x**2)*a*b** 
2*c**5*x**3 + 87040*sqrt(a + b*x + c*x**2)*a*b*c**6*x**4 + 34816*sqrt(a + 
b*x + c*x**2)*a*c**7*x**5 + 30*sqrt(a + b*x + c*x**2)*b**7*c - 20*sqrt(a + 
 b*x + c*x**2)*b**6*c**2*x + 16*sqrt(a + b*x + c*x**2)*b**5*c**3*x**2 + 70 
08*sqrt(a + b*x + c*x**2)*b**4*c**4*x**3 + 32000*sqrt(a + b*x + c*x**2)*b* 
*3*c**5*x**4 + 55808*sqrt(a + b*x + c*x**2)*b**2*c**6*x**5 + 43008*sqrt(a 
+ b*x + c*x**2)*b*c**7*x**6 + 12288*sqrt(a + b*x + c*x**2)*c**8*x**7 - 384 
0*sqrt(c)*log((2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - 
b**2))*a**4*c**4 + 3840*sqrt(c)*log((2*sqrt(c)*sqrt(a + b*x + c*x**2) + b 
+ 2*c*x)/sqrt(4*a*c - b**2))*a**3*b**2*c**3 - 1440*sqrt(c)*log((2*sqrt(c)* 
sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*a**2*b**4*c**2 + 2 
40*sqrt(c)*log((2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - 
 b**2))*a*b**6*c - 15*sqrt(c)*log((2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 
2*c*x)/sqrt(4*a*c - b**2))*b**8))/(24576*c**4)