\(\int (b d+2 c d x)^4 (a+b x+c x^2)^{5/2} \, dx\) [178]

Optimal result

Integrand size = 26, antiderivative size = 249 \[ \int (b d+2 c d x)^4 \left (a+b x+c x^2\right )^{5/2} \, dx=-\frac {3 \left (b^2-4 a c\right )^4 d^4 (b+2 c x) \sqrt {a+b x+c x^2}}{8192 c^3}-\frac {\left (b^2-4 a c\right )^3 d^4 (b+2 c x)^3 \sqrt {a+b x+c x^2}}{4096 c^3}+\frac {\left (b^2-4 a c\right )^2 d^4 (b+2 c x)^5 \sqrt {a+b x+c x^2}}{1024 c^3}-\frac {\left (b^2-4 a c\right ) d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{3/2}}{128 c^2}+\frac {d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{5/2}}{20 c}-\frac {3 \left (b^2-4 a c\right )^5 d^4 \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{16384 c^{7/2}} \] Output:

-3/8192*(-4*a*c+b^2)^4*d^4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c^3-1/4096*(-4*a* 
c+b^2)^3*d^4*(2*c*x+b)^3*(c*x^2+b*x+a)^(1/2)/c^3+1/1024*(-4*a*c+b^2)^2*d^4 
*(2*c*x+b)^5*(c*x^2+b*x+a)^(1/2)/c^3-1/128*(-4*a*c+b^2)*d^4*(2*c*x+b)^5*(c 
*x^2+b*x+a)^(3/2)/c^2+1/20*d^4*(2*c*x+b)^5*(c*x^2+b*x+a)^(5/2)/c-3/16384*( 
-4*a*c+b^2)^5*d^4*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 = 4.31 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.27 \[ \int (b d+2 c d x)^4 \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {d^4 \left (\sqrt {c} (b+2 c x) \sqrt {a+x (b+c x)} \left (15 b^8-40 b^7 c x+8 b^6 c \left (-35 a+11 c x^2\right )+32 b^5 c^2 x \left (23 a+360 c x^2\right )+128 b^3 c^3 x \left (233 a^2+1184 a c x^2+1288 c^2 x^4\right )+32 b^4 c^2 \left (64 a^2+1047 a c x^2+2084 c^2 x^4\right )+512 b c^4 x \left (5 a^3+248 a^2 c x^2+504 a c^2 x^4+256 c^3 x^6\right )+128 b^2 c^3 \left (35 a^3+729 a^2 c x^2+2272 a c^2 x^4+1624 c^3 x^6\right )+256 c^4 \left (-15 a^4+10 a^3 c x^2+248 a^2 c^2 x^4+336 a c^3 x^6+128 c^4 x^8\right )\right )-15 \left (b^2-4 a c\right )^5 \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {a}+\sqrt {a+x (b+c x)}}\right )\right )}{40960 c^{7/2}} \] Input:

Integrate[(b*d + 2*c*d*x)^4*(a + b*x + c*x^2)^(5/2),x]
 

Output:

(d^4*(Sqrt[c]*(b + 2*c*x)*Sqrt[a + x*(b + c*x)]*(15*b^8 - 40*b^7*c*x + 8*b 
^6*c*(-35*a + 11*c*x^2) + 32*b^5*c^2*x*(23*a + 360*c*x^2) + 128*b^3*c^3*x* 
(233*a^2 + 1184*a*c*x^2 + 1288*c^2*x^4) + 32*b^4*c^2*(64*a^2 + 1047*a*c*x^ 
2 + 2084*c^2*x^4) + 512*b*c^4*x*(5*a^3 + 248*a^2*c*x^2 + 504*a*c^2*x^4 + 2 
56*c^3*x^6) + 128*b^2*c^3*(35*a^3 + 729*a^2*c*x^2 + 2272*a*c^2*x^4 + 1624* 
c^3*x^6) + 256*c^4*(-15*a^4 + 10*a^3*c*x^2 + 248*a^2*c^2*x^4 + 336*a*c^3*x 
^6 + 128*c^4*x^8)) - 15*(b^2 - 4*a*c)^5*ArcTanh[(Sqrt[c]*x)/(-Sqrt[a] + Sq 
rt[a + x*(b + c*x)])]))/(40960*c^(7/2))
 

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {1109, 27, 1109, 1109, 1116, 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.

\(\displaystyle \int \left (a+b x+c x^2\right )^{5/2} (b d+2 c d x)^4 \, dx\)

\(\Big \downarrow \) 1109

\(\displaystyle \frac {d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{5/2}}{20 c}-\frac {\left (b^2-4 a c\right ) \int d^4 (b+2 c x)^4 \left (c x^2+b x+a\right )^{3/2}dx}{8 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{5/2}}{20 c}-\frac {d^4 \left (b^2-4 a c\right ) \int (b+2 c x)^4 \left (c x^2+b x+a\right )^{3/2}dx}{8 c}\)

\(\Big \downarrow \) 1109

\(\displaystyle \frac {d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{5/2}}{20 c}-\frac {d^4 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x)^5 \left (a+b x+c x^2\right )^{3/2}}{16 c}-\frac {3 \left (b^2-4 a c\right ) \int (b+2 c x)^4 \sqrt {c x^2+b x+a}dx}{32 c}\right )}{8 c}\)

\(\Big \downarrow \) 1109

\(\displaystyle \frac {d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{5/2}}{20 c}-\frac {d^4 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x)^5 \left (a+b x+c x^2\right )^{3/2}}{16 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x)^5 \sqrt {a+b x+c x^2}}{12 c}-\frac {\left (b^2-4 a c\right ) \int \frac {(b+2 c x)^4}{\sqrt {c x^2+b x+a}}dx}{24 c}\right )}{32 c}\right )}{8 c}\)

\(\Big \downarrow \) 1116

\(\displaystyle \frac {d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{5/2}}{20 c}-\frac {d^4 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x)^5 \left (a+b x+c x^2\right )^{3/2}}{16 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x)^5 \sqrt {a+b x+c x^2}}{12 c}-\frac {\left (b^2-4 a c\right ) \left (\frac {3}{4} \left (b^2-4 a c\right ) \int \frac {(b+2 c x)^2}{\sqrt {c x^2+b x+a}}dx+\frac {1}{2} (b+2 c x)^3 \sqrt {a+b x+c x^2}\right )}{24 c}\right )}{32 c}\right )}{8 c}\)

\(\Big \downarrow \) 1116

\(\displaystyle \frac {d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{5/2}}{20 c}-\frac {d^4 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x)^5 \left (a+b x+c x^2\right )^{3/2}}{16 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x)^5 \sqrt {a+b x+c x^2}}{12 c}-\frac {\left (b^2-4 a c\right ) \left (\frac {3}{4} \left (b^2-4 a c\right ) \left (\frac {1}{2} \left (b^2-4 a c\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx+(b+2 c x) \sqrt {a+b x+c x^2}\right )+\frac {1}{2} (b+2 c x)^3 \sqrt {a+b x+c x^2}\right )}{24 c}\right )}{32 c}\right )}{8 c}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{5/2}}{20 c}-\frac {d^4 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x)^5 \left (a+b x+c x^2\right )^{3/2}}{16 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x)^5 \sqrt {a+b x+c x^2}}{12 c}-\frac {\left (b^2-4 a c\right ) \left (\frac {3}{4} \left (b^2-4 a c\right ) \left (\left (b^2-4 a c\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}+(b+2 c x) \sqrt {a+b x+c x^2}\right )+\frac {1}{2} (b+2 c x)^3 \sqrt {a+b x+c x^2}\right )}{24 c}\right )}{32 c}\right )}{8 c}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {d^4 (b+2 c x)^5 \left (a+b x+c x^2\right )^{5/2}}{20 c}-\frac {d^4 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x)^5 \left (a+b x+c x^2\right )^{3/2}}{16 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x)^5 \sqrt {a+b x+c x^2}}{12 c}-\frac {\left (b^2-4 a c\right ) \left (\frac {3}{4} \left (b^2-4 a c\right ) \left (\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{2 \sqrt {c}}+(b+2 c x) \sqrt {a+b x+c x^2}\right )+\frac {1}{2} (b+2 c x)^3 \sqrt {a+b x+c x^2}\right )}{24 c}\right )}{32 c}\right )}{8 c}\)

Input:

Int[(b*d + 2*c*d*x)^4*(a + b*x + c*x^2)^(5/2),x]
 

Output:

(d^4*(b + 2*c*x)^5*(a + b*x + c*x^2)^(5/2))/(20*c) - ((b^2 - 4*a*c)*d^4*(( 
(b + 2*c*x)^5*(a + b*x + c*x^2)^(3/2))/(16*c) - (3*(b^2 - 4*a*c)*(((b + 2* 
c*x)^5*Sqrt[a + b*x + c*x^2])/(12*c) - ((b^2 - 4*a*c)*(((b + 2*c*x)^3*Sqrt 
[a + b*x + c*x^2])/2 + (3*(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])))/4))/(24*c)))/(32*c)))/(8*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 = 1.18 (sec) , antiderivative size = 427, normalized size of antiderivative = 1.71

method	result	size
risch	\(-\frac {\left (-65536 c^{9} x^{9}-294912 b \,c^{8} x^{8}-172032 a \,c^{8} x^{7}-546816 b^{2} c^{7} x^{7}-602112 a b \,c^{7} x^{6}-537600 b^{3} c^{6} x^{6}-126976 a^{2} c^{7} x^{5}-839680 a \,b^{2} c^{6} x^{5}-298240 b^{4} c^{5} x^{5}-317440 a^{2} b \,c^{6} x^{4}-593920 a \,b^{3} c^{5} x^{4}-89728 b^{5} c^{4} x^{4}-5120 a^{3} c^{6} x^{3}-313600 a^{2} b^{2} c^{5} x^{3}-218560 a \,b^{4} c^{4} x^{3}-11696 b^{6} c^{3} x^{3}-7680 a^{3} b \,c^{5} x^{2}-152960 a^{2} b^{3} c^{4} x^{2}-34976 c^{3} a \,b^{5} x^{2}-8 b^{7} c^{2} x^{2}+7680 a^{4} c^{5} x -11520 a^{3} b^{2} c^{4} x -33920 c^{3} a^{2} b^{4} x -176 a \,b^{6} c^{2} x +10 b^{8} c x +3840 a^{4} b \,c^{4}-4480 a^{3} b^{3} c^{3}-2048 a^{2} b^{5} c^{2}+280 a \,b^{7} c -15 b^{9}\right ) \sqrt {c \,x^{2}+b x +a}\, d^{4}}{40960 c^{3}}+\frac {3 \left (1024 a^{5} c^{5}-1280 a^{4} b^{2} c^{4}+640 a^{3} b^{4} c^{3}-160 a^{2} b^{6} c^{2}+20 a \,b^{8} c -b^{10}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right ) d^{4}}{16384 c^{\frac {7}{2}}}\)	\(427\)
default	\(\text {Expression too large to display}\)	\(2130\)

Input:

int((2*c*d*x+b*d)^4*(c*x^2+b*x+a)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

-1/40960/c^3*(-65536*c^9*x^9-294912*b*c^8*x^8-172032*a*c^8*x^7-546816*b^2* 
c^7*x^7-602112*a*b*c^7*x^6-537600*b^3*c^6*x^6-126976*a^2*c^7*x^5-839680*a* 
b^2*c^6*x^5-298240*b^4*c^5*x^5-317440*a^2*b*c^6*x^4-593920*a*b^3*c^5*x^4-8 
9728*b^5*c^4*x^4-5120*a^3*c^6*x^3-313600*a^2*b^2*c^5*x^3-218560*a*b^4*c^4* 
x^3-11696*b^6*c^3*x^3-7680*a^3*b*c^5*x^2-152960*a^2*b^3*c^4*x^2-34976*a*b^ 
5*c^3*x^2-8*b^7*c^2*x^2+7680*a^4*c^5*x-11520*a^3*b^2*c^4*x-33920*a^2*b^4*c 
^3*x-176*a*b^6*c^2*x+10*b^8*c*x+3840*a^4*b*c^4-4480*a^3*b^3*c^3-2048*a^2*b 
^5*c^2+280*a*b^7*c-15*b^9)*(c*x^2+b*x+a)^(1/2)*d^4+3/16384*(1024*a^5*c^5-1 
280*a^4*b^2*c^4+640*a^3*b^4*c^3-160*a^2*b^6*c^2+20*a*b^8*c-b^10)/c^(7/2)*l 
n((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d^4
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 460 vs. \(2 (219) = 438\).

Time = 0.14 (sec) , antiderivative size = 923, normalized size of antiderivative = 3.71 \[ \int (b d+2 c d x)^4 \left (a+b x+c x^2\right )^{5/2} \, dx =\text {Too large to display} \] Input:

integrate((2*c*d*x+b*d)^4*(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")
 

Output:

[-1/163840*(15*(b^10 - 20*a*b^8*c + 160*a^2*b^6*c^2 - 640*a^3*b^4*c^3 + 12 
80*a^4*b^2*c^4 - 1024*a^5*c^5)*sqrt(c)*d^4*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*(65536*c^10*d^4 
*x^9 + 294912*b*c^9*d^4*x^8 + 6144*(89*b^2*c^8 + 28*a*c^9)*d^4*x^7 + 21504 
*(25*b^3*c^7 + 28*a*b*c^8)*d^4*x^6 + 256*(1165*b^4*c^6 + 3280*a*b^2*c^7 + 
496*a^2*c^8)*d^4*x^5 + 128*(701*b^5*c^5 + 4640*a*b^3*c^6 + 2480*a^2*b*c^7) 
*d^4*x^4 + 16*(731*b^6*c^4 + 13660*a*b^4*c^5 + 19600*a^2*b^2*c^6 + 320*a^3 
*c^7)*d^4*x^3 + 8*(b^7*c^3 + 4372*a*b^5*c^4 + 19120*a^2*b^3*c^5 + 960*a^3* 
b*c^6)*d^4*x^2 - 2*(5*b^8*c^2 - 88*a*b^6*c^3 - 16960*a^2*b^4*c^4 - 5760*a^ 
3*b^2*c^5 + 3840*a^4*c^6)*d^4*x + (15*b^9*c - 280*a*b^7*c^2 + 2048*a^2*b^5 
*c^3 + 4480*a^3*b^3*c^4 - 3840*a^4*b*c^5)*d^4)*sqrt(c*x^2 + b*x + a))/c^4, 
 1/81920*(15*(b^10 - 20*a*b^8*c + 160*a^2*b^6*c^2 - 640*a^3*b^4*c^3 + 1280 
*a^4*b^2*c^4 - 1024*a^5*c^5)*sqrt(-c)*d^4*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*(65536*c^10*d^4*x^9 + 2 
94912*b*c^9*d^4*x^8 + 6144*(89*b^2*c^8 + 28*a*c^9)*d^4*x^7 + 21504*(25*b^3 
*c^7 + 28*a*b*c^8)*d^4*x^6 + 256*(1165*b^4*c^6 + 3280*a*b^2*c^7 + 496*a^2* 
c^8)*d^4*x^5 + 128*(701*b^5*c^5 + 4640*a*b^3*c^6 + 2480*a^2*b*c^7)*d^4*x^4 
 + 16*(731*b^6*c^4 + 13660*a*b^4*c^5 + 19600*a^2*b^2*c^6 + 320*a^3*c^7)*d^ 
4*x^3 + 8*(b^7*c^3 + 4372*a*b^5*c^4 + 19120*a^2*b^3*c^5 + 960*a^3*b*c^6)*d 
^4*x^2 - 2*(5*b^8*c^2 - 88*a*b^6*c^3 - 16960*a^2*b^4*c^4 - 5760*a^3*b^2...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 8670 vs. \(2 (238) = 476\).

Time = 0.91 (sec) , antiderivative size = 8670, normalized size of antiderivative = 34.82 \[ \int (b d+2 c d x)^4 \left (a+b x+c x^2\right )^{5/2} \, dx=\text {Too large to display} \] Input:

integrate((2*c*d*x+b*d)**4*(c*x**2+b*x+a)**(5/2),x)
 

Output:

Piecewise((sqrt(a + b*x + c*x**2)*(36*b*c**5*d**4*x**8/5 + 8*c**6*d**4*x** 
9/5 + x**7*(168*a*c**6*d**4/5 + 534*b**2*c**5*d**4/5)/(8*c) + x**6*(672*a* 
b*c**5*d**4/5 + 192*b**3*c**4*d**4 - 15*b*(168*a*c**6*d**4/5 + 534*b**2*c* 
*5*d**4/5)/(16*c))/(7*c) + x**5*(48*a**2*c**5*d**4 + 312*a*b**2*c**4*d**4 
- 7*a*(168*a*c**6*d**4/5 + 534*b**2*c**5*d**4/5)/(8*c) + 129*b**4*c**3*d** 
4 - 13*b*(672*a*b*c**5*d**4/5 + 192*b**3*c**4*d**4 - 15*b*(168*a*c**6*d**4 
/5 + 534*b**2*c**5*d**4/5)/(16*c))/(14*c))/(6*c) + x**4*(144*a**2*b*c**4*d 
**4 + 264*a*b**3*c**3*d**4 - 6*a*(672*a*b*c**5*d**4/5 + 192*b**3*c**4*d**4 
 - 15*b*(168*a*c**6*d**4/5 + 534*b**2*c**5*d**4/5)/(16*c))/(7*c) + 51*b**5 
*c**2*d**4 - 11*b*(48*a**2*c**5*d**4 + 312*a*b**2*c**4*d**4 - 7*a*(168*a*c 
**6*d**4/5 + 534*b**2*c**5*d**4/5)/(8*c) + 129*b**4*c**3*d**4 - 13*b*(672* 
a*b*c**5*d**4/5 + 192*b**3*c**4*d**4 - 15*b*(168*a*c**6*d**4/5 + 534*b**2* 
c**5*d**4/5)/(16*c))/(14*c))/(12*c))/(5*c) + x**3*(16*a**3*c**4*d**4 + 168 
*a**2*b**2*c**3*d**4 + 123*a*b**4*c**2*d**4 - 5*a*(48*a**2*c**5*d**4 + 312 
*a*b**2*c**4*d**4 - 7*a*(168*a*c**6*d**4/5 + 534*b**2*c**5*d**4/5)/(8*c) + 
 129*b**4*c**3*d**4 - 13*b*(672*a*b*c**5*d**4/5 + 192*b**3*c**4*d**4 - 15* 
b*(168*a*c**6*d**4/5 + 534*b**2*c**5*d**4/5)/(16*c))/(14*c))/(6*c) + 11*b* 
*6*c*d**4 - 9*b*(144*a**2*b*c**4*d**4 + 264*a*b**3*c**3*d**4 - 6*a*(672*a* 
b*c**5*d**4/5 + 192*b**3*c**4*d**4 - 15*b*(168*a*c**6*d**4/5 + 534*b**2*c* 
*5*d**4/5)/(16*c))/(7*c) + 51*b**5*c**2*d**4 - 11*b*(48*a**2*c**5*d**4 ...
 

Maxima [F(-2)]

Exception generated. \[ \int (b d+2 c d x)^4 \left (a+b x+c x^2\right )^{5/2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((2*c*d*x+b*d)^4*(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. 545 vs. \(2 (219) = 438\).

Time = 0.41 (sec) , antiderivative size = 545, normalized size of antiderivative = 2.19 \[ \int (b d+2 c d x)^4 \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {1}{40960} \, \sqrt {c x^{2} + b x + a} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, {\left (4 \, {\left (2 \, {\left (16 \, {\left (2 \, c^{6} d^{4} x + 9 \, b c^{5} d^{4}\right )} x + \frac {3 \, {\left (89 \, b^{2} c^{13} d^{4} + 28 \, a c^{14} d^{4}\right )}}{c^{9}}\right )} x + \frac {21 \, {\left (25 \, b^{3} c^{12} d^{4} + 28 \, a b c^{13} d^{4}\right )}}{c^{9}}\right )} x + \frac {1165 \, b^{4} c^{11} d^{4} + 3280 \, a b^{2} c^{12} d^{4} + 496 \, a^{2} c^{13} d^{4}}{c^{9}}\right )} x + \frac {701 \, b^{5} c^{10} d^{4} + 4640 \, a b^{3} c^{11} d^{4} + 2480 \, a^{2} b c^{12} d^{4}}{c^{9}}\right )} x + \frac {731 \, b^{6} c^{9} d^{4} + 13660 \, a b^{4} c^{10} d^{4} + 19600 \, a^{2} b^{2} c^{11} d^{4} + 320 \, a^{3} c^{12} d^{4}}{c^{9}}\right )} x + \frac {b^{7} c^{8} d^{4} + 4372 \, a b^{5} c^{9} d^{4} + 19120 \, a^{2} b^{3} c^{10} d^{4} + 960 \, a^{3} b c^{11} d^{4}}{c^{9}}\right )} x - \frac {5 \, b^{8} c^{7} d^{4} - 88 \, a b^{6} c^{8} d^{4} - 16960 \, a^{2} b^{4} c^{9} d^{4} - 5760 \, a^{3} b^{2} c^{10} d^{4} + 3840 \, a^{4} c^{11} d^{4}}{c^{9}}\right )} x + \frac {15 \, b^{9} c^{6} d^{4} - 280 \, a b^{7} c^{7} d^{4} + 2048 \, a^{2} b^{5} c^{8} d^{4} + 4480 \, a^{3} b^{3} c^{9} d^{4} - 3840 \, a^{4} b c^{10} d^{4}}{c^{9}}\right )} + \frac {3 \, {\left (b^{10} d^{4} - 20 \, a b^{8} c d^{4} + 160 \, a^{2} b^{6} c^{2} d^{4} - 640 \, a^{3} b^{4} c^{3} d^{4} + 1280 \, a^{4} b^{2} c^{4} d^{4} - 1024 \, a^{5} c^{5} d^{4}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{16384 \, c^{\frac {7}{2}}} \] Input:

integrate((2*c*d*x+b*d)^4*(c*x^2+b*x+a)^(5/2),x, algorithm="giac")
 

Output:

1/40960*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(2*(4*(2*(16*(2*c^6*d^4*x + 9*b* 
c^5*d^4)*x + 3*(89*b^2*c^13*d^4 + 28*a*c^14*d^4)/c^9)*x + 21*(25*b^3*c^12* 
d^4 + 28*a*b*c^13*d^4)/c^9)*x + (1165*b^4*c^11*d^4 + 3280*a*b^2*c^12*d^4 + 
 496*a^2*c^13*d^4)/c^9)*x + (701*b^5*c^10*d^4 + 4640*a*b^3*c^11*d^4 + 2480 
*a^2*b*c^12*d^4)/c^9)*x + (731*b^6*c^9*d^4 + 13660*a*b^4*c^10*d^4 + 19600* 
a^2*b^2*c^11*d^4 + 320*a^3*c^12*d^4)/c^9)*x + (b^7*c^8*d^4 + 4372*a*b^5*c^ 
9*d^4 + 19120*a^2*b^3*c^10*d^4 + 960*a^3*b*c^11*d^4)/c^9)*x - (5*b^8*c^7*d 
^4 - 88*a*b^6*c^8*d^4 - 16960*a^2*b^4*c^9*d^4 - 5760*a^3*b^2*c^10*d^4 + 38 
40*a^4*c^11*d^4)/c^9)*x + (15*b^9*c^6*d^4 - 280*a*b^7*c^7*d^4 + 2048*a^2*b 
^5*c^8*d^4 + 4480*a^3*b^3*c^9*d^4 - 3840*a^4*b*c^10*d^4)/c^9) + 3/16384*(b 
^10*d^4 - 20*a*b^8*c*d^4 + 160*a^2*b^6*c^2*d^4 - 640*a^3*b^4*c^3*d^4 + 128 
0*a^4*b^2*c^4*d^4 - 1024*a^5*c^5*d^4)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + 
b*x + a))*sqrt(c) + b))/c^(7/2)
 

Mupad [F(-1)]

Timed out. \[ \int (b d+2 c d x)^4 \left (a+b x+c x^2\right )^{5/2} \, dx=\int {\left (b\,d+2\,c\,d\,x\right )}^4\,{\left (c\,x^2+b\,x+a\right )}^{5/2} \,d x \] Input:

int((b*d + 2*c*d*x)^4*(a + b*x + c*x^2)^(5/2),x)
 

Output:

int((b*d + 2*c*d*x)^4*(a + b*x + c*x^2)^(5/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.41 (sec) , antiderivative size = 942, normalized size of antiderivative = 3.78 \[ \int (b d+2 c d x)^4 \left (a+b x+c x^2\right )^{5/2} \, dx =\text {Too large to display} \] Input:

int((2*c*d*x+b*d)^4*(c*x^2+b*x+a)^(5/2),x)
 

Output:

(d**4*( - 7680*sqrt(a + b*x + c*x**2)*a**4*b*c**5 - 15360*sqrt(a + b*x + c 
*x**2)*a**4*c**6*x + 8960*sqrt(a + b*x + c*x**2)*a**3*b**3*c**4 + 23040*sq 
rt(a + b*x + c*x**2)*a**3*b**2*c**5*x + 15360*sqrt(a + b*x + c*x**2)*a**3* 
b*c**6*x**2 + 10240*sqrt(a + b*x + c*x**2)*a**3*c**7*x**3 + 4096*sqrt(a + 
b*x + c*x**2)*a**2*b**5*c**3 + 67840*sqrt(a + b*x + c*x**2)*a**2*b**4*c**4 
*x + 305920*sqrt(a + b*x + c*x**2)*a**2*b**3*c**5*x**2 + 627200*sqrt(a + b 
*x + c*x**2)*a**2*b**2*c**6*x**3 + 634880*sqrt(a + b*x + c*x**2)*a**2*b*c* 
*7*x**4 + 253952*sqrt(a + b*x + c*x**2)*a**2*c**8*x**5 - 560*sqrt(a + b*x 
+ c*x**2)*a*b**7*c**2 + 352*sqrt(a + b*x + c*x**2)*a*b**6*c**3*x + 69952*s 
qrt(a + b*x + c*x**2)*a*b**5*c**4*x**2 + 437120*sqrt(a + b*x + c*x**2)*a*b 
**4*c**5*x**3 + 1187840*sqrt(a + b*x + c*x**2)*a*b**3*c**6*x**4 + 1679360* 
sqrt(a + b*x + c*x**2)*a*b**2*c**7*x**5 + 1204224*sqrt(a + b*x + c*x**2)*a 
*b*c**8*x**6 + 344064*sqrt(a + b*x + c*x**2)*a*c**9*x**7 + 30*sqrt(a + b*x 
 + c*x**2)*b**9*c - 20*sqrt(a + b*x + c*x**2)*b**8*c**2*x + 16*sqrt(a + b* 
x + c*x**2)*b**7*c**3*x**2 + 23392*sqrt(a + b*x + c*x**2)*b**6*c**4*x**3 + 
 179456*sqrt(a + b*x + c*x**2)*b**5*c**5*x**4 + 596480*sqrt(a + b*x + c*x* 
*2)*b**4*c**6*x**5 + 1075200*sqrt(a + b*x + c*x**2)*b**3*c**7*x**6 + 10936 
32*sqrt(a + b*x + c*x**2)*b**2*c**8*x**7 + 589824*sqrt(a + b*x + c*x**2)*b 
*c**9*x**8 + 131072*sqrt(a + b*x + c*x**2)*c**10*x**9 + 15360*sqrt(c)*log( 
(2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*a**5...