\(\int (d+e x)^2 (a+b x+c x^2)^{5/2} \, dx\) [594]

Optimal result

Integrand size = 22, antiderivative size = 293 \[ \int (d+e x)^2 \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {5 \left (b^2-4 a c\right )^2 \left (\left (9 b^2-4 a c\right ) e^2+32 c d (c d-b e)\right ) (b+2 c x) \sqrt {a+b x+c x^2}}{16384 c^5}-\frac {5 \left (b^2-4 a c\right ) \left (\left (9 b^2-4 a c\right ) e^2+32 c d (c d-b e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{6144 c^4}+\frac {\left (\left (9 b^2-4 a c\right ) e^2+32 c d (c d-b e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{384 c^3}+\frac {e (32 c d-9 b e+14 c e x) \left (a+b x+c x^2\right )^{7/2}}{112 c^2}-\frac {5 \left (b^2-4 a c\right )^3 \left (\left (9 b^2-4 a c\right ) e^2+32 c d (c d-b e)\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{32768 c^{11/2}} \] Output:

5/16384*(-4*a*c+b^2)^2*((-4*a*c+9*b^2)*e^2+32*c*d*(-b*e+c*d))*(2*c*x+b)*(c 
*x^2+b*x+a)^(1/2)/c^5-5/6144*(-4*a*c+b^2)*((-4*a*c+9*b^2)*e^2+32*c*d*(-b*e 
+c*d))*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c^4+1/384*((-4*a*c+9*b^2)*e^2+32*c*d* 
(-b*e+c*d))*(2*c*x+b)*(c*x^2+b*x+a)^(5/2)/c^3+1/112*e*(14*c*e*x-9*b*e+32*c 
*d)*(c*x^2+b*x+a)^(7/2)/c^2-5/32768*(-4*a*c+b^2)^3*((-4*a*c+9*b^2)*e^2+32* 
c*d*(-b*e+c*d))*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(11/2 
)
 

Mathematica [A] (verified)

Time = 7.21 (sec) , antiderivative size = 541, normalized size of antiderivative = 1.85 \[ \int (d+e x)^2 \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {\sqrt {c} \sqrt {a+x (b+c x)} \left (945 b^7 e^2-210 b^6 c e (16 d+3 e x)+28 b^5 c \left (-375 a e^2+2 c \left (60 d^2+40 d e x+9 e^2 x^2\right )\right )+8 b^4 c^2 \left (7 a e (640 d+113 e x)-2 c x \left (140 d^2+112 d e x+27 e^2 x^2\right )\right )+16 b^3 c^2 \left (2359 a^2 e^2+8 c^2 x^2 \left (14 d^2+12 d e x+3 e^2 x^2\right )-4 a c \left (560 d^2+336 d e x+71 e^2 x^2\right )\right )+32 b^2 c^3 \left (-3 a^2 e (1232 d+199 e x)+12 a c x \left (56 d^2+40 d e x+9 e^2 x^2\right )+8 c^2 x^3 \left (378 d^2+592 d e x+243 e^2 x^2\right )\right )+64 b c^3 \left (-663 a^3 e^2+6 a^2 c \left (308 d^2+152 d e x+29 e^2 x^2\right )+16 c^3 x^4 \left (140 d^2+232 d e x+99 e^2 x^2\right )+8 a c^2 x^2 \left (546 d^2+788 d e x+307 e^2 x^2\right )\right )+128 c^4 \left (3 a^3 e (256 d+35 e x)+16 c^3 x^5 \left (28 d^2+48 d e x+21 e^2 x^2\right )+8 a c^2 x^3 \left (182 d^2+288 d e x+119 e^2 x^2\right )+2 a^2 c x \left (924 d^2+1152 d e x+413 e^2 x^2\right )\right )\right )-105 \left (b^2-4 a c\right )^3 \left (32 c^2 d^2+9 b^2 e^2-4 c e (8 b d+a e)\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{-\sqrt {a}+\sqrt {a+x (b+c x)}}\right )}{344064 c^{11/2}} \] Input:

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

Output:

(Sqrt[c]*Sqrt[a + x*(b + c*x)]*(945*b^7*e^2 - 210*b^6*c*e*(16*d + 3*e*x) + 
 28*b^5*c*(-375*a*e^2 + 2*c*(60*d^2 + 40*d*e*x + 9*e^2*x^2)) + 8*b^4*c^2*( 
7*a*e*(640*d + 113*e*x) - 2*c*x*(140*d^2 + 112*d*e*x + 27*e^2*x^2)) + 16*b 
^3*c^2*(2359*a^2*e^2 + 8*c^2*x^2*(14*d^2 + 12*d*e*x + 3*e^2*x^2) - 4*a*c*( 
560*d^2 + 336*d*e*x + 71*e^2*x^2)) + 32*b^2*c^3*(-3*a^2*e*(1232*d + 199*e* 
x) + 12*a*c*x*(56*d^2 + 40*d*e*x + 9*e^2*x^2) + 8*c^2*x^3*(378*d^2 + 592*d 
*e*x + 243*e^2*x^2)) + 64*b*c^3*(-663*a^3*e^2 + 6*a^2*c*(308*d^2 + 152*d*e 
*x + 29*e^2*x^2) + 16*c^3*x^4*(140*d^2 + 232*d*e*x + 99*e^2*x^2) + 8*a*c^2 
*x^2*(546*d^2 + 788*d*e*x + 307*e^2*x^2)) + 128*c^4*(3*a^3*e*(256*d + 35*e 
*x) + 16*c^3*x^5*(28*d^2 + 48*d*e*x + 21*e^2*x^2) + 8*a*c^2*x^3*(182*d^2 + 
 288*d*e*x + 119*e^2*x^2) + 2*a^2*c*x*(924*d^2 + 1152*d*e*x + 413*e^2*x^2) 
)) - 105*(b^2 - 4*a*c)^3*(32*c^2*d^2 + 9*b^2*e^2 - 4*c*e*(8*b*d + a*e))*Ar 
cTanh[(Sqrt[c]*x)/(-Sqrt[a] + Sqrt[a + x*(b + c*x)])])/(344064*c^(11/2))
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 264, normalized size of antiderivative = 0.90, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {1166, 27, 1160, 1087, 1087, 1087, 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.

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

\(\Big \downarrow \) 1166

\(\displaystyle \frac {\int \frac {1}{2} \left (16 c d^2-e (7 b d+2 a e)+9 e (2 c d-b e) x\right ) \left (c x^2+b x+a\right )^{5/2}dx}{8 c}+\frac {e (d+e x) \left (a+b x+c x^2\right )^{7/2}}{8 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \left (16 c d^2-7 b e d-2 a e^2+9 e (2 c d-b e) x\right ) \left (c x^2+b x+a\right )^{5/2}dx}{16 c}+\frac {e (d+e x) \left (a+b x+c x^2\right )^{7/2}}{8 c}\)

\(\Big \downarrow \) 1160

\(\displaystyle \frac {\frac {\left (-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right ) \int \left (c x^2+b x+a\right )^{5/2}dx}{2 c}+\frac {9 e \left (a+b x+c x^2\right )^{7/2} (2 c d-b e)}{7 c}}{16 c}+\frac {e (d+e x) \left (a+b x+c x^2\right )^{7/2}}{8 c}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {\frac {\left (-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{12 c}-\frac {5 \left (b^2-4 a c\right ) \int \left (c x^2+b x+a\right )^{3/2}dx}{24 c}\right )}{2 c}+\frac {9 e \left (a+b x+c x^2\right )^{7/2} (2 c d-b e)}{7 c}}{16 c}+\frac {e (d+e x) \left (a+b x+c x^2\right )^{7/2}}{8 c}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {\frac {\left (-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{12 c}-\frac {5 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \int \sqrt {c x^2+b x+a}dx}{16 c}\right )}{24 c}\right )}{2 c}+\frac {9 e \left (a+b x+c x^2\right )^{7/2} (2 c d-b e)}{7 c}}{16 c}+\frac {e (d+e x) \left (a+b x+c x^2\right )^{7/2}}{8 c}\)

\(\Big \downarrow \) 1087

\(\displaystyle \frac {\frac {\left (-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{12 c}-\frac {5 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{8 c}\right )}{16 c}\right )}{24 c}\right )}{2 c}+\frac {9 e \left (a+b x+c x^2\right )^{7/2} (2 c d-b e)}{7 c}}{16 c}+\frac {e (d+e x) \left (a+b x+c x^2\right )^{7/2}}{8 c}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {\frac {\left (-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{12 c}-\frac {5 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{4 c}\right )}{16 c}\right )}{24 c}\right )}{2 c}+\frac {9 e \left (a+b x+c x^2\right )^{7/2} (2 c d-b e)}{7 c}}{16 c}+\frac {e (d+e x) \left (a+b x+c x^2\right )^{7/2}}{8 c}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{5/2}}{12 c}-\frac {5 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{8 c}-\frac {3 \left (b^2-4 a c\right ) \left (\frac {(b+2 c x) \sqrt {a+b x+c x^2}}{4 c}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{3/2}}\right )}{16 c}\right )}{24 c}\right ) \left (-4 c e (a e+8 b d)+9 b^2 e^2+32 c^2 d^2\right )}{2 c}+\frac {9 e \left (a+b x+c x^2\right )^{7/2} (2 c d-b e)}{7 c}}{16 c}+\frac {e (d+e x) \left (a+b x+c x^2\right )^{7/2}}{8 c}\)

Input:

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

Output:

(e*(d + e*x)*(a + b*x + c*x^2)^(7/2))/(8*c) + ((9*e*(2*c*d - b*e)*(a + b*x 
 + c*x^2)^(7/2))/(7*c) + ((32*c^2*d^2 + 9*b^2*e^2 - 4*c*e*(8*b*d + a*e))*( 
((b + 2*c*x)*(a + b*x + c*x^2)^(5/2))/(12*c) - (5*(b^2 - 4*a*c)*(((b + 2*c 
*x)*(a + b*x + c*x^2)^(3/2))/(8*c) - (3*(b^2 - 4*a*c)*(((b + 2*c*x)*Sqrt[a 
 + b*x + c*x^2])/(4*c) - ((b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqr 
t[a + b*x + c*x^2])])/(8*c^(3/2))))/(16*c)))/(24*c)))/(2*c))/(16*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 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 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 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 1166

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[1/(c*(m + 2*p + 1))   Int[(d + e*x)^(m - 2)*Simp[c*d^2*(m 
+ 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]* 
(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && If[Ration 
alQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadrat 
icQ[a, b, c, d, e, m, p, x]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(660\) vs. \(2(267)=534\).

Time = 1.15 (sec) , antiderivative size = 661, normalized size of antiderivative = 2.26

method	result	size
default	\(d^{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 )+e^{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 )+2 d e \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 )\)	\(661\)
risch	\(-\frac {\left (-43008 c^{7} e^{2} x^{7}-101376 b \,c^{6} e^{2} x^{6}-98304 c^{7} d e \,x^{6}-121856 a \,c^{6} e^{2} x^{5}-62208 b^{2} c^{5} e^{2} x^{5}-237568 b \,c^{6} d e \,x^{5}-57344 c^{7} d^{2} x^{5}-157184 a b \,c^{5} e^{2} x^{4}-294912 a \,c^{6} d e \,x^{4}-384 b^{3} c^{4} e^{2} x^{4}-151552 b^{2} c^{5} d e \,x^{4}-143360 b \,c^{6} d^{2} x^{4}-105728 a^{2} c^{5} e^{2} x^{3}-3456 a \,b^{2} c^{4} e^{2} x^{3}-403456 a b \,c^{5} d e \,x^{3}-186368 d^{2} c^{6} a \,x^{3}+432 b^{4} c^{3} e^{2} x^{3}-1536 b^{3} c^{4} d e \,x^{3}-96768 b^{2} c^{5} d^{2} x^{3}-11136 a^{2} b \,c^{4} e^{2} x^{2}-294912 a^{2} c^{5} d e \,x^{2}+4544 a \,b^{3} c^{3} e^{2} x^{2}-15360 a \,c^{4} e d \,b^{2} x^{2}-279552 a b \,c^{5} d^{2} x^{2}-504 b^{5} c^{2} e^{2} x^{2}+1792 b^{4} c^{3} d e \,x^{2}-1792 b^{3} c^{4} d^{2} x^{2}-13440 a^{3} c^{4} e^{2} x +19104 a^{2} b^{2} c^{3} e^{2} x -58368 a^{2} b \,c^{4} d e x -236544 a^{2} c^{5} d^{2} x -6328 a \,b^{4} c^{2} e^{2} x +21504 a \,b^{3} c^{3} d e x -21504 a \,b^{2} c^{4} d^{2} x +630 b^{6} c \,e^{2} x -2240 b^{5} c^{2} d e x +2240 b^{4} c^{3} d^{2} x +42432 a^{3} b \,c^{3} e^{2}-98304 a^{3} c^{4} d e -37744 a^{2} b^{3} c^{2} e^{2}+118272 a^{2} b^{2} c^{3} d e -118272 a^{2} b \,c^{4} d^{2}+10500 a \,b^{5} c \,e^{2}-35840 a \,b^{4} c^{2} d e +35840 a \,c^{3} d^{2} b^{3}-945 b^{7} e^{2}+3360 b^{6} c d e -3360 b^{5} c^{2} d^{2}\right ) \sqrt {c \,x^{2}+b x +a}}{344064 c^{5}}-\frac {5 \left (256 a^{4} c^{4} e^{2}-768 a^{3} b^{2} c^{3} e^{2}+2048 a^{3} b \,c^{4} d e -2048 a^{3} c^{5} d^{2}+480 a^{2} b^{4} c^{2} e^{2}-1536 a^{2} b^{3} c^{3} d e +1536 a^{2} b^{2} c^{4} d^{2}-112 a \,b^{6} c \,e^{2}+384 a \,b^{5} c^{2} d e -384 a \,b^{4} c^{3} d^{2}+9 b^{8} e^{2}-32 b^{7} c d e +32 b^{6} c^{2} d^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{32768 c^{\frac {11}{2}}}\)	\(791\)

Input:

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

Output:

d^2*(1/12*(2*c*x+b)*(c*x^2+b*x+a)^(5/2)/c+5/24*(4*a*c-b^2)/c*(1/8*(2*c*x+b 
)*(c*x^2+b*x+a)^(3/2)/c+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1 
/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))) 
))+e^2*(1/8*x*(c*x^2+b*x+a)^(7/2)/c-9/16*b/c*(1/7*(c*x^2+b*x+a)^(7/2)/c-1/ 
2*b/c*(1/12*(2*c*x+b)*(c*x^2+b*x+a)^(5/2)/c+5/24*(4*a*c-b^2)/c*(1/8*(2*c*x 
+b)*(c*x^2+b*x+a)^(3/2)/c+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^ 
(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2) 
)))))-1/8*a/c*(1/12*(2*c*x+b)*(c*x^2+b*x+a)^(5/2)/c+5/24*(4*a*c-b^2)/c*(1/ 
8*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)*(c*x^2 
+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+ 
a)^(1/2))))))+2*d*e*(1/7*(c*x^2+b*x+a)^(7/2)/c-1/2*b/c*(1/12*(2*c*x+b)*(c* 
x^2+b*x+a)^(5/2)/c+5/24*(4*a*c-b^2)/c*(1/8*(2*c*x+b)*(c*x^2+b*x+a)^(3/2)/c 
+3/16*(4*a*c-b^2)/c*(1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/8*(4*a*c-b^2)/c 
^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 711 vs. \(2 (267) = 534\).

Time = 0.21 (sec) , antiderivative size = 1425, normalized size of antiderivative = 4.86 \[ \int (d+e x)^2 \left (a+b x+c x^2\right )^{5/2} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/1376256*(105*(32*(b^6*c^2 - 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5) 
*d^2 - 32*(b^7*c - 12*a*b^5*c^2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d*e + (9* 
b^8 - 112*a*b^6*c + 480*a^2*b^4*c^2 - 768*a^3*b^2*c^3 + 256*a^4*c^4)*e^2)* 
sqrt(c)*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*(43008*c^8*e^2*x^7 + 3072*(32*c^8*d*e + 33*b*c^7*e 
^2)*x^6 + 256*(224*c^8*d^2 + 928*b*c^7*d*e + (243*b^2*c^6 + 476*a*c^7)*e^2 
)*x^5 + 128*(1120*b*c^7*d^2 + 32*(37*b^2*c^6 + 72*a*c^7)*d*e + (3*b^3*c^5 
+ 1228*a*b*c^6)*e^2)*x^4 + 16*(224*(27*b^2*c^6 + 52*a*c^7)*d^2 + 32*(3*b^3 
*c^5 + 788*a*b*c^6)*d*e - (27*b^4*c^4 - 216*a*b^2*c^5 - 6608*a^2*c^6)*e^2) 
*x^3 + 224*(15*b^5*c^3 - 160*a*b^3*c^4 + 528*a^2*b*c^5)*d^2 - 32*(105*b^6* 
c^2 - 1120*a*b^4*c^3 + 3696*a^2*b^2*c^4 - 3072*a^3*c^5)*d*e + (945*b^7*c - 
 10500*a*b^5*c^2 + 37744*a^2*b^3*c^3 - 42432*a^3*b*c^4)*e^2 + 8*(224*(b^3* 
c^5 + 156*a*b*c^6)*d^2 - 32*(7*b^4*c^4 - 60*a*b^2*c^5 - 1152*a^2*c^6)*d*e 
+ (63*b^5*c^3 - 568*a*b^3*c^4 + 1392*a^2*b*c^5)*e^2)*x^2 - 2*(224*(5*b^4*c 
^4 - 48*a*b^2*c^5 - 528*a^2*c^6)*d^2 - 32*(35*b^5*c^3 - 336*a*b^3*c^4 + 91 
2*a^2*b*c^5)*d*e + (315*b^6*c^2 - 3164*a*b^4*c^3 + 9552*a^2*b^2*c^4 - 6720 
*a^3*c^5)*e^2)*x)*sqrt(c*x^2 + b*x + a))/c^6, 1/688128*(105*(32*(b^6*c^2 - 
 12*a*b^4*c^3 + 48*a^2*b^2*c^4 - 64*a^3*c^5)*d^2 - 32*(b^7*c - 12*a*b^5*c^ 
2 + 48*a^2*b^3*c^3 - 64*a^3*b*c^4)*d*e + (9*b^8 - 112*a*b^6*c + 480*a^2*b^ 
4*c^2 - 768*a^3*b^2*c^3 + 256*a^4*c^4)*e^2)*sqrt(-c)*arctan(1/2*sqrt(c*...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5671 vs. \(2 (284) = 568\).

Time = 1.09 (sec) , antiderivative size = 5671, normalized size of antiderivative = 19.35 \[ \int (d+e x)^2 \left (a+b x+c x^2\right )^{5/2} \, dx=\text {Too large to display} \] Input:

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

Output:

Piecewise((sqrt(a + b*x + c*x**2)*(c**2*e**2*x**7/8 + x**6*(33*b*c**2*e**2 
/16 + 2*c**3*d*e)/(7*c) + x**5*(17*a*c**2*e**2/8 + 3*b**2*c*e**2 + 6*b*c** 
2*d*e - 13*b*(33*b*c**2*e**2/16 + 2*c**3*d*e)/(14*c) + c**3*d**2)/(6*c) + 
x**4*(6*a*b*c*e**2 + 6*a*c**2*d*e - 6*a*(33*b*c**2*e**2/16 + 2*c**3*d*e)/( 
7*c) + b**3*e**2 + 6*b**2*c*d*e + 3*b*c**2*d**2 - 11*b*(17*a*c**2*e**2/8 + 
 3*b**2*c*e**2 + 6*b*c**2*d*e - 13*b*(33*b*c**2*e**2/16 + 2*c**3*d*e)/(14* 
c) + c**3*d**2)/(12*c))/(5*c) + x**3*(3*a**2*c*e**2 + 3*a*b**2*e**2 + 12*a 
*b*c*d*e + 3*a*c**2*d**2 - 5*a*(17*a*c**2*e**2/8 + 3*b**2*c*e**2 + 6*b*c** 
2*d*e - 13*b*(33*b*c**2*e**2/16 + 2*c**3*d*e)/(14*c) + c**3*d**2)/(6*c) + 
2*b**3*d*e + 3*b**2*c*d**2 - 9*b*(6*a*b*c*e**2 + 6*a*c**2*d*e - 6*a*(33*b* 
c**2*e**2/16 + 2*c**3*d*e)/(7*c) + b**3*e**2 + 6*b**2*c*d*e + 3*b*c**2*d** 
2 - 11*b*(17*a*c**2*e**2/8 + 3*b**2*c*e**2 + 6*b*c**2*d*e - 13*b*(33*b*c** 
2*e**2/16 + 2*c**3*d*e)/(14*c) + c**3*d**2)/(12*c))/(10*c))/(4*c) + x**2*( 
3*a**2*b*e**2 + 6*a**2*c*d*e + 6*a*b**2*d*e + 6*a*b*c*d**2 - 4*a*(6*a*b*c* 
e**2 + 6*a*c**2*d*e - 6*a*(33*b*c**2*e**2/16 + 2*c**3*d*e)/(7*c) + b**3*e* 
*2 + 6*b**2*c*d*e + 3*b*c**2*d**2 - 11*b*(17*a*c**2*e**2/8 + 3*b**2*c*e**2 
 + 6*b*c**2*d*e - 13*b*(33*b*c**2*e**2/16 + 2*c**3*d*e)/(14*c) + c**3*d**2 
)/(12*c))/(5*c) + b**3*d**2 - 7*b*(3*a**2*c*e**2 + 3*a*b**2*e**2 + 12*a*b* 
c*d*e + 3*a*c**2*d**2 - 5*a*(17*a*c**2*e**2/8 + 3*b**2*c*e**2 + 6*b*c**2*d 
*e - 13*b*(33*b*c**2*e**2/16 + 2*c**3*d*e)/(14*c) + c**3*d**2)/(6*c) + ...
 

Maxima [F(-2)]

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

integrate((e*x+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. 770 vs. \(2 (267) = 534\).

Time = 0.42 (sec) , antiderivative size = 770, normalized size of antiderivative = 2.63 \[ \int (d+e x)^2 \left (a+b x+c x^2\right )^{5/2} \, dx =\text {Too large to display} \] Input:

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

Output:

1/344064*sqrt(c*x^2 + b*x + a)*(2*(4*(2*(8*(2*(12*(14*c^2*e^2*x + (32*c^9* 
d*e + 33*b*c^8*e^2)/c^7)*x + (224*c^9*d^2 + 928*b*c^8*d*e + 243*b^2*c^7*e^ 
2 + 476*a*c^8*e^2)/c^7)*x + (1120*b*c^8*d^2 + 1184*b^2*c^7*d*e + 2304*a*c^ 
8*d*e + 3*b^3*c^6*e^2 + 1228*a*b*c^7*e^2)/c^7)*x + (6048*b^2*c^7*d^2 + 116 
48*a*c^8*d^2 + 96*b^3*c^6*d*e + 25216*a*b*c^7*d*e - 27*b^4*c^5*e^2 + 216*a 
*b^2*c^6*e^2 + 6608*a^2*c^7*e^2)/c^7)*x + (224*b^3*c^6*d^2 + 34944*a*b*c^7 
*d^2 - 224*b^4*c^5*d*e + 1920*a*b^2*c^6*d*e + 36864*a^2*c^7*d*e + 63*b^5*c 
^4*e^2 - 568*a*b^3*c^5*e^2 + 1392*a^2*b*c^6*e^2)/c^7)*x - (1120*b^4*c^5*d^ 
2 - 10752*a*b^2*c^6*d^2 - 118272*a^2*c^7*d^2 - 1120*b^5*c^4*d*e + 10752*a* 
b^3*c^5*d*e - 29184*a^2*b*c^6*d*e + 315*b^6*c^3*e^2 - 3164*a*b^4*c^4*e^2 + 
 9552*a^2*b^2*c^5*e^2 - 6720*a^3*c^6*e^2)/c^7)*x + (3360*b^5*c^4*d^2 - 358 
40*a*b^3*c^5*d^2 + 118272*a^2*b*c^6*d^2 - 3360*b^6*c^3*d*e + 35840*a*b^4*c 
^4*d*e - 118272*a^2*b^2*c^5*d*e + 98304*a^3*c^6*d*e + 945*b^7*c^2*e^2 - 10 
500*a*b^5*c^3*e^2 + 37744*a^2*b^3*c^4*e^2 - 42432*a^3*b*c^5*e^2)/c^7) + 5/ 
32768*(32*b^6*c^2*d^2 - 384*a*b^4*c^3*d^2 + 1536*a^2*b^2*c^4*d^2 - 2048*a^ 
3*c^5*d^2 - 32*b^7*c*d*e + 384*a*b^5*c^2*d*e - 1536*a^2*b^3*c^3*d*e + 2048 
*a^3*b*c^4*d*e + 9*b^8*e^2 - 112*a*b^6*c*e^2 + 480*a^2*b^4*c^2*e^2 - 768*a 
^3*b^2*c^3*e^2 + 256*a^4*c^4*e^2)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x 
+ a))*sqrt(c) + b))/c^(11/2)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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