\(\int (d+e x)^2 (b x+c x^2)^{5/2} \, dx\) [149]

Optimal result

Integrand size = 21, antiderivative size = 392 \[ \int (d+e x)^2 \left (b x+c x^2\right )^{5/2} \, dx=\frac {5 b^5 \left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right ) \sqrt {b x+c x^2}}{16384 c^5}-\frac {5 b^4 \left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right ) x \sqrt {b x+c x^2}}{24576 c^4}+\frac {b^3 \left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right ) x^2 \sqrt {b x+c x^2}}{6144 c^3}+\frac {9 b^2 \left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right ) x^3 \sqrt {b x+c x^2}}{1024 c^2}+\frac {5 b \left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right ) x^4 \sqrt {b x+c x^2}}{384 c}+\frac {1}{192} \left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right ) x^5 \sqrt {b x+c x^2}+\frac {e (32 c d-9 b e) \left (b x+c x^2\right )^{7/2}}{112 c^2}+\frac {e^2 x \left (b x+c x^2\right )^{7/2}}{8 c}-\frac {5 b^6 \left (32 c^2 d^2-32 b c d e+9 b^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{16384 c^{11/2}} \] Output:

5/16384*b^5*(9*b^2*e^2-32*b*c*d*e+32*c^2*d^2)*(c*x^2+b*x)^(1/2)/c^5-5/2457 
6*b^4*(9*b^2*e^2-32*b*c*d*e+32*c^2*d^2)*x*(c*x^2+b*x)^(1/2)/c^4+1/6144*b^3 
*(9*b^2*e^2-32*b*c*d*e+32*c^2*d^2)*x^2*(c*x^2+b*x)^(1/2)/c^3+9/1024*b^2*(9 
*b^2*e^2-32*b*c*d*e+32*c^2*d^2)*x^3*(c*x^2+b*x)^(1/2)/c^2+5/384*b*(9*b^2*e 
^2-32*b*c*d*e+32*c^2*d^2)*x^4*(c*x^2+b*x)^(1/2)/c+1/192*(9*b^2*e^2-32*b*c* 
d*e+32*c^2*d^2)*x^5*(c*x^2+b*x)^(1/2)+1/112*e*(-9*b*e+32*c*d)*(c*x^2+b*x)^ 
(7/2)/c^2+1/8*e^2*x*(c*x^2+b*x)^(7/2)/c-5/16384*b^6*(9*b^2*e^2-32*b*c*d*e+ 
32*c^2*d^2)*arctanh(c^(1/2)*x/(c*x^2+b*x)^(1/2))/c^(11/2)
 

Mathematica [A] (verified)

Time = 1.95 (sec) , antiderivative size = 341, normalized size of antiderivative = 0.87 \[ \int (d+e x)^2 \left (b x+c x^2\right )^{5/2} \, dx=\frac {\sqrt {x} \sqrt {b+c x} \left (\sqrt {c} \sqrt {x} \sqrt {b+c x} \left (945 b^7 e^2-210 b^6 c e (16 d+3 e x)+128 b^3 c^4 x^2 \left (14 d^2+12 d e x+3 e^2 x^2\right )+56 b^5 c^2 \left (60 d^2+40 d e x+9 e^2 x^2\right )+2048 c^7 x^5 \left (28 d^2+48 d e x+21 e^2 x^2\right )-16 b^4 c^3 x \left (140 d^2+112 d e x+27 e^2 x^2\right )+1024 b c^6 x^4 \left (140 d^2+232 d e x+99 e^2 x^2\right )+256 b^2 c^5 x^3 \left (378 d^2+592 d e x+243 e^2 x^2\right )\right )+210 b^6 \left (32 c^2 d^2+9 b^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}-\sqrt {b+c x}}\right )+6720 b^7 c d e \text {arctanh}\left (\frac {\sqrt {c} \sqrt {x}}{-\sqrt {b}+\sqrt {b+c x}}\right )\right )}{344064 c^{11/2} \sqrt {x (b+c x)}} \] Input:

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

Output:

(Sqrt[x]*Sqrt[b + c*x]*(Sqrt[c]*Sqrt[x]*Sqrt[b + c*x]*(945*b^7*e^2 - 210*b 
^6*c*e*(16*d + 3*e*x) + 128*b^3*c^4*x^2*(14*d^2 + 12*d*e*x + 3*e^2*x^2) + 
56*b^5*c^2*(60*d^2 + 40*d*e*x + 9*e^2*x^2) + 2048*c^7*x^5*(28*d^2 + 48*d*e 
*x + 21*e^2*x^2) - 16*b^4*c^3*x*(140*d^2 + 112*d*e*x + 27*e^2*x^2) + 1024* 
b*c^6*x^4*(140*d^2 + 232*d*e*x + 99*e^2*x^2) + 256*b^2*c^5*x^3*(378*d^2 + 
592*d*e*x + 243*e^2*x^2)) + 210*b^6*(32*c^2*d^2 + 9*b^2*e^2)*ArcTanh[(Sqrt 
[c]*Sqrt[x])/(Sqrt[b] - Sqrt[b + c*x])] + 6720*b^7*c*d*e*ArcTanh[(Sqrt[c]* 
Sqrt[x])/(-Sqrt[b] + Sqrt[b + c*x])]))/(344064*c^(11/2)*Sqrt[x*(b + c*x)])
 

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.58, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {1166, 27, 1160, 1087, 1087, 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.

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

\(\Big \downarrow \) 1166

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1160

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

\(\Big \downarrow \) 1087

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

\(\Big \downarrow \) 1087

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

\(\Big \downarrow \) 1087

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

\(\Big \downarrow \) 1091

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

(e*(d + e*x)*(b*x + c*x^2)^(7/2))/(8*c) + ((9*e*(2*c*d - b*e)*(b*x + c*x^2 
)^(7/2))/(7*c) + ((32*c^2*d^2 - 32*b*c*d*e + 9*b^2*e^2)*(((b + 2*c*x)*(b*x 
 + c*x^2)^(5/2))/(12*c) - (5*b^2*(((b + 2*c*x)*(b*x + c*x^2)^(3/2))/(8*c) 
- (3*b^2*(((b + 2*c*x)*Sqrt[b*x + c*x^2])/(4*c) - (b^2*ArcTanh[(Sqrt[c]*x) 
/Sqrt[b*x + c*x^2]])/(4*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 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 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 [A] (verified)

Time = 0.65 (sec) , antiderivative size = 328, normalized size of antiderivative = 0.84

method	result	size
risch	\(\frac {\left (43008 c^{7} e^{2} x^{7}+101376 b \,c^{6} e^{2} x^{6}+98304 c^{7} d e \,x^{6}+62208 b^{2} c^{5} e^{2} x^{5}+237568 b \,c^{6} d e \,x^{5}+57344 c^{7} d^{2} x^{5}+384 b^{3} c^{4} e^{2} x^{4}+151552 b^{2} c^{5} d e \,x^{4}+143360 b \,c^{6} d^{2} x^{4}-432 b^{4} c^{3} e^{2} x^{3}+1536 b^{3} c^{4} d e \,x^{3}+96768 b^{2} c^{5} d^{2} x^{3}+504 b^{5} c^{2} e^{2} x^{2}-1792 b^{4} c^{3} d e \,x^{2}+1792 b^{3} c^{4} d^{2} x^{2}-630 b^{6} c \,e^{2} x +2240 b^{5} c^{2} d e x -2240 b^{4} c^{3} d^{2} x +945 b^{7} e^{2}-3360 b^{6} c d e +3360 b^{5} c^{2} d^{2}\right ) x \left (c x +b \right )}{344064 c^{5} \sqrt {x \left (c x +b \right )}}-\frac {5 b^{6} \left (9 b^{2} e^{2}-32 b c d e +32 c^{2} d^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{32768 c^{\frac {11}{2}}}\)	\(328\)
default	\(d^{2} \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{12 c}-\frac {5 b^{2} \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )+e^{2} \left (\frac {x \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{8 c}-\frac {9 b \left (\frac {\left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{7 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{12 c}-\frac {5 b^{2} \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )}{2 c}\right )}{16 c}\right )+2 d e \left (\frac {\left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{7 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{12 c}-\frac {5 b^{2} \left (\frac {\left (2 c x +b \right ) \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{8 c}-\frac {3 b^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{16 c}\right )}{24 c}\right )}{2 c}\right )\)	\(435\)

Input:

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

Output:

1/344064/c^5*(43008*c^7*e^2*x^7+101376*b*c^6*e^2*x^6+98304*c^7*d*e*x^6+622 
08*b^2*c^5*e^2*x^5+237568*b*c^6*d*e*x^5+57344*c^7*d^2*x^5+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^4-432*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+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^2-630*b^6*c*e^2*x+2240*b^5*c^2*d*e*x-2240*b^4*c^3*d 
^2*x+945*b^7*e^2-3360*b^6*c*d*e+3360*b^5*c^2*d^2)*x*(c*x+b)/(x*(c*x+b))^(1 
/2)-5/32768*b^6*(9*b^2*e^2-32*b*c*d*e+32*c^2*d^2)/c^(11/2)*ln((1/2*b+c*x)/ 
c^(1/2)+(c*x^2+b*x)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 644, normalized size of antiderivative = 1.64 \[ \int (d+e x)^2 \left (b x+c x^2\right )^{5/2} \, dx=\left [\frac {105 \, {\left (32 \, b^{6} c^{2} d^{2} - 32 \, b^{7} c d e + 9 \, b^{8} e^{2}\right )} \sqrt {c} \log \left (2 \, c x + b - 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) + 2 \, {\left (43008 \, c^{8} e^{2} x^{7} + 3360 \, b^{5} c^{3} d^{2} - 3360 \, b^{6} c^{2} d e + 945 \, b^{7} c e^{2} + 3072 \, {\left (32 \, c^{8} d e + 33 \, b c^{7} e^{2}\right )} x^{6} + 256 \, {\left (224 \, c^{8} d^{2} + 928 \, b c^{7} d e + 243 \, b^{2} c^{6} e^{2}\right )} x^{5} + 128 \, {\left (1120 \, b c^{7} d^{2} + 1184 \, b^{2} c^{6} d e + 3 \, b^{3} c^{5} e^{2}\right )} x^{4} + 48 \, {\left (2016 \, b^{2} c^{6} d^{2} + 32 \, b^{3} c^{5} d e - 9 \, b^{4} c^{4} e^{2}\right )} x^{3} + 56 \, {\left (32 \, b^{3} c^{5} d^{2} - 32 \, b^{4} c^{4} d e + 9 \, b^{5} c^{3} e^{2}\right )} x^{2} - 70 \, {\left (32 \, b^{4} c^{4} d^{2} - 32 \, b^{5} c^{3} d e + 9 \, b^{6} c^{2} e^{2}\right )} x\right )} \sqrt {c x^{2} + b x}}{688128 \, c^{6}}, \frac {105 \, {\left (32 \, b^{6} c^{2} d^{2} - 32 \, b^{7} c d e + 9 \, b^{8} e^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x + b}\right ) + {\left (43008 \, c^{8} e^{2} x^{7} + 3360 \, b^{5} c^{3} d^{2} - 3360 \, b^{6} c^{2} d e + 945 \, b^{7} c e^{2} + 3072 \, {\left (32 \, c^{8} d e + 33 \, b c^{7} e^{2}\right )} x^{6} + 256 \, {\left (224 \, c^{8} d^{2} + 928 \, b c^{7} d e + 243 \, b^{2} c^{6} e^{2}\right )} x^{5} + 128 \, {\left (1120 \, b c^{7} d^{2} + 1184 \, b^{2} c^{6} d e + 3 \, b^{3} c^{5} e^{2}\right )} x^{4} + 48 \, {\left (2016 \, b^{2} c^{6} d^{2} + 32 \, b^{3} c^{5} d e - 9 \, b^{4} c^{4} e^{2}\right )} x^{3} + 56 \, {\left (32 \, b^{3} c^{5} d^{2} - 32 \, b^{4} c^{4} d e + 9 \, b^{5} c^{3} e^{2}\right )} x^{2} - 70 \, {\left (32 \, b^{4} c^{4} d^{2} - 32 \, b^{5} c^{3} d e + 9 \, b^{6} c^{2} e^{2}\right )} x\right )} \sqrt {c x^{2} + b x}}{344064 \, c^{6}}\right ] \] Input:

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

Output:

[1/688128*(105*(32*b^6*c^2*d^2 - 32*b^7*c*d*e + 9*b^8*e^2)*sqrt(c)*log(2*c 
*x + b - 2*sqrt(c*x^2 + b*x)*sqrt(c)) + 2*(43008*c^8*e^2*x^7 + 3360*b^5*c^ 
3*d^2 - 3360*b^6*c^2*d*e + 945*b^7*c*e^2 + 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*e^2)*x^5 + 128*(112 
0*b*c^7*d^2 + 1184*b^2*c^6*d*e + 3*b^3*c^5*e^2)*x^4 + 48*(2016*b^2*c^6*d^2 
 + 32*b^3*c^5*d*e - 9*b^4*c^4*e^2)*x^3 + 56*(32*b^3*c^5*d^2 - 32*b^4*c^4*d 
*e + 9*b^5*c^3*e^2)*x^2 - 70*(32*b^4*c^4*d^2 - 32*b^5*c^3*d*e + 9*b^6*c^2* 
e^2)*x)*sqrt(c*x^2 + b*x))/c^6, 1/344064*(105*(32*b^6*c^2*d^2 - 32*b^7*c*d 
*e + 9*b^8*e^2)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x + b)) + (4 
3008*c^8*e^2*x^7 + 3360*b^5*c^3*d^2 - 3360*b^6*c^2*d*e + 945*b^7*c*e^2 + 3 
072*(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 + 2 
43*b^2*c^6*e^2)*x^5 + 128*(1120*b*c^7*d^2 + 1184*b^2*c^6*d*e + 3*b^3*c^5*e 
^2)*x^4 + 48*(2016*b^2*c^6*d^2 + 32*b^3*c^5*d*e - 9*b^4*c^4*e^2)*x^3 + 56* 
(32*b^3*c^5*d^2 - 32*b^4*c^4*d*e + 9*b^5*c^3*e^2)*x^2 - 70*(32*b^4*c^4*d^2 
 - 32*b^5*c^3*d*e + 9*b^6*c^2*e^2)*x)*sqrt(c*x^2 + b*x))/c^6]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 986 vs. \(2 (393) = 786\).

Time = 0.56 (sec) , antiderivative size = 986, normalized size of antiderivative = 2.52 \[ \int (d+e x)^2 \left (b x+c x^2\right )^{5/2} \, dx =\text {Too large to display} \] Input:

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

Output:

Piecewise((-5*b**3*(b**3*d**2 - 7*b*(2*b**3*d*e + 3*b**2*c*d**2 - 9*b*(b** 
3*e**2 + 6*b**2*c*d*e + 3*b*c**2*d**2 - 11*b*(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 
))/(8*c))*Piecewise((log(b + 2*sqrt(c)*sqrt(b*x + c*x**2) + 2*c*x)/sqrt(c) 
, Ne(b**2/c, 0)), ((b/(2*c) + x)*log(b/(2*c) + x)/sqrt(c*(b/(2*c) + x)**2) 
, True))/(16*c**3) + sqrt(b*x + c*x**2)*(5*b**2*(b**3*d**2 - 7*b*(2*b**3*d 
*e + 3*b**2*c*d**2 - 9*b*(b**3*e**2 + 6*b**2*c*d*e + 3*b*c**2*d**2 - 11*b* 
(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))/(8*c))/(8*c**3) - 5*b*x*(b**3*d**2 - 7*b*( 
2*b**3*d*e + 3*b**2*c*d**2 - 9*b*(b**3*e**2 + 6*b**2*c*d*e + 3*b*c**2*d**2 
 - 11*b*(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))/(8*c))/(12*c**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*(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*(b**3*e**2 + 6*b**2*c*d*e + 3*b*c**2*d**2 - 11*b*(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*(2*b**3*d*e + 3*b**2*c*d**2 - 9*b*(b**3*e**2 + 6*b** 
2*c*d*e + 3*b*c**2*d**2 - 11*b*(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*(b**3*d**2 - 7*b*(2*b**3*d*e + 3*b**2*c*d**2 - 9*b*(b**3*e**2 + 6*b**...
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 549, normalized size of antiderivative = 1.40 \[ \int (d+e x)^2 \left (b x+c x^2\right )^{5/2} \, dx=\frac {1}{6} \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} d^{2} x + \frac {5 \, \sqrt {c x^{2} + b x} b^{4} d^{2} x}{256 \, c^{2}} - \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{2} d^{2} x}{96 \, c} - \frac {5 \, \sqrt {c x^{2} + b x} b^{5} d e x}{256 \, c^{3}} + \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{3} d e x}{96 \, c^{2}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} b d e x}{6 \, c} + \frac {45 \, \sqrt {c x^{2} + b x} b^{6} e^{2} x}{8192 \, c^{4}} - \frac {15 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{4} e^{2} x}{1024 \, c^{3}} + \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} b^{2} e^{2} x}{64 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {7}{2}} e^{2} x}{8 \, c} - \frac {5 \, b^{6} d^{2} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{1024 \, c^{\frac {7}{2}}} + \frac {5 \, b^{7} d e \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{1024 \, c^{\frac {9}{2}}} - \frac {45 \, b^{8} e^{2} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{32768 \, c^{\frac {11}{2}}} + \frac {5 \, \sqrt {c x^{2} + b x} b^{5} d^{2}}{512 \, c^{3}} - \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{3} d^{2}}{192 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} b d^{2}}{12 \, c} - \frac {5 \, \sqrt {c x^{2} + b x} b^{6} d e}{512 \, c^{4}} + \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{4} d e}{192 \, c^{3}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} b^{2} d e}{12 \, c^{2}} + \frac {2 \, {\left (c x^{2} + b x\right )}^{\frac {7}{2}} d e}{7 \, c} + \frac {45 \, \sqrt {c x^{2} + b x} b^{7} e^{2}}{16384 \, c^{5}} - \frac {15 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{5} e^{2}}{2048 \, c^{4}} + \frac {3 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} b^{3} e^{2}}{128 \, c^{3}} - \frac {9 \, {\left (c x^{2} + b x\right )}^{\frac {7}{2}} b e^{2}}{112 \, c^{2}} \] Input:

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

Output:

1/6*(c*x^2 + b*x)^(5/2)*d^2*x + 5/256*sqrt(c*x^2 + b*x)*b^4*d^2*x/c^2 - 5/ 
96*(c*x^2 + b*x)^(3/2)*b^2*d^2*x/c - 5/256*sqrt(c*x^2 + b*x)*b^5*d*e*x/c^3 
 + 5/96*(c*x^2 + b*x)^(3/2)*b^3*d*e*x/c^2 - 1/6*(c*x^2 + b*x)^(5/2)*b*d*e* 
x/c + 45/8192*sqrt(c*x^2 + b*x)*b^6*e^2*x/c^4 - 15/1024*(c*x^2 + b*x)^(3/2 
)*b^4*e^2*x/c^3 + 3/64*(c*x^2 + b*x)^(5/2)*b^2*e^2*x/c^2 + 1/8*(c*x^2 + b* 
x)^(7/2)*e^2*x/c - 5/1024*b^6*d^2*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt 
(c))/c^(7/2) + 5/1024*b^7*d*e*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)) 
/c^(9/2) - 45/32768*b^8*e^2*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c 
^(11/2) + 5/512*sqrt(c*x^2 + b*x)*b^5*d^2/c^3 - 5/192*(c*x^2 + b*x)^(3/2)* 
b^3*d^2/c^2 + 1/12*(c*x^2 + b*x)^(5/2)*b*d^2/c - 5/512*sqrt(c*x^2 + b*x)*b 
^6*d*e/c^4 + 5/192*(c*x^2 + b*x)^(3/2)*b^4*d*e/c^3 - 1/12*(c*x^2 + b*x)^(5 
/2)*b^2*d*e/c^2 + 2/7*(c*x^2 + b*x)^(7/2)*d*e/c + 45/16384*sqrt(c*x^2 + b* 
x)*b^7*e^2/c^5 - 15/2048*(c*x^2 + b*x)^(3/2)*b^5*e^2/c^4 + 3/128*(c*x^2 + 
b*x)^(5/2)*b^3*e^2/c^3 - 9/112*(c*x^2 + b*x)^(7/2)*b*e^2/c^2
 

Giac [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 349, normalized size of antiderivative = 0.89 \[ \int (d+e x)^2 \left (b x+c x^2\right )^{5/2} \, dx=\frac {1}{344064} \, \sqrt {c x^{2} + b x} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, {\left (12 \, {\left (14 \, c^{2} e^{2} x + \frac {32 \, c^{9} d e + 33 \, b c^{8} e^{2}}{c^{7}}\right )} x + \frac {224 \, c^{9} d^{2} + 928 \, b c^{8} d e + 243 \, b^{2} c^{7} e^{2}}{c^{7}}\right )} x + \frac {1120 \, b c^{8} d^{2} + 1184 \, b^{2} c^{7} d e + 3 \, b^{3} c^{6} e^{2}}{c^{7}}\right )} x + \frac {3 \, {\left (2016 \, b^{2} c^{7} d^{2} + 32 \, b^{3} c^{6} d e - 9 \, b^{4} c^{5} e^{2}\right )}}{c^{7}}\right )} x + \frac {7 \, {\left (32 \, b^{3} c^{6} d^{2} - 32 \, b^{4} c^{5} d e + 9 \, b^{5} c^{4} e^{2}\right )}}{c^{7}}\right )} x - \frac {35 \, {\left (32 \, b^{4} c^{5} d^{2} - 32 \, b^{5} c^{4} d e + 9 \, b^{6} c^{3} e^{2}\right )}}{c^{7}}\right )} x + \frac {105 \, {\left (32 \, b^{5} c^{4} d^{2} - 32 \, b^{6} c^{3} d e + 9 \, b^{7} c^{2} e^{2}\right )}}{c^{7}}\right )} + \frac {5 \, {\left (32 \, b^{6} c^{2} d^{2} - 32 \, b^{7} c d e + 9 \, b^{8} e^{2}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} + b \right |}\right )}{32768 \, c^{\frac {11}{2}}} \] Input:

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

Output:

1/344064*sqrt(c*x^2 + b*x)*(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)/c 
^7)*x + (1120*b*c^8*d^2 + 1184*b^2*c^7*d*e + 3*b^3*c^6*e^2)/c^7)*x + 3*(20 
16*b^2*c^7*d^2 + 32*b^3*c^6*d*e - 9*b^4*c^5*e^2)/c^7)*x + 7*(32*b^3*c^6*d^ 
2 - 32*b^4*c^5*d*e + 9*b^5*c^4*e^2)/c^7)*x - 35*(32*b^4*c^5*d^2 - 32*b^5*c 
^4*d*e + 9*b^6*c^3*e^2)/c^7)*x + 105*(32*b^5*c^4*d^2 - 32*b^6*c^3*d*e + 9* 
b^7*c^2*e^2)/c^7) + 5/32768*(32*b^6*c^2*d^2 - 32*b^7*c*d*e + 9*b^8*e^2)*lo 
g(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) + b))/c^(11/2)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.66 (sec) , antiderivative size = 516, normalized size of antiderivative = 1.32 \[ \int (d+e x)^2 \left (b x+c x^2\right )^{5/2} \, dx=\frac {945 \sqrt {x}\, \sqrt {c x +b}\, b^{7} c \,e^{2}-3360 \sqrt {x}\, \sqrt {c x +b}\, b^{6} c^{2} d e -630 \sqrt {x}\, \sqrt {c x +b}\, b^{6} c^{2} e^{2} x +3360 \sqrt {x}\, \sqrt {c x +b}\, b^{5} c^{3} d^{2}+2240 \sqrt {x}\, \sqrt {c x +b}\, b^{5} c^{3} d e x +504 \sqrt {x}\, \sqrt {c x +b}\, b^{5} c^{3} e^{2} x^{2}-2240 \sqrt {x}\, \sqrt {c x +b}\, b^{4} c^{4} d^{2} x -1792 \sqrt {x}\, \sqrt {c x +b}\, b^{4} c^{4} d e \,x^{2}-432 \sqrt {x}\, \sqrt {c x +b}\, b^{4} c^{4} e^{2} x^{3}+1792 \sqrt {x}\, \sqrt {c x +b}\, b^{3} c^{5} d^{2} x^{2}+1536 \sqrt {x}\, \sqrt {c x +b}\, b^{3} c^{5} d e \,x^{3}+384 \sqrt {x}\, \sqrt {c x +b}\, b^{3} c^{5} e^{2} x^{4}+96768 \sqrt {x}\, \sqrt {c x +b}\, b^{2} c^{6} d^{2} x^{3}+151552 \sqrt {x}\, \sqrt {c x +b}\, b^{2} c^{6} d e \,x^{4}+62208 \sqrt {x}\, \sqrt {c x +b}\, b^{2} c^{6} e^{2} x^{5}+143360 \sqrt {x}\, \sqrt {c x +b}\, b \,c^{7} d^{2} x^{4}+237568 \sqrt {x}\, \sqrt {c x +b}\, b \,c^{7} d e \,x^{5}+101376 \sqrt {x}\, \sqrt {c x +b}\, b \,c^{7} e^{2} x^{6}+57344 \sqrt {x}\, \sqrt {c x +b}\, c^{8} d^{2} x^{5}+98304 \sqrt {x}\, \sqrt {c x +b}\, c^{8} d e \,x^{6}+43008 \sqrt {x}\, \sqrt {c x +b}\, c^{8} e^{2} x^{7}-945 \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {c x +b}+\sqrt {x}\, \sqrt {c}}{\sqrt {b}}\right ) b^{8} e^{2}+3360 \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {c x +b}+\sqrt {x}\, \sqrt {c}}{\sqrt {b}}\right ) b^{7} c d e -3360 \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {c x +b}+\sqrt {x}\, \sqrt {c}}{\sqrt {b}}\right ) b^{6} c^{2} d^{2}}{344064 c^{6}} \] Input:

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

Output:

(945*sqrt(x)*sqrt(b + c*x)*b**7*c*e**2 - 3360*sqrt(x)*sqrt(b + c*x)*b**6*c 
**2*d*e - 630*sqrt(x)*sqrt(b + c*x)*b**6*c**2*e**2*x + 3360*sqrt(x)*sqrt(b 
 + c*x)*b**5*c**3*d**2 + 2240*sqrt(x)*sqrt(b + c*x)*b**5*c**3*d*e*x + 504* 
sqrt(x)*sqrt(b + c*x)*b**5*c**3*e**2*x**2 - 2240*sqrt(x)*sqrt(b + c*x)*b** 
4*c**4*d**2*x - 1792*sqrt(x)*sqrt(b + c*x)*b**4*c**4*d*e*x**2 - 432*sqrt(x 
)*sqrt(b + c*x)*b**4*c**4*e**2*x**3 + 1792*sqrt(x)*sqrt(b + c*x)*b**3*c**5 
*d**2*x**2 + 1536*sqrt(x)*sqrt(b + c*x)*b**3*c**5*d*e*x**3 + 384*sqrt(x)*s 
qrt(b + c*x)*b**3*c**5*e**2*x**4 + 96768*sqrt(x)*sqrt(b + c*x)*b**2*c**6*d 
**2*x**3 + 151552*sqrt(x)*sqrt(b + c*x)*b**2*c**6*d*e*x**4 + 62208*sqrt(x) 
*sqrt(b + c*x)*b**2*c**6*e**2*x**5 + 143360*sqrt(x)*sqrt(b + c*x)*b*c**7*d 
**2*x**4 + 237568*sqrt(x)*sqrt(b + c*x)*b*c**7*d*e*x**5 + 101376*sqrt(x)*s 
qrt(b + c*x)*b*c**7*e**2*x**6 + 57344*sqrt(x)*sqrt(b + c*x)*c**8*d**2*x**5 
 + 98304*sqrt(x)*sqrt(b + c*x)*c**8*d*e*x**6 + 43008*sqrt(x)*sqrt(b + c*x) 
*c**8*e**2*x**7 - 945*sqrt(c)*log((sqrt(b + c*x) + sqrt(x)*sqrt(c))/sqrt(b 
))*b**8*e**2 + 3360*sqrt(c)*log((sqrt(b + c*x) + sqrt(x)*sqrt(c))/sqrt(b)) 
*b**7*c*d*e - 3360*sqrt(c)*log((sqrt(b + c*x) + sqrt(x)*sqrt(c))/sqrt(b))* 
b**6*c**2*d**2)/(344064*c**6)