\(\int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^5} \, dx\) [581]

Optimal result

Integrand size = 22, antiderivative size = 308 \[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^5} \, dx=\frac {\left (16 c^2 d^2+5 b^2 e^2-4 c e (4 b d+a e)\right ) (b d-2 a e+(2 c d-b e) x) \sqrt {a+b x+c x^2}}{64 \left (c d^2-b d e+a e^2\right )^3 (d+e x)^2}-\frac {e \left (a+b x+c x^2\right )^{3/2}}{4 \left (c d^2-b d e+a e^2\right ) (d+e x)^4}-\frac {5 e (2 c d-b e) \left (a+b x+c x^2\right )^{3/2}}{24 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^3}-\frac {\left (b^2-4 a c\right ) \left (16 c^2 d^2+5 b^2 e^2-4 c e (4 b d+a e)\right ) \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{128 \left (c d^2-b d e+a e^2\right )^{7/2}} \] Output:

1/64*(16*c^2*d^2+5*b^2*e^2-4*c*e*(a*e+4*b*d))*(b*d-2*a*e+(-b*e+2*c*d)*x)*( 
c*x^2+b*x+a)^(1/2)/(a*e^2-b*d*e+c*d^2)^3/(e*x+d)^2-1/4*e*(c*x^2+b*x+a)^(3/ 
2)/(a*e^2-b*d*e+c*d^2)/(e*x+d)^4-5/24*e*(-b*e+2*c*d)*(c*x^2+b*x+a)^(3/2)/( 
a*e^2-b*d*e+c*d^2)^2/(e*x+d)^3-1/128*(-4*a*c+b^2)*(16*c^2*d^2+5*b^2*e^2-4* 
c*e*(a*e+4*b*d))*arctanh(1/2*(b*d-2*a*e+(-b*e+2*c*d)*x)/(a*e^2-b*d*e+c*d^2 
)^(1/2)/(c*x^2+b*x+a)^(1/2))/(a*e^2-b*d*e+c*d^2)^(7/2)
 

Mathematica [A] (verified)

Time = 10.53 (sec) , antiderivative size = 276, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^5} \, dx=\frac {-\frac {6 e \left (c d^2+e (-b d+a e)\right ) (a+x (b+c x))^{3/2}}{(d+e x)^4}-\frac {5 e (2 c d-b e) (a+x (b+c x))^{3/2}}{(d+e x)^3}+3 \left (8 c^2 d^2+\frac {5 b^2 e^2}{2}-2 c e (4 b d+a e)\right ) \left (\frac {\sqrt {a+x (b+c x)} (-2 a e+2 c d x+b (d-e x))}{4 \left (c d^2+e (-b d+a e)\right ) (d+e x)^2}+\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )}{8 \left (c d^2+e (-b d+a e)\right )^{3/2}}\right )}{24 \left (c d^2+e (-b d+a e)\right )^2} \] Input:

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

Output:

((-6*e*(c*d^2 + e*(-(b*d) + a*e))*(a + x*(b + c*x))^(3/2))/(d + e*x)^4 - ( 
5*e*(2*c*d - b*e)*(a + x*(b + c*x))^(3/2))/(d + e*x)^3 + 3*(8*c^2*d^2 + (5 
*b^2*e^2)/2 - 2*c*e*(4*b*d + a*e))*((Sqrt[a + x*(b + c*x)]*(-2*a*e + 2*c*d 
*x + b*(d - e*x)))/(4*(c*d^2 + e*(-(b*d) + a*e))*(d + e*x)^2) + ((b^2 - 4* 
a*c)*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) 
+ a*e)]*Sqrt[a + x*(b + c*x)])])/(8*(c*d^2 + e*(-(b*d) + a*e))^(3/2))))/(2 
4*(c*d^2 + e*(-(b*d) + a*e))^2)
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.06, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1167, 27, 1228, 1152, 1154, 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.

\(\displaystyle \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^5} \, dx\)

\(\Big \downarrow \) 1167

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1228

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

\(\Big \downarrow \) 1152

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

\(\Big \downarrow \) 1154

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

-1/4*(e*(a + b*x + c*x^2)^(3/2))/((c*d^2 - b*d*e + a*e^2)*(d + e*x)^4) + ( 
(-5*e*(2*c*d - b*e)*(a + b*x + c*x^2)^(3/2))/(3*(c*d^2 - b*d*e + a*e^2)*(d 
 + e*x)^3) + ((16*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(4*b*d + a*e))*(((b*d - 2*a* 
e + (2*c*d - b*e)*x)*Sqrt[a + b*x + c*x^2])/(4*(c*d^2 - b*d*e + a*e^2)*(d 
+ e*x)^2) - ((b^2 - 4*a*c)*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt 
[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])])/(8*(c*d^2 - b*d*e + a*e^2 
)^(3/2))))/(2*(c*d^2 - b*d*e + a*e^2)))/(8*(c*d^2 - b*d*e + a*e^2))
 

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 1152

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

rule 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-2   Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 
2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c 
, d, e}, x]
 

rule 1167

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)/((m + 1)*(c*d 
^2 - b*d*e + a*e^2))), x] + Simp[1/((m + 1)*(c*d^2 - b*d*e + a*e^2))   Int[ 
(d + e*x)^(m + 1)*Simp[c*d*(m + 1) - b*e*(m + p + 2) - c*e*(m + 2*p + 3)*x, 
 x]*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[m 
, -1] && ((LtQ[m, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]) || (SumSimp 
lerQ[m, 1] && IntegerQ[p]) || ILtQ[Simplify[m + 2*p + 3], 0])
 

rule 1228

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + 
 b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Simp[(b*(e 
*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^ 
(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x 
] && EqQ[Simplify[m + 2*p + 3], 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2524\) vs. \(2(286)=572\).

Time = 1.36 (sec) , antiderivative size = 2525, normalized size of antiderivative = 8.20

method	result	size
default	\(\text {Expression too large to display}\)	\(2525\)

Input:

int((c*x^2+b*x+a)^(1/2)/(e*x+d)^5,x,method=_RETURNVERBOSE)
 

Output:

1/e^5*(-1/4/(a*e^2-b*d*e+c*d^2)*e^2/(x+d/e)^4*(c*(x+d/e)^2+(b*e-2*c*d)/e*( 
x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)-5/8*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2 
)*(-1/3/(a*e^2-b*d*e+c*d^2)*e^2/(x+d/e)^3*(c*(x+d/e)^2+(b*e-2*c*d)/e*(x+d/ 
e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)-1/2*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)*(- 
1/2/(a*e^2-b*d*e+c*d^2)*e^2/(x+d/e)^2*(c*(x+d/e)^2+(b*e-2*c*d)/e*(x+d/e)+( 
a*e^2-b*d*e+c*d^2)/e^2)^(3/2)-1/4*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)*(-1/(a 
*e^2-b*d*e+c*d^2)*e^2/(x+d/e)*(c*(x+d/e)^2+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b* 
d*e+c*d^2)/e^2)^(3/2)+1/2*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)*((c*(x+d/e)^2+ 
(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)+1/2*(b*e-2*c*d)/e*ln( 
(1/2*(b*e-2*c*d)/e+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2+(b*e-2*c*d)/e*(x+d/e)+( 
a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/c^(1/2)-(a*e^2-b*d*e+c*d^2)/e^2/((a*e^2-b*d 
*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/e)+2 
*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(x+d/e)^2+(b*e-2*c*d)/e*(x+d/e)+(a*e^2 
-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e)))+2*c/(a*e^2-b*d*e+c*d^2)*e^2*(1/4*(2*c* 
(x+d/e)+(b*e-2*c*d)/e)/c*(c*(x+d/e)^2+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c 
*d^2)/e^2)^(1/2)+1/8*(4*c*(a*e^2-b*d*e+c*d^2)/e^2-(b*e-2*c*d)^2/e^2)/c^(3/ 
2)*ln((1/2*(b*e-2*c*d)/e+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2+(b*e-2*c*d)/e*(x+ 
d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))))+1/2*c/(a*e^2-b*d*e+c*d^2)*e^2*((c*( 
x+d/e)^2+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)+1/2*(b*e-2*c 
*d)/e*ln((1/2*(b*e-2*c*d)/e+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2+(b*e-2*c*d)...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1869 vs. \(2 (286) = 572\).

Time = 10.98 (sec) , antiderivative size = 3780, normalized size of antiderivative = 12.27 \[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^5} \, dx=\text {Too large to display} \] Input:

integrate((c*x^2+b*x+a)^(1/2)/(e*x+d)^5,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F]

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

integrate((c*x**2+b*x+a)**(1/2)/(e*x+d)**5,x)
 

Output:

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

Maxima [F(-2)]

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

integrate((c*x^2+b*x+a)^(1/2)/(e*x+d)^5,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(a*e^2-b*d*e>0)', see `assume?` f 
or more de
 

Giac [F(-1)]

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

integrate((c*x^2+b*x+a)^(1/2)/(e*x+d)^5,x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

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

int((a + b*x + c*x^2)^(1/2)/(d + e*x)^5,x)
 

Output:

int((a + b*x + c*x^2)^(1/2)/(d + e*x)^5, x)
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 11.38 (sec) , antiderivative size = 6410, normalized size of antiderivative = 20.81 \[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^5} \, dx =\text {Too large to display} \] Input:

int((c*x^2+b*x+a)^(1/2)/(e*x+d)^5,x)
 

Output:

(48*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x**2)*sqrt(a*e 
**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**2*c**2*d**4*e**2 
 + 192*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x**2)*sqrt( 
a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**2*c**2*d**3*e 
**3*x + 288*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x**2)* 
sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**2*c**2*d 
**2*e**4*x**2 + 192*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + 
c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a** 
2*c**2*d*e**5*x**3 + 48*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b* 
x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x) 
*a**2*c**2*e**6*x**4 - 72*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + 
b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d* 
x)*a*b**2*c*d**4*e**2 - 288*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a 
+ b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c* 
d*x)*a*b**2*c*d**3*e**3*x - 432*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqr 
t(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 
2*c*d*x)*a*b**2*c*d**2*e**4*x**2 - 288*sqrt(a*e**2 - b*d*e + c*d**2)*log( 
- 2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b 
*e*x + 2*c*d*x)*a*b**2*c*d*e**5*x**3 - 72*sqrt(a*e**2 - b*d*e + c*d**2)*lo 
g( - 2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b...