\(\int \frac {1}{(d+e x)^2 (a+b x+c x^2)^{3/2}} \, dx\) [622]

Optimal result

Integrand size = 22, antiderivative size = 254 \[ \int \frac {1}{(d+e x)^2 \left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {e}{\left (c d^2-b d e+a e^2\right ) (d+e x) \sqrt {a+b x+c x^2}}+\frac {5 b^2 c d e-12 a c^2 d e-3 b^3 e^2-2 b c \left (c d^2-5 a e^2\right )-c \left (4 c^2 d^2+3 b^2 e^2-4 c e (b d+2 a e)\right ) x}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \sqrt {a+b x+c x^2}}+\frac {3 e^2 (2 c d-b e) \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{2 \left (c d^2-b d e+a e^2\right )^{5/2}} \] Output:

-e/(a*e^2-b*d*e+c*d^2)/(e*x+d)/(c*x^2+b*x+a)^(1/2)+(5*b^2*c*d*e-12*a*c^2*d 
*e-3*b^3*e^2-2*b*c*(-5*a*e^2+c*d^2)-c*(4*c^2*d^2+3*b^2*e^2-4*c*e*(2*a*e+b* 
d))*x)/(-4*a*c+b^2)/(a*e^2-b*d*e+c*d^2)^2/(c*x^2+b*x+a)^(1/2)+3/2*e^2*(-b* 
e+2*c*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)^(5/2)
 

Mathematica [A] (verified)

Time = 10.27 (sec) , antiderivative size = 250, normalized size of antiderivative = 0.98 \[ \int \frac {1}{(d+e x)^2 \left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {2 \left (-\frac {e \left (4 c^2 d^2+3 b^2 e^2-4 c e (b d+2 a e)\right ) \sqrt {a+x (b+c x)}}{2 \left (c d^2+e (-b d+a e)\right ) (d+e x)}+\frac {b^2 e-2 c (a e+c d x)+b c (-d+e x)}{(d+e x) \sqrt {a+x (b+c x)}}+\frac {3 \left (b^2-4 a c\right ) e^2 (-2 c d+b e) \text {arctanh}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )}{4 \left (c d^2+e (-b d+a e)\right )^{3/2}}\right )}{\left (b^2-4 a c\right ) \left (c d^2+e (-b d+a e)\right )} \] Input:

Integrate[1/((d + e*x)^2*(a + b*x + c*x^2)^(3/2)),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.11, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {1165, 27, 1228, 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.

\(\displaystyle \int \frac {1}{(d+e x)^2 \left (a+b x+c x^2\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 1165

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1228

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

\(\Big \downarrow \) 1154

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

\(\Big \downarrow \) 219

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

Input:

Int[1/((d + e*x)^2*(a + b*x + c*x^2)^(3/2)),x]
 

Output:

(-2*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/((b^2 - 4*a*c)*(c*d^2 - 
 b*d*e + a*e^2)*(d + e*x)*Sqrt[a + b*x + c*x^2]) - (e*(((4*c^2*d^2 + 3*b^2 
*e^2 - 4*c*e*(b*d + 2*a*e))*Sqrt[a + b*x + c*x^2])/((c*d^2 - b*d*e + a*e^2 
)*(d + e*x)) - (3*(b^2 - 4*a*c)*e*(2*c*d - b*e)*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])])/(2*( 
c*d^2 - b*d*e + a*e^2)^(3/2))))/((b^2 - 4*a*c)*(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 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 1165

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

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. \(654\) vs. \(2(240)=480\).

Time = 1.16 (sec) , antiderivative size = 655, normalized size of antiderivative = 2.58

method	result	size
default	\(\frac {-\frac {e^{2}}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (x +\frac {d}{e}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}-\frac {3 \left (b e -2 c d \right ) e \left (\frac {e^{2}}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}-\frac {\left (b e -2 c d \right ) e \left (2 c \left (x +\frac {d}{e}\right )+\frac {b e -2 c d}{e}\right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (\frac {4 c \left (a \,e^{2}-b d e +c \,d^{2}\right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}-\frac {e^{2} \ln \left (\frac {\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}\right )}{2 \left (a \,e^{2}-b d e +c \,d^{2}\right )}-\frac {4 c \,e^{2} \left (2 c \left (x +\frac {d}{e}\right )+\frac {b e -2 c d}{e}\right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (\frac {4 c \left (a \,e^{2}-b d e +c \,d^{2}\right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}}{e^{2}}\)	\(655\)

Input:

int(1/(e*x+d)^2/(c*x^2+b*x+a)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1481 vs. \(2 (240) = 480\).

Time = 0.96 (sec) , antiderivative size = 3004, normalized size of antiderivative = 11.83 \[ \int \frac {1}{(d+e x)^2 \left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

integrate(1/(e*x+d)^2/(c*x^2+b*x+a)^(3/2),x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F]

\[ \int \frac {1}{(d+e x)^2 \left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {1}{\left (d + e x\right )^{2} \left (a + b x + c x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(1/(e*x+d)**2/(c*x**2+b*x+a)**(3/2),x)
 

Output:

Integral(1/((d + e*x)**2*(a + b*x + c*x**2)**(3/2)), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{(d+e x)^2 \left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(1/(e*x+d)^2/(c*x^2+b*x+a)^(3/2),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(a*e^2-b*d*e>0)', see `assume?` f 
or more de
 

Giac [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x)^2 \left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:

integrate(1/(e*x+d)^2/(c*x^2+b*x+a)^(3/2),x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x)^2 \left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {1}{{\left (d+e\,x\right )}^2\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,d x \] Input:

int(1/((d + e*x)^2*(a + b*x + c*x^2)^(3/2)),x)
 

Output:

int(1/((d + e*x)^2*(a + b*x + c*x^2)^(3/2)), x)
 

Reduce [B] (verification not implemented)

Time = 1.61 (sec) , antiderivative size = 4195, normalized size of antiderivative = 16.52 \[ \int \frac {1}{(d+e x)^2 \left (a+b x+c x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

int(1/(e*x+d)^2/(c*x^2+b*x+a)^(3/2),x)
 

Output:

(12*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*b*c*d*e**3 + 1 
2*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*b*c*e**4*x - 24* 
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**2 - 2 
4*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**3*x - 
3*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**3*d*e**3 - 3*sqr 
t(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**3*e**4*x + 6*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**2*e**2 + 18*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**3*x + 12*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*e**4*x**2 - 24*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*c**2*d**2*e**2*x - 12*...