\(\int \frac {1}{a b^2+b^3 x+c x^2+b^2 d x^3+a d^2 x^4} \, dx\) [29]

Optimal result

Integrand size = 34, antiderivative size = 479 \[ \int \frac {1}{a b^2+b^3 x+c x^2+b^2 d x^3+a d^2 x^4} \, dx=\frac {\left (b^2-\sqrt {b^4-4 a c+8 a^2 b d}\right ) \text {arctanh}\left (\frac {b^2-\sqrt {b^4-4 a c+8 a^2 b d}+4 a d x}{\sqrt {2} \sqrt {b^4-2 a c-4 a^2 b d-b^2 \sqrt {b^4-4 a c+8 a^2 b d}}}\right )}{\sqrt {2} b \sqrt {b^4-4 a c+8 a^2 b d} \sqrt {b^4-2 a c-4 a^2 b d-b^2 \sqrt {b^4-4 a c+8 a^2 b d}}}-\frac {\left (b^2+\sqrt {b^4-4 a c+8 a^2 b d}\right ) \text {arctanh}\left (\frac {b^2+\sqrt {b^4-4 a c+8 a^2 b d}+4 a d x}{\sqrt {2} \sqrt {b^4-2 a c-4 a^2 b d+b^2 \sqrt {b^4-4 a c+8 a^2 b d}}}\right )}{\sqrt {2} b \sqrt {b^4-4 a c+8 a^2 b d} \sqrt {b^4-2 a c-4 a^2 b d+b^2 \sqrt {b^4-4 a c+8 a^2 b d}}}-\frac {\log \left (2 a b+b^2 x-\sqrt {b^4-4 a c+8 a^2 b d} x+2 a d x^2\right )}{2 b \sqrt {b^4-4 a c+8 a^2 b d}}+\frac {\log \left (2 a b+b^2 x+\sqrt {b^4-4 a c+8 a^2 b d} x+2 a d x^2\right )}{2 b \sqrt {b^4-4 a c+8 a^2 b d}} \] Output:

1/2*(b^2-(8*a^2*b*d+b^4-4*a*c)^(1/2))*arctanh(1/2*(b^2-(8*a^2*b*d+b^4-4*a* 
c)^(1/2)+4*a*d*x)*2^(1/2)/(b^4-2*a*c-4*a^2*b*d-b^2*(8*a^2*b*d+b^4-4*a*c)^( 
1/2))^(1/2))*2^(1/2)/b/(8*a^2*b*d+b^4-4*a*c)^(1/2)/(b^4-2*a*c-4*a^2*b*d-b^ 
2*(8*a^2*b*d+b^4-4*a*c)^(1/2))^(1/2)-1/2*(b^2+(8*a^2*b*d+b^4-4*a*c)^(1/2)) 
*arctanh(1/2*(b^2+(8*a^2*b*d+b^4-4*a*c)^(1/2)+4*a*d*x)*2^(1/2)/(b^4-2*a*c- 
4*a^2*b*d+b^2*(8*a^2*b*d+b^4-4*a*c)^(1/2))^(1/2))*2^(1/2)/b/(8*a^2*b*d+b^4 
-4*a*c)^(1/2)/(b^4-2*a*c-4*a^2*b*d+b^2*(8*a^2*b*d+b^4-4*a*c)^(1/2))^(1/2)- 
1/2*ln(2*a*b+b^2*x-(8*a^2*b*d+b^4-4*a*c)^(1/2)*x+2*a*d*x^2)/b/(8*a^2*b*d+b 
^4-4*a*c)^(1/2)+1/2*ln(2*a*b+b^2*x+(8*a^2*b*d+b^4-4*a*c)^(1/2)*x+2*a*d*x^2 
)/b/(8*a^2*b*d+b^4-4*a*c)^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 0.04 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.16 \[ \int \frac {1}{a b^2+b^3 x+c x^2+b^2 d x^3+a d^2 x^4} \, dx=\text {RootSum}\left [a b^2+b^3 \text {$\#$1}+c \text {$\#$1}^2+b^2 d \text {$\#$1}^3+a d^2 \text {$\#$1}^4\&,\frac {\log (x-\text {$\#$1})}{b^3+2 c \text {$\#$1}+3 b^2 d \text {$\#$1}^2+4 a d^2 \text {$\#$1}^3}\&\right ] \] Input:

Integrate[(a*b^2 + b^3*x + c*x^2 + b^2*d*x^3 + a*d^2*x^4)^(-1),x]
 

Output:

RootSum[a*b^2 + b^3*#1 + c*#1^2 + b^2*d*#1^3 + a*d^2*#1^4 & , Log[x - #1]/ 
(b^3 + 2*c*#1 + 3*b^2*d*#1^2 + 4*a*d^2*#1^3) & ]
 

Rubi [A] (verified)

Time = 2.14 (sec) , antiderivative size = 613, normalized size of antiderivative = 1.28, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2492, 2009}

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.

\(\displaystyle \int \frac {1}{a b^2+a d^2 x^4+b^3 x+b^2 d x^3+c x^2} \, dx\)

\(\Big \downarrow \) 2492

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

\(\Big \downarrow \) 2009

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

Input:

Int[(a*b^2 + b^3*x + c*x^2 + b^2*d*x^3 + a*d^2*x^4)^(-1),x]
 

Output:

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

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2492

Int[(Px_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2 + (d_.)*(x_)^3 + (e_.)*(x_)^4) 
^(p_), x_Symbol] :> Simp[e^p   Int[ExpandIntegrand[Px*(b/d + ((d + Sqrt[e*( 
(b^2 - 4*a*c)/a) + 8*a*d*(e/b)])/(2*e))*x + x^2)^p*(b/d + ((d - Sqrt[e*((b^ 
2 - 4*a*c)/a) + 8*a*d*(e/b)])/(2*e))*x + x^2)^p, x], x], x] /; FreeQ[{a, b, 
 c, d, e}, x] && PolyQ[Px, x] && ILtQ[p, 0] && EqQ[a*d^2 - b^2*e, 0]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.09 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.15

method	result	size
default	\(\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,d^{2} \textit {\_Z}^{4}+d \,b^{2} \textit {\_Z}^{3}+b^{3} \textit {\_Z} +\textit {\_Z}^{2} c +b^{2} a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{4 \textit {\_R}^{3} a \,d^{2}+3 \textit {\_R}^{2} b^{2} d +b^{3}+2 \textit {\_R} c}\)	\(72\)
risch	\(\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,d^{2} \textit {\_Z}^{4}+d \,b^{2} \textit {\_Z}^{3}+b^{3} \textit {\_Z} +\textit {\_Z}^{2} c +b^{2} a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{4 \textit {\_R}^{3} a \,d^{2}+3 \textit {\_R}^{2} b^{2} d +b^{3}+2 \textit {\_R} c}\)	\(72\)

Input:

int(1/(a*d^2*x^4+b^2*d*x^3+b^3*x+a*b^2+c*x^2),x,method=_RETURNVERBOSE)
 

Output:

sum(1/(4*_R^3*a*d^2+3*_R^2*b^2*d+b^3+2*_R*c)*ln(x-_R),_R=RootOf(_Z^4*a*d^2 
+_Z^3*b^2*d+_Z*b^3+_Z^2*c+a*b^2))
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{a b^2+b^3 x+c x^2+b^2 d x^3+a d^2 x^4} \, dx=\text {Timed out} \] Input:

integrate(1/(a*d^2*x^4+b^2*d*x^3+b^3*x+a*b^2+c*x^2),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{a b^2+b^3 x+c x^2+b^2 d x^3+a d^2 x^4} \, dx=\text {Timed out} \] Input:

integrate(1/(a*d**2*x**4+b**2*d*x**3+b**3*x+a*b**2+c*x**2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {1}{a b^2+b^3 x+c x^2+b^2 d x^3+a d^2 x^4} \, dx=\int { \frac {1}{a d^{2} x^{4} + b^{2} d x^{3} + b^{3} x + a b^{2} + c x^{2}} \,d x } \] Input:

integrate(1/(a*d^2*x^4+b^2*d*x^3+b^3*x+a*b^2+c*x^2),x, algorithm="maxima")
 

Output:

integrate(1/(a*d^2*x^4 + b^2*d*x^3 + b^3*x + a*b^2 + c*x^2), x)
 

Giac [F]

\[ \int \frac {1}{a b^2+b^3 x+c x^2+b^2 d x^3+a d^2 x^4} \, dx=\int { \frac {1}{a d^{2} x^{4} + b^{2} d x^{3} + b^{3} x + a b^{2} + c x^{2}} \,d x } \] Input:

integrate(1/(a*d^2*x^4+b^2*d*x^3+b^3*x+a*b^2+c*x^2),x, algorithm="giac")
 

Output:

integrate(1/(a*d^2*x^4 + b^2*d*x^3 + b^3*x + a*b^2 + c*x^2), x)
                                                                                    
                                                                                    
 

Mupad [B] (verification not implemented)

Time = 23.62 (sec) , antiderivative size = 5563, normalized size of antiderivative = 11.61 \[ \int \frac {1}{a b^2+b^3 x+c x^2+b^2 d x^3+a d^2 x^4} \, dx=\text {Too large to display} \] Input:

int(1/(a*b^2 + b^3*x + c*x^2 + a*d^2*x^4 + b^2*d*x^3),x)
 

Output:

symsum(log(4*root(288*a^3*b^8*c*d^2*z^4 - 80*a^2*b^7*c^2*d*z^4 - 128*a^4*b 
^4*c^2*d^2*z^4 + 36*a*b^11*c*d*z^4 + 256*a^6*b^6*d^4*z^4 - 192*a^4*b^9*d^3 
*z^4 - 60*a^2*b^12*d^2*z^4 + 16*a^2*b^2*c^4*z^4 - 8*a*b^6*c^3*z^4 - 4*b^15 
*d*z^4 + b^10*c^2*z^4 - 16*a^3*b^2*c*d^2*z^2 - 14*a*b^5*c*d*z^2 + 24*a^2*b 
^6*d^2*z^2 - b^4*c^2*z^2 + 3*b^9*d*z^2 + 4*a*c^3*z^2 - 4*a*b^2*c*d*z + 8*a 
^2*b^3*d^2*z + b^6*d*z + a^2*d^2, z, k)^2*a^2*c^2*d^4 - 16*root(288*a^3*b^ 
8*c*d^2*z^4 - 80*a^2*b^7*c^2*d*z^4 - 128*a^4*b^4*c^2*d^2*z^4 + 36*a*b^11*c 
*d*z^4 + 256*a^6*b^6*d^4*z^4 - 192*a^4*b^9*d^3*z^4 - 60*a^2*b^12*d^2*z^4 + 
 16*a^2*b^2*c^4*z^4 - 8*a*b^6*c^3*z^4 - 4*b^15*d*z^4 + b^10*c^2*z^4 - 16*a 
^3*b^2*c*d^2*z^2 - 14*a*b^5*c*d*z^2 + 24*a^2*b^6*d^2*z^2 - b^4*c^2*z^2 + 3 
*b^9*d*z^2 + 4*a*c^3*z^2 - 4*a*b^2*c*d*z + 8*a^2*b^3*d^2*z + b^6*d*z + a^2 
*d^2, z, k)^2*a^4*b^2*d^6 - 6*root(288*a^3*b^8*c*d^2*z^4 - 80*a^2*b^7*c^2* 
d*z^4 - 128*a^4*b^4*c^2*d^2*z^4 + 36*a*b^11*c*d*z^4 + 256*a^6*b^6*d^4*z^4 
- 192*a^4*b^9*d^3*z^4 - 60*a^2*b^12*d^2*z^4 + 16*a^2*b^2*c^4*z^4 - 8*a*b^6 
*c^3*z^4 - 4*b^15*d*z^4 + b^10*c^2*z^4 - 16*a^3*b^2*c*d^2*z^2 - 14*a*b^5*c 
*d*z^2 + 24*a^2*b^6*d^2*z^2 - b^4*c^2*z^2 + 3*b^9*d*z^2 + 4*a*c^3*z^2 - 4* 
a*b^2*c*d*z + 8*a^2*b^3*d^2*z + b^6*d*z + a^2*d^2, z, k)^3*a^2*b^8*d^5 - 4 
8*root(288*a^3*b^8*c*d^2*z^4 - 80*a^2*b^7*c^2*d*z^4 - 128*a^4*b^4*c^2*d^2* 
z^4 + 36*a*b^11*c*d*z^4 + 256*a^6*b^6*d^4*z^4 - 192*a^4*b^9*d^3*z^4 - 60*a 
^2*b^12*d^2*z^4 + 16*a^2*b^2*c^4*z^4 - 8*a*b^6*c^3*z^4 - 4*b^15*d*z^4 +...
 

Reduce [F]

\[ \int \frac {1}{a b^2+b^3 x+c x^2+b^2 d x^3+a d^2 x^4} \, dx=\int \frac {1}{a \,d^{2} x^{4}+b^{2} d \,x^{3}+b^{3} x +a \,b^{2}+c \,x^{2}}d x \] Input:

int(1/(a*d^2*x^4+b^2*d*x^3+b^3*x+a*b^2+c*x^2),x)
 

Output:

int(1/(a*b**2 + a*d**2*x**4 + b**3*x + b**2*d*x**3 + c*x**2),x)