\(\int \frac {A+B x^2+C x^4}{x (c+d x^2) \sqrt {a+c x^4}} \, dx\) [24]

Optimal result

Integrand size = 36, antiderivative size = 143 \[ \int \frac {A+B x^2+C x^4}{x \left (c+d x^2\right ) \sqrt {a+c x^4}} \, dx=\frac {C \text {arctanh}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right )}{2 \sqrt {c} d}+\frac {\left (c^2 C-B c d+A d^2\right ) \text {arctanh}\left (\frac {a d-c^2 x^2}{\sqrt {c^3+a d^2} \sqrt {a+c x^4}}\right )}{2 c d \sqrt {c^3+a d^2}}-\frac {A \text {arctanh}\left (\frac {\sqrt {a+c x^4}}{\sqrt {a}}\right )}{2 \sqrt {a} c} \] Output:

1/2*C*arctanh(c^(1/2)*x^2/(c*x^4+a)^(1/2))/c^(1/2)/d+1/2*(A*d^2-B*c*d+C*c^ 
2)*arctanh((-c^2*x^2+a*d)/(a*d^2+c^3)^(1/2)/(c*x^4+a)^(1/2))/c/d/(a*d^2+c^ 
3)^(1/2)-1/2*A*arctanh((c*x^4+a)^(1/2)/a^(1/2))/a^(1/2)/c
 

Mathematica [A] (verified)

Time = 1.01 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.15 \[ \int \frac {A+B x^2+C x^4}{x \left (c+d x^2\right ) \sqrt {a+c x^4}} \, dx=\frac {\frac {2 A \text {arctanh}\left (\frac {\sqrt {c} x^2-\sqrt {a+c x^4}}{\sqrt {a}}\right )}{\sqrt {a}}+\frac {\frac {2 \left (c^2 C-B c d+A d^2\right ) \arctan \left (\frac {c^{3/2}+\sqrt {c} d x^2-d \sqrt {a+c x^4}}{\sqrt {-c^3-a d^2}}\right )}{\sqrt {-c^3-a d^2}}-\sqrt {c} C \log \left (-\sqrt {c} x^2+\sqrt {a+c x^4}\right )}{d}}{2 c} \] Input:

Integrate[(A + B*x^2 + C*x^4)/(x*(c + d*x^2)*Sqrt[a + c*x^4]),x]
 

Output:

((2*A*ArcTanh[(Sqrt[c]*x^2 - Sqrt[a + c*x^4])/Sqrt[a]])/Sqrt[a] + ((2*(c^2 
*C - B*c*d + A*d^2)*ArcTan[(c^(3/2) + Sqrt[c]*d*x^2 - d*Sqrt[a + c*x^4])/S 
qrt[-c^3 - a*d^2]])/Sqrt[-c^3 - a*d^2] - Sqrt[c]*C*Log[-(Sqrt[c]*x^2) + Sq 
rt[a + c*x^4]])/d)/(2*c)
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {2249, 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.

\(\displaystyle \int \frac {A+B x^2+C x^4}{x \sqrt {a+c x^4} \left (c+d x^2\right )} \, dx\)

\(\Big \downarrow \) 2249

\(\displaystyle \int \left (-\frac {x \left (A d^2-B c d+c^2 C\right )}{c d \sqrt {a+c x^4} \left (c+d x^2\right )}+\frac {A}{c x \sqrt {a+c x^4}}+\frac {C x}{d \sqrt {a+c x^4}}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\text {arctanh}\left (\frac {a d-c^2 x^2}{\sqrt {a d^2+c^3} \sqrt {a+c x^4}}\right ) \left (A d^2-B c d+c^2 C\right )}{2 c d \sqrt {a d^2+c^3}}-\frac {A \text {arctanh}\left (\frac {\sqrt {a+c x^4}}{\sqrt {a}}\right )}{2 \sqrt {a} c}+\frac {C \text {arctanh}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right )}{2 \sqrt {c} d}\)

Input:

Int[(A + B*x^2 + C*x^4)/(x*(c + d*x^2)*Sqrt[a + c*x^4]),x]
 

Output:

(C*ArcTanh[(Sqrt[c]*x^2)/Sqrt[a + c*x^4]])/(2*Sqrt[c]*d) + ((c^2*C - B*c*d 
 + A*d^2)*ArcTanh[(a*d - c^2*x^2)/(Sqrt[c^3 + a*d^2]*Sqrt[a + c*x^4])])/(2 
*c*d*Sqrt[c^3 + a*d^2]) - (A*ArcTanh[Sqrt[a + c*x^4]/Sqrt[a]])/(2*Sqrt[a]* 
c)
 

Defintions of rubi rules used

rule 2009

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

rule 2249

Int[(Px_)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_) 
^4)^(p_), x_Symbol] :> Int[ExpandIntegrand[1/Sqrt[a + c*x^4], Px*(f*x)^m*(d 
 + e*x^2)^q*(a + c*x^4)^(p + 1/2), x], x] /; FreeQ[{a, c, d, e, f, m}, x] & 
& PolyQ[Px, x] && IntegerQ[p + 1/2] && IntegerQ[q]
 

Maple [A] (verified)

Time = 0.52 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.45

method	result	size
default	\(-\frac {A \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{4}+a}}{x^{2}}\right )}{2 c \sqrt {a}}+\frac {C \ln \left (\sqrt {c}\, x^{2}+\sqrt {c \,x^{4}+a}\right )}{2 d \sqrt {c}}+\frac {\left (A \,d^{2}-B c d +C \,c^{2}\right ) \ln \left (\frac {\frac {2 a \,d^{2}+2 c^{3}}{d^{2}}-\frac {2 c^{2} \left (x^{2}+\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+c^{3}}{d^{2}}}\, \sqrt {\left (x^{2}+\frac {c}{d}\right )^{2} c -\frac {2 c^{2} \left (x^{2}+\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+c^{3}}{d^{2}}}}{x^{2}+\frac {c}{d}}\right )}{2 c \,d^{2} \sqrt {\frac {a \,d^{2}+c^{3}}{d^{2}}}}\)	\(208\)
elliptic	\(-\frac {A \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{4}+a}}{x^{2}}\right )}{2 c \sqrt {a}}+\frac {C \ln \left (\sqrt {c}\, x^{2}+\sqrt {c \,x^{4}+a}\right )}{2 d \sqrt {c}}+\frac {\left (A \,d^{2}-B c d +C \,c^{2}\right ) \ln \left (\frac {\frac {2 a \,d^{2}+2 c^{3}}{d^{2}}-\frac {2 c^{2} \left (x^{2}+\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+c^{3}}{d^{2}}}\, \sqrt {\left (x^{2}+\frac {c}{d}\right )^{2} c -\frac {2 c^{2} \left (x^{2}+\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+c^{3}}{d^{2}}}}{x^{2}+\frac {c}{d}}\right )}{2 c \,d^{2} \sqrt {\frac {a \,d^{2}+c^{3}}{d^{2}}}}\)	\(208\)

Input:

int((C*x^4+B*x^2+A)/x/(d*x^2+c)/(c*x^4+a)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/2*A/c/a^(1/2)*ln((2*a+2*a^(1/2)*(c*x^4+a)^(1/2))/x^2)+1/2*C/d*ln(c^(1/2 
)*x^2+(c*x^4+a)^(1/2))/c^(1/2)+1/2*(A*d^2-B*c*d+C*c^2)/c/d^2/((a*d^2+c^3)/ 
d^2)^(1/2)*ln((2*(a*d^2+c^3)/d^2-2*c^2/d*(x^2+c/d)+2*((a*d^2+c^3)/d^2)^(1/ 
2)*((x^2+c/d)^2*c-2*c^2/d*(x^2+c/d)+(a*d^2+c^3)/d^2)^(1/2))/(x^2+c/d))
 

Fricas [F(-1)]

Timed out. \[ \int \frac {A+B x^2+C x^4}{x \left (c+d x^2\right ) \sqrt {a+c x^4}} \, dx=\text {Timed out} \] Input:

integrate((C*x^4+B*x^2+A)/x/(d*x^2+c)/(c*x^4+a)^(1/2),x, algorithm="fricas 
")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {A+B x^2+C x^4}{x \left (c+d x^2\right ) \sqrt {a+c x^4}} \, dx=\int \frac {A + B x^{2} + C x^{4}}{x \sqrt {a + c x^{4}} \left (c + d x^{2}\right )}\, dx \] Input:

integrate((C*x**4+B*x**2+A)/x/(d*x**2+c)/(c*x**4+a)**(1/2),x)
 

Output:

Integral((A + B*x**2 + C*x**4)/(x*sqrt(a + c*x**4)*(c + d*x**2)), x)
 

Maxima [F]

\[ \int \frac {A+B x^2+C x^4}{x \left (c+d x^2\right ) \sqrt {a+c x^4}} \, dx=\int { \frac {C x^{4} + B x^{2} + A}{\sqrt {c x^{4} + a} {\left (d x^{2} + c\right )} x} \,d x } \] Input:

integrate((C*x^4+B*x^2+A)/x/(d*x^2+c)/(c*x^4+a)^(1/2),x, algorithm="maxima 
")
 

Output:

integrate((C*x^4 + B*x^2 + A)/(sqrt(c*x^4 + a)*(d*x^2 + c)*x), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {A+B x^2+C x^4}{x \left (c+d x^2\right ) \sqrt {a+c x^4}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((C*x^4+B*x^2+A)/x/(d*x^2+c)/(c*x^4+a)^(1/2),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x^2+C x^4}{x \left (c+d x^2\right ) \sqrt {a+c x^4}} \, dx=\int \frac {C\,x^4+B\,x^2+A}{x\,\sqrt {c\,x^4+a}\,\left (d\,x^2+c\right )} \,d x \] Input:

int((A + B*x^2 + C*x^4)/(x*(a + c*x^4)^(1/2)*(c + d*x^2)),x)
 

Output:

int((A + B*x^2 + C*x^4)/(x*(a + c*x^4)^(1/2)*(c + d*x^2)), x)
 

Reduce [F]

\[ \int \frac {A+B x^2+C x^4}{x \left (c+d x^2\right ) \sqrt {a+c x^4}} \, dx=\left (\int \frac {x^{3}}{\sqrt {c \,x^{4}+a}\, c +\sqrt {c \,x^{4}+a}\, d \,x^{2}}d x \right ) c +\left (\int \frac {x}{\sqrt {c \,x^{4}+a}\, c +\sqrt {c \,x^{4}+a}\, d \,x^{2}}d x \right ) b +\left (\int \frac {1}{\sqrt {c \,x^{4}+a}\, c x +\sqrt {c \,x^{4}+a}\, d \,x^{3}}d x \right ) a \] Input:

int((C*x^4+B*x^2+A)/x/(d*x^2+c)/(c*x^4+a)^(1/2),x)
 

Output:

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