\(\int \frac {A+B x^2+C x^4}{\sqrt {d+e x^2} (a+b x^2+c x^4)} \, dx\) [157]

Optimal result

Integrand size = 38, antiderivative size = 337 \[ \int \frac {A+B x^2+C x^4}{\sqrt {d+e x^2} \left (a+b x^2+c x^4\right )} \, dx=\frac {\left (B c-b C-\frac {b B c-b^2 C-2 c (A c-a C)}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{c \sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {\left (B c-b C+\frac {b B c-b^2 C-2 c (A c-a C)}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{c \sqrt {b+\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}+\frac {C \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{c \sqrt {e}} \] Output:

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

Mathematica [A] (verified)

Time = 11.60 (sec) , antiderivative size = 329, normalized size of antiderivative = 0.98 \[ \int \frac {A+B x^2+C x^4}{\sqrt {d+e x^2} \left (a+b x^2+c x^4\right )} \, dx=\frac {\frac {\left (B c-b C+\frac {-b B c+b^2 C+2 c (A c-a C)}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2 c d-b e+\sqrt {b^2-4 a c} e} x}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d+\left (-b+\sqrt {b^2-4 a c}\right ) e}}+\frac {\left (B c-b C+\frac {b B c-b^2 C+2 c (-A c+a C)}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}+\frac {C \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{\sqrt {e}}}{c} \] Input:

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

Output:

(((B*c - b*C + (-(b*B*c) + b^2*C + 2*c*(A*c - a*C))/Sqrt[b^2 - 4*a*c])*Arc 
Tan[(Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]*x)/(Sqrt[b - Sqrt[b^2 - 4*a*c 
]]*Sqrt[d + e*x^2])])/(Sqrt[b - Sqrt[b^2 - 4*a*c]]*Sqrt[2*c*d + (-b + Sqrt 
[b^2 - 4*a*c])*e]) + ((B*c - b*C + (b*B*c - b^2*C + 2*c*(-(A*c) + a*C))/Sq 
rt[b^2 - 4*a*c])*ArcTan[(Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]*x)/(Sqrt[ 
b + Sqrt[b^2 - 4*a*c]]*Sqrt[d + e*x^2])])/(Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqr 
t[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]) + (C*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e* 
x^2]])/Sqrt[e])/c
 

Rubi [A] (verified)

Time = 1.00 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2256, 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.

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

\(\Big \downarrow \) 2256

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2009

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

rule 2256

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

Maple [A] (verified)

Time = 0.30 (sec) , antiderivative size = 327, normalized size of antiderivative = 0.97

method	result	size
default	\(\frac {C \ln \left (\sqrt {e \,x^{2}+d}+x \sqrt {e}\right )}{c \sqrt {e}}-\frac {\sqrt {2}\, \left (-\frac {\left (-A b c d +2 a B c d -C a b d +A \sqrt {-d^{2} \left (4 a c -b^{2}\right )}\, c -C \sqrt {-d^{2} \left (4 a c -b^{2}\right )}\, a \right ) \operatorname {arctanh}\left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (2 a e -b d +\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\right ) a}}\right )}{\sqrt {\left (2 a e -b d +\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\right ) a}}+\frac {\left (A b c d -2 a B c d +C a b d +A \sqrt {-d^{2} \left (4 a c -b^{2}\right )}\, c -C \sqrt {-d^{2} \left (4 a c -b^{2}\right )}\, a \right ) \arctan \left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (-2 a e +b d +\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\right ) a}}\right )}{\sqrt {\left (-2 a e +b d +\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\right ) a}}\right )}{2 c \sqrt {-d^{2} \left (4 a c -b^{2}\right )}}\)	\(327\)
pseudoelliptic	\(-\frac {-\sqrt {e}\, \sqrt {\left (-2 a e +b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}\, \left (\left (A c -C a \right ) \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}-d \left (\left (-2 B c +b C \right ) a +A b c \right )\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (2 a e -b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}}\right )+\sqrt {\left (2 a e -b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}\, \left (\sqrt {e}\, \sqrt {2}\, \left (\left (A c -C a \right ) \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}+d \left (\left (-2 B c +b C \right ) a +A b c \right )\right ) \arctan \left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (-2 a e +b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}}\right )-2 \sqrt {\left (-2 a e +b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}\, \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\, \operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}}{x \sqrt {e}}\right ) C \right )}{2 \sqrt {\left (-2 a e +b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}\, \sqrt {e}\, \sqrt {\left (2 a e -b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}\, \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\, c}\)	\(401\)

Input:

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

Output:

C/c*ln((e*x^2+d)^(1/2)+x*e^(1/2))/e^(1/2)-1/2/c*2^(1/2)/(-d^2*(4*a*c-b^2)) 
^(1/2)*(-(-A*b*c*d+2*a*B*c*d-C*a*b*d+A*(-d^2*(4*a*c-b^2))^(1/2)*c-C*(-d^2* 
(4*a*c-b^2))^(1/2)*a)/((2*a*e-b*d+(-d^2*(4*a*c-b^2))^(1/2))*a)^(1/2)*arcta 
nh(a*(e*x^2+d)^(1/2)/x*2^(1/2)/((2*a*e-b*d+(-d^2*(4*a*c-b^2))^(1/2))*a)^(1 
/2))+(A*b*c*d-2*a*B*c*d+C*a*b*d+A*(-d^2*(4*a*c-b^2))^(1/2)*c-C*(-d^2*(4*a* 
c-b^2))^(1/2)*a)/((-2*a*e+b*d+(-d^2*(4*a*c-b^2))^(1/2))*a)^(1/2)*arctan(a* 
(e*x^2+d)^(1/2)/x*2^(1/2)/((-2*a*e+b*d+(-d^2*(4*a*c-b^2))^(1/2))*a)^(1/2)) 
)
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E 
rror: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.07 \[ \int \frac {A+B x^2+C x^4}{\sqrt {d+e x^2} \left (a+b x^2+c x^4\right )} \, dx=\frac {\sqrt {e}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {e}\, x}{\sqrt {d}}\right )}{e} \] Input:

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

Output:

(sqrt(e)*log((sqrt(d + e*x**2) + sqrt(e)*x)/sqrt(d)))/e