\(\int \frac {\sqrt {d+e x^2} (A+B x^2+C x^4)}{a+b x^2+c x^4} \, dx\) [156]

Optimal result

Integrand size = 38, antiderivative size = 504 \[ \int \frac {\sqrt {d+e x^2} \left (A+B x^2+C x^4\right )}{a+b x^2+c x^4} \, dx=\frac {C x \sqrt {d+e x^2}}{2 c}-\frac {\left (b c C d-b^2 C e-c (A c-a C) e-B c (c d-b e)+\frac {b^3 C e-b^2 c (C d+B e)-2 c^2 (A c d-a C d-a B e)+b c (B c d+A c e-3 a C e)}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{c^2 \sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {\left (b c C d-b^2 C e-c (A c-a C) e-B c (c d-b e)-\frac {b^3 C e-b^2 c (C d+B e)+b c (B c d+A c e-3 a C e)-2 c^2 (A c d-a (C d+B e))}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{c^2 \sqrt {b+\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}+\frac {(c C d+2 B c e-2 b C e) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 c^2 \sqrt {e}} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(22359\) vs. \(2(504)=1008\).

Time = 17.05 (sec) , antiderivative size = 22359, normalized size of antiderivative = 44.36 \[ \int \frac {\sqrt {d+e x^2} \left (A+B x^2+C x^4\right )}{a+b x^2+c x^4} \, dx=\text {Result too large to show} \] Input:

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

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 1.30 (sec) , antiderivative size = 516, normalized size of antiderivative = 1.02, 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.

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

\(\Big \downarrow \) 2256

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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 = 0.76 (sec) , antiderivative size = 490, normalized size of antiderivative = 0.97

method	result	size
risch	\(\frac {C x \sqrt {e \,x^{2}+d}}{2 c}+\frac {\frac {\left (2 B c e -2 C b e +C c d \right ) \ln \left (\sqrt {e \,x^{2}+d}+x \sqrt {e}\right )}{c \sqrt {e}}+\frac {\left (\left (\left (\left (-C c d -e \left (B c -b C \right )\right ) a +A \,c^{2} d \right ) \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}+2 d \left (-C \,a^{2} c e +\left (c \left (B c -\frac {b C}{2}\right ) d +e \left (A \,c^{2}-\frac {1}{2} b B c +\frac {1}{2} b^{2} C \right )\right ) a -\frac {A b \,c^{2} d}{2}\right )\right ) \sqrt {\left (-2 a e +b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}\, \operatorname {arctanh}\left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (2 a e -b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}}\right )-\arctan \left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (-2 a e +b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}}\right ) \left (\left (\left (-C c d -e \left (B c -b C \right )\right ) a +A \,c^{2} d \right ) \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}-2 d \left (-C \,a^{2} c e +\left (c \left (B c -\frac {b C}{2}\right ) d +e \left (A \,c^{2}-\frac {1}{2} b B c +\frac {1}{2} b^{2} C \right )\right ) a -\frac {A b \,c^{2} d}{2}\right )\right ) \sqrt {\left (2 a e -b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}\right ) \sqrt {2}}{c \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\, \sqrt {\left (-2 a e +b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}\, \sqrt {\left (2 a e -b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}}}{2 c}\)	\(490\)
pseudoelliptic	\(\frac {\left (\frac {\left (\left (\left (-B c +b C \right ) e -C c d \right ) a +A \,c^{2} d \right ) \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}}{2}+d \left (-C \,a^{2} c e +\left (c \left (B c -\frac {b C}{2}\right ) d +e \left (A \,c^{2}-\frac {1}{2} b B c +\frac {1}{2} b^{2} C \right )\right ) a -\frac {A b \,c^{2} d}{2}\right )\right ) \sqrt {e}\, \sqrt {\left (-2 a e +b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (2 a e -b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}}\right )+\sqrt {\left (2 a e -b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}\, \left (\left (\frac {\left (\left (C c d +e \left (B c -b C \right )\right ) a -A \,c^{2} d \right ) \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}}{2}+d \left (-C \,a^{2} c e +\left (c \left (B c -\frac {b C}{2}\right ) d +e \left (A \,c^{2}-\frac {1}{2} b B c +\frac {1}{2} b^{2} C \right )\right ) a -\frac {A b \,c^{2} d}{2}\right )\right ) \sqrt {e}\, \sqrt {2}\, \arctan \left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (-2 a e +b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}}\right )+\frac {\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\, \sqrt {\left (-2 a e +b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}\, \left (\left (2 e \left (B c -b C \right )+C c d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}}{x \sqrt {e}}\right )+C \sqrt {e \,x^{2}+d}\, c x \sqrt {e}\right )}{2}\right )}{\sqrt {\left (-2 a e +b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}\, \sqrt {e}\, \sqrt {\left (2 a e -b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}\, \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\, c^{2}}\)	\(537\)
default	\(\frac {C \left (\frac {x \sqrt {e \,x^{2}+d}}{2}+\frac {d \ln \left (\sqrt {e \,x^{2}+d}+x \sqrt {e}\right )}{2 \sqrt {e}}\right )}{c}+\frac {\frac {\left (\left (\left (-C c d -e \left (B c -b C \right )\right ) a +A \,c^{2} d \right ) \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}+2 d \left (-C \,a^{2} c e +\left (c \left (B c -\frac {b C}{2}\right ) d +e \left (A \,c^{2}-\frac {1}{2} b B c +\frac {1}{2} b^{2} C \right )\right ) a -\frac {A b \,c^{2} d}{2}\right )\right ) \sqrt {\left (-2 a e +b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (2 a e -b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}}\right )}{2}+\sqrt {\left (2 a e -b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}\, \left (-\frac {\left (\left (\left (-C c d -e \left (B c -b C \right )\right ) a +A \,c^{2} d \right ) \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}-2 d \left (-C \,a^{2} c e +\left (c \left (B c -\frac {b C}{2}\right ) d +e \left (A \,c^{2}-\frac {1}{2} b B c +\frac {1}{2} b^{2} C \right )\right ) a -\frac {A b \,c^{2} d}{2}\right )\right ) \sqrt {2}\, \arctan \left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (-2 a e +b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}}\right )}{2}+\operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}}{x \sqrt {e}}\right ) \sqrt {e}\, \left (B c -b C \right ) \sqrt {\left (-2 a e +b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}\, \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right )}{c^{2} \sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\, \sqrt {\left (-2 a e +b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}\, \sqrt {\left (2 a e -b d +\sqrt {-4 \left (a c -\frac {b^{2}}{4}\right ) d^{2}}\right ) a}}\)	\(549\)

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.08 \[ \int \frac {\sqrt {d+e x^2} \left (A+B x^2+C x^4\right )}{a+b x^2+c x^4} \, dx=\frac {\sqrt {e \,x^{2}+d}\, e x +\sqrt {e}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {e}\, x}{\sqrt {d}}\right ) d}{2 e} \] Input:

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

Output:

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