Integrand size = 33, antiderivative size = 497 \[ \int \frac {\sqrt {d+e x^2} \left (A+B x^2+C x^4\right )}{a+c x^4} \, dx=\frac {C x \sqrt {d+e x^2}}{2 c}+\frac {\sqrt {\sqrt {a} e+\sqrt {c d^2+a e^2}} \left (d (B c d+A c e-a C e)-(A c d-a (C d+B e)) \left (e-\frac {\sqrt {c d^2+a e^2}}{\sqrt {a}}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt {\sqrt {a} e+\sqrt {c d^2+a e^2}} x \sqrt {d+e x^2}}{\sqrt {a} \left (\sqrt {a} e+\sqrt {c d^2+a e^2}\right )-c d x^2}\right )}{2 \sqrt {2} \sqrt [4]{a} c^{3/2} d \sqrt {c d^2+a e^2}}+\frac {(C d+2 B e) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 c \sqrt {e}}+\frac {\sqrt {-\sqrt {a} e+\sqrt {c d^2+a e^2}} \left (d (B c d+A c e-a C e)-(A c d-a (C d+B e)) \left (e+\frac {\sqrt {c d^2+a e^2}}{\sqrt {a}}\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt {-\sqrt {a} e+\sqrt {c d^2+a e^2}} x \sqrt {d+e x^2}}{\sqrt {a} \left (\sqrt {a} e-\sqrt {c d^2+a e^2}\right )-c d x^2}\right )}{2 \sqrt {2} \sqrt [4]{a} c^{3/2} d \sqrt {c d^2+a e^2}} \] Output:
1/2*C*x*(e*x^2+d)^(1/2)/c+1/4*(a^(1/2)*e+(a*e^2+c*d^2)^(1/2))^(1/2)*(d*(A* c*e+B*c*d-C*a*e)-(A*c*d-a*(B*e+C*d))*(e-(a*e^2+c*d^2)^(1/2)/a^(1/2)))*arct an(2^(1/2)*a^(1/4)*c^(1/2)*(a^(1/2)*e+(a*e^2+c*d^2)^(1/2))^(1/2)*x*(e*x^2+ d)^(1/2)/(a^(1/2)*(a^(1/2)*e+(a*e^2+c*d^2)^(1/2))-c*d*x^2))*2^(1/2)/a^(1/4 )/c^(3/2)/d/(a*e^2+c*d^2)^(1/2)+1/2*(2*B*e+C*d)*arctanh(e^(1/2)*x/(e*x^2+d )^(1/2))/c/e^(1/2)+1/4*(-a^(1/2)*e+(a*e^2+c*d^2)^(1/2))^(1/2)*(d*(A*c*e+B* c*d-C*a*e)-(A*c*d-a*(B*e+C*d))*(e+(a*e^2+c*d^2)^(1/2)/a^(1/2)))*arctanh(2^ (1/2)*a^(1/4)*c^(1/2)*(-a^(1/2)*e+(a*e^2+c*d^2)^(1/2))^(1/2)*x*(e*x^2+d)^( 1/2)/(a^(1/2)*(a^(1/2)*e-(a*e^2+c*d^2)^(1/2))-c*d*x^2))*2^(1/2)/a^(1/4)/c^ (3/2)/d/(a*e^2+c*d^2)^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.52 (sec) , antiderivative size = 557, normalized size of antiderivative = 1.12 \[ \int \frac {\sqrt {d+e x^2} \left (A+B x^2+C x^4\right )}{a+c x^4} \, dx=\frac {C \sqrt {e} x \sqrt {d+e x^2}-(C d+2 B e) \log \left (-\sqrt {e} x+\sqrt {d+e x^2}\right )+e \text {RootSum}\left [c d^4-4 c d^3 \text {$\#$1}+6 c d^2 \text {$\#$1}^2+16 a e^2 \text {$\#$1}^2-4 c d \text {$\#$1}^3+c \text {$\#$1}^4\&,\frac {B c d^3 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )+A c d^2 e \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )-a C d^2 e \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )-2 B c d^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}+2 A c d e \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}-2 a C d e \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}-4 a B e^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}+B c d \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+A c e \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}^2-a C e \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{c d^3-3 c d^2 \text {$\#$1}-8 a e^2 \text {$\#$1}+3 c d \text {$\#$1}^2-c \text {$\#$1}^3}\&\right ]}{2 c \sqrt {e}} \] Input:
Integrate[(Sqrt[d + e*x^2]*(A + B*x^2 + C*x^4))/(a + c*x^4),x]
Output:
(C*Sqrt[e]*x*Sqrt[d + e*x^2] - (C*d + 2*B*e)*Log[-(Sqrt[e]*x) + Sqrt[d + e *x^2]] + e*RootSum[c*d^4 - 4*c*d^3*#1 + 6*c*d^2*#1^2 + 16*a*e^2*#1^2 - 4*c *d*#1^3 + c*#1^4 & , (B*c*d^3*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2 ] - #1] + A*c*d^2*e*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] - a*C*d^2*e*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] - 2*B*c*d^2* Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 + 2*A*c*d*e*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 - 2*a*C*d*e*Log[d + 2*e*x^ 2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 - 4*a*B*e^2*Log[d + 2*e*x^2 - 2*S qrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 + B*c*d*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sq rt[d + e*x^2] - #1]*#1^2 + A*c*e*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e* x^2] - #1]*#1^2 - a*C*e*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1 ]*#1^2)/(c*d^3 - 3*c*d^2*#1 - 8*a*e^2*#1 + 3*c*d*#1^2 - c*#1^3) & ])/(2*c* Sqrt[e])
Time = 0.82 (sec) , antiderivative size = 382, normalized size of antiderivative = 0.77, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {2257, 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.
| \(\displaystyle \int \frac {\sqrt {d+e x^2} \left (A+B x^2+C x^4\right )}{a+c x^4} \, dx\) |
| \(\Big \downarrow \) 2257 |
| \(\displaystyle \int \left (\frac {\sqrt {d+e x^2} \left (-a C+A c+B c x^2\right )}{c \left (a+c x^4\right )}+\frac {C \sqrt {d+e x^2}}{c}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {\sqrt {c} d-\sqrt {-a} e} \left (\sqrt {-a} B \sqrt {c}+a C-A c\right ) \arctan \left (\frac {x \sqrt {\sqrt {c} d-\sqrt {-a} e}}{\sqrt [4]{-a} \sqrt {d+e x^2}}\right )}{2 (-a)^{3/4} c^{3/2}}+\frac {\sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \left (\sqrt {-a} B \sqrt {c}-a C+A c\right )}{2 \sqrt {-a} c^{3/2}}+\frac {\sqrt {e} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right ) \left (\sqrt {-a} B \sqrt {c}+a C-A c\right )}{2 \sqrt {-a} c^{3/2}}-\frac {\sqrt {\sqrt {-a} e+\sqrt {c} d} \left (\sqrt {-a} B \sqrt {c}-a C+A c\right ) \text {arctanh}\left (\frac {x \sqrt {\sqrt {-a} e+\sqrt {c} d}}{\sqrt [4]{-a} \sqrt {d+e x^2}}\right )}{2 (-a)^{3/4} c^{3/2}}+\frac {C d \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 c \sqrt {e}}+\frac {C x \sqrt {d+e x^2}}{2 c}\) |
Input:
Int[(Sqrt[d + e*x^2]*(A + B*x^2 + C*x^4))/(a + c*x^4),x]
Output:
(C*x*Sqrt[d + e*x^2])/(2*c) + ((Sqrt[-a]*B*Sqrt[c] - A*c + a*C)*Sqrt[Sqrt[ c]*d - Sqrt[-a]*e]*ArcTan[(Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*x)/((-a)^(1/4)*Sqr t[d + e*x^2])])/(2*(-a)^(3/4)*c^(3/2)) + (C*d*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(2*c*Sqrt[e]) + ((Sqrt[-a]*B*Sqrt[c] + A*c - a*C)*Sqrt[e]*ArcTan h[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(2*Sqrt[-a]*c^(3/2)) + ((Sqrt[-a]*B*Sqrt[c ] - A*c + a*C)*Sqrt[e]*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(2*Sqrt[-a]*c ^(3/2)) - ((Sqrt[-a]*B*Sqrt[c] + A*c - a*C)*Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*A rcTanh[(Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*x)/((-a)^(1/4)*Sqrt[d + e*x^2])])/(2* (-a)^(3/4)*c^(3/2))
Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol ] :> Int[ExpandIntegrand[Px*(d + e*x^2)^q*(a + c*x^4)^p, x], x] /; FreeQ[{a , c, d, e, q}, x] && PolyQ[Px, x] && IntegerQ[p]
Leaf count of result is larger than twice the leaf count of optimal. \(831\) vs. \(2(405)=810\).
Time = 0.39 (sec) , antiderivative size = 832, normalized size of antiderivative = 1.67
| method | result | size |
| pseudoelliptic | \(-\frac {4 c \,d^{2} \left (a^{\frac {3}{2}} \sqrt {e}\, B \sqrt {a \,e^{2}+c \,d^{2}}+\left (-B a \,e^{\frac {3}{2}}+\sqrt {e}\, d \left (A c -C a \right )\right ) a \right ) \arctan \left (\frac {2 \sqrt {a}\, \sqrt {e \,x^{2}+d}+\sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, x}{x \sqrt {4 \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {a}-2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}-2 a e}}\right )-4 c \,d^{2} \left (a^{\frac {3}{2}} \sqrt {e}\, B \sqrt {a \,e^{2}+c \,d^{2}}+\left (-B a \,e^{\frac {3}{2}}+\sqrt {e}\, d \left (A c -C a \right )\right ) a \right ) \arctan \left (\frac {\sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, x -2 \sqrt {a}\, \sqrt {e \,x^{2}+d}}{x \sqrt {4 \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {a}-2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}-2 a e}}\right )+\sqrt {4 \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {a}-2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}-2 a e}\, \left (\left (\left (-\sqrt {a}\, \sqrt {e}\, B \sqrt {a \,e^{2}+c \,d^{2}}-B a \,e^{\frac {3}{2}}+\sqrt {e}\, d \left (A c -C a \right )\right ) \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+a^{\frac {3}{2}} e^{\frac {3}{2}} \sqrt {a \,e^{2}+c \,d^{2}}\, B -a \left (d \left (A c -C a \right ) e^{\frac {3}{2}}-e^{\frac {5}{2}} B a \right )\right ) \sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, \ln \left (\frac {\sqrt {e \,x^{2}+d}\, \sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, x -x^{2} \sqrt {a \,e^{2}+c \,d^{2}}-\sqrt {a}\, \left (e \,x^{2}+d \right )}{x^{2}}\right )-\left (\left (-\sqrt {a}\, \sqrt {e}\, B \sqrt {a \,e^{2}+c \,d^{2}}-B a \,e^{\frac {3}{2}}+\sqrt {e}\, d \left (A c -C a \right )\right ) \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+a^{\frac {3}{2}} e^{\frac {3}{2}} \sqrt {a \,e^{2}+c \,d^{2}}\, B -a \left (d \left (A c -C a \right ) e^{\frac {3}{2}}-e^{\frac {5}{2}} B a \right )\right ) \sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, \ln \left (\frac {\sqrt {a}\, \left (e \,x^{2}+d \right )+x^{2} \sqrt {a \,e^{2}+c \,d^{2}}+\sqrt {e \,x^{2}+d}\, \sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, x}{x^{2}}\right )-4 c \left (\left (2 B e +C d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}}{x \sqrt {e}}\right )+\sqrt {e}\, \sqrt {e \,x^{2}+d}\, C x \right ) d^{2} a^{\frac {3}{2}}\right )}{8 \sqrt {e}\, a^{\frac {3}{2}} \sqrt {4 \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {a}-2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}-2 a e}\, d^{2} c^{2}}\) | \(832\) |
| risch | \(\text {Expression too large to display}\) | \(1044\) |
| default | \(\text {Expression too large to display}\) | \(1952\) |
Input:
int((e*x^2+d)^(1/2)*(C*x^4+B*x^2+A)/(c*x^4+a),x,method=_RETURNVERBOSE)
Output:
-1/8/e^(1/2)/a^(3/2)/(4*(a*e^2+c*d^2)^(1/2)*a^(1/2)-2*(a*(a*e^2+c*d^2))^(1 /2)-2*a*e)^(1/2)*(4*c*d^2*(a^(3/2)*e^(1/2)*B*(a*e^2+c*d^2)^(1/2)+(-B*a*e^( 3/2)+e^(1/2)*d*(A*c-C*a))*a)*arctan((2*a^(1/2)*(e*x^2+d)^(1/2)+(2*(a*(a*e^ 2+c*d^2))^(1/2)+2*a*e)^(1/2)*x)/x/(4*(a*e^2+c*d^2)^(1/2)*a^(1/2)-2*(a*(a*e ^2+c*d^2))^(1/2)-2*a*e)^(1/2))-4*c*d^2*(a^(3/2)*e^(1/2)*B*(a*e^2+c*d^2)^(1 /2)+(-B*a*e^(3/2)+e^(1/2)*d*(A*c-C*a))*a)*arctan(((2*(a*(a*e^2+c*d^2))^(1/ 2)+2*a*e)^(1/2)*x-2*a^(1/2)*(e*x^2+d)^(1/2))/x/(4*(a*e^2+c*d^2)^(1/2)*a^(1 /2)-2*(a*(a*e^2+c*d^2))^(1/2)-2*a*e)^(1/2))+(4*(a*e^2+c*d^2)^(1/2)*a^(1/2) -2*(a*(a*e^2+c*d^2))^(1/2)-2*a*e)^(1/2)*(((-a^(1/2)*e^(1/2)*B*(a*e^2+c*d^2 )^(1/2)-B*a*e^(3/2)+e^(1/2)*d*(A*c-C*a))*(a*(a*e^2+c*d^2))^(1/2)+a^(3/2)*e ^(3/2)*(a*e^2+c*d^2)^(1/2)*B-a*(d*(A*c-C*a)*e^(3/2)-e^(5/2)*B*a))*(2*(a*(a *e^2+c*d^2))^(1/2)+2*a*e)^(1/2)*ln(((e*x^2+d)^(1/2)*(2*(a*(a*e^2+c*d^2))^( 1/2)+2*a*e)^(1/2)*x-x^2*(a*e^2+c*d^2)^(1/2)-a^(1/2)*(e*x^2+d))/x^2)-((-a^( 1/2)*e^(1/2)*B*(a*e^2+c*d^2)^(1/2)-B*a*e^(3/2)+e^(1/2)*d*(A*c-C*a))*(a*(a* e^2+c*d^2))^(1/2)+a^(3/2)*e^(3/2)*(a*e^2+c*d^2)^(1/2)*B-a*(d*(A*c-C*a)*e^( 3/2)-e^(5/2)*B*a))*(2*(a*(a*e^2+c*d^2))^(1/2)+2*a*e)^(1/2)*ln((a^(1/2)*(e* x^2+d)+x^2*(a*e^2+c*d^2)^(1/2)+(e*x^2+d)^(1/2)*(2*(a*(a*e^2+c*d^2))^(1/2)+ 2*a*e)^(1/2)*x)/x^2)-4*c*((2*B*e+C*d)*arctanh((e*x^2+d)^(1/2)/x/e^(1/2))+e ^(1/2)*(e*x^2+d)^(1/2)*C*x)*d^2*a^(3/2)))/d^2/c^2
Timed out. \[ \int \frac {\sqrt {d+e x^2} \left (A+B x^2+C x^4\right )}{a+c x^4} \, dx=\text {Timed out} \] Input:
integrate((e*x^2+d)^(1/2)*(C*x^4+B*x^2+A)/(c*x^4+a),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {\sqrt {d+e x^2} \left (A+B x^2+C x^4\right )}{a+c x^4} \, dx=\int \frac {\sqrt {d + e x^{2}} \left (A + B x^{2} + C x^{4}\right )}{a + c x^{4}}\, dx \] Input:
integrate((e*x**2+d)**(1/2)*(C*x**4+B*x**2+A)/(c*x**4+a),x)
Output:
Integral(sqrt(d + e*x**2)*(A + B*x**2 + C*x**4)/(a + c*x**4), x)
\[ \int \frac {\sqrt {d+e x^2} \left (A+B x^2+C x^4\right )}{a+c x^4} \, dx=\int { \frac {{\left (C x^{4} + B x^{2} + A\right )} \sqrt {e x^{2} + d}}{c x^{4} + a} \,d x } \] Input:
integrate((e*x^2+d)^(1/2)*(C*x^4+B*x^2+A)/(c*x^4+a),x, algorithm="maxima")
Output:
integrate((C*x^4 + B*x^2 + A)*sqrt(e*x^2 + d)/(c*x^4 + a), x)
Exception generated. \[ \int \frac {\sqrt {d+e x^2} \left (A+B x^2+C x^4\right )}{a+c x^4} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((e*x^2+d)^(1/2)*(C*x^4+B*x^2+A)/(c*x^4+a),x, algorithm="giac")
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
Timed out. \[ \int \frac {\sqrt {d+e x^2} \left (A+B x^2+C x^4\right )}{a+c x^4} \, dx=\int \frac {\sqrt {e\,x^2+d}\,\left (C\,x^4+B\,x^2+A\right )}{c\,x^4+a} \,d x \] Input:
int(((d + e*x^2)^(1/2)*(A + B*x^2 + C*x^4))/(a + c*x^4),x)
Output:
int(((d + e*x^2)^(1/2)*(A + B*x^2 + C*x^4))/(a + c*x^4), x)
\[ \int \frac {\sqrt {d+e x^2} \left (A+B x^2+C x^4\right )}{a+c x^4} \, dx=\left (\int \frac {\sqrt {e \,x^{2}+d}}{c \,x^{4}+a}d x \right ) a +\left (\int \frac {\sqrt {e \,x^{2}+d}\, x^{4}}{c \,x^{4}+a}d x \right ) c +\left (\int \frac {\sqrt {e \,x^{2}+d}\, x^{2}}{c \,x^{4}+a}d x \right ) b \] Input:
int((e*x^2+d)^(1/2)*(C*x^4+B*x^2+A)/(c*x^4+a),x)
Output:
int(sqrt(d + e*x**2)/(a + c*x**4),x)*a + int((sqrt(d + e*x**2)*x**4)/(a + c*x**4),x)*c + int((sqrt(d + e*x**2)*x**2)/(a + c*x**4),x)*b