\(\int \frac {(d+e x^2)^{3/2}}{(a+c x^4)^3} \, dx\) [401]

Optimal result

Integrand size = 21, antiderivative size = 435 \[ \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a+c x^4\right )^3} \, dx=\frac {x \left (d+e x^2\right )^{3/2}}{8 a \left (a+c x^4\right )^2}+\frac {x \sqrt {d+e x^2} \left (7 d+4 e x^2\right )}{32 a^2 \left (a+c x^4\right )}-\frac {3 d \left (\sqrt {a} e-7 \sqrt {c d^2+a e^2}\right ) \sqrt {\sqrt {a} e+\sqrt {c d^2+a e^2}} \arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt {\sqrt {a} e+\sqrt {c d^2+a e^2}} x \sqrt {d+e x^2}}{\sqrt {a} \left (\sqrt {a} e+\sqrt {c d^2+a e^2}\right )-c d x^2}\right )}{64 \sqrt {2} a^{11/4} \sqrt {c} \sqrt {c d^2+a e^2}}-\frac {3 d \sqrt {-\sqrt {a} e+\sqrt {c d^2+a e^2}} \left (\sqrt {a} e+7 \sqrt {c d^2+a e^2}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt {-\sqrt {a} e+\sqrt {c d^2+a e^2}} x \sqrt {d+e x^2}}{\sqrt {a} \left (\sqrt {a} e-\sqrt {c d^2+a e^2}\right )-c d x^2}\right )}{64 \sqrt {2} a^{11/4} \sqrt {c} \sqrt {c d^2+a e^2}} \] Output:

1/8*x*(e*x^2+d)^(3/2)/a/(c*x^4+a)^2+1/32*x*(e*x^2+d)^(1/2)*(4*e*x^2+7*d)/a 
^2/(c*x^4+a)-3/128*d*(a^(1/2)*e-7*(a*e^2+c*d^2)^(1/2))*(a^(1/2)*e+(a*e^2+c 
*d^2)^(1/2))^(1/2)*arctan(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^(11/4)/c^(1/2)/(a*e^2+c*d^2)^(1/2)-3/128*d*(-a^(1/2)*e+ 
(a*e^2+c*d^2)^(1/2))^(1/2)*(a^(1/2)*e+7*(a*e^2+c*d^2)^(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^(11/4)/c^(1 
/2)/(a*e^2+c*d^2)^(1/2)
 

Mathematica [C] (verified)

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

Time = 1.59 (sec) , antiderivative size = 1310, normalized size of antiderivative = 3.01 \[ \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a+c x^4\right )^3} \, dx =\text {Too large to display} \] Input:

Integrate[(d + e*x^2)^(3/2)/(a + c*x^4)^3,x]
 

Output:

((x*Sqrt[d + e*x^2]*(11*a*d + 8*a*e*x^2 + 7*c*d*x^4 + 4*c*e*x^6))/(a + c*x 
^4)^2 + (4*a*e^(7/2)*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 & , (3329*c^3*d^6*Log[d + 2*e*x^2 - 2*Sqrt[e]*x 
*Sqrt[d + e*x^2] - #1] + 8432*a*c^2*d^4*e^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x* 
Sqrt[d + e*x^2] - #1] - 19712*a^2*c*d^2*e^4*Log[d + 2*e*x^2 - 2*Sqrt[e]*x* 
Sqrt[d + e*x^2] - #1] - 24576*a^3*e^6*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d 
 + e*x^2] - #1] - 2422*c^3*d^5*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^ 
2] - #1]*#1 + 2288*a*c^2*d^3*e^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e* 
x^2] - #1]*#1 + 6144*a^2*c*d*e^4*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e* 
x^2] - #1]*#1 + 641*c^3*d^4*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] 
- #1]*#1^2 - 656*a*c^2*d^2*e^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^ 
2] - #1]*#1^2 - 1536*a^2*c*e^4*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) & ] 
)/(c^3*d^3*(c*d^2 + a*e^2)) + (e^(3/2)*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 & , (9*c^4*d^8*Log[d + 2*e*x^ 
2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] - 13307*a*c^3*d^6*e^2*Log[d + 2*e*x^ 
2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] - 33728*a^2*c^2*d^4*e^4*Log[d + 2*e* 
x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] + 78848*a^3*c*d^2*e^6*Log[d + 2*e* 
x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] + 98304*a^4*e^8*Log[d + 2*e*x^2 - 
2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] + 24*c^4*d^7*Log[d + 2*e*x^2 - 2*Sqrt...
 

Rubi [F]

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.

\(\displaystyle \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a+c x^4\right )^3} \, dx\)

\(\Big \downarrow \) 1571

\(\displaystyle \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a+c x^4\right )^3}dx\)

Input:

Int[(d + e*x^2)^(3/2)/(a + c*x^4)^3,x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 1571

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> U 
nintegrable[(d + e*x^2)^q*(a + c*x^4)^p, x] /; FreeQ[{a, c, d, e, p, q}, x]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(924\) vs. \(2(347)=694\).

Time = 1.64 (sec) , antiderivative size = 925, normalized size of antiderivative = 2.13

method	result	size
pseudoelliptic	\(-\frac {21 \left (-\frac {\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 (c \,x^{4}+a \right )^{2} \sqrt {a \,e^{2}+c \,d^{2}}-\frac {e \left (c^{2} x^{8} \sqrt {a}+2 c \,x^{4} a^{\frac {3}{2}}+a^{\frac {5}{2}}\right )}{7}\right ) \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+\left (a \left (c \,x^{4}+a \right )^{2} \sqrt {a \,e^{2}+c \,d^{2}}+\frac {e \left (c^{2} x^{8} a^{\frac {3}{2}}+2 c \,x^{4} a^{\frac {5}{2}}+a^{\frac {7}{2}}\right )}{7}\right ) e \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 )-\sqrt {e \,x^{2}+d}\, \sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, x +\sqrt {a \,e^{2}+c \,d^{2}}\, x^{2}}{x^{2}}\right )}{4}+\frac {\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 (c \,x^{4}+a \right )^{2} \sqrt {a \,e^{2}+c \,d^{2}}-\frac {e \left (c^{2} x^{8} \sqrt {a}+2 c \,x^{4} a^{\frac {3}{2}}+a^{\frac {5}{2}}\right )}{7}\right ) \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+\left (a \left (c \,x^{4}+a \right )^{2} \sqrt {a \,e^{2}+c \,d^{2}}+\frac {e \left (c^{2} x^{8} a^{\frac {3}{2}}+2 c \,x^{4} a^{\frac {5}{2}}+a^{\frac {7}{2}}\right )}{7}\right ) e \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 )+\sqrt {e \,x^{2}+d}\, \sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, x +\sqrt {a \,e^{2}+c \,d^{2}}\, x^{2}}{x^{2}}\right )}{4}+\left (-\frac {2 x \sqrt {e \,x^{2}+d}\, \sqrt {a \,e^{2}+c \,d^{2}}\, \left (x^{4} \left (\frac {4 e \,x^{2}}{7}+d \right ) c \,a^{\frac {3}{2}}+\frac {11 \left (\frac {8 e \,x^{2}}{11}+d \right ) a^{\frac {5}{2}}}{7}\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}}{3}+\left (a \left (c \,x^{4}+a \right )^{2} \sqrt {a \,e^{2}+c \,d^{2}}-\frac {e \left (c^{2} x^{8} a^{\frac {3}{2}}+2 c \,x^{4} a^{\frac {5}{2}}+a^{\frac {7}{2}}\right )}{7}\right ) d^{2} \left (\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 )-\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 )\right )\right ) c \right )}{64 a^{\frac {7}{2}} \sqrt {a \,e^{2}+c \,d^{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}\, c \left (c \,x^{4}+a \right )^{2}}\)	\(925\)
default	\(\text {Expression too large to display}\)	\(18058\)

Input:

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

Output:

-21/64*(-1/4*(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)*((-(c*x^4+a)^2*(a*e^2+c*d^2)^(1/2)-1/7*e*(c^2*x^8*a^(1/2)+2*c*x^4 
*a^(3/2)+a^(5/2)))*(a*(a*e^2+c*d^2))^(1/2)+(a*(c*x^4+a)^2*(a*e^2+c*d^2)^(1 
/2)+1/7*e*(c^2*x^8*a^(3/2)+2*c*x^4*a^(5/2)+a^(7/2)))*e)*(2*(a*(a*e^2+c*d^2 
))^(1/2)+2*a*e)^(1/2)*ln((a^(1/2)*(e*x^2+d)-(e*x^2+d)^(1/2)*(2*(a*(a*e^2+c 
*d^2))^(1/2)+2*a*e)^(1/2)*x+(a*e^2+c*d^2)^(1/2)*x^2)/x^2)+1/4*(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)*((-(c*x^4+a)^2*( 
a*e^2+c*d^2)^(1/2)-1/7*e*(c^2*x^8*a^(1/2)+2*c*x^4*a^(3/2)+a^(5/2)))*(a*(a* 
e^2+c*d^2))^(1/2)+(a*(c*x^4+a)^2*(a*e^2+c*d^2)^(1/2)+1/7*e*(c^2*x^8*a^(3/2 
)+2*c*x^4*a^(5/2)+a^(7/2)))*e)*(2*(a*(a*e^2+c*d^2))^(1/2)+2*a*e)^(1/2)*ln( 
(a^(1/2)*(e*x^2+d)+(e*x^2+d)^(1/2)*(2*(a*(a*e^2+c*d^2))^(1/2)+2*a*e)^(1/2) 
*x+(a*e^2+c*d^2)^(1/2)*x^2)/x^2)+(-2/3*x*(e*x^2+d)^(1/2)*(a*e^2+c*d^2)^(1/ 
2)*(x^4*(4/7*e*x^2+d)*c*a^(3/2)+11/7*(8/11*e*x^2+d)*a^(5/2))*(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*(c*x^4+a)^2*(a 
*e^2+c*d^2)^(1/2)-1/7*e*(c^2*x^8*a^(3/2)+2*c*x^4*a^(5/2)+a^(7/2)))*d^2*(ar 
ctan((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))- 
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) 
)))*c)/a^(7/2)/(a*e^2+c*d^2)^(1/2)/(4*(a*e^2+c*d^2)^(1/2)*a^(1/2)-2*(a*...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2534 vs. \(2 (349) = 698\).

Time = 11.14 (sec) , antiderivative size = 2534, normalized size of antiderivative = 5.83 \[ \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a+c x^4\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

1/256*(3*(a^2*c^2*x^8 + 2*a^3*c*x^4 + a^4)*sqrt(-(35*c*d^4*e + 36*a*d^2*e^ 
3 + (a^5*c^2*d^2 + a^6*c*e^2)*sqrt(-(2401*c^2*d^10 + 4704*a*c*d^8*e^2 + 23 
04*a^2*d^6*e^4)/(a^11*c^3*d^4 + 2*a^12*c^2*d^2*e^2 + a^13*c*e^4)))/(a^5*c^ 
2*d^2 + a^6*c*e^2))*log(27*(2401*c^2*d^9 + 4116*a*c*d^7*e^2 + 1728*a^2*d^5 
*e^4 + (49*a^5*c^3*d^6 + 85*a^6*c^2*d^4*e^2 + 36*a^7*c*d^2*e^4)*x^2*sqrt(- 
(2401*c^2*d^10 + 4704*a*c*d^8*e^2 + 2304*a^2*d^6*e^4)/(a^11*c^3*d^4 + 2*a^ 
12*c^2*d^2*e^2 + a^13*c*e^4)) + 2*(2401*c^2*d^8*e + 4116*a*c*d^6*e^3 + 172 
8*a^2*d^4*e^5)*x^2 + 2*sqrt(e*x^2 + d)*((7*a^8*c^3*d^4 + 13*a^9*c^2*d^2*e^ 
2 + 6*a^10*c*e^4)*x*sqrt(-(2401*c^2*d^10 + 4704*a*c*d^8*e^2 + 2304*a^2*d^6 
*e^4)/(a^11*c^3*d^4 + 2*a^12*c^2*d^2*e^2 + a^13*c*e^4)) + (49*a^3*c^2*d^6* 
e + 48*a^4*c*d^4*e^3)*x)*sqrt(-(35*c*d^4*e + 36*a*d^2*e^3 + (a^5*c^2*d^2 + 
 a^6*c*e^2)*sqrt(-(2401*c^2*d^10 + 4704*a*c*d^8*e^2 + 2304*a^2*d^6*e^4)/(a 
^11*c^3*d^4 + 2*a^12*c^2*d^2*e^2 + a^13*c*e^4)))/(a^5*c^2*d^2 + a^6*c*e^2) 
))/x^2) - 3*(a^2*c^2*x^8 + 2*a^3*c*x^4 + a^4)*sqrt(-(35*c*d^4*e + 36*a*d^2 
*e^3 + (a^5*c^2*d^2 + a^6*c*e^2)*sqrt(-(2401*c^2*d^10 + 4704*a*c*d^8*e^2 + 
 2304*a^2*d^6*e^4)/(a^11*c^3*d^4 + 2*a^12*c^2*d^2*e^2 + a^13*c*e^4)))/(a^5 
*c^2*d^2 + a^6*c*e^2))*log(27*(2401*c^2*d^9 + 4116*a*c*d^7*e^2 + 1728*a^2* 
d^5*e^4 + (49*a^5*c^3*d^6 + 85*a^6*c^2*d^4*e^2 + 36*a^7*c*d^2*e^4)*x^2*sqr 
t(-(2401*c^2*d^10 + 4704*a*c*d^8*e^2 + 2304*a^2*d^6*e^4)/(a^11*c^3*d^4 + 2 
*a^12*c^2*d^2*e^2 + a^13*c*e^4)) + 2*(2401*c^2*d^8*e + 4116*a*c*d^6*e^3...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a+c x^4\right )^3} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a+c x^4\right )^3} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}}}{{\left (c x^{4} + a\right )}^{3}} \,d x } \] Input:

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

Output:

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

Giac [F(-1)]

Timed out. \[ \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a+c x^4\right )^3} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a+c x^4\right )^3} \, dx=\int \frac {{\left (e\,x^2+d\right )}^{3/2}}{{\left (c\,x^4+a\right )}^3} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a+c x^4\right )^3} \, dx=\int \frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}}}{\left (c \,x^{4}+a \right )^{3}}d x \] Input:

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

Output:

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