\(\int \frac {(d+e x^2)^{5/2}}{(a+c x^4)^3} \, dx\) [400]

Optimal result

Integrand size = 21, antiderivative size = 512 \[ \int \frac {\left (d+e x^2\right )^{5/2}}{\left (a+c x^4\right )^3} \, dx=-\frac {e^2 x \sqrt {d+e x^2}}{16 a^2 c}+\frac {x \left (d+e x^2\right )^{5/2}}{8 a \left (a+c x^4\right )^2}+\frac {x \left (d+e x^2\right )^{3/2} \left (7 d+2 e x^2\right )}{32 a^2 \left (a+c x^4\right )}+\frac {\sqrt {\sqrt {a} e+\sqrt {c d^2+a e^2}} \left (2 \sqrt {a} c d^2 e+2 a^{3/2} e^3+\sqrt {c d^2+a e^2} \left (21 c d^2+2 a e^2\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt {\sqrt {a} e+\sqrt {c d^2+a e^2}} x \sqrt {d+e x^2}}{\sqrt {a} \left (\sqrt {a} e+\sqrt {c d^2+a e^2}\right )-c d x^2}\right )}{64 \sqrt {2} a^{11/4} c^{3/2} \sqrt {c d^2+a e^2}}+\frac {\sqrt {-\sqrt {a} e+\sqrt {c d^2+a e^2}} \left (23 c d^2 e+4 a e^3-\left (21 c d^2+2 a e^2\right ) \left (e+\frac {\sqrt {c d^2+a e^2}}{\sqrt {a}}\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt {-\sqrt {a} e+\sqrt {c d^2+a e^2}} x \sqrt {d+e x^2}}{\sqrt {a} \left (\sqrt {a} e-\sqrt {c d^2+a e^2}\right )-c d x^2}\right )}{64 \sqrt {2} a^{9/4} c^{3/2} \sqrt {c d^2+a e^2}} \] Output:

-1/16*e^2*x*(e*x^2+d)^(1/2)/a^2/c+1/8*x*(e*x^2+d)^(5/2)/a/(c*x^4+a)^2+1/32 
*x*(e*x^2+d)^(3/2)*(2*e*x^2+7*d)/a^2/(c*x^4+a)+1/128*(a^(1/2)*e+(a*e^2+c*d 
^2)^(1/2))^(1/2)*(2*a^(1/2)*c*d^2*e+2*a^(3/2)*e^3+(a*e^2+c*d^2)^(1/2)*(2*a 
*e^2+21*c*d^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^(3/2)/(a*e^2+c*d^2)^(1/2)+1/128*(-a^(1/2)*e+(a*e^ 
2+c*d^2)^(1/2))^(1/2)*(23*c*d^2*e+4*a*e^3-(2*a*e^2+21*c*d^2)*(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^(9/4)/c^(3/2)/(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 = 2.27 (sec) , antiderivative size = 1763, normalized size of antiderivative = 3.44 \[ \int \frac {\left (d+e x^2\right )^{5/2}}{\left (a+c x^4\right )^3} \, dx =\text {Too large to display} \] Input:

Integrate[(d + e*x^2)^(5/2)/(a + c*x^4)^3,x]
 

Output:

((2*c^3*Sqrt[d + e*x^2]*(-2*a^2*e^2*x + c^2*d*x^5*(7*d + 9*e*x^2) + a*c*x* 
(11*d^2 + 17*d*e*x^2 + 2*e^2*x^4)))/(a^2*(a + c*x^4)^2) + 512*c^2*e^(11/2) 
*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 & , Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]/(c*d^3 - 3* 
c*d^2*#1 - 8*a*e^2*#1 + 3*c*d*#1^2 - c*#1^3) & ] + (16*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 & , ( 
1934*c^4*d^8*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] + 1709*a* 
c^3*d^6*e^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] - 24784*a^ 
2*c^2*d^4*e^4*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] - 20224* 
a^3*c*d^2*e^6*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] + 4096*a 
^4*e^8*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] - 1370*c^4*d^7* 
Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 + 3724*a*c^3*d^5*e^ 
2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 + 5504*a^2*c^2*d^ 
3*e^4*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 - 1024*a^3*c* 
d*e^6*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 + 366*c^4*d^6 
*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1^2 - 1011*a*c^3*d^4 
*e^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1^2 - 1360*a^2*c 
^2*d^2*e^4*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1^2 + 256* 
a^3*c*e^6*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) & ])/(a*c*d^6 + a^2*d...
 

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.

\(\displaystyle \int \frac {\left (d+e x^2\right )^{5/2}}{\left (a+c x^4\right )^3} \, dx\)

\(\Big \downarrow \) 1571

\(\displaystyle \int \frac {\left (d+e x^2\right )^{5/2}}{\left (a+c x^4\right )^3}dx\)

Input:

Int[(d + e*x^2)^(5/2)/(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. \(981\) vs. \(2(418)=836\).

Time = 1.41 (sec) , antiderivative size = 982, normalized size of antiderivative = 1.92

method	result	size
pseudoelliptic	\(\frac {\frac {\sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, \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 (e \left (c^{2} x^{8} \sqrt {a}+2 c \,x^{4} a^{\frac {3}{2}}+a^{\frac {5}{2}}\right ) \sqrt {a \,e^{2}+c \,d^{2}}-\left (a \,e^{2}+\frac {21 c \,d^{2}}{2}\right ) \left (c \,x^{4}+a \right )^{2}\right ) \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+\left (-e \left (c^{2} x^{8} a^{\frac {3}{2}}+2 c \,x^{4} a^{\frac {5}{2}}+a^{\frac {7}{2}}\right ) \sqrt {a \,e^{2}+c \,d^{2}}+\left (a \,e^{2}+\frac {21 c \,d^{2}}{2}\right ) a \left (c \,x^{4}+a \right )^{2}\right ) e \right ) \ln \left (\frac {\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}-\sqrt {a}\, \left (e \,x^{2}+d \right )}{x^{2}}\right )}{4}-\frac {\sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, \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 (e \left (c^{2} x^{8} \sqrt {a}+2 c \,x^{4} a^{\frac {3}{2}}+a^{\frac {5}{2}}\right ) \sqrt {a \,e^{2}+c \,d^{2}}-\left (a \,e^{2}+\frac {21 c \,d^{2}}{2}\right ) \left (c \,x^{4}+a \right )^{2}\right ) \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+\left (-e \left (c^{2} x^{8} a^{\frac {3}{2}}+2 c \,x^{4} a^{\frac {5}{2}}+a^{\frac {7}{2}}\right ) \sqrt {a \,e^{2}+c \,d^{2}}+\left (a \,e^{2}+\frac {21 c \,d^{2}}{2}\right ) a \left (c \,x^{4}+a \right )^{2}\right ) e \right ) \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}+d \left (-2 x \sqrt {e \,x^{2}+d}\, \left (-\frac {11 \left (\frac {2}{11} e^{2} x^{4}+\frac {17}{11} d e \,x^{2}+d^{2}\right ) c \,a^{\frac {5}{2}}}{2}-\frac {7 \left (\frac {9 e \,x^{2}}{7}+d \right ) x^{4} d \,c^{2} a^{\frac {3}{2}}}{2}+a^{\frac {7}{2}} e^{2}\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 (e \left (c^{2} x^{8} a^{\frac {3}{2}}+2 c \,x^{4} a^{\frac {5}{2}}+a^{\frac {7}{2}}\right ) \sqrt {a \,e^{2}+c \,d^{2}}+\left (a \,e^{2}+\frac {21 c \,d^{2}}{2}\right ) a \left (c \,x^{4}+a \right )^{2}\right ) d \left (\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 )-\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 )\right )\right ) c}{32 a^{\frac {7}{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 \,c^{2} \left (c \,x^{4}+a \right )^{2}}\)	\(982\)
default	\(\text {Expression too large to display}\)	\(27422\)

Input:

int((e*x^2+d)^(5/2)/(c*x^4+a)^3,x,method=_RETURNVERBOSE)
 

Output:

1/32/a^(7/2)/(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)*(1/4*(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)*((e*(c^2*x^8*a^(1/2)+2*c 
*x^4*a^(3/2)+a^(5/2))*(a*e^2+c*d^2)^(1/2)-(a*e^2+21/2*c*d^2)*(c*x^4+a)^2)* 
(a*(a*e^2+c*d^2))^(1/2)+(-e*(c^2*x^8*a^(3/2)+2*c*x^4*a^(5/2)+a^(7/2))*(a*e 
^2+c*d^2)^(1/2)+(a*e^2+21/2*c*d^2)*a*(c*x^4+a)^2)*e)*ln(((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-a^(1/2)*( 
e*x^2+d))/x^2)-1/4*(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)*((e*(c^2*x^8*a^(1/2 
)+2*c*x^4*a^(3/2)+a^(5/2))*(a*e^2+c*d^2)^(1/2)-(a*e^2+21/2*c*d^2)*(c*x^4+a 
)^2)*(a*(a*e^2+c*d^2))^(1/2)+(-e*(c^2*x^8*a^(3/2)+2*c*x^4*a^(5/2)+a^(7/2)) 
*(a*e^2+c*d^2)^(1/2)+(a*e^2+21/2*c*d^2)*a*(c*x^4+a)^2)*e)*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)+d*(-2*x*(e*x^2+d)^(1/2)*(-11/2*(2/11*e^2*x^4+17/11*d*e 
*x^2+d^2)*c*a^(5/2)-7/2*(9/7*e*x^2+d)*x^4*d*c^2*a^(3/2)+a^(7/2)*e^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)+(e*(c^2*x 
^8*a^(3/2)+2*c*x^4*a^(5/2)+a^(7/2))*(a*e^2+c*d^2)^(1/2)+(a*e^2+21/2*c*d^2) 
*a*(c*x^4+a)^2)*d*(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))-arctan((2*a^(1/2)*(e*x^2+d)^(1/2)+(2*(a*(a*e^2+c*d...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2045 vs. \(2 (420) = 840\).

Time = 21.00 (sec) , antiderivative size = 2045, normalized size of antiderivative = 3.99 \[ \int \frac {\left (d+e x^2\right )^{5/2}}{\left (a+c x^4\right )^3} \, dx=\text {Too large to display} \] Input:

integrate((e*x^2+d)^(5/2)/(c*x^4+a)^3,x, algorithm="fricas")
 

Output:

1/256*((a^2*c^3*x^8 + 2*a^3*c^2*x^4 + a^4*c)*sqrt(-(a^5*c^3*sqrt(-(194481* 
c^2*d^10 + 70560*a*c*d^8*e^2 + 6400*a^2*d^6*e^4)/(a^11*c^3)) + 525*c^2*d^4 
*e + 180*a*c*d^2*e^3 + 16*a^2*e^5)/(a^5*c^3))*log((194481*c^3*d^10 + 11113 
2*a*c^2*d^8*e^2 + 20816*a^2*c*d^6*e^4 + 1280*a^3*d^4*e^6 + (441*a^5*c^4*d^ 
5 + 172*a^6*c^3*d^3*e^2 + 16*a^7*c^2*d*e^4)*x^2*sqrt(-(194481*c^2*d^10 + 7 
0560*a*c*d^8*e^2 + 6400*a^2*d^6*e^4)/(a^11*c^3)) + 2*(194481*c^3*d^9*e + 1 
11132*a*c^2*d^7*e^3 + 20816*a^2*c*d^5*e^5 + 1280*a^3*d^3*e^7)*x^2 + 2*sqrt 
(e*x^2 + d)*((21*a^8*c^4*d^2 + 4*a^9*c^3*e^2)*x*sqrt(-(194481*c^2*d^10 + 7 
0560*a*c*d^8*e^2 + 6400*a^2*d^6*e^4)/(a^11*c^3)) - 2*(441*a^3*c^3*d^6*e + 
80*a^4*c^2*d^4*e^3)*x)*sqrt(-(a^5*c^3*sqrt(-(194481*c^2*d^10 + 70560*a*c*d 
^8*e^2 + 6400*a^2*d^6*e^4)/(a^11*c^3)) + 525*c^2*d^4*e + 180*a*c*d^2*e^3 + 
 16*a^2*e^5)/(a^5*c^3)))/x^2) - (a^2*c^3*x^8 + 2*a^3*c^2*x^4 + a^4*c)*sqrt 
(-(a^5*c^3*sqrt(-(194481*c^2*d^10 + 70560*a*c*d^8*e^2 + 6400*a^2*d^6*e^4)/ 
(a^11*c^3)) + 525*c^2*d^4*e + 180*a*c*d^2*e^3 + 16*a^2*e^5)/(a^5*c^3))*log 
((194481*c^3*d^10 + 111132*a*c^2*d^8*e^2 + 20816*a^2*c*d^6*e^4 + 1280*a^3* 
d^4*e^6 + (441*a^5*c^4*d^5 + 172*a^6*c^3*d^3*e^2 + 16*a^7*c^2*d*e^4)*x^2*s 
qrt(-(194481*c^2*d^10 + 70560*a*c*d^8*e^2 + 6400*a^2*d^6*e^4)/(a^11*c^3)) 
+ 2*(194481*c^3*d^9*e + 111132*a*c^2*d^7*e^3 + 20816*a^2*c*d^5*e^5 + 1280* 
a^3*d^3*e^7)*x^2 - 2*sqrt(e*x^2 + d)*((21*a^8*c^4*d^2 + 4*a^9*c^3*e^2)*x*s 
qrt(-(194481*c^2*d^10 + 70560*a*c*d^8*e^2 + 6400*a^2*d^6*e^4)/(a^11*c^3...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (d+e x^2\right )^{5/2}}{\left (a+c x^4\right )^3} \, dx=\text {Timed out} \] Input:

integrate((e*x**2+d)**(5/2)/(c*x**4+a)**3,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (d+e x^2\right )^{5/2}}{\left (a+c x^4\right )^3} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{\frac {5}{2}}}{{\left (c x^{4} + a\right )}^{3}} \,d x } \] Input:

integrate((e*x^2+d)^(5/2)/(c*x^4+a)^3,x, algorithm="maxima")
 

Output:

integrate((e*x^2 + d)^(5/2)/(c*x^4 + a)^3, x)
 

Giac [F(-1)]

Timed out. \[ \int \frac {\left (d+e x^2\right )^{5/2}}{\left (a+c x^4\right )^3} \, dx=\text {Timed out} \] Input:

integrate((e*x^2+d)^(5/2)/(c*x^4+a)^3,x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d+e x^2\right )^{5/2}}{\left (a+c x^4\right )^3} \, dx=\int \frac {{\left (e\,x^2+d\right )}^{5/2}}{{\left (c\,x^4+a\right )}^3} \,d x \] Input:

int((d + e*x^2)^(5/2)/(a + c*x^4)^3,x)
 

Output:

int((d + e*x^2)^(5/2)/(a + c*x^4)^3, x)
 

Reduce [F]

\[ \int \frac {\left (d+e x^2\right )^{5/2}}{\left (a+c x^4\right )^3} \, dx=\int \frac {\left (e \,x^{2}+d \right )^{\frac {5}{2}}}{\left (c \,x^{4}+a \right )^{3}}d x \] Input:

int((e*x^2+d)^(5/2)/(c*x^4+a)^3,x)
 

Output:

int((e*x^2+d)^(5/2)/(c*x^4+a)^3,x)