\(\int \frac {(d+e x^2)^{7/2}}{(a+c x^4)^3} \, dx\) [399]

Optimal result

Integrand size = 21, antiderivative size = 505 \[ \int \frac {\left (d+e x^2\right )^{7/2}}{\left (a+c x^4\right )^3} \, dx=-\frac {7 d e^2 x \sqrt {d+e x^2}}{32 a^2 c}+\frac {x \left (d+e x^2\right )^{7/2}}{8 a \left (a+c x^4\right )^2}+\frac {7 d x \left (d+e x^2\right )^{5/2}}{32 a^2 \left (a+c x^4\right )}+\frac {7 d \sqrt {\sqrt {a} e+\sqrt {c d^2+a e^2}} \left (4 c d^2 e+2 a e^3-\left (3 c d^2+a e^2\right ) \left (e-\frac {\sqrt {c d^2+a e^2}}{\sqrt {a}}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt {\sqrt {a} e+\sqrt {c d^2+a e^2}} x \sqrt {d+e x^2}}{\sqrt {a} \left (\sqrt {a} e+\sqrt {c d^2+a e^2}\right )-c d x^2}\right )}{64 \sqrt {2} a^{9/4} c^{3/2} \sqrt {c d^2+a e^2}}+\frac {7 d \sqrt {-\sqrt {a} e+\sqrt {c d^2+a e^2}} \left (4 c d^2 e+2 a e^3-\left (3 c d^2+a e^2\right ) \left (e+\frac {\sqrt {c d^2+a e^2}}{\sqrt {a}}\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt {-\sqrt {a} e+\sqrt {c d^2+a e^2}} x \sqrt {d+e x^2}}{\sqrt {a} \left (\sqrt {a} e-\sqrt {c d^2+a e^2}\right )-c d x^2}\right )}{64 \sqrt {2} a^{9/4} c^{3/2} \sqrt {c d^2+a e^2}} \] Output:

-7/32*d*e^2*x*(e*x^2+d)^(1/2)/a^2/c+1/8*x*(e*x^2+d)^(7/2)/a/(c*x^4+a)^2+7/ 
32*d*x*(e*x^2+d)^(5/2)/a^2/(c*x^4+a)+7/128*d*(a^(1/2)*e+(a*e^2+c*d^2)^(1/2 
))^(1/2)*(4*c*d^2*e+2*a*e^3-(a*e^2+3*c*d^2)*(e-(a*e^2+c*d^2)^(1/2)/a^(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^(9/4)/c^(3/2)/(a*e^2+c*d^2)^(1/2)+7/128*d*(-a^(1/2)*e+(a*e^2+c*d^2)^(1/ 
2))^(1/2)*(4*c*d^2*e+2*a*e^3-(a*e^2+3*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 = 2.42 (sec) , antiderivative size = 1814, normalized size of antiderivative = 3.59 \[ \int \frac {\left (d+e x^2\right )^{7/2}}{\left (a+c x^4\right )^3} \, dx =\text {Too large to display} \] Input:

Integrate[(d + e*x^2)^(7/2)/(a + c*x^4)^3,x]
 

Output:

((c^3*Sqrt[d + e*x^2]*(-7*a^2*d*e^2*x + 7*c^2*d^2*x^5*(d + 2*e*x^2) + a*c* 
x*(11*d^3 + 26*d^2*e*x^2 + 5*d*e^2*x^4 + 4*e^3*x^6)))/(a^2*(a + c*x^4)^2) 
+ 64*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 & , (16*d*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e 
*x^2] - #1] + Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1)/(c*d 
^3 - 3*c*d^2*#1 - 8*a*e^2*#1 + 3*c*d*#1^2 - c*#1^3) & ] + (8*e^(7/2)*RootS 
um[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 
 & , (2205*c^4*d^8*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] - 2 
525*a*c^3*d^6*e^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] - 43 
992*a^2*c^2*d^4*e^4*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] - 
18304*a^3*c*d^2*e^6*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] + 
20480*a^4*e^8*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] - 1526*c 
^4*d^7*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 + 7510*a*c^3 
*d^5*e^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 + 6784*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 - 5120 
*a^3*c*d*e^6*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 + 413* 
c^4*d^6*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1^2 - 2013*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 - 162 
4*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 
 + 1280*a^3*c*e^6*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#...
 

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.

\(\displaystyle \int \frac {\left (d+e x^2\right )^{7/2}}{\left (a+c x^4\right )^3} \, dx\)

\(\Big \downarrow \) 1571

\(\displaystyle \int \frac {\left (d+e x^2\right )^{7/2}}{\left (a+c x^4\right )^3}dx\)

Input:

Int[(d + e*x^2)^(7/2)/(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. \(992\) vs. \(2(413)=826\).

Time = 1.43 (sec) , antiderivative size = 993, normalized size of antiderivative = 1.97

method	result	size
pseudoelliptic	\(-\frac {7 \left (-\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 (c \,x^{4}+a \right )^{2} \left (a \,e^{2}+3 c \,d^{2}\right )\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}}+a \left (c \,x^{4}+a \right )^{2} \left (a \,e^{2}+3 c \,d^{2}\right )\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}+\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 (c \,x^{4}+a \right )^{2} \left (a \,e^{2}+3 c \,d^{2}\right )\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}}+a \left (c \,x^{4}+a \right )^{2} \left (a \,e^{2}+3 c \,d^{2}\right )\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}+\left (2 x \sqrt {e \,x^{2}+d}\, \left (-c^{2} d^{2} x^{4} \left (2 e \,x^{2}+d \right ) a^{\frac {3}{2}}-\frac {11 \left (\frac {4}{11} e^{3} x^{6}+\frac {5}{11} d \,e^{2} x^{4}+\frac {26}{11} d^{2} e \,x^{2}+d^{3}\right ) c \,a^{\frac {5}{2}}}{7}+a^{\frac {7}{2}} d \,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}}+a \left (c \,x^{4}+a \right )^{2} \left (a \,e^{2}+3 c \,d^{2}\right )\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 {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^{2} \left (c \,x^{4}+a \right )^{2}}\)	\(993\)
default	\(\text {Expression too large to display}\)	\(39194\)

Input:

int((e*x^2+d)^(7/2)/(c*x^4+a)^3,x,method=_RETURNVERBOSE)
 

Output:

-7/64*(-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)-(c*x^4+a)^2*(a*e^2+3*c*d^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*(c*x^4+a)^2*(a*e^2+3*c*d^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)+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)-(c*x^4+a)^2*(a*e^2+3*c*d^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*(c*x^4+a)^2*(a*e^2+3*c*d^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)+(2*x*(e*x^2+d)^(1/2)*(-c^2*d^2*x^4*(2*e*x^2+d)*a^(3/2)-11/7*(4/11*e^ 
3*x^6+5/11*d*e^2*x^4+26/11*d^2*e*x^2+d^3)*c*a^(5/2)+a^(7/2)*d*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*(c*x^4+a)^2*(a*e^2 
+3*c*d^2))*d^2*(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))-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...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2105 vs. \(2 (415) = 830\).

Time = 28.54 (sec) , antiderivative size = 2105, normalized size of antiderivative = 4.17 \[ \int \frac {\left (d+e x^2\right )^{7/2}}{\left (a+c x^4\right )^3} \, dx=\text {Too large to display} \] Input:

integrate((e*x^2+d)^(7/2)/(c*x^4+a)^3,x, algorithm="fricas")
 

Output:

1/256*(7*(a^2*c^3*x^8 + 2*a^3*c^2*x^4 + a^4*c)*sqrt(-(15*c^2*d^6*e + 15*a* 
c*d^4*e^3 + 4*a^2*d^2*e^5 + a^5*c^3*sqrt(-(81*c^2*d^14 + 90*a*c*d^12*e^2 + 
 25*a^2*d^10*e^4)/(a^11*c^3)))/(a^5*c^3))*log(343*(81*c^3*d^13 + 162*a*c^2 
*d^11*e^2 + 101*a^2*c*d^9*e^4 + 20*a^3*d^7*e^6 + (9*a^5*c^4*d^6 + 13*a^6*c 
^3*d^4*e^2 + 4*a^7*c^2*d^2*e^4)*x^2*sqrt(-(81*c^2*d^14 + 90*a*c*d^12*e^2 + 
 25*a^2*d^10*e^4)/(a^11*c^3)) + 2*(81*c^3*d^12*e + 162*a*c^2*d^10*e^3 + 10 
1*a^2*c*d^8*e^5 + 20*a^3*d^6*e^7)*x^2 + 2*sqrt(e*x^2 + d)*((3*a^8*c^4*d^2 
+ 2*a^9*c^3*e^2)*x*sqrt(-(81*c^2*d^14 + 90*a*c*d^12*e^2 + 25*a^2*d^10*e^4) 
/(a^11*c^3)) - (9*a^3*c^3*d^8*e + 5*a^4*c^2*d^6*e^3)*x)*sqrt(-(15*c^2*d^6* 
e + 15*a*c*d^4*e^3 + 4*a^2*d^2*e^5 + a^5*c^3*sqrt(-(81*c^2*d^14 + 90*a*c*d 
^12*e^2 + 25*a^2*d^10*e^4)/(a^11*c^3)))/(a^5*c^3)))/x^2) - 7*(a^2*c^3*x^8 
+ 2*a^3*c^2*x^4 + a^4*c)*sqrt(-(15*c^2*d^6*e + 15*a*c*d^4*e^3 + 4*a^2*d^2* 
e^5 + a^5*c^3*sqrt(-(81*c^2*d^14 + 90*a*c*d^12*e^2 + 25*a^2*d^10*e^4)/(a^1 
1*c^3)))/(a^5*c^3))*log(343*(81*c^3*d^13 + 162*a*c^2*d^11*e^2 + 101*a^2*c* 
d^9*e^4 + 20*a^3*d^7*e^6 + (9*a^5*c^4*d^6 + 13*a^6*c^3*d^4*e^2 + 4*a^7*c^2 
*d^2*e^4)*x^2*sqrt(-(81*c^2*d^14 + 90*a*c*d^12*e^2 + 25*a^2*d^10*e^4)/(a^1 
1*c^3)) + 2*(81*c^3*d^12*e + 162*a*c^2*d^10*e^3 + 101*a^2*c*d^8*e^5 + 20*a 
^3*d^6*e^7)*x^2 - 2*sqrt(e*x^2 + d)*((3*a^8*c^4*d^2 + 2*a^9*c^3*e^2)*x*sqr 
t(-(81*c^2*d^14 + 90*a*c*d^12*e^2 + 25*a^2*d^10*e^4)/(a^11*c^3)) - (9*a^3* 
c^3*d^8*e + 5*a^4*c^2*d^6*e^3)*x)*sqrt(-(15*c^2*d^6*e + 15*a*c*d^4*e^3 ...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (d+e x^2\right )^{7/2}}{\left (a+c x^4\right )^3} \, dx=\text {Timed out} \] Input:

integrate((e*x**2+d)**(7/2)/(c*x**4+a)**3,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (d+e x^2\right )^{7/2}}{\left (a+c x^4\right )^3} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{\frac {7}{2}}}{{\left (c x^{4} + a\right )}^{3}} \,d x } \] Input:

integrate((e*x^2+d)^(7/2)/(c*x^4+a)^3,x, algorithm="maxima")
 

Output:

integrate((e*x^2 + d)^(7/2)/(c*x^4 + a)^3, x)
 

Giac [F(-1)]

Timed out. \[ \int \frac {\left (d+e x^2\right )^{7/2}}{\left (a+c x^4\right )^3} \, dx=\text {Timed out} \] Input:

integrate((e*x^2+d)^(7/2)/(c*x^4+a)^3,x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d+e x^2\right )^{7/2}}{\left (a+c x^4\right )^3} \, dx=\int \frac {{\left (e\,x^2+d\right )}^{7/2}}{{\left (c\,x^4+a\right )}^3} \,d x \] Input:

int((d + e*x^2)^(7/2)/(a + c*x^4)^3,x)
 

Output:

int((d + e*x^2)^(7/2)/(a + c*x^4)^3, x)
 

Reduce [F]

\[ \int \frac {\left (d+e x^2\right )^{7/2}}{\left (a+c x^4\right )^3} \, dx=\int \frac {\left (e \,x^{2}+d \right )^{\frac {7}{2}}}{\left (c \,x^{4}+a \right )^{3}}d x \] Input:

int((e*x^2+d)^(7/2)/(c*x^4+a)^3,x)
 

Output:

int((e*x^2+d)^(7/2)/(c*x^4+a)^3,x)