\(\int \frac {(d+e x^2)^{11/2}}{(a-c x^4)^3} \, dx\) [357]

Optimal result

Integrand size = 22, antiderivative size = 420 \[ \int \frac {\left (d+e x^2\right )^{11/2}}{\left (a-c x^4\right )^3} \, dx=\frac {d e^2 \left (17 c d^2-21 a e^2\right ) x \sqrt {d+e x^2}}{32 a^2 c^2}+\frac {e^2 \left (5 c d^2-4 a e^2\right ) x \left (d+e x^2\right )^{3/2}}{16 a^2 c^2}+\frac {3 d e^2 x \left (d+e x^2\right )^{5/2}}{32 a^2 c}-\frac {e^2 x \left (d+e x^2\right )^{7/2}}{8 a^2 c}+\frac {x \left (d+e x^2\right )^{11/2}}{8 a \left (a-c x^4\right )^2}+\frac {x \left (7 d-4 e x^2\right ) \left (d+e x^2\right )^{9/2}}{32 a^2 \left (a-c x^4\right )}+\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^{7/2} \left (21 c d^2+46 \sqrt {a} \sqrt {c} d e+32 a e^2\right ) \arctan \left (\frac {\sqrt {\sqrt {c} d-\sqrt {a} e} x}{\sqrt [4]{a} \sqrt {d+e x^2}}\right )}{64 a^{11/4} c^3}-\frac {e^{11/2} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{c^3}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right )^{7/2} \left (21 c d^2-46 \sqrt {a} \sqrt {c} d e+32 a e^2\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {a} e} x}{\sqrt [4]{a} \sqrt {d+e x^2}}\right )}{64 a^{11/4} c^3} \] Output:

1/32*d*e^2*(-21*a*e^2+17*c*d^2)*x*(e*x^2+d)^(1/2)/a^2/c^2+1/16*e^2*(-4*a*e 
^2+5*c*d^2)*x*(e*x^2+d)^(3/2)/a^2/c^2+3/32*d*e^2*x*(e*x^2+d)^(5/2)/a^2/c-1 
/8*e^2*x*(e*x^2+d)^(7/2)/a^2/c+1/8*x*(e*x^2+d)^(11/2)/a/(-c*x^4+a)^2+1/32* 
x*(-4*e*x^2+7*d)*(e*x^2+d)^(9/2)/a^2/(-c*x^4+a)+1/64*(c^(1/2)*d-a^(1/2)*e) 
^(7/2)*(21*c*d^2+46*a^(1/2)*c^(1/2)*d*e+32*a*e^2)*arctan((c^(1/2)*d-a^(1/2 
)*e)^(1/2)*x/a^(1/4)/(e*x^2+d)^(1/2))/a^(11/4)/c^3-e^(11/2)*arctanh(e^(1/2 
)*x/(e*x^2+d)^(1/2))/c^3+1/64*(c^(1/2)*d+a^(1/2)*e)^(7/2)*(21*c*d^2-46*a^( 
1/2)*c^(1/2)*d*e+32*a*e^2)*arctanh((c^(1/2)*d+a^(1/2)*e)^(1/2)*x/a^(1/4)/( 
e*x^2+d)^(1/2))/a^(11/4)/c^3
 

Mathematica [C] (verified)

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

Time = 3.12 (sec) , antiderivative size = 2319, normalized size of antiderivative = 5.52 \[ \int \frac {\left (d+e x^2\right )^{11/2}}{\left (a-c x^4\right )^3} \, dx=\text {Result too large to show} \] Input:

Integrate[(d + e*x^2)^(11/2)/(a - c*x^4)^3,x]
 

Output:

(-((c^3*x*Sqrt[d + e*x^2]*(a^3*e^4*(29*d + 8*e*x^2) + c^3*d^4*x^4*(7*d + 2 
4*e*x^2) - a^2*c*e^2*(26*d^3 + 4*d^2*e*x^2 + 49*d*e^2*x^4 + 12*e^3*x^6) - 
a*c^2*d^2*(11*d^3 + 44*d^2*e*x^2 + 14*d*e^2*x^4 + 36*e^3*x^6)))/(a^2*(a - 
c*x^4)^2)) + 32*c^2*e^(11/2)*Log[-(Sqrt[e]*x) + Sqrt[d + e*x^2]] - 32*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 & , (163*c*d^3*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2 
] - #1] + 96*a*d*e^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] + 
 24*c*d^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 + 6*a*e^2 
*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 + 3*c*d*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) & ] + (4*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 & , (5507*c^5*d^10*Lo 
g[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] + 34770*a*c^4*d^8*e^2*Lo 
g[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] - 166869*a^2*c^3*d^6*e^4 
*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] - 158576*a^3*c^2*d^4* 
e^6*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] + 224000*a^4*c*d^2 
*e^8*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] + 57344*a^5*e^10* 
Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] - 3666*c^5*d^9*Log[d + 
 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 - 38556*a*c^4*d^7*e^2*Log[ 
d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 - 11362*a^2*c^3*d^5*...
 

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.

\(\displaystyle \int \frac {\left (d+e x^2\right )^{11/2}}{\left (a-c x^4\right )^3} \, dx\)

\(\Big \downarrow \) 1571

\(\displaystyle \int \frac {\left (d+e x^2\right )^{11/2}}{\left (a-c x^4\right )^3}dx\)

Input:

Int[(d + e*x^2)^(11/2)/(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 [A] (verified)

Time = 1.38 (sec) , antiderivative size = 538, normalized size of antiderivative = 1.28

method	result	size
pseudoelliptic	\(\frac {\frac {41 \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}\, \left (\frac {\left (-16 a^{3} e^{6}-\frac {29}{2} a^{2} c \,d^{2} e^{4}+13 a \,c^{2} d^{4} e^{2}-\frac {21}{2} c^{3} d^{6}\right ) \sqrt {d^{2} a c}}{41}+a c \,d^{2} e \left (a^{2} e^{4}-\frac {32}{41} a c \,d^{2} e^{2}+\frac {19}{41} c^{2} d^{4}\right )\right ) \left (-c \,x^{4}+a \right )^{2} \arctan \left (\frac {\sqrt {e \,x^{2}+d}\, a}{x \sqrt {\left (-a e +\sqrt {d^{2} a c}\right ) a}}\right )}{32}+\frac {41 \left (\left (\frac {\left (16 a^{3} e^{6}+\frac {29}{2} a^{2} c \,d^{2} e^{4}-13 a \,c^{2} d^{4} e^{2}+\frac {21}{2} c^{3} d^{6}\right ) \sqrt {d^{2} a c}}{41}+a c \,d^{2} e \left (a^{2} e^{4}-\frac {32}{41} a c \,d^{2} e^{2}+\frac {19}{41} c^{2} d^{4}\right )\right ) \left (-c \,x^{4}+a \right )^{2} \operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}\, a}{x \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}}\right )-\frac {32 \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}\, \sqrt {d^{2} a c}\, \left (a^{2} e^{\frac {11}{2}} \left (-c \,x^{4}+a \right )^{2} \operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}}{x \sqrt {e}}\right )+\frac {29 x \left (e^{4} \left (\frac {8 e \,x^{2}}{29}+d \right ) a^{3}-\frac {26 \left (\frac {6}{13} e^{3} x^{6}+\frac {49}{26} d \,e^{2} x^{4}+\frac {2}{13} d^{2} e \,x^{2}+d^{3}\right ) c \,e^{2} a^{2}}{29}-\frac {11 \left (\frac {36}{11} e^{3} x^{6}+\frac {14}{11} d \,e^{2} x^{4}+4 d^{2} e \,x^{2}+d^{3}\right ) d^{2} c^{2} a}{29}+\frac {7 x^{4} d^{4} c^{3} \left (\frac {24 e \,x^{2}}{7}+d \right )}{29}\right ) c \sqrt {e \,x^{2}+d}}{32}\right )}{41}\right ) \sqrt {\left (-a e +\sqrt {d^{2} a c}\right ) a}}{32}}{\sqrt {d^{2} a c}\, \sqrt {\left (-a e +\sqrt {d^{2} a c}\right ) a}\, \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}\, \left (-c \,x^{4}+a \right )^{2} c^{3} a^{2}}\)	\(538\)
default	\(\text {Expression too large to display}\)	\(62196\)

Input:

int((e*x^2+d)^(11/2)/(-c*x^4+a)^3,x,method=_RETURNVERBOSE)
 

Output:

41/32/(d^2*a*c)^(1/2)/((a*e+(d^2*a*c)^(1/2))*a)^(1/2)/((-a*e+(d^2*a*c)^(1/ 
2))*a)^(1/2)*(((a*e+(d^2*a*c)^(1/2))*a)^(1/2)*(1/41*(-16*a^3*e^6-29/2*a^2* 
c*d^2*e^4+13*a*c^2*d^4*e^2-21/2*c^3*d^6)*(d^2*a*c)^(1/2)+a*c*d^2*e*(a^2*e^ 
4-32/41*a*c*d^2*e^2+19/41*c^2*d^4))*(-c*x^4+a)^2*arctan((e*x^2+d)^(1/2)/x* 
a/((-a*e+(d^2*a*c)^(1/2))*a)^(1/2))+((1/41*(16*a^3*e^6+29/2*a^2*c*d^2*e^4- 
13*a*c^2*d^4*e^2+21/2*c^3*d^6)*(d^2*a*c)^(1/2)+a*c*d^2*e*(a^2*e^4-32/41*a* 
c*d^2*e^2+19/41*c^2*d^4))*(-c*x^4+a)^2*arctanh((e*x^2+d)^(1/2)/x*a/((a*e+( 
d^2*a*c)^(1/2))*a)^(1/2))-32/41*((a*e+(d^2*a*c)^(1/2))*a)^(1/2)*(d^2*a*c)^ 
(1/2)*(a^2*e^(11/2)*(-c*x^4+a)^2*arctanh((e*x^2+d)^(1/2)/x/e^(1/2))+29/32* 
x*(e^4*(8/29*e*x^2+d)*a^3-26/29*(6/13*e^3*x^6+49/26*d*e^2*x^4+2/13*d^2*e*x 
^2+d^3)*c*e^2*a^2-11/29*(36/11*e^3*x^6+14/11*d*e^2*x^4+4*d^2*e*x^2+d^3)*d^ 
2*c^2*a+7/29*x^4*d^4*c^3*(24/7*e*x^2+d))*c*(e*x^2+d)^(1/2)))*((-a*e+(d^2*a 
*c)^(1/2))*a)^(1/2))/(-c*x^4+a)^2/c^3/a^2
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\left (d+e x^2\right )^{11/2}}{\left (a-c x^4\right )^3} \, dx=\text {Timed out} \] Input:

integrate((e*x^2+d)^(11/2)/(-c*x^4+a)^3,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (d+e x^2\right )^{11/2}}{\left (a-c x^4\right )^3} \, dx=\text {Timed out} \] Input:

integrate((e*x**2+d)**(11/2)/(-c*x**4+a)**3,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (d+e x^2\right )^{11/2}}{\left (a-c x^4\right )^3} \, dx=\int { -\frac {{\left (e x^{2} + d\right )}^{\frac {11}{2}}}{{\left (c x^{4} - a\right )}^{3}} \,d x } \] Input:

integrate((e*x^2+d)^(11/2)/(-c*x^4+a)^3,x, algorithm="maxima")
 

Output:

-integrate((e*x^2 + d)^(11/2)/(c*x^4 - a)^3, x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1035 vs. \(2 (344) = 688\).

Time = 0.22 (sec) , antiderivative size = 1035, normalized size of antiderivative = 2.46 \[ \int \frac {\left (d+e x^2\right )^{11/2}}{\left (a-c x^4\right )^3} \, dx=\text {Too large to display} \] Input:

integrate((e*x^2+d)^(11/2)/(-c*x^4+a)^3,x, algorithm="giac")
 

Output:

1/2*e^(11/2)*log((sqrt(e)*x - sqrt(e*x^2 + d))^2)/c^3 + 1/8*(19*(sqrt(e)*x 
 - sqrt(e*x^2 + d))^14*c^4*d^5*e^(3/2) - 32*(sqrt(e)*x - sqrt(e*x^2 + d))^ 
14*a*c^3*d^3*e^(7/2) - 55*(sqrt(e)*x - sqrt(e*x^2 + d))^14*a^2*c^2*d*e^(11 
/2) - 112*(sqrt(e)*x - sqrt(e*x^2 + d))^12*c^4*d^6*e^(3/2) + 198*(sqrt(e)* 
x - sqrt(e*x^2 + d))^12*a*c^3*d^4*e^(7/2) - 66*(sqrt(e)*x - sqrt(e*x^2 + d 
))^12*a^2*c^2*d^2*e^(11/2) - 64*(sqrt(e)*x - sqrt(e*x^2 + d))^12*a^3*c*e^( 
15/2) + 287*(sqrt(e)*x - sqrt(e*x^2 + d))^10*c^4*d^7*e^(3/2) - 1040*(sqrt( 
e)*x - sqrt(e*x^2 + d))^10*a*c^3*d^5*e^(7/2) + 397*(sqrt(e)*x - sqrt(e*x^2 
 + d))^10*a^2*c^2*d^3*e^(11/2) + 784*(sqrt(e)*x - sqrt(e*x^2 + d))^10*a^3* 
c*d*e^(15/2) - 420*(sqrt(e)*x - sqrt(e*x^2 + d))^8*c^4*d^8*e^(3/2) + 1782* 
(sqrt(e)*x - sqrt(e*x^2 + d))^8*a*c^3*d^6*e^(7/2) - 3054*(sqrt(e)*x - sqrt 
(e*x^2 + d))^8*a^2*c^2*d^4*e^(11/2) + 1728*(sqrt(e)*x - sqrt(e*x^2 + d))^8 
*a^3*c*d^2*e^(15/2) + 768*(sqrt(e)*x - sqrt(e*x^2 + d))^8*a^4*e^(19/2) + 3 
85*(sqrt(e)*x - sqrt(e*x^2 + d))^6*c^4*d^9*e^(3/2) - 1504*(sqrt(e)*x - sqr 
t(e*x^2 + d))^6*a*c^3*d^7*e^(7/2) + 1571*(sqrt(e)*x - sqrt(e*x^2 + d))^6*a 
^2*c^2*d^5*e^(11/2) - 16*(sqrt(e)*x - sqrt(e*x^2 + d))^6*a^3*c*d^3*e^(15/2 
) - 224*(sqrt(e)*x - sqrt(e*x^2 + d))^4*c^4*d^10*e^(3/2) + 690*(sqrt(e)*x 
- sqrt(e*x^2 + d))^4*a*c^3*d^8*e^(7/2) - 342*(sqrt(e)*x - sqrt(e*x^2 + d)) 
^4*a^2*c^2*d^6*e^(11/2) - 128*(sqrt(e)*x - sqrt(e*x^2 + d))^4*a^3*c*d^4*e^ 
(15/2) + 77*(sqrt(e)*x - sqrt(e*x^2 + d))^2*c^4*d^11*e^(3/2) - 112*(sqr...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d+e x^2\right )^{11/2}}{\left (a-c x^4\right )^3} \, dx=\int \frac {{\left (e\,x^2+d\right )}^{11/2}}{{\left (a-c\,x^4\right )}^3} \,d x \] Input:

int((d + e*x^2)^(11/2)/(a - c*x^4)^3,x)
 

Output:

int((d + e*x^2)^(11/2)/(a - c*x^4)^3, x)
 

Reduce [F]

\[ \int \frac {\left (d+e x^2\right )^{11/2}}{\left (a-c x^4\right )^3} \, dx=\int \frac {\left (e \,x^{2}+d \right )^{\frac {11}{2}}}{\left (-c \,x^{4}+a \right )^{3}}d x \] Input:

int((e*x^2+d)^(11/2)/(-c*x^4+a)^3,x)
 

Output:

int((e*x^2+d)^(11/2)/(-c*x^4+a)^3,x)