Integrand size = 21, antiderivative size = 748 \[ \int \frac {\left (d+e x^2\right )^{11/2}}{\left (a+c x^4\right )^3} \, dx=-\frac {d e^2 \left (17 c d^2+21 a e^2\right ) x \sqrt {d+e x^2}}{32 a^2 c^2}-\frac {e^2 \left (5 c d^2+4 a e^2\right ) x \left (d+e x^2\right )^{3/2}}{16 a^2 c^2}-\frac {3 d e^2 x \left (d+e x^2\right )^{5/2}}{32 a^2 c}+\frac {e^2 x \left (d+e x^2\right )^{7/2}}{8 a^2 c}+\frac {x \left (d+e x^2\right )^{11/2}}{8 a \left (a+c x^4\right )^2}+\frac {x \left (7 d-4 e x^2\right ) \left (d+e x^2\right )^{9/2}}{32 a^2 \left (a+c x^4\right )}+\frac {\sqrt {\sqrt {a} e+\sqrt {c d^2+a e^2}} \left (2 c d^2 e \left (19 c^2 d^4+32 a c d^2 e^2+41 a^2 e^4\right )-\left (21 c^3 d^6+26 a c^2 d^4 e^2+29 a^2 c d^2 e^4-32 a^3 e^6\right ) \left (e-\frac {\sqrt {c d^2+a e^2}}{\sqrt {a}}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt {\sqrt {a} e+\sqrt {c d^2+a e^2}} x \sqrt {d+e x^2}}{\sqrt {a} \left (\sqrt {a} e+\sqrt {c d^2+a e^2}\right )-c d x^2}\right )}{64 \sqrt {2} a^{9/4} c^{7/2} d \sqrt {c d^2+a e^2}}+\frac {e^{11/2} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{c^3}+\frac {\sqrt {-\sqrt {a} e+\sqrt {c d^2+a e^2}} \left (2 c d^2 e \left (19 c^2 d^4+32 a c d^2 e^2+41 a^2 e^4\right )-\left (21 c^3 d^6+26 a c^2 d^4 e^2+29 a^2 c d^2 e^4-32 a^3 e^6\right ) \left (e+\frac {\sqrt {c d^2+a e^2}}{\sqrt {a}}\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt {-\sqrt {a} e+\sqrt {c d^2+a e^2}} x \sqrt {d+e x^2}}{\sqrt {a} \left (\sqrt {a} e-\sqrt {c d^2+a e^2}\right )-c d x^2}\right )}{64 \sqrt {2} a^{9/4} c^{7/2} d \sqrt {c d^2+a e^2}} \] 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)+1/128*(a^(1/2)*e+(a*e^2+c*d^2 )^(1/2))^(1/2)*(2*c*d^2*e*(41*a^2*e^4+32*a*c*d^2*e^2+19*c^2*d^4)-(-32*a^3* e^6+29*a^2*c*d^2*e^4+26*a*c^2*d^4*e^2+21*c^3*d^6)*(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^(7/2)/d/(a*e^2+c*d^2)^(1/2)+e^(11/2)*arctanh(e^(1/2)*x/(e *x^2+d)^(1/2))/c^3+1/128*(-a^(1/2)*e+(a*e^2+c*d^2)^(1/2))^(1/2)*(2*c*d^2*e *(41*a^2*e^4+32*a*c*d^2*e^2+19*c^2*d^4)-(-32*a^3*e^6+29*a^2*c*d^2*e^4+26*a *c^2*d^4*e^2+21*c^3*d^6)*(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^(7/2)/ 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 = 3.29 (sec) , antiderivative size = 2315, normalized size of antiderivative = 3.09 \[ \int \frac {\left (d+e x^2\right )^{11/2}}{\left (a+c x^4\right )^3} \, dx=\text {Result too large to show} \] 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 + 24*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*...
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
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]
Leaf count of result is larger than twice the leaf count of optimal. \(1516\) vs. \(2(638)=1276\).
Time = 2.52 (sec) , antiderivative size = 1517, normalized size of antiderivative = 2.03
| method | result | size |
| pseudoelliptic | \(\text {Expression too large to display}\) | \(1517\) |
| default | \(\text {Expression too large to display}\) | \(69866\) |
Input:
int((e*x^2+d)^(11/2)/(c*x^4+a)^3,x,method=_RETURNVERBOSE)
Output:
-1/2*(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)*(2*(a*(a*e^2+c*d^2))^(1/2)+2*a*e)^(1/2)*(c*x^4+a)^2*((-1/32*a^(7/2)* (32*a^2*e^4+21*a*c*d^2*e^2+17*c^2*d^4)*e*(a*e^2+c*d^2)^(1/2)-a^3*(a^3*e^6- 29/32*a^2*c*d^2*e^4-13/16*a*c^2*d^4*e^2-21/32*c^3*d^6))*(a*(a*e^2+c*d^2))^ (1/2)+(1/32*a^(9/2)*(32*a^2*e^4+21*a*c*d^2*e^2+17*c^2*d^4)*e*(a*e^2+c*d^2) ^(1/2)+a^4*(a^3*e^6-29/32*a^2*c*d^2*e^4-13/16*a*c^2*d^4*e^2-21/32*c^3*d^6) )*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*(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)*(2*(a*(a*e^2+c*d^2))^(1/2)+2*a*e) ^(1/2)*(c*x^4+a)^2*((-1/32*a^(7/2)*(32*a^2*e^4+21*a*c*d^2*e^2+17*c^2*d^4)* e*(a*e^2+c*d^2)^(1/2)-a^3*(a^3*e^6-29/32*a^2*c*d^2*e^4-13/16*a*c^2*d^4*e^2 -21/32*c^3*d^6))*(a*(a*e^2+c*d^2))^(1/2)+(1/32*a^(9/2)*(32*a^2*e^4+21*a*c* d^2*e^2+17*c^2*d^4)*e*(a*e^2+c*d^2)^(1/2)+a^4*(a^3*e^6-29/32*a^2*c*d^2*e^4 -13/16*a*c^2*d^4*e^2-21/32*c^3*d^6))*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*c*(-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)*(c*x^4+a)^2*a^(13/2)*e^(11/2)*arctanh((e*x^2+d)^(1/2)/x/e^(1/2))+1 /32*(c*x^4+a)^2*a^(9/2)*(32*a^2*e^4+21*a*c*d^2*e^2+17*c^2*d^4)*(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))-arcta...
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
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
\[ \int \frac {\left (d+e x^2\right )^{11/2}}{\left (a+c x^4\right )^3} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{\frac {11}{2}}}{{\left (c x^{4} + a\right )}^{3}} \,d x } \] 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)
Time = 0.23 (sec) , antiderivative size = 1035, normalized size of antiderivative = 1.38 \[ \int \frac {\left (d+e x^2\right )^{11/2}}{\left (a+c x^4\right )^3} \, dx=\text {Too large to display} \] 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^(1 1/2) - 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 - sqr t(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) + 385*(sqrt(e)*x - sqrt(e*x^2 + d))^6*c^4*d^9*e^(3/2) + 1504*(sqrt(e)*x - sq rt(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*(sq...
Timed out. \[ \int \frac {\left (d+e x^2\right )^{11/2}}{\left (a+c x^4\right )^3} \, dx=\int \frac {{\left (e\,x^2+d\right )}^{11/2}}{{\left (c\,x^4+a\right )}^3} \,d x \] 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)
\[ \int \frac {\left (d+e x^2\right )^{11/2}}{\left (a+c x^4\right )^3} \, dx=\int \frac {\left (e \,x^{2}+d \right )^{\frac {11}{2}}}{\left (c \,x^{4}+a \right )^{3}}d x \] 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)