Integrand size = 21, antiderivative size = 627 \[ \int \frac {\left (d+e x^2\right )^{9/2}}{\left (a+c x^4\right )^2} \, dx=-\frac {3 e^2 \left (2 c d^2-a e^2\right ) x \sqrt {d+e x^2}}{4 a c^2}-\frac {d e^3 x^3 \sqrt {d+e x^2}}{a c}-\frac {e^4 x^5 \sqrt {d+e x^2}}{4 a c}+\frac {x \left (d+e x^2\right )^{9/2}}{4 a \left (a+c x^4\right )}+\frac {3 \sqrt {\sqrt {a} e+\sqrt {c d^2+a e^2}} \left (2 c^2 d^4 e+8 a c d^2 e^3-2 a^2 e^5-\left (c^2 d^4+2 a c d^2 e^2-7 a^2 e^4\right ) \left (e-\frac {\sqrt {c d^2+a e^2}}{\sqrt {a}}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt {\sqrt {a} e+\sqrt {c d^2+a e^2}} x \sqrt {d+e x^2}}{\sqrt {a} \left (\sqrt {a} e+\sqrt {c d^2+a e^2}\right )-c d x^2}\right )}{8 \sqrt {2} a^{5/4} c^{5/2} \sqrt {c d^2+a e^2}}+\frac {9 d e^{7/2} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 c^2}+\frac {3 \sqrt {-\sqrt {a} e+\sqrt {c d^2+a e^2}} \left (2 c^2 d^4 e+8 a c d^2 e^3-2 a^2 e^5-\left (c^2 d^4+2 a c d^2 e^2-7 a^2 e^4\right ) \left (e+\frac {\sqrt {c d^2+a e^2}}{\sqrt {a}}\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt {-\sqrt {a} e+\sqrt {c d^2+a e^2}} x \sqrt {d+e x^2}}{\sqrt {a} \left (\sqrt {a} e-\sqrt {c d^2+a e^2}\right )-c d x^2}\right )}{8 \sqrt {2} a^{5/4} c^{5/2} \sqrt {c d^2+a e^2}} \] Output:
-3/4*e^2*(-a*e^2+2*c*d^2)*x*(e*x^2+d)^(1/2)/a/c^2-d*e^3*x^3*(e*x^2+d)^(1/2 )/a/c-1/4*e^4*x^5*(e*x^2+d)^(1/2)/a/c+1/4*x*(e*x^2+d)^(9/2)/a/(c*x^4+a)+3/ 16*(a^(1/2)*e+(a*e^2+c*d^2)^(1/2))^(1/2)*(2*c^2*d^4*e+8*a*c*d^2*e^3-2*a^2* e^5-(-7*a^2*e^4+2*a*c*d^2*e^2+c^2*d^4)*(e-(a*e^2+c*d^2)^(1/2)/a^(1/2)))*ar ctan(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^(5 /4)/c^(5/2)/(a*e^2+c*d^2)^(1/2)+9/2*d*e^(7/2)*arctanh(e^(1/2)*x/(e*x^2+d)^ (1/2))/c^2+3/16*(-a^(1/2)*e+(a*e^2+c*d^2)^(1/2))^(1/2)*(2*c^2*d^4*e+8*a*c* d^2*e^3-2*a^2*e^5-(-7*a^2*e^4+2*a*c*d^2*e^2+c^2*d^4)*(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^(5/4)/c^(5/2)/(a*e^2+c*d^2)^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.95 (sec) , antiderivative size = 1002, normalized size of antiderivative = 1.60 \[ \int \frac {\left (d+e x^2\right )^{9/2}}{\left (a+c x^4\right )^2} \, dx =\text {Too large to display} \] Input:
Integrate[(d + e*x^2)^(9/2)/(a + c*x^4)^2,x]
Output:
((c*x*Sqrt[d + e*x^2]*(3*a^2*e^4 + c^2*d^3*(d + 4*e*x^2) + 2*a*c*e^2*(-3*d ^2 - 2*d*e*x^2 + e^2*x^4)))/(a*(a + c*x^4)) - 18*c*d*e^(7/2)*Log[-(Sqrt[e] *x) + Sqrt[d + e*x^2]] + 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 & , (45*c^2*d^4*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] - 81*a*c*d^2*e^2*Log[d + 2*e*x^2 - 2*Sq rt[e]*x*Sqrt[d + e*x^2] - #1] + 8*a^2*e^4*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sq rt[d + e*x^2] - #1] + 10*c^2*d^3*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e* x^2] - #1]*#1 - 18*a*c*d*e^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 + 5*c^2*d^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]* #1^2 - a*c*e^2*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) & ] + (e^(3/2)*Root Sum[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 & , (3*c^3*d^6*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] - 168 *a*c^2*d^4*e^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] + 321*a ^2*c*d^2*e^4*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] - 32*a^3* e^6*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] - 52*a*c^2*d^3*e^2 *Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 + 36*a^2*c*d*e^4*L og[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 + 3*c^3*d^4*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1^2 - 8*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 + a^2*c*e^4*Log[d + 2...
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.
| \(\displaystyle \int \frac {\left (d+e x^2\right )^{9/2}}{\left (a+c x^4\right )^2} \, dx\) |
| \(\Big \downarrow \) 1571 |
| \(\displaystyle \int \frac {\left (d+e x^2\right )^{9/2}}{\left (a+c x^4\right )^2}dx\) |
Input:
Int[(d + e*x^2)^(9/2)/(a + c*x^4)^2,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. \(1174\) vs. \(2(525)=1050\).
Time = 1.66 (sec) , antiderivative size = 1175, normalized size of antiderivative = 1.87
| method | result | size |
| pseudoelliptic | \(\text {Expression too large to display}\) | \(1175\) |
| risch | \(\text {Expression too large to display}\) | \(18791\) |
| default | \(\text {Expression too large to display}\) | \(26470\) |
Input:
int((e*x^2+d)^(9/2)/(c*x^4+a)^2,x,method=_RETURNVERBOSE)
Output:
-15/8/(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^(9/2)*(1/4*((-e*(a^(9/2)*e^2+1/5*c*((5*e^2*x^4+d^2)*a^(7/2)+a^(5/2)*c* d^2*x^4))*(a*e^2+c*d^2)^(1/2)-7/5*(a^2*e^4-2/7*a*c*d^2*e^2-1/7*c^2*d^4)*a^ 2*(c*x^4+a))*(a*(a*e^2+c*d^2))^(1/2)+((1/5*c*(5*e^2*x^4+d^2)*a^(9/2)+a^(11 /2)*e^2+1/5*a^(7/2)*c^2*d^2*x^4)*e*(a*e^2+c*d^2)^(1/2)+7/5*(a^2*e^4-2/7*a* c*d^2*e^2-1/7*c^2*d^4)*a^3*(c*x^4+a))*e)*(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)*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*((-e*(a^(9/2)*e^2+1/5*c*((5*e^2 *x^4+d^2)*a^(7/2)+a^(5/2)*c*d^2*x^4))*(a*e^2+c*d^2)^(1/2)-7/5*(a^2*e^4-2/7 *a*c*d^2*e^2-1/7*c^2*d^4)*a^2*(c*x^4+a))*(a*(a*e^2+c*d^2))^(1/2)+((1/5*c*( 5*e^2*x^4+d^2)*a^(9/2)+a^(11/2)*e^2+1/5*a^(7/2)*c^2*d^2*x^4)*e*(a*e^2+c*d^ 2)^(1/2)+7/5*(a^2*e^4-2/7*a*c*d^2*e^2-1/7*c^2*d^4)*a^3*(c*x^4+a))*e)*(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)*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*c*(2/ 5*(-6*d*e^(7/2)*(a^(11/2)+a^(9/2)*c*x^4)*arctanh((e*x^2+d)^(1/2)/x/e^(1/2) )-x*(e*x^2+d)^(1/2)*(-2*(e*x^2+d)*(-1/3*e*x^2+d)*c*e^2*a^(9/2)+1/3*c^2*d^3 *(4*e*x^2+d)*a^(7/2)+a^(11/2)*e^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)+(arctan((2*a^(1/2)*(e*x^2+d)^(1/2)+(2*(...
Timed out. \[ \int \frac {\left (d+e x^2\right )^{9/2}}{\left (a+c x^4\right )^2} \, dx=\text {Timed out} \] Input:
integrate((e*x^2+d)^(9/2)/(c*x^4+a)^2,x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {\left (d+e x^2\right )^{9/2}}{\left (a+c x^4\right )^2} \, dx=\int \frac {\left (d + e x^{2}\right )^{\frac {9}{2}}}{\left (a + c x^{4}\right )^{2}}\, dx \] Input:
integrate((e*x**2+d)**(9/2)/(c*x**4+a)**2,x)
Output:
Integral((d + e*x**2)**(9/2)/(a + c*x**4)**2, x)
\[ \int \frac {\left (d+e x^2\right )^{9/2}}{\left (a+c x^4\right )^2} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{\frac {9}{2}}}{{\left (c x^{4} + a\right )}^{2}} \,d x } \] Input:
integrate((e*x^2+d)^(9/2)/(c*x^4+a)^2,x, algorithm="maxima")
Output:
integrate((e*x^2 + d)^(9/2)/(c*x^4 + a)^2, x)
Time = 0.32 (sec) , antiderivative size = 437, normalized size of antiderivative = 0.70 \[ \int \frac {\left (d+e x^2\right )^{9/2}}{\left (a+c x^4\right )^2} \, dx=\frac {\sqrt {e x^{2} + d} e^{4} x}{2 \, c^{2}} - \frac {9 \, d e^{\frac {7}{2}} \log \left ({\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{2}\right )}{4 \, c^{2}} - \frac {3 \, {\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{6} c^{2} d^{4} e^{\frac {3}{2}} - 8 \, {\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{6} a c d^{2} e^{\frac {7}{2}} + {\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{6} a^{2} e^{\frac {11}{2}} - 6 \, {\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{4} c^{2} d^{5} e^{\frac {3}{2}} - 10 \, {\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{4} a c d^{3} e^{\frac {7}{2}} + 16 \, {\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{4} a^{2} d e^{\frac {11}{2}} + 5 \, {\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{2} c^{2} d^{6} e^{\frac {3}{2}} - {\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{2} a^{2} d^{2} e^{\frac {11}{2}} - 2 \, c^{2} d^{7} e^{\frac {3}{2}} + 2 \, a c d^{5} e^{\frac {7}{2}}}{{\left ({\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{8} c - 4 \, {\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{6} c d + 6 \, {\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{4} c d^{2} + 16 \, {\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{4} a e^{2} - 4 \, {\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{2} c d^{3} + c d^{4}\right )} a c^{2}} \] Input:
integrate((e*x^2+d)^(9/2)/(c*x^4+a)^2,x, algorithm="giac")
Output:
1/2*sqrt(e*x^2 + d)*e^4*x/c^2 - 9/4*d*e^(7/2)*log((sqrt(e)*x - sqrt(e*x^2 + d))^2)/c^2 - (3*(sqrt(e)*x - sqrt(e*x^2 + d))^6*c^2*d^4*e^(3/2) - 8*(sqr t(e)*x - sqrt(e*x^2 + d))^6*a*c*d^2*e^(7/2) + (sqrt(e)*x - sqrt(e*x^2 + d) )^6*a^2*e^(11/2) - 6*(sqrt(e)*x - sqrt(e*x^2 + d))^4*c^2*d^5*e^(3/2) - 10* (sqrt(e)*x - sqrt(e*x^2 + d))^4*a*c*d^3*e^(7/2) + 16*(sqrt(e)*x - sqrt(e*x ^2 + d))^4*a^2*d*e^(11/2) + 5*(sqrt(e)*x - sqrt(e*x^2 + d))^2*c^2*d^6*e^(3 /2) - (sqrt(e)*x - sqrt(e*x^2 + d))^2*a^2*d^2*e^(11/2) - 2*c^2*d^7*e^(3/2) + 2*a*c*d^5*e^(7/2))/(((sqrt(e)*x - sqrt(e*x^2 + d))^8*c - 4*(sqrt(e)*x - sqrt(e*x^2 + d))^6*c*d + 6*(sqrt(e)*x - sqrt(e*x^2 + d))^4*c*d^2 + 16*(sq rt(e)*x - sqrt(e*x^2 + d))^4*a*e^2 - 4*(sqrt(e)*x - sqrt(e*x^2 + d))^2*c*d ^3 + c*d^4)*a*c^2)
Timed out. \[ \int \frac {\left (d+e x^2\right )^{9/2}}{\left (a+c x^4\right )^2} \, dx=\int \frac {{\left (e\,x^2+d\right )}^{9/2}}{{\left (c\,x^4+a\right )}^2} \,d x \] Input:
int((d + e*x^2)^(9/2)/(a + c*x^4)^2,x)
Output:
int((d + e*x^2)^(9/2)/(a + c*x^4)^2, x)
\[ \int \frac {\left (d+e x^2\right )^{9/2}}{\left (a+c x^4\right )^2} \, dx=\left (\int \frac {\sqrt {e \,x^{2}+d}}{c^{2} x^{8}+2 a c \,x^{4}+a^{2}}d x \right ) d^{4}+\left (\int \frac {\sqrt {e \,x^{2}+d}\, x^{8}}{c^{2} x^{8}+2 a c \,x^{4}+a^{2}}d x \right ) e^{4}+4 \left (\int \frac {\sqrt {e \,x^{2}+d}\, x^{6}}{c^{2} x^{8}+2 a c \,x^{4}+a^{2}}d x \right ) d \,e^{3}+6 \left (\int \frac {\sqrt {e \,x^{2}+d}\, x^{4}}{c^{2} x^{8}+2 a c \,x^{4}+a^{2}}d x \right ) d^{2} e^{2}+4 \left (\int \frac {\sqrt {e \,x^{2}+d}\, x^{2}}{c^{2} x^{8}+2 a c \,x^{4}+a^{2}}d x \right ) d^{3} e \] Input:
int((e*x^2+d)^(9/2)/(c*x^4+a)^2,x)
Output:
int(sqrt(d + e*x**2)/(a**2 + 2*a*c*x**4 + c**2*x**8),x)*d**4 + int((sqrt(d + e*x**2)*x**8)/(a**2 + 2*a*c*x**4 + c**2*x**8),x)*e**4 + 4*int((sqrt(d + e*x**2)*x**6)/(a**2 + 2*a*c*x**4 + c**2*x**8),x)*d*e**3 + 6*int((sqrt(d + e*x**2)*x**4)/(a**2 + 2*a*c*x**4 + c**2*x**8),x)*d**2*e**2 + 4*int((sqrt( d + e*x**2)*x**2)/(a**2 + 2*a*c*x**4 + c**2*x**8),x)*d**3*e