Integrand size = 22, antiderivative size = 327 \[ \int \frac {\left (d+e x^2\right )^{9/2}}{\left (a-c x^4\right )^2} \, dx=\frac {3 e^2 \left (2 c d^2+a e^2\right ) x \sqrt {d+e x^2}}{4 a c^2}+\frac {d e^3 x^3 \sqrt {d+e x^2}}{a c}+\frac {e^4 x^5 \sqrt {d+e x^2}}{4 a c}+\frac {x \left (d+e x^2\right )^{9/2}}{4 a \left (a-c x^4\right )}+\frac {3 \left (\sqrt {c} d-\sqrt {a} e\right )^{7/2} \left (\sqrt {c} d+2 \sqrt {a} e\right ) \arctan \left (\frac {\sqrt {\sqrt {c} d-\sqrt {a} e} x}{\sqrt [4]{a} \sqrt {d+e x^2}}\right )}{8 a^{7/4} c^{5/2}}+\frac {9 d e^{7/2} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 c^2}+\frac {3 \left (\sqrt {c} d-2 \sqrt {a} e\right ) \left (\sqrt {c} d+\sqrt {a} e\right )^{7/2} \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {a} e} x}{\sqrt [4]{a} \sqrt {d+e x^2}}\right )}{8 a^{7/4} c^{5/2}} \] 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)+3/8 *(c^(1/2)*d-a^(1/2)*e)^(7/2)*(c^(1/2)*d+2*a^(1/2)*e)*arctan((c^(1/2)*d-a^( 1/2)*e)^(1/2)*x/a^(1/4)/(e*x^2+d)^(1/2))/a^(7/4)/c^(5/2)+9/2*d*e^(7/2)*arc tanh(e^(1/2)*x/(e*x^2+d)^(1/2))/c^2+3/8*(c^(1/2)*d-2*a^(1/2)*e)*(c^(1/2)*d +a^(1/2)*e)^(7/2)*arctanh((c^(1/2)*d+a^(1/2)*e)^(1/2)*x/a^(1/4)/(e*x^2+d)^ (1/2))/a^(7/4)/c^(5/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 1.04 (sec) , antiderivative size = 1004, normalized size of antiderivative = 3.07 \[ \int \frac {\left (d+e x^2\right )^{9/2}}{\left (a-c x^4\right )^2} \, dx =\text {Too large to display} \] 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*Sqr t[e]*x*Sqrt[d + e*x^2] - #1] + 8*a^2*e^4*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqr t[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)*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 & , (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*Lo g[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]
Time = 0.98 (sec) , antiderivative size = 426, normalized size of antiderivative = 1.30
| method | result | size |
| pseudoelliptic | \(-\frac {3 \left (\sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}\, d \left (-c \,x^{4}+a \right ) \left (\left (-\frac {7}{2} a^{2} e^{4}-a c \,d^{2} e^{2}+\frac {1}{2} c^{2} d^{4}\right ) \sqrt {d^{2} a c}+a e \left (a^{2} e^{4}+4 a c \,d^{2} e^{2}-c^{2} d^{4}\right )\right ) \arctan \left (\frac {\sqrt {e \,x^{2}+d}\, a}{x \sqrt {\left (-a e +\sqrt {d^{2} a c}\right ) a}}\right )+\left (\left (\left (\frac {7}{2} a^{2} e^{4}+a c \,d^{2} e^{2}-\frac {1}{2} c^{2} d^{4}\right ) \sqrt {d^{2} a c}+a e \left (a^{2} e^{4}+4 a c \,d^{2} e^{2}-c^{2} d^{4}\right )\right ) d \left (-c \,x^{4}+a \right ) \operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}\, a}{x \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}}\right )-\sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}\, \left (6 a d \,e^{\frac {7}{2}} \left (-c \,x^{4}+a \right ) \operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}}{x \sqrt {e}}\right )+x \sqrt {e \,x^{2}+d}\, \left (a^{2} e^{4}+2 \left (-\frac {e \,x^{2}}{3}+d \right ) \left (e \,x^{2}+d \right ) c \,e^{2} a +\frac {c^{2} d^{3} \left (4 e \,x^{2}+d \right )}{3}\right )\right ) \sqrt {d^{2} a c}\right ) \sqrt {\left (-a e +\sqrt {d^{2} a c}\right ) a}\right )}{4 \sqrt {d^{2} a c}\, \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}\, \sqrt {\left (-a e +\sqrt {d^{2} a c}\right ) a}\, a \left (-c \,x^{4}+a \right ) c^{2}}\) | \(426\) |
| risch | \(\text {Expression too large to display}\) | \(2816\) |
| default | \(\text {Expression too large to display}\) | \(23548\) |
Input:
int((e*x^2+d)^(9/2)/(-c*x^4+a)^2,x,method=_RETURNVERBOSE)
Output:
-3/4/(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)*d*(-c*x^4+a)*((-7/2*a^2*e^4-a *c*d^2*e^2+1/2*c^2*d^4)*(d^2*a*c)^(1/2)+a*e*(a^2*e^4+4*a*c*d^2*e^2-c^2*d^4 ))*arctan((e*x^2+d)^(1/2)/x*a/((-a*e+(d^2*a*c)^(1/2))*a)^(1/2))+(((7/2*a^2 *e^4+a*c*d^2*e^2-1/2*c^2*d^4)*(d^2*a*c)^(1/2)+a*e*(a^2*e^4+4*a*c*d^2*e^2-c ^2*d^4))*d*(-c*x^4+a)*arctanh((e*x^2+d)^(1/2)/x*a/((a*e+(d^2*a*c)^(1/2))*a )^(1/2))-((a*e+(d^2*a*c)^(1/2))*a)^(1/2)*(6*a*d*e^(7/2)*(-c*x^4+a)*arctanh ((e*x^2+d)^(1/2)/x/e^(1/2))+x*(e*x^2+d)^(1/2)*(a^2*e^4+2*(-1/3*e*x^2+d)*(e *x^2+d)*c*e^2*a+1/3*c^2*d^3*(4*e*x^2+d)))*(d^2*a*c)^(1/2))*((-a*e+(d^2*a*c )^(1/2))*a)^(1/2))/a/(-c*x^4+a)/c^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 = 0.19 (sec) , antiderivative size = 436, normalized size of antiderivative = 1.33 \[ \int \frac {\left (d+e x^2\right )^{9/2}}{\left (a-c x^4\right )^2} \, dx=\frac {\sqrt {e x^{2} + d} e^{4} x}{2 \, c^{2}} - \frac {9 \, d e^{\frac {7}{2}} \log \left ({\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{2}\right )}{4 \, c^{2}} + \frac {3 \, {\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{6} c^{2} d^{4} e^{\frac {3}{2}} + 8 \, {\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{6} a c d^{2} e^{\frac {7}{2}} + {\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{6} a^{2} e^{\frac {11}{2}} - 6 \, {\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{4} c^{2} d^{5} e^{\frac {3}{2}} + 10 \, {\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{4} a c d^{3} e^{\frac {7}{2}} + 16 \, {\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{4} a^{2} d e^{\frac {11}{2}} + 5 \, {\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{2} c^{2} d^{6} e^{\frac {3}{2}} - {\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{2} a^{2} d^{2} e^{\frac {11}{2}} - 2 \, c^{2} d^{7} e^{\frac {3}{2}} - 2 \, a c d^{5} e^{\frac {7}{2}}}{{\left ({\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{8} c - 4 \, {\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{6} c d + 6 \, {\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{4} c d^{2} - 16 \, {\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{4} a e^{2} - 4 \, {\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{2} c d^{3} + c d^{4}\right )} a c^{2}} \] 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. \[ \int \frac {\left (d+e x^2\right )^{9/2}}{\left (a-c x^4\right )^2} \, dx=\int \frac {{\left (e\,x^2+d\right )}^{9/2}}{{\left (a-c\,x^4\right )}^2} \,d x \] 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