Integrand size = 22, antiderivative size = 285 \[ \int \frac {\left (d+e x^2\right )^{7/2}}{\left (a-c x^4\right )^2} \, dx=\frac {3 d e^2 x \sqrt {d+e x^2}}{4 a c}+\frac {e^3 x^3 \sqrt {d+e x^2}}{4 a c}+\frac {x \left (d+e x^2\right )^{7/2}}{4 a \left (a-c x^4\right )}+\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^{5/2} \left (3 \sqrt {c} d+4 \sqrt {a} e\right ) \arctan \left (\frac {\sqrt {\sqrt {c} d-\sqrt {a} e} x}{\sqrt [4]{a} \sqrt {d+e x^2}}\right )}{8 a^{7/4} c^2}+\frac {e^{7/2} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{c^2}+\frac {\left (3 \sqrt {c} d-4 \sqrt {a} e\right ) \left (\sqrt {c} d+\sqrt {a} e\right )^{5/2} \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {a} e} x}{\sqrt [4]{a} \sqrt {d+e x^2}}\right )}{8 a^{7/4} c^2} \] Output:
3/4*d*e^2*x*(e*x^2+d)^(1/2)/a/c+1/4*e^3*x^3*(e*x^2+d)^(1/2)/a/c+1/4*x*(e*x ^2+d)^(7/2)/a/(-c*x^4+a)+1/8*(c^(1/2)*d-a^(1/2)*e)^(5/2)*(3*c^(1/2)*d+4*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^2+e^(7/2)*arctanh(e^(1/2)*x/(e*x^2+d)^(1/2))/c^2+1/8*(3*c^(1/2)*d-4 *a^(1/2)*e)*(c^(1/2)*d+a^(1/2)*e)^(5/2)*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^2
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.78 (sec) , antiderivative size = 828, normalized size of antiderivative = 2.91 \[ \int \frac {\left (d+e x^2\right )^{7/2}}{\left (a-c x^4\right )^2} \, dx=\frac {\frac {2 c x \sqrt {d+e x^2} \left (a e^2 \left (3 d+e x^2\right )+c d^2 \left (d+3 e x^2\right )\right )}{a \left (a-c x^4\right )}-8 e^{7/2} \log \left (-\sqrt {e} x+\sqrt {d+e x^2}\right )+16 e^{7/2} \text {RootSum}\left [c d^4-4 c d^3 \text {$\#$1}+6 c d^2 \text {$\#$1}^2-16 a e^2 \text {$\#$1}^2-4 c d \text {$\#$1}^3+c \text {$\#$1}^4\&,\frac {17 c d^3 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )+16 a d e^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )+4 c d^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}+2 a e^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}+c d \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{c d^3-3 c d^2 \text {$\#$1}+8 a e^2 \text {$\#$1}+3 c d \text {$\#$1}^2-c \text {$\#$1}^3}\&\right ]-\frac {e^{3/2} \text {RootSum}\left [c d^4-4 c d^3 \text {$\#$1}+6 c d^2 \text {$\#$1}^2-16 a e^2 \text {$\#$1}^2-4 c d \text {$\#$1}^3+c \text {$\#$1}^4\&,\frac {5 c^2 d^5 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )+263 a c d^3 e^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )+256 a^2 d e^4 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )+2 c^2 d^4 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}+70 a c d^2 e^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}+16 a^2 e^4 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}+5 c^2 d^3 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}^2+7 a c d e^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{c d^3-3 c d^2 \text {$\#$1}+8 a e^2 \text {$\#$1}+3 c d \text {$\#$1}^2-c \text {$\#$1}^3}\&\right ]}{a}}{8 c^2} \] Input:
Integrate[(d + e*x^2)^(7/2)/(a - c*x^4)^2,x]
Output:
((2*c*x*Sqrt[d + e*x^2]*(a*e^2*(3*d + e*x^2) + c*d^2*(d + 3*e*x^2)))/(a*(a - c*x^4)) - 8*e^(7/2)*Log[-(Sqrt[e]*x) + Sqrt[d + e*x^2]] + 16*e^(7/2)*Ro otSum[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 & , (17*c*d^3*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] + 16 *a*d*e^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] + 4*c*d^2*Log [d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 + 2*a*e^2*Log[d + 2*e* x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 + 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) & ] - (e^(3/2)*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 & , (5*c^2*d^5*Log[d + 2*e*x^2 - 2*Sqr t[e]*x*Sqrt[d + e*x^2] - #1] + 263*a*c*d^3*e^2*Log[d + 2*e*x^2 - 2*Sqrt[e] *x*Sqrt[d + e*x^2] - #1] + 256*a^2*d*e^4*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqr t[d + e*x^2] - #1] + 2*c^2*d^4*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^ 2] - #1]*#1 + 70*a*c*d^2*e^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 + 16*a^2*e^4*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] *#1 + 5*c^2*d^3*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1^2 + 7*a*c*d*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) & ])/a)/(8*c^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.
| \(\displaystyle \int \frac {\left (d+e x^2\right )^{7/2}}{\left (a-c x^4\right )^2} \, dx\) |
| \(\Big \downarrow \) 1571 |
| \(\displaystyle \int \frac {\left (d+e x^2\right )^{7/2}}{\left (a-c x^4\right )^2}dx\) |
Input:
Int[(d + e*x^2)^(7/2)/(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 = 0.76 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.36
| method | result | size |
| pseudoelliptic | \(-\frac {9 \left (\sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}\, \left (\frac {\left (-\frac {4}{3} a^{2} e^{4}-a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \sqrt {d^{2} a c}}{3}+a c \,d^{2} e \left (a \,e^{2}-\frac {5 c \,d^{2}}{9}\right )\right ) \left (-c \,x^{4}+a \right ) \arctan \left (\frac {\sqrt {e \,x^{2}+d}\, a}{x \sqrt {\left (-a e +\sqrt {d^{2} a c}\right ) a}}\right )+\left (\left (\frac {\left (\frac {4}{3} a^{2} e^{4}+a c \,d^{2} e^{2}-c^{2} d^{4}\right ) \sqrt {d^{2} a c}}{3}+a c \,d^{2} e \left (a \,e^{2}-\frac {5 c \,d^{2}}{9}\right )\right ) \left (-c \,x^{4}+a \right ) \operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}\, a}{x \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}}\right )-\frac {8 \left (a \,e^{\frac {7}{2}} \left (-c \,x^{4}+a \right ) \operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}}{x \sqrt {e}}\right )+\frac {3 x \left (\left (d^{2} e \,x^{2}+\frac {1}{3} d^{3}\right ) c +e^{2} \left (\frac {e \,x^{2}}{3}+d \right ) a \right ) c \sqrt {e \,x^{2}+d}}{4}\right ) \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}\, \sqrt {d^{2} a c}}{9}\right ) \sqrt {\left (-a e +\sqrt {d^{2} a c}\right ) a}\right )}{8 \sqrt {d^{2} a c}\, \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}\, \sqrt {\left (-a e +\sqrt {d^{2} a c}\right ) a}\, a \left (-c \,x^{4}+a \right ) c^{2}}\) | \(389\) |
| default | \(\text {Expression too large to display}\) | \(17244\) |
Input:
int((e*x^2+d)^(7/2)/(-c*x^4+a)^2,x,method=_RETURNVERBOSE)
Output:
-9/8/(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)*(1/3*(-4/3*a^2*e^4-a*c*d^2*e^ 2+c^2*d^4)*(d^2*a*c)^(1/2)+a*c*d^2*e*(a*e^2-5/9*c*d^2))*(-c*x^4+a)*arctan( (e*x^2+d)^(1/2)/x*a/((-a*e+(d^2*a*c)^(1/2))*a)^(1/2))+((1/3*(4/3*a^2*e^4+a *c*d^2*e^2-c^2*d^4)*(d^2*a*c)^(1/2)+a*c*d^2*e*(a*e^2-5/9*c*d^2))*(-c*x^4+a )*arctanh((e*x^2+d)^(1/2)/x*a/((a*e+(d^2*a*c)^(1/2))*a)^(1/2))-8/9*(a*e^(7 /2)*(-c*x^4+a)*arctanh((e*x^2+d)^(1/2)/x/e^(1/2))+3/4*x*((d^2*e*x^2+1/3*d^ 3)*c+e^2*(1/3*e*x^2+d)*a)*c*(e*x^2+d)^(1/2))*((a*e+(d^2*a*c)^(1/2))*a)^(1/ 2)*(d^2*a*c)^(1/2))*((-a*e+(d^2*a*c)^(1/2))*a)^(1/2))/a/(-c*x^4+a)/c^2
Leaf count of result is larger than twice the leaf count of optimal. 3538 vs. \(2 (220) = 440\).
Time = 69.08 (sec) , antiderivative size = 7084, normalized size of antiderivative = 24.86 \[ \int \frac {\left (d+e x^2\right )^{7/2}}{\left (a-c x^4\right )^2} \, dx=\text {Too large to display} \] Input:
integrate((e*x^2+d)^(7/2)/(-c*x^4+a)^2,x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {\left (d+e x^2\right )^{7/2}}{\left (a-c x^4\right )^2} \, dx=\int \frac {\left (d + e x^{2}\right )^{\frac {7}{2}}}{\left (- a + c x^{4}\right )^{2}}\, dx \] Input:
integrate((e*x**2+d)**(7/2)/(-c*x**4+a)**2,x)
Output:
Integral((d + e*x**2)**(7/2)/(-a + c*x**4)**2, x)
\[ \int \frac {\left (d+e x^2\right )^{7/2}}{\left (a-c x^4\right )^2} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{\frac {7}{2}}}{{\left (c x^{4} - a\right )}^{2}} \,d x } \] Input:
integrate((e*x^2+d)^(7/2)/(-c*x^4+a)^2,x, algorithm="maxima")
Output:
integrate((e*x^2 + d)^(7/2)/(c*x^4 - a)^2, x)
Time = 0.19 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.36 \[ \int \frac {\left (d+e x^2\right )^{7/2}}{\left (a-c x^4\right )^2} \, dx=-\frac {e^{\frac {7}{2}} \log \left ({\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{2}\right )}{2 \, c^{2}} + \frac {5 \, {\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{6} c^{2} d^{3} e^{\frac {3}{2}} + 7 \, {\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{6} a c d e^{\frac {7}{2}} - 9 \, {\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{4} c^{2} d^{4} e^{\frac {3}{2}} + 21 \, {\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{4} a c d^{2} e^{\frac {7}{2}} + 8 \, {\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{4} a^{2} e^{\frac {11}{2}} + 7 \, {\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{2} c^{2} d^{5} e^{\frac {3}{2}} - 3 \, {\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{2} a c d^{3} e^{\frac {7}{2}} - 3 \, c^{2} d^{6} e^{\frac {3}{2}} - a c d^{4} e^{\frac {7}{2}}}{2 \, {\left ({\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{8} c - 4 \, {\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{6} c d + 6 \, {\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{4} c d^{2} - 16 \, {\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{4} a e^{2} - 4 \, {\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{2} c d^{3} + c d^{4}\right )} a c^{2}} \] Input:
integrate((e*x^2+d)^(7/2)/(-c*x^4+a)^2,x, algorithm="giac")
Output:
-1/2*e^(7/2)*log((sqrt(e)*x - sqrt(e*x^2 + d))^2)/c^2 + 1/2*(5*(sqrt(e)*x - sqrt(e*x^2 + d))^6*c^2*d^3*e^(3/2) + 7*(sqrt(e)*x - sqrt(e*x^2 + d))^6*a *c*d*e^(7/2) - 9*(sqrt(e)*x - sqrt(e*x^2 + d))^4*c^2*d^4*e^(3/2) + 21*(sqr t(e)*x - sqrt(e*x^2 + d))^4*a*c*d^2*e^(7/2) + 8*(sqrt(e)*x - sqrt(e*x^2 + d))^4*a^2*e^(11/2) + 7*(sqrt(e)*x - sqrt(e*x^2 + d))^2*c^2*d^5*e^(3/2) - 3 *(sqrt(e)*x - sqrt(e*x^2 + d))^2*a*c*d^3*e^(7/2) - 3*c^2*d^6*e^(3/2) - a*c *d^4*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*(sqrt(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. \[ \int \frac {\left (d+e x^2\right )^{7/2}}{\left (a-c x^4\right )^2} \, dx=\int \frac {{\left (e\,x^2+d\right )}^{7/2}}{{\left (a-c\,x^4\right )}^2} \,d x \] Input:
int((d + e*x^2)^(7/2)/(a - c*x^4)^2,x)
Output:
int((d + e*x^2)^(7/2)/(a - c*x^4)^2, x)
\[ \int \frac {\left (d+e x^2\right )^{7/2}}{\left (a-c x^4\right )^2} \, dx=\left (\int \frac {\sqrt {e \,x^{2}+d}}{c^{2} x^{8}-2 a c \,x^{4}+a^{2}}d x \right ) d^{3}+\left (\int \frac {\sqrt {e \,x^{2}+d}\, x^{6}}{c^{2} x^{8}-2 a c \,x^{4}+a^{2}}d x \right ) e^{3}+3 \left (\int \frac {\sqrt {e \,x^{2}+d}\, x^{4}}{c^{2} x^{8}-2 a c \,x^{4}+a^{2}}d x \right ) d \,e^{2}+3 \left (\int \frac {\sqrt {e \,x^{2}+d}\, x^{2}}{c^{2} x^{8}-2 a c \,x^{4}+a^{2}}d x \right ) d^{2} e \] Input:
int((e*x^2+d)^(7/2)/(-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**3 + int((sqrt(d + e*x**2)*x**6)/(a**2 - 2*a*c*x**4 + c**2*x**8),x)*e**3 + 3*int((sqrt(d + e*x**2)*x**4)/(a**2 - 2*a*c*x**4 + c**2*x**8),x)*d*e**2 + 3*int((sqrt(d + e*x**2)*x**2)/(a**2 - 2*a*c*x**4 + c**2*x**8),x)*d**2*e