Integrand size = 21, antiderivative size = 573 \[ \int \frac {\left (d+e x^2\right )^{7/2}}{\left (a+c x^4\right )^2} \, dx=-\frac {3 d e^2 x \sqrt {d+e x^2}}{4 a c}-\frac {e^3 x^3 \sqrt {d+e x^2}}{4 a c}+\frac {x \left (d+e x^2\right )^{7/2}}{4 a \left (a+c x^4\right )}+\frac {\sqrt {\sqrt {a} e+\sqrt {c d^2+a e^2}} \left (c d^2 e \left (5 c d^2+9 a e^2\right )+\left (e-\frac {\sqrt {c d^2+a e^2}}{\sqrt {a}}\right ) \left (4 a^2 e^4-3 c d^2 \left (c d^2+a e^2\right )\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt {\sqrt {a} e+\sqrt {c d^2+a e^2}} x \sqrt {d+e x^2}}{\sqrt {a} \left (\sqrt {a} e+\sqrt {c d^2+a e^2}\right )-c d x^2}\right )}{8 \sqrt {2} a^{5/4} c^{5/2} d \sqrt {c d^2+a e^2}}+\frac {e^{7/2} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{c^2}+\frac {\sqrt {-\sqrt {a} e+\sqrt {c d^2+a e^2}} \left (c d^2 e \left (5 c d^2+9 a e^2\right )+\left (e+\frac {\sqrt {c d^2+a e^2}}{\sqrt {a}}\right ) \left (4 a^2 e^4-3 c d^2 \left (c d^2+a e^2\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt {-\sqrt {a} e+\sqrt {c d^2+a e^2}} x \sqrt {d+e x^2}}{\sqrt {a} \left (\sqrt {a} e-\sqrt {c d^2+a e^2}\right )-c d x^2}\right )}{8 \sqrt {2} a^{5/4} c^{5/2} d \sqrt {c d^2+a e^2}} \] 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)+1/16*(a^(1/2)*e+(a*e^2+c*d^2)^(1/2))^(1/2)*(c*d^2 *e*(9*a*e^2+5*c*d^2)+(e-(a*e^2+c*d^2)^(1/2)/a^(1/2))*(4*a^2*e^4-3*c*d^2*(a *e^2+c*d^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^(5/4)/c^(5/2)/d/(a*e^2+c*d^2)^(1/2)+e^(7/2)*arctanh(e^(1/2)* x/(e*x^2+d)^(1/2))/c^2+1/16*(-a^(1/2)*e+(a*e^2+c*d^2)^(1/2))^(1/2)*(c*d^2* e*(9*a*e^2+5*c*d^2)+(e+(a*e^2+c*d^2)^(1/2)/a^(1/2))*(4*a^2*e^4-3*c*d^2*(a* e^2+c*d^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)/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 = 0.70 (sec) , antiderivative size = 827, normalized size of antiderivative = 1.44 \[ \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) *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 & , (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*Sq rt[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* Sqrt[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* Sqrt[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]
Leaf count of result is larger than twice the leaf count of optimal. \(1154\) vs. \(2(475)=950\).
Time = 1.46 (sec) , antiderivative size = 1155, normalized size of antiderivative = 2.02
| method | result | size |
| pseudoelliptic | \(\text {Expression too large to display}\) | \(1155\) |
| default | \(\text {Expression too large to display}\) | \(19398\) |
Input:
int((e*x^2+d)^(7/2)/(c*x^4+a)^2,x,method=_RETURNVERBOSE)
Output:
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)/ a^(9/2)*(-1/4*(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)*((-(a^(9/2)*e^2+1/2*((2* e^2*x^4+d^2)*a^(7/2)+a^(5/2)*c*d^2*x^4)*c)*e*(a*e^2+c*d^2)^(1/2)-a^2*(c*x^ 4+a)*(a^2*e^4-3/4*a*c*d^2*e^2-3/4*c^2*d^4))*(a*(a*e^2+c*d^2))^(1/2)+((1/2* c*(2*e^2*x^4+d^2)*a^(9/2)+a^(11/2)*e^2+1/2*a^(7/2)*c^2*d^2*x^4)*e*(a*e^2+c *d^2)^(1/2)+a^3*(c*x^4+a)*(a^2*e^4-3/4*a*c*d^2*e^2-3/4*c^2*d^4))*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*(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)*((-( a^(9/2)*e^2+1/2*((2*e^2*x^4+d^2)*a^(7/2)+a^(5/2)*c*d^2*x^4)*c)*e*(a*e^2+c* d^2)^(1/2)-a^2*(c*x^4+a)*(a^2*e^4-3/4*a*c*d^2*e^2-3/4*c^2*d^4))*(a*(a*e^2+ c*d^2))^(1/2)+((1/2*c*(2*e^2*x^4+d^2)*a^(9/2)+a^(11/2)*e^2+1/2*a^(7/2)*c^2 *d^2*x^4)*e*(a*e^2+c*d^2)^(1/2)+a^3*(c*x^4+a)*(a^2*e^4-3/4*a*c*d^2*e^2-3/4 *c^2*d^4))*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*((2*e^(7/2)*(a^(11/2 )+a^(9/2)*c*x^4)*arctanh((e*x^2+d)^(1/2)/x/e^(1/2))-3/2*x*(e*x^2+d)^(1/2)* (e^2*(1/3*e*x^2+d)*a^(9/2)-1/3*a^(7/2)*c*d^2*(3*e*x^2+d))*c)*(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)+((1/2*c*(2*e^2*x^ 4+d^2)*a^(9/2)+a^(11/2)*e^2+1/2*a^(7/2)*c^2*d^2*x^4)*e*(a*e^2+c*d^2)^(1...
Leaf count of result is larger than twice the leaf count of optimal. 3557 vs. \(2 (477) = 954\).
Time = 68.68 (sec) , antiderivative size = 7122, normalized size of antiderivative = 12.43 \[ \int \frac {\left (d+e x^2\right )^{7/2}}{\left (a+c x^4\right )^2} \, dx=\text {Too large to display} \] 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 = 0.82 (sec) , antiderivative size = 387, normalized size of antiderivative = 0.68 \[ \int \frac {\left (d+e x^2\right )^{7/2}}{\left (a+c x^4\right )^2} \, dx=-\frac {e^{\frac {7}{2}} \log \left ({\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{2}\right )}{2 \, c^{2}} - \frac {5 \, {\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{6} c^{2} d^{3} e^{\frac {3}{2}} - 7 \, {\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{6} a c d e^{\frac {7}{2}} - 9 \, {\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{4} c^{2} d^{4} e^{\frac {3}{2}} - 21 \, {\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{4} a c d^{2} e^{\frac {7}{2}} + 8 \, {\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{4} a^{2} e^{\frac {11}{2}} + 7 \, {\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{2} c^{2} d^{5} e^{\frac {3}{2}} + 3 \, {\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{2} a c d^{3} e^{\frac {7}{2}} - 3 \, c^{2} d^{6} e^{\frac {3}{2}} + a c d^{4} e^{\frac {7}{2}}}{2 \, {\left ({\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{8} c - 4 \, {\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{6} c d + 6 \, {\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{4} c d^{2} + 16 \, {\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{4} a e^{2} - 4 \, {\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{2} c d^{3} + c d^{4}\right )} a c^{2}} \] 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. \[ \int \frac {\left (d+e x^2\right )^{7/2}}{\left (a+c x^4\right )^2} \, dx=\int \frac {{\left (e\,x^2+d\right )}^{7/2}}{{\left (c\,x^4+a\right )}^2} \,d x \] 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