Integrand size = 22, antiderivative size = 292 \[ \int \frac {\left (d+e x^2\right )^{5/2}}{\left (a-c x^4\right )^3} \, dx=\frac {e^2 x \sqrt {d+e x^2}}{16 a^2 c}+\frac {x \left (d+e x^2\right )^{5/2}}{8 a \left (a-c x^4\right )^2}+\frac {x \left (d+e x^2\right )^{3/2} \left (7 d+2 e x^2\right )}{32 a^2 \left (a-c x^4\right )}+\frac {\sqrt {\sqrt {c} d-\sqrt {a} e} \left (21 c d^2-2 \sqrt {a} \sqrt {c} d e-4 a e^2\right ) \arctan \left (\frac {\sqrt {\sqrt {c} d-\sqrt {a} e} x}{\sqrt [4]{a} \sqrt {d+e x^2}}\right )}{64 a^{11/4} c^{3/2}}+\frac {\sqrt {\sqrt {c} d+\sqrt {a} e} \left (21 c d^2+2 \sqrt {a} \sqrt {c} d e-4 a e^2\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {a} e} x}{\sqrt [4]{a} \sqrt {d+e x^2}}\right )}{64 a^{11/4} c^{3/2}} \] Output:
1/16*e^2*x*(e*x^2+d)^(1/2)/a^2/c+1/8*x*(e*x^2+d)^(5/2)/a/(-c*x^4+a)^2+1/32 *x*(e*x^2+d)^(3/2)*(2*e*x^2+7*d)/a^2/(-c*x^4+a)+1/64*(c^(1/2)*d-a^(1/2)*e) ^(1/2)*(21*c*d^2-2*a^(1/2)*c^(1/2)*d*e-4*a*e^2)*arctan((c^(1/2)*d-a^(1/2)* e)^(1/2)*x/a^(1/4)/(e*x^2+d)^(1/2))/a^(11/4)/c^(3/2)+1/64*(c^(1/2)*d+a^(1/ 2)*e)^(1/2)*(21*c*d^2+2*a^(1/2)*c^(1/2)*d*e-4*a*e^2)*arctanh((c^(1/2)*d+a^ (1/2)*e)^(1/2)*x/a^(1/4)/(e*x^2+d)^(1/2))/a^(11/4)/c^(3/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 2.19 (sec) , antiderivative size = 1766, normalized size of antiderivative = 6.05 \[ \int \frac {\left (d+e x^2\right )^{5/2}}{\left (a-c x^4\right )^3} \, dx =\text {Too large to display} \] Input:
Integrate[(d + e*x^2)^(5/2)/(a - c*x^4)^3,x]
Output:
((-2*c^3*x*Sqrt[d + e*x^2]*(-2*a^2*e^2 + c^2*d*x^4*(7*d + 9*e*x^2) - a*c*( 11*d^2 + 17*d*e*x^2 + 2*e^2*x^4)))/(a^2*(a - c*x^4)^2) - 512*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 & , Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]/(c*d^3 - 3*c *d^2*#1 + 8*a*e^2*#1 + 3*c*d*#1^2 - c*#1^3) & ] + (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 & , (1 934*c^4*d^8*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] - 1709*a*c ^3*d^6*e^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] - 24784*a^2 *c^2*d^4*e^4*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] + 20224*a ^3*c*d^2*e^6*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] + 4096*a^ 4*e^8*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] - 1370*c^4*d^7*L og[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 - 3724*a*c^3*d^5*e^2 *Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 + 5504*a^2*c^2*d^3 *e^4*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 + 1024*a^3*c*d *e^6*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 + 366*c^4*d^6* Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1^2 + 1011*a*c^3*d^4* e^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1^2 - 1360*a^2*c^ 2*d^2*e^4*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1^2 - 256*a ^3*c*e^6*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*c*d^6 - a^2*d^...
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.
\(\displaystyle \int \frac {\left (d+e x^2\right )^{5/2}}{\left (a-c x^4\right )^3} \, dx\) |
\(\Big \downarrow \) 1571 |
\(\displaystyle \int \frac {\left (d+e x^2\right )^{5/2}}{\left (a-c x^4\right )^3}dx\) |
Input:
Int[(d + e*x^2)^(5/2)/(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]
Time = 1.14 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.18
method | result | size |
pseudoelliptic | \(\frac {-d \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}\, \left (\left (-\frac {a \,e^{2}}{2}+\frac {21 c \,d^{2}}{4}\right ) \sqrt {d^{2} a c}+a e \left (a \,e^{2}-\frac {23 c \,d^{2}}{4}\right )\right ) \left (-c \,x^{4}+a \right )^{2} \arctan \left (\frac {\sqrt {e \,x^{2}+d}\, a}{x \sqrt {\left (-a e +\sqrt {d^{2} a c}\right ) a}}\right )+\left (-\left (\left (\frac {a \,e^{2}}{2}-\frac {21 c \,d^{2}}{4}\right ) \sqrt {d^{2} a c}+a e \left (a \,e^{2}-\frac {23 c \,d^{2}}{4}\right )\right ) d \left (-c \,x^{4}+a \right )^{2} \operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}\, a}{x \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}}\right )+\left (a^{2} e^{2}+\frac {11 \left (\frac {2}{11} e^{2} x^{4}+\frac {17}{11} d e \,x^{2}+d^{2}\right ) c a}{2}-\frac {7 x^{4} d \left (\frac {9 e \,x^{2}}{7}+d \right ) c^{2}}{2}\right ) x \sqrt {e \,x^{2}+d}\, \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}\, \sqrt {d^{2} a c}\right ) \sqrt {\left (-a e +\sqrt {d^{2} a c}\right ) a}}{16 \sqrt {d^{2} a c}\, \sqrt {\left (-a e +\sqrt {d^{2} a c}\right ) a}\, \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}\, a^{2} c \left (-c \,x^{4}+a \right )^{2}}\) | \(344\) |
default | \(\text {Expression too large to display}\) | \(24372\) |
Input:
int((e*x^2+d)^(5/2)/(-c*x^4+a)^3,x,method=_RETURNVERBOSE)
Output:
1/16*(-d*((a*e+(d^2*a*c)^(1/2))*a)^(1/2)*((-1/2*a*e^2+21/4*c*d^2)*(d^2*a*c )^(1/2)+a*e*(a*e^2-23/4*c*d^2))*(-c*x^4+a)^2*arctan((e*x^2+d)^(1/2)/x*a/(( -a*e+(d^2*a*c)^(1/2))*a)^(1/2))+(-((1/2*a*e^2-21/4*c*d^2)*(d^2*a*c)^(1/2)+ a*e*(a*e^2-23/4*c*d^2))*d*(-c*x^4+a)^2*arctanh((e*x^2+d)^(1/2)/x*a/((a*e+( d^2*a*c)^(1/2))*a)^(1/2))+(a^2*e^2+11/2*(2/11*e^2*x^4+17/11*d*e*x^2+d^2)*c *a-7/2*x^4*d*(9/7*e*x^2+d)*c^2)*x*(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))/(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^2/c/(- c*x^4+a)^2
Leaf count of result is larger than twice the leaf count of optimal. 2034 vs. \(2 (230) = 460\).
Time = 21.98 (sec) , antiderivative size = 2034, normalized size of antiderivative = 6.97 \[ \int \frac {\left (d+e x^2\right )^{5/2}}{\left (a-c x^4\right )^3} \, dx=\text {Too large to display} \] Input:
integrate((e*x^2+d)^(5/2)/(-c*x^4+a)^3,x, algorithm="fricas")
Output:
1/256*((a^2*c^3*x^8 - 2*a^3*c^2*x^4 + a^4*c)*sqrt((a^5*c^3*sqrt((194481*c^ 2*d^10 - 70560*a*c*d^8*e^2 + 6400*a^2*d^6*e^4)/(a^11*c^3)) + 525*c^2*d^4*e - 180*a*c*d^2*e^3 + 16*a^2*e^5)/(a^5*c^3))*log(-(194481*c^3*d^10 - 111132 *a*c^2*d^8*e^2 + 20816*a^2*c*d^6*e^4 - 1280*a^3*d^4*e^6 + (441*a^5*c^4*d^5 - 172*a^6*c^3*d^3*e^2 + 16*a^7*c^2*d*e^4)*x^2*sqrt((194481*c^2*d^10 - 705 60*a*c*d^8*e^2 + 6400*a^2*d^6*e^4)/(a^11*c^3)) + 2*(194481*c^3*d^9*e - 111 132*a*c^2*d^7*e^3 + 20816*a^2*c*d^5*e^5 - 1280*a^3*d^3*e^7)*x^2 + 2*sqrt(e *x^2 + d)*((21*a^8*c^4*d^2 - 4*a^9*c^3*e^2)*x*sqrt((194481*c^2*d^10 - 7056 0*a*c*d^8*e^2 + 6400*a^2*d^6*e^4)/(a^11*c^3)) - 2*(441*a^3*c^3*d^6*e - 80* a^4*c^2*d^4*e^3)*x)*sqrt((a^5*c^3*sqrt((194481*c^2*d^10 - 70560*a*c*d^8*e^ 2 + 6400*a^2*d^6*e^4)/(a^11*c^3)) + 525*c^2*d^4*e - 180*a*c*d^2*e^3 + 16*a ^2*e^5)/(a^5*c^3)))/x^2) - (a^2*c^3*x^8 - 2*a^3*c^2*x^4 + a^4*c)*sqrt((a^5 *c^3*sqrt((194481*c^2*d^10 - 70560*a*c*d^8*e^2 + 6400*a^2*d^6*e^4)/(a^11*c ^3)) + 525*c^2*d^4*e - 180*a*c*d^2*e^3 + 16*a^2*e^5)/(a^5*c^3))*log(-(1944 81*c^3*d^10 - 111132*a*c^2*d^8*e^2 + 20816*a^2*c*d^6*e^4 - 1280*a^3*d^4*e^ 6 + (441*a^5*c^4*d^5 - 172*a^6*c^3*d^3*e^2 + 16*a^7*c^2*d*e^4)*x^2*sqrt((1 94481*c^2*d^10 - 70560*a*c*d^8*e^2 + 6400*a^2*d^6*e^4)/(a^11*c^3)) + 2*(19 4481*c^3*d^9*e - 111132*a*c^2*d^7*e^3 + 20816*a^2*c*d^5*e^5 - 1280*a^3*d^3 *e^7)*x^2 - 2*sqrt(e*x^2 + d)*((21*a^8*c^4*d^2 - 4*a^9*c^3*e^2)*x*sqrt((19 4481*c^2*d^10 - 70560*a*c*d^8*e^2 + 6400*a^2*d^6*e^4)/(a^11*c^3)) - 2*(...
Timed out. \[ \int \frac {\left (d+e x^2\right )^{5/2}}{\left (a-c x^4\right )^3} \, dx=\text {Timed out} \] Input:
integrate((e*x**2+d)**(5/2)/(-c*x**4+a)**3,x)
Output:
Timed out
\[ \int \frac {\left (d+e x^2\right )^{5/2}}{\left (a-c x^4\right )^3} \, dx=\int { -\frac {{\left (e x^{2} + d\right )}^{\frac {5}{2}}}{{\left (c x^{4} - a\right )}^{3}} \,d x } \] Input:
integrate((e*x^2+d)^(5/2)/(-c*x^4+a)^3,x, algorithm="maxima")
Output:
-integrate((e*x^2 + d)^(5/2)/(c*x^4 - a)^3, x)
Timed out. \[ \int \frac {\left (d+e x^2\right )^{5/2}}{\left (a-c x^4\right )^3} \, dx=\text {Timed out} \] Input:
integrate((e*x^2+d)^(5/2)/(-c*x^4+a)^3,x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {\left (d+e x^2\right )^{5/2}}{\left (a-c x^4\right )^3} \, dx=\int \frac {{\left (e\,x^2+d\right )}^{5/2}}{{\left (a-c\,x^4\right )}^3} \,d x \] Input:
int((d + e*x^2)^(5/2)/(a - c*x^4)^3,x)
Output:
int((d + e*x^2)^(5/2)/(a - c*x^4)^3, x)
\[ \int \frac {\left (d+e x^2\right )^{5/2}}{\left (a-c x^4\right )^3} \, dx=\int \frac {\left (e \,x^{2}+d \right )^{\frac {5}{2}}}{\left (-c \,x^{4}+a \right )^{3}}d x \] Input:
int((e*x^2+d)^(5/2)/(-c*x^4+a)^3,x)
Output:
int((e*x^2+d)^(5/2)/(-c*x^4+a)^3,x)