Integrand size = 22, antiderivative size = 249 \[ \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a-c x^4\right )^3} \, dx=\frac {x \left (d+e x^2\right )^{3/2}}{8 a \left (a-c x^4\right )^2}+\frac {x \sqrt {d+e x^2} \left (7 d+4 e x^2\right )}{32 a^2 \left (a-c x^4\right )}+\frac {3 d \left (7 \sqrt {c} d-6 \sqrt {a} e\right ) \arctan \left (\frac {\sqrt {\sqrt {c} d-\sqrt {a} e} x}{\sqrt [4]{a} \sqrt {d+e x^2}}\right )}{64 a^{11/4} \sqrt {c} \sqrt {\sqrt {c} d-\sqrt {a} e}}+\frac {3 d \left (7 \sqrt {c} d+6 \sqrt {a} e\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {a} e} x}{\sqrt [4]{a} \sqrt {d+e x^2}}\right )}{64 a^{11/4} \sqrt {c} \sqrt {\sqrt {c} d+\sqrt {a} e}} \] Output:
1/8*x*(e*x^2+d)^(3/2)/a/(-c*x^4+a)^2+1/32*x*(e*x^2+d)^(1/2)*(4*e*x^2+7*d)/ a^2/(-c*x^4+a)+3/64*d*(7*c^(1/2)*d-6*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^(11/4)/c^(1/2)/(c^(1/2)*d-a^(1/2)*e) ^(1/2)+3/64*d*(7*c^(1/2)*d+6*a^(1/2)*e)*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^(1/2)/(c^(1/2)*d+a^(1/2)*e)^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 1.49 (sec) , antiderivative size = 1315, normalized size of antiderivative = 5.28 \[ \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a-c x^4\right )^3} \, dx =\text {Too large to display} \] Input:
Integrate[(d + e*x^2)^(3/2)/(a - c*x^4)^3,x]
Output:
((x*Sqrt[d + e*x^2]*(11*a*d + 8*a*e*x^2 - 7*c*d*x^4 - 4*c*e*x^6))/(a - c*x ^4)^2 + (4*a*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 & , (3329*c^3*d^6*Log[d + 2*e*x^2 - 2*Sqrt[e]*x *Sqrt[d + e*x^2] - #1] - 8432*a*c^2*d^4*e^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x* Sqrt[d + e*x^2] - #1] - 19712*a^2*c*d^2*e^4*Log[d + 2*e*x^2 - 2*Sqrt[e]*x* Sqrt[d + e*x^2] - #1] + 24576*a^3*e^6*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] - 2422*c^3*d^5*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^ 2] - #1]*#1 - 2288*a*c^2*d^3*e^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e* x^2] - #1]*#1 + 6144*a^2*c*d*e^4*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e* x^2] - #1]*#1 + 641*c^3*d^4*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1^2 + 656*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 - 1536*a^2*c*e^4*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) & ] )/(c^4*d^5 - a*c^3*d^3*e^2) + (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 & , (9*c^4*d^8*Log[d + 2*e*x^ 2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] + 13307*a*c^3*d^6*e^2*Log[d + 2*e*x^ 2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] - 33728*a^2*c^2*d^4*e^4*Log[d + 2*e* x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] - 78848*a^3*c*d^2*e^6*Log[d + 2*e* x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] + 98304*a^4*e^8*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] + 24*c^4*d^7*Log[d + 2*e*x^2 - 2*Sqrt...
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.
\(\displaystyle \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a-c x^4\right )^3} \, dx\) |
\(\Big \downarrow \) 1571 |
\(\displaystyle \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a-c x^4\right )^3}dx\) |
Input:
Int[(d + e*x^2)^(3/2)/(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 = 0.64 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.10
method | result | size |
pseudoelliptic | \(\frac {\frac {9 d^{2} \left (a e -\frac {7 \sqrt {d^{2} a c}}{6}\right ) \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}\, \left (-c \,x^{4}+a \right )^{2} \arctan \left (\frac {\sqrt {e \,x^{2}+d}\, a}{x \sqrt {\left (-a e +\sqrt {d^{2} a c}\right ) a}}\right )}{32}+\frac {11 \left (\frac {9 \left (a e +\frac {7 \sqrt {d^{2} a c}}{6}\right ) d^{2} \left (-c \,x^{4}+a \right )^{2} \operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}\, a}{x \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}}\right )}{11}+x \sqrt {e \,x^{2}+d}\, \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}\, \left (\left (\frac {8 e \,x^{2}}{11}+d \right ) a -\frac {7 x^{4} c \left (\frac {4 e \,x^{2}}{7}+d \right )}{11}\right ) \sqrt {d^{2} a c}\right ) \sqrt {\left (-a e +\sqrt {d^{2} a c}\right ) a}}{32}}{a^{2} \left (-c \,x^{4}+a \right )^{2} \sqrt {d^{2} a c}\, \sqrt {\left (-a e +\sqrt {d^{2} a c}\right ) a}\, \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}}\) | \(274\) |
default | \(\text {Expression too large to display}\) | \(16020\) |
Input:
int((e*x^2+d)^(3/2)/(-c*x^4+a)^3,x,method=_RETURNVERBOSE)
Output:
11/32*(9/11*d^2*(a*e-7/6*(d^2*a*c)^(1/2))*((a*e+(d^2*a*c)^(1/2))*a)^(1/2)* (-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))+ (9/11*(a*e+7/6*(d^2*a*c)^(1/2))*d^2*(-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))+x*(e*x^2+d)^(1/2)*((a*e+(d^2*a*c)^(1/2 ))*a)^(1/2)*((8/11*e*x^2+d)*a-7/11*x^4*c*(4/7*e*x^2+d))*(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*x^4+a)^2
Leaf count of result is larger than twice the leaf count of optimal. 2526 vs. \(2 (191) = 382\).
Time = 11.70 (sec) , antiderivative size = 2526, normalized size of antiderivative = 10.14 \[ \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a-c x^4\right )^3} \, dx=\text {Too large to display} \] Input:
integrate((e*x^2+d)^(3/2)/(-c*x^4+a)^3,x, algorithm="fricas")
Output:
1/256*(3*(a^2*c^2*x^8 - 2*a^3*c*x^4 + a^4)*sqrt((35*c*d^4*e - 36*a*d^2*e^3 + (a^5*c^2*d^2 - a^6*c*e^2)*sqrt((2401*c^2*d^10 - 4704*a*c*d^8*e^2 + 2304 *a^2*d^6*e^4)/(a^11*c^3*d^4 - 2*a^12*c^2*d^2*e^2 + a^13*c*e^4)))/(a^5*c^2* d^2 - a^6*c*e^2))*log(27*(2401*c^2*d^9 - 4116*a*c*d^7*e^2 + 1728*a^2*d^5*e ^4 + (49*a^5*c^3*d^6 - 85*a^6*c^2*d^4*e^2 + 36*a^7*c*d^2*e^4)*x^2*sqrt((24 01*c^2*d^10 - 4704*a*c*d^8*e^2 + 2304*a^2*d^6*e^4)/(a^11*c^3*d^4 - 2*a^12* c^2*d^2*e^2 + a^13*c*e^4)) + 2*(2401*c^2*d^8*e - 4116*a*c*d^6*e^3 + 1728*a ^2*d^4*e^5)*x^2 + 2*sqrt(e*x^2 + d)*((7*a^8*c^3*d^4 - 13*a^9*c^2*d^2*e^2 + 6*a^10*c*e^4)*x*sqrt((2401*c^2*d^10 - 4704*a*c*d^8*e^2 + 2304*a^2*d^6*e^4 )/(a^11*c^3*d^4 - 2*a^12*c^2*d^2*e^2 + a^13*c*e^4)) + (49*a^3*c^2*d^6*e - 48*a^4*c*d^4*e^3)*x)*sqrt((35*c*d^4*e - 36*a*d^2*e^3 + (a^5*c^2*d^2 - a^6* c*e^2)*sqrt((2401*c^2*d^10 - 4704*a*c*d^8*e^2 + 2304*a^2*d^6*e^4)/(a^11*c^ 3*d^4 - 2*a^12*c^2*d^2*e^2 + a^13*c*e^4)))/(a^5*c^2*d^2 - a^6*c*e^2)))/x^2 ) - 3*(a^2*c^2*x^8 - 2*a^3*c*x^4 + a^4)*sqrt((35*c*d^4*e - 36*a*d^2*e^3 + (a^5*c^2*d^2 - a^6*c*e^2)*sqrt((2401*c^2*d^10 - 4704*a*c*d^8*e^2 + 2304*a^ 2*d^6*e^4)/(a^11*c^3*d^4 - 2*a^12*c^2*d^2*e^2 + a^13*c*e^4)))/(a^5*c^2*d^2 - a^6*c*e^2))*log(27*(2401*c^2*d^9 - 4116*a*c*d^7*e^2 + 1728*a^2*d^5*e^4 + (49*a^5*c^3*d^6 - 85*a^6*c^2*d^4*e^2 + 36*a^7*c*d^2*e^4)*x^2*sqrt((2401* c^2*d^10 - 4704*a*c*d^8*e^2 + 2304*a^2*d^6*e^4)/(a^11*c^3*d^4 - 2*a^12*c^2 *d^2*e^2 + a^13*c*e^4)) + 2*(2401*c^2*d^8*e - 4116*a*c*d^6*e^3 + 1728*a...
Timed out. \[ \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a-c x^4\right )^3} \, dx=\text {Timed out} \] Input:
integrate((e*x**2+d)**(3/2)/(-c*x**4+a)**3,x)
Output:
Timed out
\[ \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a-c x^4\right )^3} \, dx=\int { -\frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}}}{{\left (c x^{4} - a\right )}^{3}} \,d x } \] Input:
integrate((e*x^2+d)^(3/2)/(-c*x^4+a)^3,x, algorithm="maxima")
Output:
-integrate((e*x^2 + d)^(3/2)/(c*x^4 - a)^3, x)
Timed out. \[ \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a-c x^4\right )^3} \, dx=\text {Timed out} \] Input:
integrate((e*x^2+d)^(3/2)/(-c*x^4+a)^3,x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a-c x^4\right )^3} \, dx=\int \frac {{\left (e\,x^2+d\right )}^{3/2}}{{\left (a-c\,x^4\right )}^3} \,d x \] Input:
int((d + e*x^2)^(3/2)/(a - c*x^4)^3,x)
Output:
int((d + e*x^2)^(3/2)/(a - c*x^4)^3, x)
\[ \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a-c x^4\right )^3} \, dx=\int \frac {\left (e \,x^{2}+d \right )^{\frac {3}{2}}}{\left (-c \,x^{4}+a \right )^{3}}d x \] Input:
int((e*x^2+d)^(3/2)/(-c*x^4+a)^3,x)
Output:
int((e*x^2+d)^(3/2)/(-c*x^4+a)^3,x)