Integrand size = 22, antiderivative size = 356 \[ \int \frac {\left (d+e x^2\right )^{9/2}}{\left (a-c x^4\right )^3} \, dx=\frac {3 e^2 \left (2 c d^2-a e^2\right ) x \sqrt {d+e x^2}}{16 a^2 c^2}+\frac {5 d e^2 x \left (d+e x^2\right )^{3/2}}{32 a^2 c}-\frac {e^2 x \left (d+e x^2\right )^{5/2}}{16 a^2 c}+\frac {x \left (d+e x^2\right )^{9/2}}{8 a \left (a-c x^4\right )^2}+\frac {x \left (7 d-2 e x^2\right ) \left (d+e x^2\right )^{7/2}}{32 a^2 \left (a-c x^4\right )}+\frac {3 \left (\sqrt {c} d-\sqrt {a} e\right )^{5/2} \left (7 c d^2+10 \sqrt {a} \sqrt {c} d e+4 a e^2\right ) \arctan \left (\frac {\sqrt {\sqrt {c} d-\sqrt {a} e} x}{\sqrt [4]{a} \sqrt {d+e x^2}}\right )}{64 a^{11/4} c^{5/2}}+\frac {3 \left (\sqrt {c} d+\sqrt {a} e\right )^{5/2} \left (7 c d^2-10 \sqrt {a} \sqrt {c} d e+4 a e^2\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {a} e} x}{\sqrt [4]{a} \sqrt {d+e x^2}}\right )}{64 a^{11/4} c^{5/2}} \] Output:
3/16*e^2*(-a*e^2+2*c*d^2)*x*(e*x^2+d)^(1/2)/a^2/c^2+5/32*d*e^2*x*(e*x^2+d) ^(3/2)/a^2/c-1/16*e^2*x*(e*x^2+d)^(5/2)/a^2/c+1/8*x*(e*x^2+d)^(9/2)/a/(-c* x^4+a)^2+1/32*x*(-2*e*x^2+7*d)*(e*x^2+d)^(7/2)/a^2/(-c*x^4+a)+3/64*(c^(1/2 )*d-a^(1/2)*e)^(5/2)*(7*c*d^2+10*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^(5/2)+3/64*(c ^(1/2)*d+a^(1/2)*e)^(5/2)*(7*c*d^2-10*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^(5/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 2.95 (sec) , antiderivative size = 2220, normalized size of antiderivative = 6.24 \[ \int \frac {\left (d+e x^2\right )^{9/2}}{\left (a-c x^4\right )^3} \, dx=\text {Result too large to show} \] Input:
Integrate[(d + e*x^2)^(9/2)/(a - c*x^4)^3,x]
Output:
((-2*c^3*x*Sqrt[d + e*x^2]*(6*a^3*e^4 + c^3*d^3*x^4*(7*d + 19*e*x^2) - a^2 *c*e^2*(15*d^2 + d*e*x^2 + 10*e^2*x^4) - a*c^2*d*(11*d^3 + 35*d^2*e*x^2 + 9*d*e^2*x^4 + 15*e^3*x^6)))/(a^2*(a - c*x^4)^2) - 32*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 & , (161*c*d^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] + 32*a*e^ 2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] + 18*c*d*Log[d + 2*e *x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 + c*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) & ] + (32*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 & , (1239*c^5*d^10*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] + 4394*a*c^4*d^8*e^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] - 32487*a^2*c^3*d^6*e^4*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] - 4216*a^3*c^2*d^4*e^6*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] + 28544*a^4*c*d^2*e^8*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] + 2048*a^5*e^10*Log[d + 2*e*x^2 - 2*S qrt[e]*x*Sqrt[d + e*x^2] - #1] - 840*c^5*d^9*Log[d + 2*e*x^2 - 2*Sqrt[e]*x *Sqrt[d + e*x^2] - #1]*#1 - 6324*a*c^4*d^7*e^2*Log[d + 2*e*x^2 - 2*Sqrt[e] *x*Sqrt[d + e*x^2] - #1]*#1 + 2160*a^2*c^3*d^5*e^4*Log[d + 2*e*x^2 - 2*Sqr t[e]*x*Sqrt[d + e*x^2] - #1]*#1 + 7360*a^3*c^2*d^3*e^6*Log[d + 2*e*x^2 - 2 *Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 + 512*a^4*c*d*e^8*Log[d + 2*e*x^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.
\(\displaystyle \int \frac {\left (d+e x^2\right )^{9/2}}{\left (a-c x^4\right )^3} \, dx\) |
\(\Big \downarrow \) 1571 |
\(\displaystyle \int \frac {\left (d+e x^2\right )^{9/2}}{\left (a-c x^4\right )^3}dx\) |
Input:
Int[(d + e*x^2)^(9/2)/(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 = 1.20 (sec) , antiderivative size = 444, normalized size of antiderivative = 1.25
method | result | size |
pseudoelliptic | \(-\frac {3 \left (-d \left (\left (-\frac {1}{2} a^{2} e^{4}+\frac {5}{4} a c \,d^{2} e^{2}-\frac {7}{4} c^{2} d^{4}\right ) \sqrt {d^{2} a c}+a e \left (a^{2} e^{4}-\frac {11}{4} a c \,d^{2} e^{2}+\frac {11}{4} c^{2} d^{4}\right )\right ) \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}\, \left (-c \,x^{4}+a \right )^{2} \arctan \left (\frac {\sqrt {e \,x^{2}+d}\, a}{x \sqrt {\left (-a e +\sqrt {d^{2} a c}\right ) a}}\right )+\left (-d \left (\left (\frac {1}{2} a^{2} e^{4}-\frac {5}{4} a c \,d^{2} e^{2}+\frac {7}{4} c^{2} d^{4}\right ) \sqrt {d^{2} a c}+a e \left (a^{2} e^{4}-\frac {11}{4} a c \,d^{2} e^{2}+\frac {11}{4} c^{2} d^{4}\right )\right ) \left (-c \,x^{4}+a \right )^{2} \operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}\, a}{x \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}}\right )+x \sqrt {e \,x^{2}+d}\, \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}\, \left (a^{3} e^{4}-\frac {5 \left (\frac {2}{3} e^{2} x^{4}+\frac {1}{15} d e \,x^{2}+d^{2}\right ) c \,e^{2} a^{2}}{2}-\frac {11 \left (\frac {15}{11} e^{3} x^{6}+\frac {9}{11} d \,e^{2} x^{4}+\frac {35}{11} d^{2} e \,x^{2}+d^{3}\right ) d \,c^{2} a}{6}+\frac {7 x^{4} d^{3} \left (\frac {19 e \,x^{2}}{7}+d \right ) c^{3}}{6}\right ) \sqrt {d^{2} a c}\right ) \sqrt {\left (-a e +\sqrt {d^{2} a c}\right ) a}\right )}{16 \sqrt {d^{2} a c}\, \sqrt {\left (-a e +\sqrt {d^{2} a c}\right ) a}\, \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}\, a^{2} c^{2} \left (-c \,x^{4}+a \right )^{2}}\) | \(444\) |
default | \(\text {Expression too large to display}\) | \(47460\) |
Input:
int((e*x^2+d)^(9/2)/(-c*x^4+a)^3,x,method=_RETURNVERBOSE)
Output:
-3/16*(-d*((-1/2*a^2*e^4+5/4*a*c*d^2*e^2-7/4*c^2*d^4)*(d^2*a*c)^(1/2)+a*e* (a^2*e^4-11/4*a*c*d^2*e^2+11/4*c^2*d^4))*((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))+( -d*((1/2*a^2*e^4-5/4*a*c*d^2*e^2+7/4*c^2*d^4)*(d^2*a*c)^(1/2)+a*e*(a^2*e^4 -11/4*a*c*d^2*e^2+11/4*c^2*d^4))*(-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)*(a^3*e^4-5/2*(2/3*e^2*x^4+1/15*d*e*x^2+d^2)*c*e^2*a^2-11/6*(15/11 *e^3*x^6+9/11*d*e^2*x^4+35/11*d^2*e*x^2+d^3)*d*c^2*a+7/6*x^4*d^3*(19/7*e*x ^2+d)*c^3)*(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 ^2/(-c*x^4+a)^2
Leaf count of result is larger than twice the leaf count of optimal. 3392 vs. \(2 (286) = 572\).
Time = 143.23 (sec) , antiderivative size = 3392, normalized size of antiderivative = 9.53 \[ \int \frac {\left (d+e x^2\right )^{9/2}}{\left (a-c x^4\right )^3} \, dx=\text {Too large to display} \] Input:
integrate((e*x^2+d)^(9/2)/(-c*x^4+a)^3,x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {\left (d+e x^2\right )^{9/2}}{\left (a-c x^4\right )^3} \, dx=\text {Timed out} \] Input:
integrate((e*x**2+d)**(9/2)/(-c*x**4+a)**3,x)
Output:
Timed out
\[ \int \frac {\left (d+e x^2\right )^{9/2}}{\left (a-c x^4\right )^3} \, dx=\int { -\frac {{\left (e x^{2} + d\right )}^{\frac {9}{2}}}{{\left (c x^{4} - a\right )}^{3}} \,d x } \] Input:
integrate((e*x^2+d)^(9/2)/(-c*x^4+a)^3,x, algorithm="maxima")
Output:
-integrate((e*x^2 + d)^(9/2)/(c*x^4 - a)^3, x)
Timed out. \[ \int \frac {\left (d+e x^2\right )^{9/2}}{\left (a-c x^4\right )^3} \, dx=\text {Timed out} \] Input:
integrate((e*x^2+d)^(9/2)/(-c*x^4+a)^3,x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {\left (d+e x^2\right )^{9/2}}{\left (a-c x^4\right )^3} \, dx=\int \frac {{\left (e\,x^2+d\right )}^{9/2}}{{\left (a-c\,x^4\right )}^3} \,d x \] Input:
int((d + e*x^2)^(9/2)/(a - c*x^4)^3,x)
Output:
int((d + e*x^2)^(9/2)/(a - c*x^4)^3, x)
\[ \int \frac {\left (d+e x^2\right )^{9/2}}{\left (a-c x^4\right )^3} \, dx=\int \frac {\left (e \,x^{2}+d \right )^{\frac {9}{2}}}{\left (-c \,x^{4}+a \right )^{3}}d x \] Input:
int((e*x^2+d)^(9/2)/(-c*x^4+a)^3,x)
Output:
int((e*x^2+d)^(9/2)/(-c*x^4+a)^3,x)