Integrand size = 22, antiderivative size = 266 \[ \int \frac {\left (d+e x^2\right )^{7/2}}{\left (a-c x^4\right )^3} \, dx=\frac {7 d e^2 x \sqrt {d+e x^2}}{32 a^2 c}+\frac {x \left (d+e x^2\right )^{7/2}}{8 a \left (a-c x^4\right )^2}+\frac {7 d x \left (d+e x^2\right )^{5/2}}{32 a^2 \left (a-c x^4\right )}+\frac {7 d \left (\sqrt {c} d-\sqrt {a} e\right )^{3/2} \left (3 \sqrt {c} d+2 \sqrt {a} e\right ) \arctan \left (\frac {\sqrt {\sqrt {c} d-\sqrt {a} e} x}{\sqrt [4]{a} \sqrt {d+e x^2}}\right )}{64 a^{11/4} c^{3/2}}+\frac {7 d \left (3 \sqrt {c} d-2 \sqrt {a} e\right ) \left (\sqrt {c} d+\sqrt {a} e\right )^{3/2} \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {a} e} x}{\sqrt [4]{a} \sqrt {d+e x^2}}\right )}{64 a^{11/4} c^{3/2}} \] Output:
7/32*d*e^2*x*(e*x^2+d)^(1/2)/a^2/c+1/8*x*(e*x^2+d)^(7/2)/a/(-c*x^4+a)^2+7/ 32*d*x*(e*x^2+d)^(5/2)/a^2/(-c*x^4+a)+7/64*d*(c^(1/2)*d-a^(1/2)*e)^(3/2)*( 3*c^(1/2)*d+2*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^(3/2)+7/64*d*(3*c^(1/2)*d-2*a^(1/2)*e)*(c^(1/2)*d+ a^(1/2)*e)^(3/2)*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 = 2.35 (sec) , antiderivative size = 1818, normalized size of antiderivative = 6.83 \[ \int \frac {\left (d+e x^2\right )^{7/2}}{\left (a-c x^4\right )^3} \, dx =\text {Too large to display} \] Input:
Integrate[(d + e*x^2)^(7/2)/(a - c*x^4)^3,x]
Output:
(-((c^3*x*Sqrt[d + e*x^2]*(-7*a^2*d*e^2 + 7*c^2*d^2*x^4*(d + 2*e*x^2) - a* c*(11*d^3 + 26*d^2*e*x^2 + 5*d*e^2*x^4 + 4*e^3*x^6)))/(a^2*(a - c*x^4)^2)) - 64*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 & , (16*d*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] + Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1)/(c* d^3 - 3*c*d^2*#1 + 8*a*e^2*#1 + 3*c*d*#1^2 - c*#1^3) & ] + (8*e^(7/2)*Root Sum[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 & , (2205*c^4*d^8*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] + 2525*a*c^3*d^6*e^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] - 4 3992*a^2*c^2*d^4*e^4*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] + 18304*a^3*c*d^2*e^6*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] + 20480*a^4*e^8*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] - 1526* c^4*d^7*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 - 7510*a*c^ 3*d^5*e^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 + 6784*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 + 512 0*a^3*c*d*e^6*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 + 413 *c^4*d^6*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1^2 + 2013*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 - 16 24*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 - 1280*a^3*c*e^6*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*...
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.
\(\displaystyle \int \frac {\left (d+e x^2\right )^{7/2}}{\left (a-c x^4\right )^3} \, dx\) |
\(\Big \downarrow \) 1571 |
\(\displaystyle \int \frac {\left (d+e x^2\right )^{7/2}}{\left (a-c x^4\right )^3}dx\) |
Input:
Int[(d + e*x^2)^(7/2)/(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 = 1.06 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.36
method | result | size |
pseudoelliptic | \(\frac {-\frac {7 \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}\, d^{2} \left (\frac {\left (-a \,e^{2}+3 c \,d^{2}\right ) \sqrt {d^{2} a c}}{2}+a e \left (a \,e^{2}-2 c \,d^{2}\right )\right ) \left (-c \,x^{4}+a \right )^{2} \arctan \left (\frac {\sqrt {e \,x^{2}+d}\, a}{x \sqrt {\left (-a e +\sqrt {d^{2} a c}\right ) a}}\right )}{32}+\frac {7 \left (-d^{2} \left (\frac {\left (a \,e^{2}-3 c \,d^{2}\right ) \sqrt {d^{2} a c}}{2}+a e \left (a \,e^{2}-2 c \,d^{2}\right )\right ) \left (-c \,x^{4}+a \right )^{2} \operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}\, a}{x \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}}\right )+\sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}\, x \sqrt {e \,x^{2}+d}\, \left (d \,e^{2} a^{2}+\frac {11 \left (\frac {4}{11} e^{3} x^{6}+\frac {5}{11} d \,e^{2} x^{4}+\frac {26}{11} d^{2} e \,x^{2}+d^{3}\right ) c a}{7}-c^{2} d^{2} x^{4} \left (2 e \,x^{2}+d \right )\right ) \sqrt {d^{2} a c}\right ) \sqrt {\left (-a e +\sqrt {d^{2} a c}\right ) a}}{32}}{a^{2} c \left (-c \,x^{4}+a \right )^{2} \sqrt {d^{2} a c}\, \sqrt {\left (-a e +\sqrt {d^{2} a c}\right ) a}\, \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}}\) | \(363\) |
default | \(\text {Expression too large to display}\) | \(34852\) |
Input:
int((e*x^2+d)^(7/2)/(-c*x^4+a)^3,x,method=_RETURNVERBOSE)
Output:
7/32/(d^2*a*c)^(1/2)*(-((a*e+(d^2*a*c)^(1/2))*a)^(1/2)*d^2*(1/2*(-a*e^2+3* c*d^2)*(d^2*a*c)^(1/2)+a*e*(a*e^2-2*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))+(-d^2*(1/2*(a*e^2-3*c*d^2)*(d^ 2*a*c)^(1/2)+a*e*(a*e^2-2*c*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))+((a*e+(d^2*a*c)^(1/2))*a)^(1/2)*x*(e*x^2 +d)^(1/2)*(d*e^2*a^2+11/7*(4/11*e^3*x^6+5/11*d*e^2*x^4+26/11*d^2*e*x^2+d^3 )*c*a-c^2*d^2*x^4*(2*e*x^2+d))*(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)/a ^2/c/(-c*x^4+a)^2
Leaf count of result is larger than twice the leaf count of optimal. 2082 vs. \(2 (204) = 408\).
Time = 28.79 (sec) , antiderivative size = 2082, normalized size of antiderivative = 7.83 \[ \int \frac {\left (d+e x^2\right )^{7/2}}{\left (a-c x^4\right )^3} \, dx=\text {Too large to display} \] Input:
integrate((e*x^2+d)^(7/2)/(-c*x^4+a)^3,x, algorithm="fricas")
Output:
1/256*(7*(a^2*c^3*x^8 - 2*a^3*c^2*x^4 + a^4*c)*sqrt((15*c^2*d^6*e - 15*a*c *d^4*e^3 + 4*a^2*d^2*e^5 + a^5*c^3*sqrt((81*c^2*d^14 - 90*a*c*d^12*e^2 + 2 5*a^2*d^10*e^4)/(a^11*c^3)))/(a^5*c^3))*log(-343*(81*c^3*d^13 - 162*a*c^2* d^11*e^2 + 101*a^2*c*d^9*e^4 - 20*a^3*d^7*e^6 + (9*a^5*c^4*d^6 - 13*a^6*c^ 3*d^4*e^2 + 4*a^7*c^2*d^2*e^4)*x^2*sqrt((81*c^2*d^14 - 90*a*c*d^12*e^2 + 2 5*a^2*d^10*e^4)/(a^11*c^3)) + 2*(81*c^3*d^12*e - 162*a*c^2*d^10*e^3 + 101* a^2*c*d^8*e^5 - 20*a^3*d^6*e^7)*x^2 + 2*sqrt(e*x^2 + d)*((3*a^8*c^4*d^2 - 2*a^9*c^3*e^2)*x*sqrt((81*c^2*d^14 - 90*a*c*d^12*e^2 + 25*a^2*d^10*e^4)/(a ^11*c^3)) - (9*a^3*c^3*d^8*e - 5*a^4*c^2*d^6*e^3)*x)*sqrt((15*c^2*d^6*e - 15*a*c*d^4*e^3 + 4*a^2*d^2*e^5 + a^5*c^3*sqrt((81*c^2*d^14 - 90*a*c*d^12*e ^2 + 25*a^2*d^10*e^4)/(a^11*c^3)))/(a^5*c^3)))/x^2) - 7*(a^2*c^3*x^8 - 2*a ^3*c^2*x^4 + a^4*c)*sqrt((15*c^2*d^6*e - 15*a*c*d^4*e^3 + 4*a^2*d^2*e^5 + a^5*c^3*sqrt((81*c^2*d^14 - 90*a*c*d^12*e^2 + 25*a^2*d^10*e^4)/(a^11*c^3)) )/(a^5*c^3))*log(-343*(81*c^3*d^13 - 162*a*c^2*d^11*e^2 + 101*a^2*c*d^9*e^ 4 - 20*a^3*d^7*e^6 + (9*a^5*c^4*d^6 - 13*a^6*c^3*d^4*e^2 + 4*a^7*c^2*d^2*e ^4)*x^2*sqrt((81*c^2*d^14 - 90*a*c*d^12*e^2 + 25*a^2*d^10*e^4)/(a^11*c^3)) + 2*(81*c^3*d^12*e - 162*a*c^2*d^10*e^3 + 101*a^2*c*d^8*e^5 - 20*a^3*d^6* e^7)*x^2 - 2*sqrt(e*x^2 + d)*((3*a^8*c^4*d^2 - 2*a^9*c^3*e^2)*x*sqrt((81*c ^2*d^14 - 90*a*c*d^12*e^2 + 25*a^2*d^10*e^4)/(a^11*c^3)) - (9*a^3*c^3*d^8* e - 5*a^4*c^2*d^6*e^3)*x)*sqrt((15*c^2*d^6*e - 15*a*c*d^4*e^3 + 4*a^2*d...
Timed out. \[ \int \frac {\left (d+e x^2\right )^{7/2}}{\left (a-c x^4\right )^3} \, dx=\text {Timed out} \] Input:
integrate((e*x**2+d)**(7/2)/(-c*x**4+a)**3,x)
Output:
Timed out
\[ \int \frac {\left (d+e x^2\right )^{7/2}}{\left (a-c x^4\right )^3} \, dx=\int { -\frac {{\left (e x^{2} + d\right )}^{\frac {7}{2}}}{{\left (c x^{4} - a\right )}^{3}} \,d x } \] Input:
integrate((e*x^2+d)^(7/2)/(-c*x^4+a)^3,x, algorithm="maxima")
Output:
-integrate((e*x^2 + d)^(7/2)/(c*x^4 - a)^3, x)
Timed out. \[ \int \frac {\left (d+e x^2\right )^{7/2}}{\left (a-c x^4\right )^3} \, dx=\text {Timed out} \] Input:
integrate((e*x^2+d)^(7/2)/(-c*x^4+a)^3,x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {\left (d+e x^2\right )^{7/2}}{\left (a-c x^4\right )^3} \, dx=\int \frac {{\left (e\,x^2+d\right )}^{7/2}}{{\left (a-c\,x^4\right )}^3} \,d x \] Input:
int((d + e*x^2)^(7/2)/(a - c*x^4)^3,x)
Output:
int((d + e*x^2)^(7/2)/(a - c*x^4)^3, x)
\[ \int \frac {\left (d+e x^2\right )^{7/2}}{\left (a-c x^4\right )^3} \, dx=\int \frac {\left (e \,x^{2}+d \right )^{\frac {7}{2}}}{\left (-c \,x^{4}+a \right )^{3}}d x \] Input:
int((e*x^2+d)^(7/2)/(-c*x^4+a)^3,x)
Output:
int((e*x^2+d)^(7/2)/(-c*x^4+a)^3,x)