Integrand size = 22, antiderivative size = 176 \[ \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a-c x^4\right )^2} \, dx=\frac {x \left (d+e x^2\right )^{3/2}}{4 a \left (a-c x^4\right )}+\frac {3 d \sqrt {\sqrt {c} d-\sqrt {a} e} \arctan \left (\frac {\sqrt {\sqrt {c} d-\sqrt {a} e} x}{\sqrt [4]{a} \sqrt {d+e x^2}}\right )}{8 a^{7/4} \sqrt {c}}+\frac {3 d \sqrt {\sqrt {c} d+\sqrt {a} e} \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {a} e} x}{\sqrt [4]{a} \sqrt {d+e x^2}}\right )}{8 a^{7/4} \sqrt {c}} \] Output:
1/4*x*(e*x^2+d)^(3/2)/a/(-c*x^4+a)+3/8*d*(c^(1/2)*d-a^(1/2)*e)^(1/2)*arcta n((c^(1/2)*d-a^(1/2)*e)^(1/2)*x/a^(1/4)/(e*x^2+d)^(1/2))/a^(7/4)/c^(1/2)+3 /8*d*(c^(1/2)*d+a^(1/2)*e)^(1/2)*arctanh((c^(1/2)*d+a^(1/2)*e)^(1/2)*x/a^( 1/4)/(e*x^2+d)^(1/2))/a^(7/4)/c^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.43 (sec) , antiderivative size = 516, normalized size of antiderivative = 2.93 \[ \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a-c x^4\right )^2} \, dx=-\frac {\sqrt {d+e x^2} \left (-d x-e x^3\right )}{4 a \left (a-c x^4\right )}+\frac {2 e^{7/2} \text {RootSum}\left [c d^4-4 c d^3 \text {$\#$1}+6 c d^2 \text {$\#$1}^2-16 a e^2 \text {$\#$1}^2-4 c d \text {$\#$1}^3+c \text {$\#$1}^4\&,\frac {8 d \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )+\log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}}{c d^3-3 c d^2 \text {$\#$1}+8 a e^2 \text {$\#$1}+3 c d \text {$\#$1}^2-c \text {$\#$1}^3}\&\right ]}{c}-\frac {e^{3/2} \text {RootSum}\left [c d^4-4 c d^3 \text {$\#$1}+6 c d^2 \text {$\#$1}^2-16 a e^2 \text {$\#$1}^2-4 c d \text {$\#$1}^3+c \text {$\#$1}^4\&,\frac {3 c d^3 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )+128 a d e^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )+6 c d^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}+16 a e^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}+3 c d \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{c d^3-3 c d^2 \text {$\#$1}+8 a e^2 \text {$\#$1}+3 c d \text {$\#$1}^2-c \text {$\#$1}^3}\&\right ]}{8 a c} \] Input:
Integrate[(d + e*x^2)^(3/2)/(a - c*x^4)^2,x]
Output:
-1/4*(Sqrt[d + e*x^2]*(-(d*x) - e*x^3))/(a*(a - c*x^4)) + (2*e^(7/2)*RootS um[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 & , (8*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) & ])/c - (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 & , (3*c*d^3*Log[d + 2 *e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] + 128*a*d*e^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] + 6*c*d^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x *Sqrt[d + e*x^2] - #1]*#1 + 16*a*e^2*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 + 3*c*d*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) & ])/(8* a*c)
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.
| \(\displaystyle \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a-c x^4\right )^2} \, dx\) |
| \(\Big \downarrow \) 1571 |
| \(\displaystyle \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a-c x^4\right )^2}dx\) |
Input:
Int[(d + e*x^2)^(3/2)/(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]
Time = 0.57 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.00
| method | result | size |
| pseudoelliptic | \(-\frac {d^{2} \left (-\frac {x \left (e \,x^{2}+d \right )^{\frac {3}{2}}}{d^{2} \left (-c \,x^{4}+a \right )}+\frac {3 \left (-a e +\sqrt {d^{2} a c}\right ) \arctan \left (\frac {\sqrt {e \,x^{2}+d}\, a}{x \sqrt {\left (-a e +\sqrt {d^{2} a c}\right ) a}}\right )}{2 \sqrt {d^{2} a c}\, \sqrt {\left (-a e +\sqrt {d^{2} a c}\right ) a}}-\frac {3 \left (a e +\sqrt {d^{2} a c}\right ) \operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}\, a}{x \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}}\right )}{2 \sqrt {d^{2} a c}\, \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}}\right )}{4 a}\) | \(176\) |
| default | \(\text {Expression too large to display}\) | \(7828\) |
Input:
int((e*x^2+d)^(3/2)/(-c*x^4+a)^2,x,method=_RETURNVERBOSE)
Output:
-1/4*d^2/a*(-1/d^2/(-c*x^4+a)*x*(e*x^2+d)^(3/2)+3/2*(-a*e+(d^2*a*c)^(1/2)) /(d^2*a*c)^(1/2)/((-a*e+(d^2*a*c)^(1/2))*a)^(1/2)*arctan((e*x^2+d)^(1/2)/x *a/((-a*e+(d^2*a*c)^(1/2))*a)^(1/2))-3/2*(a*e+(d^2*a*c)^(1/2))/(d^2*a*c)^( 1/2)/((a*e+(d^2*a*c)^(1/2))*a)^(1/2)*arctanh((e*x^2+d)^(1/2)/x*a/((a*e+(d^ 2*a*c)^(1/2))*a)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 641 vs. \(2 (129) = 258\).
Time = 0.74 (sec) , antiderivative size = 641, normalized size of antiderivative = 3.64 \[ \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a-c x^4\right )^2} \, dx=\frac {3 \, {\left (a c x^{4} - a^{2}\right )} \sqrt {\frac {a^{3} c \sqrt {\frac {d^{6}}{a^{7} c}} + d^{2} e}{a^{3} c}} \log \left (\frac {27 \, {\left (a^{3} c d^{2} x^{2} \sqrt {\frac {d^{6}}{a^{7} c}} + 2 \, \sqrt {e x^{2} + d} a^{5} c x \sqrt {\frac {d^{6}}{a^{7} c}} \sqrt {\frac {a^{3} c \sqrt {\frac {d^{6}}{a^{7} c}} + d^{2} e}{a^{3} c}} + 2 \, d^{4} e x^{2} + d^{5}\right )}}{x^{2}}\right ) - 3 \, {\left (a c x^{4} - a^{2}\right )} \sqrt {\frac {a^{3} c \sqrt {\frac {d^{6}}{a^{7} c}} + d^{2} e}{a^{3} c}} \log \left (\frac {27 \, {\left (a^{3} c d^{2} x^{2} \sqrt {\frac {d^{6}}{a^{7} c}} - 2 \, \sqrt {e x^{2} + d} a^{5} c x \sqrt {\frac {d^{6}}{a^{7} c}} \sqrt {\frac {a^{3} c \sqrt {\frac {d^{6}}{a^{7} c}} + d^{2} e}{a^{3} c}} + 2 \, d^{4} e x^{2} + d^{5}\right )}}{x^{2}}\right ) + 3 \, {\left (a c x^{4} - a^{2}\right )} \sqrt {-\frac {a^{3} c \sqrt {\frac {d^{6}}{a^{7} c}} - d^{2} e}{a^{3} c}} \log \left (-\frac {27 \, {\left (a^{3} c d^{2} x^{2} \sqrt {\frac {d^{6}}{a^{7} c}} + 2 \, \sqrt {e x^{2} + d} a^{5} c x \sqrt {\frac {d^{6}}{a^{7} c}} \sqrt {-\frac {a^{3} c \sqrt {\frac {d^{6}}{a^{7} c}} - d^{2} e}{a^{3} c}} - 2 \, d^{4} e x^{2} - d^{5}\right )}}{x^{2}}\right ) - 3 \, {\left (a c x^{4} - a^{2}\right )} \sqrt {-\frac {a^{3} c \sqrt {\frac {d^{6}}{a^{7} c}} - d^{2} e}{a^{3} c}} \log \left (-\frac {27 \, {\left (a^{3} c d^{2} x^{2} \sqrt {\frac {d^{6}}{a^{7} c}} - 2 \, \sqrt {e x^{2} + d} a^{5} c x \sqrt {\frac {d^{6}}{a^{7} c}} \sqrt {-\frac {a^{3} c \sqrt {\frac {d^{6}}{a^{7} c}} - d^{2} e}{a^{3} c}} - 2 \, d^{4} e x^{2} - d^{5}\right )}}{x^{2}}\right ) - 8 \, {\left (e x^{3} + d x\right )} \sqrt {e x^{2} + d}}{32 \, {\left (a c x^{4} - a^{2}\right )}} \] Input:
integrate((e*x^2+d)^(3/2)/(-c*x^4+a)^2,x, algorithm="fricas")
Output:
1/32*(3*(a*c*x^4 - a^2)*sqrt((a^3*c*sqrt(d^6/(a^7*c)) + d^2*e)/(a^3*c))*lo g(27*(a^3*c*d^2*x^2*sqrt(d^6/(a^7*c)) + 2*sqrt(e*x^2 + d)*a^5*c*x*sqrt(d^6 /(a^7*c))*sqrt((a^3*c*sqrt(d^6/(a^7*c)) + d^2*e)/(a^3*c)) + 2*d^4*e*x^2 + d^5)/x^2) - 3*(a*c*x^4 - a^2)*sqrt((a^3*c*sqrt(d^6/(a^7*c)) + d^2*e)/(a^3* c))*log(27*(a^3*c*d^2*x^2*sqrt(d^6/(a^7*c)) - 2*sqrt(e*x^2 + d)*a^5*c*x*sq rt(d^6/(a^7*c))*sqrt((a^3*c*sqrt(d^6/(a^7*c)) + d^2*e)/(a^3*c)) + 2*d^4*e* x^2 + d^5)/x^2) + 3*(a*c*x^4 - a^2)*sqrt(-(a^3*c*sqrt(d^6/(a^7*c)) - d^2*e )/(a^3*c))*log(-27*(a^3*c*d^2*x^2*sqrt(d^6/(a^7*c)) + 2*sqrt(e*x^2 + d)*a^ 5*c*x*sqrt(d^6/(a^7*c))*sqrt(-(a^3*c*sqrt(d^6/(a^7*c)) - d^2*e)/(a^3*c)) - 2*d^4*e*x^2 - d^5)/x^2) - 3*(a*c*x^4 - a^2)*sqrt(-(a^3*c*sqrt(d^6/(a^7*c) ) - d^2*e)/(a^3*c))*log(-27*(a^3*c*d^2*x^2*sqrt(d^6/(a^7*c)) - 2*sqrt(e*x^ 2 + d)*a^5*c*x*sqrt(d^6/(a^7*c))*sqrt(-(a^3*c*sqrt(d^6/(a^7*c)) - d^2*e)/( a^3*c)) - 2*d^4*e*x^2 - d^5)/x^2) - 8*(e*x^3 + d*x)*sqrt(e*x^2 + d))/(a*c* x^4 - a^2)
Timed out. \[ \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a-c x^4\right )^2} \, dx=\text {Timed out} \] Input:
integrate((e*x**2+d)**(3/2)/(-c*x**4+a)**2,x)
Output:
Timed out
\[ \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a-c x^4\right )^2} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}}}{{\left (c x^{4} - a\right )}^{2}} \,d x } \] Input:
integrate((e*x^2+d)^(3/2)/(-c*x^4+a)^2,x, algorithm="maxima")
Output:
integrate((e*x^2 + d)^(3/2)/(c*x^4 - a)^2, x)
Timed out. \[ \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a-c x^4\right )^2} \, dx=\text {Timed out} \] Input:
integrate((e*x^2+d)^(3/2)/(-c*x^4+a)^2,x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a-c x^4\right )^2} \, dx=\int \frac {{\left (e\,x^2+d\right )}^{3/2}}{{\left (a-c\,x^4\right )}^2} \,d x \] Input:
int((d + e*x^2)^(3/2)/(a - c*x^4)^2,x)
Output:
int((d + e*x^2)^(3/2)/(a - c*x^4)^2, x)
\[ \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a-c x^4\right )^2} \, dx=\left (\int \frac {\sqrt {e \,x^{2}+d}}{c^{2} x^{8}-2 a c \,x^{4}+a^{2}}d x \right ) d +\left (\int \frac {\sqrt {e \,x^{2}+d}\, x^{2}}{c^{2} x^{8}-2 a c \,x^{4}+a^{2}}d x \right ) e \] Input:
int((e*x^2+d)^(3/2)/(-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 + int((sqrt(d + e*x**2)*x**2)/(a**2 - 2*a*c*x**4 + c**2*x**8),x)*e