\(\int \frac {(d+e x^2)^{3/2}}{(a+c x^4)^2} \, dx\) [392]

Optimal result

Integrand size = 21, antiderivative size = 317 \[ \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a+c x^4\right )^2} \, dx=\frac {x \left (d+e x^2\right )^{3/2}}{4 a \left (a+c x^4\right )}+\frac {3 d \sqrt {\sqrt {a} e+\sqrt {c d^2+a e^2}} \arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt {\sqrt {a} e+\sqrt {c d^2+a e^2}} x \sqrt {d+e x^2}}{\sqrt {a} \left (\sqrt {a} e+\sqrt {c d^2+a e^2}\right )-c d x^2}\right )}{8 \sqrt {2} a^{7/4} \sqrt {c}}-\frac {3 d \sqrt {-\sqrt {a} e+\sqrt {c d^2+a e^2}} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt {-\sqrt {a} e+\sqrt {c d^2+a e^2}} x \sqrt {d+e x^2}}{\sqrt {a} \left (\sqrt {a} e-\sqrt {c d^2+a e^2}\right )-c d x^2}\right )}{8 \sqrt {2} a^{7/4} \sqrt {c}} \] Output:

1/4*x*(e*x^2+d)^(3/2)/a/(c*x^4+a)+3/16*d*(a^(1/2)*e+(a*e^2+c*d^2)^(1/2))^( 
1/2)*arctan(2^(1/2)*a^(1/4)*c^(1/2)*(a^(1/2)*e+(a*e^2+c*d^2)^(1/2))^(1/2)* 
x*(e*x^2+d)^(1/2)/(a^(1/2)*(a^(1/2)*e+(a*e^2+c*d^2)^(1/2))-c*d*x^2))*2^(1/ 
2)/a^(7/4)/c^(1/2)-3/16*d*(-a^(1/2)*e+(a*e^2+c*d^2)^(1/2))^(1/2)*arctanh(2 
^(1/2)*a^(1/4)*c^(1/2)*(-a^(1/2)*e+(a*e^2+c*d^2)^(1/2))^(1/2)*x*(e*x^2+d)^ 
(1/2)/(a^(1/2)*(a^(1/2)*e-(a*e^2+c*d^2)^(1/2))-c*d*x^2))*2^(1/2)/a^(7/4)/c 
^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 0.45 (sec) , antiderivative size = 502, normalized size of antiderivative = 1.58 \[ \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a+c x^4\right )^2} \, dx=\frac {2 e^{7/2} \text {RootSum}\left [c d^4-4 c d^3 \text {$\#$1}+6 c d^2 \text {$\#$1}^2+16 a e^2 \text {$\#$1}^2-4 c d \text {$\#$1}^3+c \text {$\#$1}^4\&,\frac {8 d \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )+\log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}}{c d^3-3 c d^2 \text {$\#$1}-8 a e^2 \text {$\#$1}+3 c d \text {$\#$1}^2-c \text {$\#$1}^3}\&\right ]}{c}+\frac {\frac {2 x \left (d+e x^2\right )^{3/2}}{a+c x^4}+\frac {e^{3/2} \text {RootSum}\left [c d^4-4 c d^3 \text {$\#$1}+6 c d^2 \text {$\#$1}^2+16 a e^2 \text {$\#$1}^2-4 c d \text {$\#$1}^3+c \text {$\#$1}^4\&,\frac {3 c d^3 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )-128 a d e^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )+6 c d^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}-16 a e^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}+3 c d \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{c d^3-3 c d^2 \text {$\#$1}-8 a e^2 \text {$\#$1}+3 c d \text {$\#$1}^2-c \text {$\#$1}^3}\&\right ]}{c}}{8 a} \] Input:

Integrate[(d + e*x^2)^(3/2)/(a + c*x^4)^2,x]
 

Output:

(2*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 & , (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 + ((2*x*(d + e*x^2)^(3/ 
2))/(a + c*x^4) + (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) & ])/c)/(8*a)
 

Rubi [F]

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
 

Defintions of rubi rules used

rule 1571

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(552\) vs. \(2(245)=490\).

Time = 0.88 (sec) , antiderivative size = 553, normalized size of antiderivative = 1.74

method	result	size
pseudoelliptic	\(\frac {\frac {3 \sqrt {4 \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {a}-2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}-2 a e}\, a \left (\ln \left (\frac {\sqrt {a}\, \left (e \,x^{2}+d \right )-\sqrt {e \,x^{2}+d}\, \sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, x +\sqrt {a \,e^{2}+c \,d^{2}}\, x^{2}}{x^{2}}\right )-\ln \left (\frac {\sqrt {a}\, \left (e \,x^{2}+d \right )+\sqrt {e \,x^{2}+d}\, \sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, x +\sqrt {a \,e^{2}+c \,d^{2}}\, x^{2}}{x^{2}}\right )\right ) \left (c \,x^{4}+a \right ) \left (a e -\sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}\right ) \sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}}{32}+\frac {\left (x \sqrt {4 \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {a}-2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}-2 a e}\, a^{\frac {5}{2}} \left (e \,x^{2}+d \right )^{\frac {3}{2}}-\frac {3 \left (\arctan \left (\frac {2 \sqrt {a}\, \sqrt {e \,x^{2}+d}+\sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, x}{x \sqrt {4 \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {a}-2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}-2 a e}}\right )-\arctan \left (\frac {\sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, x -2 \sqrt {a}\, \sqrt {e \,x^{2}+d}}{x \sqrt {4 \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {a}-2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}-2 a e}}\right )\right ) d^{2} a^{2} \left (c \,x^{4}+a \right )}{2}\right ) c}{4}}{\left (c \,x^{4}+a \right ) a^{\frac {7}{2}} c \sqrt {4 \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {a}-2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}-2 a e}}\)	\(553\)
default	\(\text {Expression too large to display}\)	\(8830\)

Input:

int((e*x^2+d)^(3/2)/(c*x^4+a)^2,x,method=_RETURNVERBOSE)
 

Output:

3/32/a^(7/2)*((4*(a*e^2+c*d^2)^(1/2)*a^(1/2)-2*(a*(a*e^2+c*d^2))^(1/2)-2*a 
*e)^(1/2)*a*(ln((a^(1/2)*(e*x^2+d)-(e*x^2+d)^(1/2)*(2*(a*(a*e^2+c*d^2))^(1 
/2)+2*a*e)^(1/2)*x+(a*e^2+c*d^2)^(1/2)*x^2)/x^2)-ln((a^(1/2)*(e*x^2+d)+(e* 
x^2+d)^(1/2)*(2*(a*(a*e^2+c*d^2))^(1/2)+2*a*e)^(1/2)*x+(a*e^2+c*d^2)^(1/2) 
*x^2)/x^2))*(c*x^4+a)*(a*e-(a*(a*e^2+c*d^2))^(1/2))*(2*(a*(a*e^2+c*d^2))^( 
1/2)+2*a*e)^(1/2)+8/3*(x*(4*(a*e^2+c*d^2)^(1/2)*a^(1/2)-2*(a*(a*e^2+c*d^2) 
)^(1/2)-2*a*e)^(1/2)*a^(5/2)*(e*x^2+d)^(3/2)-3/2*(arctan((2*a^(1/2)*(e*x^2 
+d)^(1/2)+(2*(a*(a*e^2+c*d^2))^(1/2)+2*a*e)^(1/2)*x)/x/(4*(a*e^2+c*d^2)^(1 
/2)*a^(1/2)-2*(a*(a*e^2+c*d^2))^(1/2)-2*a*e)^(1/2))-arctan(((2*(a*(a*e^2+c 
*d^2))^(1/2)+2*a*e)^(1/2)*x-2*a^(1/2)*(e*x^2+d)^(1/2))/x/(4*(a*e^2+c*d^2)^ 
(1/2)*a^(1/2)-2*(a*(a*e^2+c*d^2))^(1/2)-2*a*e)^(1/2)))*d^2*a^2*(c*x^4+a))* 
c)/(4*(a*e^2+c*d^2)^(1/2)*a^(1/2)-2*(a*(a*e^2+c*d^2))^(1/2)-2*a*e)^(1/2)/( 
c*x^4+a)/c
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 647 vs. \(2 (247) = 494\).

Time = 0.75 (sec) , antiderivative size = 647, normalized size of antiderivative = 2.04 \[ \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a+c x^4\right )^2} \, dx=\frac {3 \, {\left (a c x^{4} + a^{2}\right )} \sqrt {-\frac {a^{3} c \sqrt {-\frac {d^{6}}{a^{7} c}} + d^{2} e}{a^{3} c}} \log \left (\frac {27 \, {\left (a^{3} c d^{2} x^{2} \sqrt {-\frac {d^{6}}{a^{7} c}} + 2 \, \sqrt {e x^{2} + d} a^{5} c x \sqrt {-\frac {d^{6}}{a^{7} c}} \sqrt {-\frac {a^{3} c \sqrt {-\frac {d^{6}}{a^{7} c}} + d^{2} e}{a^{3} c}} + 2 \, d^{4} e x^{2} + d^{5}\right )}}{x^{2}}\right ) - 3 \, {\left (a c x^{4} + a^{2}\right )} \sqrt {-\frac {a^{3} c \sqrt {-\frac {d^{6}}{a^{7} c}} + d^{2} e}{a^{3} c}} \log \left (\frac {27 \, {\left (a^{3} c d^{2} x^{2} \sqrt {-\frac {d^{6}}{a^{7} c}} - 2 \, \sqrt {e x^{2} + d} a^{5} c x \sqrt {-\frac {d^{6}}{a^{7} c}} \sqrt {-\frac {a^{3} c \sqrt {-\frac {d^{6}}{a^{7} c}} + d^{2} e}{a^{3} c}} + 2 \, d^{4} e x^{2} + d^{5}\right )}}{x^{2}}\right ) + 3 \, {\left (a c x^{4} + a^{2}\right )} \sqrt {\frac {a^{3} c \sqrt {-\frac {d^{6}}{a^{7} c}} - d^{2} e}{a^{3} c}} \log \left (-\frac {27 \, {\left (a^{3} c d^{2} x^{2} \sqrt {-\frac {d^{6}}{a^{7} c}} + 2 \, \sqrt {e x^{2} + d} a^{5} c x \sqrt {-\frac {d^{6}}{a^{7} c}} \sqrt {\frac {a^{3} c \sqrt {-\frac {d^{6}}{a^{7} c}} - d^{2} e}{a^{3} c}} - 2 \, d^{4} e x^{2} - d^{5}\right )}}{x^{2}}\right ) - 3 \, {\left (a c x^{4} + a^{2}\right )} \sqrt {\frac {a^{3} c \sqrt {-\frac {d^{6}}{a^{7} c}} - d^{2} e}{a^{3} c}} \log \left (-\frac {27 \, {\left (a^{3} c d^{2} x^{2} \sqrt {-\frac {d^{6}}{a^{7} c}} - 2 \, \sqrt {e x^{2} + d} a^{5} c x \sqrt {-\frac {d^{6}}{a^{7} c}} \sqrt {\frac {a^{3} c \sqrt {-\frac {d^{6}}{a^{7} c}} - d^{2} e}{a^{3} c}} - 2 \, d^{4} e x^{2} - d^{5}\right )}}{x^{2}}\right ) + 8 \, {\left (e x^{3} + d x\right )} \sqrt {e x^{2} + d}}{32 \, {\left (a c x^{4} + a^{2}\right )}} \] 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))* 
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) + 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*sqr 
t(-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)
 

Sympy [F(-1)]

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
 

Maxima [F]

\[ \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a+c x^4\right )^2} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}}}{{\left (c x^{4} + a\right )}^{2}} \,d x } \] 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)
 

Giac [F(-1)]

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
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a+c x^4\right )^2} \, dx=\int \frac {{\left (e\,x^2+d\right )}^{3/2}}{{\left (c\,x^4+a\right )}^2} \,d x \] 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)
 

Reduce [F]

\[ \int \frac {\left (d+e x^2\right )^{3/2}}{\left (a+c x^4\right )^2} \, dx=\left (\int \frac {\sqrt {e \,x^{2}+d}}{c^{2} x^{8}+2 a c \,x^{4}+a^{2}}d x \right ) d +\left (\int \frac {\sqrt {e \,x^{2}+d}\, x^{2}}{c^{2} x^{8}+2 a c \,x^{4}+a^{2}}d x \right ) e \] 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