\(\int \frac {(d+e x^2)^{3/2}}{a-c x^4} \, dx\) [343]

Optimal result

Integrand size = 22, antiderivative size = 170 \[ \int \frac {\left (d+e x^2\right )^{3/2}}{a-c x^4} \, dx=\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^{3/2} \arctan \left (\frac {\sqrt {\sqrt {c} d-\sqrt {a} e} x}{\sqrt [4]{a} \sqrt {d+e x^2}}\right )}{2 a^{3/4} c}-\frac {e^{3/2} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{c}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right )^{3/2} \text {arctanh}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {a} e} x}{\sqrt [4]{a} \sqrt {d+e x^2}}\right )}{2 a^{3/4} c} \] Output:

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

Mathematica [C] (verified)

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

Time = 0.17 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.42 \[ \int \frac {\left (d+e x^2\right )^{3/2}}{a-c x^4} \, dx=\frac {e^{3/2} \left (\log \left (-\sqrt {e} x+\sqrt {d+e x^2}\right )-\text {RootSum}\left [c d^4-4 c d^3 \text {$\#$1}+6 c d^2 \text {$\#$1}^2-16 a e^2 \text {$\#$1}^2-4 c d \text {$\#$1}^3+c \text {$\#$1}^4\&,\frac {c d^3 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )+2 a e^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}+c d \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{c d^3-3 c d^2 \text {$\#$1}+8 a e^2 \text {$\#$1}+3 c d \text {$\#$1}^2-c \text {$\#$1}^3}\&\right ]\right )}{c} \] Input:

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

Output:

(e^(3/2)*(Log[-(Sqrt[e]*x) + Sqrt[d + e*x^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 & , (c*d^3*Log[d + 2 
*e*x^2 - 2*Sqrt[e]*x*Sqrt[d + e*x^2] - #1] + 2*a*e^2*Log[d + 2*e*x^2 - 2*S 
qrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 + 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
 

Rubi [A] (verified)

Time = 0.83 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.82, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {1489, 27, 318, 25, 398, 224, 219, 291, 218, 221}

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.

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

\(\Big \downarrow \) 1489

\(\displaystyle \frac {\sqrt {c} \int \frac {\left (e x^2+d\right )^{3/2}}{\sqrt {c} \left (\sqrt {a}-\sqrt {c} x^2\right )}dx}{2 \sqrt {a}}+\frac {\sqrt {c} \int \frac {\left (e x^2+d\right )^{3/2}}{\sqrt {c} \left (\sqrt {c} x^2+\sqrt {a}\right )}dx}{2 \sqrt {a}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\left (e x^2+d\right )^{3/2}}{\sqrt {a}-\sqrt {c} x^2}dx}{2 \sqrt {a}}+\frac {\int \frac {\left (e x^2+d\right )^{3/2}}{\sqrt {c} x^2+\sqrt {a}}dx}{2 \sqrt {a}}\)

\(\Big \downarrow \) 318

\(\displaystyle \frac {\frac {\int \frac {e \left (3 \sqrt {c} d-2 \sqrt {a} e\right ) x^2+d \left (2 \sqrt {c} d-\sqrt {a} e\right )}{\left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {e x^2+d}}dx}{2 \sqrt {c}}+\frac {e x \sqrt {d+e x^2}}{2 \sqrt {c}}}{2 \sqrt {a}}+\frac {-\frac {\int -\frac {e \left (3 \sqrt {c} d+2 \sqrt {a} e\right ) x^2+d \left (2 \sqrt {c} d+\sqrt {a} e\right )}{\left (\sqrt {a}-\sqrt {c} x^2\right ) \sqrt {e x^2+d}}dx}{2 \sqrt {c}}-\frac {e x \sqrt {d+e x^2}}{2 \sqrt {c}}}{2 \sqrt {a}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {e \left (3 \sqrt {c} d-2 \sqrt {a} e\right ) x^2+d \left (2 \sqrt {c} d-\sqrt {a} e\right )}{\left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {e x^2+d}}dx}{2 \sqrt {c}}+\frac {e x \sqrt {d+e x^2}}{2 \sqrt {c}}}{2 \sqrt {a}}+\frac {\frac {\int \frac {e \left (3 \sqrt {c} d+2 \sqrt {a} e\right ) x^2+d \left (2 \sqrt {c} d+\sqrt {a} e\right )}{\left (\sqrt {a}-\sqrt {c} x^2\right ) \sqrt {e x^2+d}}dx}{2 \sqrt {c}}-\frac {e x \sqrt {d+e x^2}}{2 \sqrt {c}}}{2 \sqrt {a}}\)

\(\Big \downarrow \) 398

\(\displaystyle \frac {\frac {\frac {2 \left (\sqrt {a} e+\sqrt {c} d\right )^2 \int \frac {1}{\left (\sqrt {a}-\sqrt {c} x^2\right ) \sqrt {e x^2+d}}dx}{\sqrt {c}}-e \left (\frac {2 \sqrt {a} e}{\sqrt {c}}+3 d\right ) \int \frac {1}{\sqrt {e x^2+d}}dx}{2 \sqrt {c}}-\frac {e x \sqrt {d+e x^2}}{2 \sqrt {c}}}{2 \sqrt {a}}+\frac {\frac {\frac {2 \left (\sqrt {c} d-\sqrt {a} e\right )^2 \int \frac {1}{\left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {e x^2+d}}dx}{\sqrt {c}}+e \left (3 d-\frac {2 \sqrt {a} e}{\sqrt {c}}\right ) \int \frac {1}{\sqrt {e x^2+d}}dx}{2 \sqrt {c}}+\frac {e x \sqrt {d+e x^2}}{2 \sqrt {c}}}{2 \sqrt {a}}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\frac {\frac {2 \left (\sqrt {c} d-\sqrt {a} e\right )^2 \int \frac {1}{\left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {e x^2+d}}dx}{\sqrt {c}}+e \left (3 d-\frac {2 \sqrt {a} e}{\sqrt {c}}\right ) \int \frac {1}{1-\frac {e x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}}{2 \sqrt {c}}+\frac {e x \sqrt {d+e x^2}}{2 \sqrt {c}}}{2 \sqrt {a}}+\frac {\frac {\frac {2 \left (\sqrt {a} e+\sqrt {c} d\right )^2 \int \frac {1}{\left (\sqrt {a}-\sqrt {c} x^2\right ) \sqrt {e x^2+d}}dx}{\sqrt {c}}-e \left (\frac {2 \sqrt {a} e}{\sqrt {c}}+3 d\right ) \int \frac {1}{1-\frac {e x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}}{2 \sqrt {c}}-\frac {e x \sqrt {d+e x^2}}{2 \sqrt {c}}}{2 \sqrt {a}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {2 \left (\sqrt {a} e+\sqrt {c} d\right )^2 \int \frac {1}{\left (\sqrt {a}-\sqrt {c} x^2\right ) \sqrt {e x^2+d}}dx}{\sqrt {c}}-\sqrt {e} \left (\frac {2 \sqrt {a} e}{\sqrt {c}}+3 d\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {c}}-\frac {e x \sqrt {d+e x^2}}{2 \sqrt {c}}}{2 \sqrt {a}}+\frac {\frac {\frac {2 \left (\sqrt {c} d-\sqrt {a} e\right )^2 \int \frac {1}{\left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {e x^2+d}}dx}{\sqrt {c}}+\sqrt {e} \left (3 d-\frac {2 \sqrt {a} e}{\sqrt {c}}\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {c}}+\frac {e x \sqrt {d+e x^2}}{2 \sqrt {c}}}{2 \sqrt {a}}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {\frac {\frac {2 \left (\sqrt {c} d-\sqrt {a} e\right )^2 \int \frac {1}{\sqrt {a}-\frac {\left (\sqrt {a} e-\sqrt {c} d\right ) x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}}{\sqrt {c}}+\sqrt {e} \left (3 d-\frac {2 \sqrt {a} e}{\sqrt {c}}\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {c}}+\frac {e x \sqrt {d+e x^2}}{2 \sqrt {c}}}{2 \sqrt {a}}+\frac {\frac {\frac {2 \left (\sqrt {a} e+\sqrt {c} d\right )^2 \int \frac {1}{\sqrt {a}-\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}}{\sqrt {c}}-\sqrt {e} \left (\frac {2 \sqrt {a} e}{\sqrt {c}}+3 d\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {c}}-\frac {e x \sqrt {d+e x^2}}{2 \sqrt {c}}}{2 \sqrt {a}}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {\frac {\frac {2 \left (\sqrt {a} e+\sqrt {c} d\right )^2 \int \frac {1}{\sqrt {a}-\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}}{\sqrt {c}}-\sqrt {e} \left (\frac {2 \sqrt {a} e}{\sqrt {c}}+3 d\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {c}}-\frac {e x \sqrt {d+e x^2}}{2 \sqrt {c}}}{2 \sqrt {a}}+\frac {\frac {\frac {2 \left (\sqrt {c} d-\sqrt {a} e\right )^{3/2} \arctan \left (\frac {x \sqrt {\sqrt {c} d-\sqrt {a} e}}{\sqrt [4]{a} \sqrt {d+e x^2}}\right )}{\sqrt [4]{a} \sqrt {c}}+\sqrt {e} \left (3 d-\frac {2 \sqrt {a} e}{\sqrt {c}}\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {c}}+\frac {e x \sqrt {d+e x^2}}{2 \sqrt {c}}}{2 \sqrt {a}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {\frac {2 \left (\sqrt {c} d-\sqrt {a} e\right )^{3/2} \arctan \left (\frac {x \sqrt {\sqrt {c} d-\sqrt {a} e}}{\sqrt [4]{a} \sqrt {d+e x^2}}\right )}{\sqrt [4]{a} \sqrt {c}}+\sqrt {e} \left (3 d-\frac {2 \sqrt {a} e}{\sqrt {c}}\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {c}}+\frac {e x \sqrt {d+e x^2}}{2 \sqrt {c}}}{2 \sqrt {a}}+\frac {\frac {\frac {2 \left (\sqrt {a} e+\sqrt {c} d\right )^{3/2} \text {arctanh}\left (\frac {x \sqrt {\sqrt {a} e+\sqrt {c} d}}{\sqrt [4]{a} \sqrt {d+e x^2}}\right )}{\sqrt [4]{a} \sqrt {c}}-\sqrt {e} \left (\frac {2 \sqrt {a} e}{\sqrt {c}}+3 d\right ) \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {c}}-\frac {e x \sqrt {d+e x^2}}{2 \sqrt {c}}}{2 \sqrt {a}}\)

Input:

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

Output:

((e*x*Sqrt[d + e*x^2])/(2*Sqrt[c]) + ((2*(Sqrt[c]*d - Sqrt[a]*e)^(3/2)*Arc 
Tan[(Sqrt[Sqrt[c]*d - Sqrt[a]*e]*x)/(a^(1/4)*Sqrt[d + e*x^2])])/(a^(1/4)*S 
qrt[c]) + Sqrt[e]*(3*d - (2*Sqrt[a]*e)/Sqrt[c])*ArcTanh[(Sqrt[e]*x)/Sqrt[d 
 + e*x^2]])/(2*Sqrt[c]))/(2*Sqrt[a]) + (-1/2*(e*x*Sqrt[d + e*x^2])/Sqrt[c] 
 + (-(Sqrt[e]*(3*d + (2*Sqrt[a]*e)/Sqrt[c])*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e 
*x^2]]) + (2*(Sqrt[c]*d + Sqrt[a]*e)^(3/2)*ArcTanh[(Sqrt[Sqrt[c]*d + Sqrt[ 
a]*e]*x)/(a^(1/4)*Sqrt[d + e*x^2])])/(a^(1/4)*Sqrt[c]))/(2*Sqrt[c]))/(2*Sq 
rt[a])
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 0]
 

rule 291

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst 
[Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, 
d}, x] && NeQ[b*c - a*d, 0]
 

rule 318

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[d*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(b*(2*(p + q) + 1))), x] + S 
imp[1/(b*(2*(p + q) + 1))   Int[(a + b*x^2)^p*(c + d*x^2)^(q - 2)*Simp[c*(b 
*c*(2*(p + q) + 1) - a*d) + d*(b*c*(2*(p + 2*q - 1) + 1) - a*d*(2*(q - 1) + 
 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && G 
tQ[q, 1] && NeQ[2*(p + q) + 1, 0] &&  !IGtQ[p, 1] && IntBinomialQ[a, b, c, 
d, 2, p, q, x]
 

rule 398

Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]) 
, x_Symbol] :> Simp[f/b   Int[1/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/ 
b   Int[1/((a + b*x^2)*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f} 
, x]
 

rule 1489

Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{r 
= Rt[(-a)*c, 2]}, Simp[-c/(2*r)   Int[(d + e*x^2)^q/(r - c*x^2), x], x] - S 
imp[c/(2*r)   Int[(d + e*x^2)^q/(r + c*x^2), x], x]] /; FreeQ[{a, c, d, e, 
q}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[q]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(256\) vs. \(2(124)=248\).

Time = 0.51 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.51

method	result	size
pseudoelliptic	\(-\frac {-\left (\frac {\left (-a \,e^{2}-c \,d^{2}\right ) \sqrt {d^{2} a c}}{2}+d^{2} e a c \right ) \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}\, \arctan \left (\frac {\sqrt {e \,x^{2}+d}\, a}{x \sqrt {\left (-a e +\sqrt {d^{2} a c}\right ) a}}\right )+\sqrt {\left (-a e +\sqrt {d^{2} a c}\right ) a}\, \left (\left (\frac {\left (-a \,e^{2}-c \,d^{2}\right ) \sqrt {d^{2} a c}}{2}-d^{2} e a c \right ) \operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}\, a}{x \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}}\right )+\operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}}{x \sqrt {e}}\right ) \sqrt {d^{2} a c}\, \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}\, e^{\frac {3}{2}}\right )}{\sqrt {d^{2} a c}\, \sqrt {\left (-a e +\sqrt {d^{2} a c}\right ) a}\, \sqrt {\left (a e +\sqrt {d^{2} a c}\right ) a}\, c}\)	\(257\)
default	\(\text {Expression too large to display}\)	\(2912\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1393 vs. \(2 (124) = 248\).

Time = 1.23 (sec) , antiderivative size = 2793, normalized size of antiderivative = 16.43 \[ \int \frac {\left (d+e x^2\right )^{3/2}}{a-c x^4} \, dx=\text {Too large to display} \] Input:

integrate((e*x^2+d)^(3/2)/(-c*x^4+a),x, algorithm="fricas")
 

Output:

[1/8*(4*e^(3/2)*log(-2*e*x^2 + 2*sqrt(e*x^2 + d)*sqrt(e)*x - d) + c*sqrt(( 
3*c*d^2*e + a*e^3 + a*c^2*sqrt((c^2*d^6 + 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/( 
a^3*c^3)))/(a*c^2))*log(-(c^2*d^5 + 2*a*c*d^3*e^2 - 3*a^2*d*e^4 + (a*c^3*d 
^2 - a^2*c^2*e^2)*x^2*sqrt((c^2*d^6 + 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^3* 
c^3)) + 2*(c^2*d^4*e + 2*a*c*d^2*e^3 - 3*a^2*e^5)*x^2 + 2*(a^2*c^3*x*sqrt( 
(c^2*d^6 + 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^3*c^3)) - (a*c^2*d^2*e + 3*a^ 
2*c*e^3)*x)*sqrt(e*x^2 + d)*sqrt((3*c*d^2*e + a*e^3 + a*c^2*sqrt((c^2*d^6 
+ 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^3*c^3)))/(a*c^2)))/x^2) - c*sqrt((3*c* 
d^2*e + a*e^3 + a*c^2*sqrt((c^2*d^6 + 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^3* 
c^3)))/(a*c^2))*log(-(c^2*d^5 + 2*a*c*d^3*e^2 - 3*a^2*d*e^4 + (a*c^3*d^2 - 
 a^2*c^2*e^2)*x^2*sqrt((c^2*d^6 + 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^3*c^3) 
) + 2*(c^2*d^4*e + 2*a*c*d^2*e^3 - 3*a^2*e^5)*x^2 - 2*(a^2*c^3*x*sqrt((c^2 
*d^6 + 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^3*c^3)) - (a*c^2*d^2*e + 3*a^2*c* 
e^3)*x)*sqrt(e*x^2 + d)*sqrt((3*c*d^2*e + a*e^3 + a*c^2*sqrt((c^2*d^6 + 6* 
a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^3*c^3)))/(a*c^2)))/x^2) - c*sqrt((3*c*d^2* 
e + a*e^3 - a*c^2*sqrt((c^2*d^6 + 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^3*c^3) 
))/(a*c^2))*log(-(c^2*d^5 + 2*a*c*d^3*e^2 - 3*a^2*d*e^4 - (a*c^3*d^2 - a^2 
*c^2*e^2)*x^2*sqrt((c^2*d^6 + 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^3*c^3)) + 
2*(c^2*d^4*e + 2*a*c*d^2*e^3 - 3*a^2*e^5)*x^2 + 2*(a^2*c^3*x*sqrt((c^2*d^6 
 + 6*a*c*d^4*e^2 + 9*a^2*d^2*e^4)/(a^3*c^3)) + (a*c^2*d^2*e + 3*a^2*c*e...
 

Sympy [F]

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

integrate((e*x**2+d)**(3/2)/(-c*x**4+a),x)
 

Output:

-Integral(d*sqrt(d + e*x**2)/(-a + c*x**4), x) - Integral(e*x**2*sqrt(d + 
e*x**2)/(-a + c*x**4), x)
 

Maxima [F]

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

integrate((e*x^2+d)^(3/2)/(-c*x^4+a),x, algorithm="maxima")
 

Output:

-integrate((e*x^2 + d)^(3/2)/(c*x^4 - a), x)
 

Giac [F]

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

integrate((e*x^2+d)^(3/2)/(-c*x^4+a),x, algorithm="giac")
 

Output:

sage0*x
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(sqrt(d + e*x**2)/(a - c*x**4),x)*d + int((sqrt(d + e*x**2)*x**2)/(a - 
c*x**4),x)*e