\(\int \frac {(d+e x^2)^{3/2}}{a+c x^4} \, dx\) [384]

Optimal result

Integrand size = 21, antiderivative size = 442 \[ \int \frac {\left (d+e x^2\right )^{3/2}}{a+c x^4} \, dx=\frac {\sqrt {\sqrt {a} e+\sqrt {c d^2+a e^2}} \left (2 c d^2 e-\left (c d^2-a e^2\right ) \left (e-\frac {\sqrt {c d^2+a e^2}}{\sqrt {a}}\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt {\sqrt {a} e+\sqrt {c d^2+a e^2}} x \sqrt {d+e x^2}}{\sqrt {a} \left (\sqrt {a} e+\sqrt {c d^2+a e^2}\right )-c d x^2}\right )}{2 \sqrt {2} \sqrt [4]{a} c^{3/2} d \sqrt {c d^2+a e^2}}+\frac {e^{3/2} \text {arctanh}\left (\frac {\sqrt {e} x}{\sqrt {d+e x^2}}\right )}{c}+\frac {\sqrt {-\sqrt {a} e+\sqrt {c d^2+a e^2}} \left (2 c d^2 e-\left (c d^2-a e^2\right ) \left (e+\frac {\sqrt {c d^2+a e^2}}{\sqrt {a}}\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {c} \sqrt {-\sqrt {a} e+\sqrt {c d^2+a e^2}} x \sqrt {d+e x^2}}{\sqrt {a} \left (\sqrt {a} e-\sqrt {c d^2+a e^2}\right )-c d x^2}\right )}{2 \sqrt {2} \sqrt [4]{a} c^{3/2} d \sqrt {c d^2+a e^2}} \] Output:

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

Mathematica [C] (verified)

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

Time = 0.16 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.55 \[ \int \frac {\left (d+e x^2\right )^{3/2}}{a+c x^4} \, dx=\frac {e^{3/2} \left (-\log \left (-\sqrt {e} x+\sqrt {d+e x^2}\right )+\text {RootSum}\left [c d^4-4 c d^3 \text {$\#$1}+6 c d^2 \text {$\#$1}^2+16 a e^2 \text {$\#$1}^2-4 c d \text {$\#$1}^3+c \text {$\#$1}^4\&,\frac {c d^3 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right )-2 a e^2 \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}+c d \log \left (d+2 e x^2-2 \sqrt {e} x \sqrt {d+e x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{c d^3-3 c d^2 \text {$\#$1}-8 a e^2 \text {$\#$1}+3 c d \text {$\#$1}^2-c \text {$\#$1}^3}\&\right ]\right )}{c} \] 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* 
Sqrt[e]*x*Sqrt[d + e*x^2] - #1]*#1 + c*d*Log[d + 2*e*x^2 - 2*Sqrt[e]*x*Sqr 
t[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 = 1.04 (sec) , antiderivative size = 390, normalized size of antiderivative = 0.88, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, 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

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 218

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

-1/2*((e*x*Sqrt[d + e*x^2])/(2*Sqrt[c]) + ((2*(c*d^2 - 2*Sqrt[-a]*Sqrt[c]* 
d*e - a*e^2)*ArcTan[(Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*x)/((-a)^(1/4)*Sqrt[d + 
e*x^2])])/((-a)^(1/4)*Sqrt[c]*Sqrt[Sqrt[c]*d - Sqrt[-a]*e]) + Sqrt[e]*(3*d 
 - (2*Sqrt[-a]*e)/Sqrt[c])*ArcTanh[(Sqrt[e]*x)/Sqrt[d + e*x^2]])/(2*Sqrt[c 
]))/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*(c*d^2 + 2 
*Sqrt[-a]*Sqrt[c]*d*e - a*e^2)*ArcTanh[(Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*x)/(( 
-a)^(1/4)*Sqrt[d + e*x^2])])/((-a)^(1/4)*Sqrt[c]*Sqrt[Sqrt[c]*d + Sqrt[-a] 
*e]))/(2*Sqrt[c]))/(2*Sqrt[-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. \(782\) vs. \(2(356)=712\).

Time = 0.78 (sec) , antiderivative size = 783, normalized size of antiderivative = 1.77

method	result	size
pseudoelliptic	\(-\frac {\frac {\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}\, \sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, \left (\left (-a \,e^{2}-e \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {a}+c \,d^{2}\right ) \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+e \left (a^{2} e^{2}-d^{2} a c +a^{\frac {3}{2}} \sqrt {a \,e^{2}+c \,d^{2}}\, e \right )\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 )}{4}-\frac {\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}\, \sqrt {2 \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+2 a e}\, \left (\left (-a \,e^{2}-e \sqrt {a \,e^{2}+c \,d^{2}}\, \sqrt {a}+c \,d^{2}\right ) \sqrt {a \left (a \,e^{2}+c \,d^{2}\right )}+e \left (a^{2} e^{2}-d^{2} a c +a^{\frac {3}{2}} \sqrt {a \,e^{2}+c \,d^{2}}\, e \right )\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 )}{4}+d^{2} c \left (-2 e^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}}{x \sqrt {e}}\right ) \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 {3}{2}}+\left (a^{2} e^{2}-d^{2} a c -a^{\frac {3}{2}} \sqrt {a \,e^{2}+c \,d^{2}}\, e \right ) \left (\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 )-\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 )\right )\right )}{2 a^{\frac {3}{2}} \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}\, d^{2} c^{2}}\)	\(783\)
default	\(\text {Expression too large to display}\)	\(3292\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1421 vs. \(2 (358) = 716\).

Time = 1.30 (sec) , antiderivative size = 2849, normalized size of antiderivative = 6.45 \[ \int \frac {\left (d+e x^2\right )^{3/2}}{a+c x^4} \, dx=\text {Too large to display} \] 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*sq 
rt(-(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*sq 
rt(-(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...
 

Sympy [F]

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

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

Output:

Integral((d + e*x**2)**(3/2)/(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(-2)]

Exception generated. \[ \int \frac {\left (d+e x^2\right )^{3/2}}{a+c x^4} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E 
rror: Bad Argument Value
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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