\(\int \frac {(d+e x^2)^3}{(d^2-e^2 x^4)^{3/2}} \, dx\) [61]

Optimal result

Integrand size = 26, antiderivative size = 147 \[ \int \frac {\left (d+e x^2\right )^3}{\left (d^2-e^2 x^4\right )^{3/2}} \, dx=\frac {2 x \left (d+e x^2\right )}{\sqrt {d^2-e^2 x^4}}-\frac {3 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d^2-e^2 x^4}}+\frac {2 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{\sqrt {e} \sqrt {d^2-e^2 x^4}} \] Output:

2*x*(e*x^2+d)/(-e^2*x^4+d^2)^(1/2)-3*d^(3/2)*(1-e^2*x^4/d^2)^(1/2)*Ellipti 
cE(e^(1/2)*x/d^(1/2),I)/e^(1/2)/(-e^2*x^4+d^2)^(1/2)+2*d^(3/2)*(1-e^2*x^4/ 
d^2)^(1/2)*EllipticF(e^(1/2)*x/d^(1/2),I)/e^(1/2)/(-e^2*x^4+d^2)^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.10 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.76 \[ \int \frac {\left (d+e x^2\right )^3}{\left (d^2-e^2 x^4\right )^{3/2}} \, dx=\frac {2 d x-e x^3-d x \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {e^2 x^4}{d^2}\right )+2 e x^3 \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{2},\frac {7}{4},\frac {e^2 x^4}{d^2}\right )}{\sqrt {d^2-e^2 x^4}} \] Input:

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

Output:

(2*d*x - e*x^3 - d*x*Sqrt[1 - (e^2*x^4)/d^2]*Hypergeometric2F1[1/4, 1/2, 5 
/4, (e^2*x^4)/d^2] + 2*e*x^3*Sqrt[1 - (e^2*x^4)/d^2]*Hypergeometric2F1[3/4 
, 3/2, 7/4, (e^2*x^4)/d^2])/Sqrt[d^2 - e^2*x^4]
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.37, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {1396, 315, 27, 399, 289, 329, 327, 765, 762}

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.

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

\(\Big \downarrow \) 1396

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

\(\Big \downarrow \) 315

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 399

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (2 d \int \frac {1}{\sqrt {d-e x^2} \sqrt {e x^2+d}}dx-3 \int \frac {\sqrt {e x^2+d}}{\sqrt {d-e x^2}}dx+\frac {2 x \sqrt {d+e x^2}}{\sqrt {d-e x^2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 289

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {2 d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}-3 \int \frac {\sqrt {e x^2+d}}{\sqrt {d-e x^2}}dx+\frac {2 x \sqrt {d+e x^2}}{\sqrt {d-e x^2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 329

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (-\frac {3 d \sqrt {1-\frac {e^2 x^4}{d^2}} \int \frac {\sqrt {\frac {e x^2}{d}+1}}{\sqrt {1-\frac {e x^2}{d}}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}+\frac {2 d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}+\frac {2 x \sqrt {d+e x^2}}{\sqrt {d-e x^2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {2 d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {3 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}+\frac {2 x \sqrt {d+e x^2}}{\sqrt {d-e x^2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 765

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {2 d \sqrt {1-\frac {e^2 x^4}{d^2}} \int \frac {1}{\sqrt {1-\frac {e^2 x^4}{d^2}}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {3 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}+\frac {2 x \sqrt {d+e x^2}}{\sqrt {d-e x^2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 762

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {2 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {3 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}+\frac {2 x \sqrt {d+e x^2}}{\sqrt {d-e x^2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

Input:

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

Output:

(Sqrt[d - e*x^2]*Sqrt[d + e*x^2]*((2*x*Sqrt[d + e*x^2])/Sqrt[d - e*x^2] - 
(3*d^(3/2)*Sqrt[1 - (e^2*x^4)/d^2]*EllipticE[ArcSin[(Sqrt[e]*x)/Sqrt[d]], 
-1])/(Sqrt[e]*Sqrt[d - e*x^2]*Sqrt[d + e*x^2]) + (2*d^(3/2)*Sqrt[1 - (e^2* 
x^4)/d^2]*EllipticF[ArcSin[(Sqrt[e]*x)/Sqrt[d]], -1])/(Sqrt[e]*Sqrt[d - e* 
x^2]*Sqrt[d + e*x^2])))/Sqrt[d^2 - e^2*x^4]
 

Defintions of rubi rules used

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 289

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> Sim 
p[(a + b*x^2)^FracPart[p]*((c + d*x^2)^FracPart[p]/(a*c + b*d*x^4)^FracPart 
[p])   Int[(a*c + b*d*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && EqQ[b* 
c + a*d, 0] &&  !IntegerQ[p]
 

rule 315

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[(a*d - c*b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(2*a*b*(p + 1))), 
x] - Simp[1/(2*a*b*(p + 1))   Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 2)*S 
imp[c*(a*d - c*b*(2*p + 3)) + d*(a*d*(2*(q - 1) + 1) - b*c*(2*(p + q) + 1)) 
*x^2, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, - 
1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, 2, p, q, x]
 

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 329

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
a*(Sqrt[1 - b^2*(x^4/a^2)]/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]))   Int[Sqrt[1 
+ b*(x^2/a)]/Sqrt[1 - b*(x^2/a)], x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b 
*c + a*d, 0] &&  !(LtQ[a*c, 0] && GtQ[a*b, 0])
 

rule 399

Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_) 
^2]), x_Symbol] :> Simp[f/b   Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] + 
 Simp[(b*e - a*f)/b   Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; Fr 
eeQ[{a, b, c, d, e, f}, x] &&  !((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && 
(PosQ[d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c])))))
 

rule 762

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) 
)*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] 
 && GtQ[a, 0]
 

rule 765

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[Sqrt[1 + b*(x^4/a)]/Sqrt 
[a + b*x^4]   Int[1/Sqrt[1 + b*(x^4/a)], x], x] /; FreeQ[{a, b}, x] && NegQ 
[b/a] &&  !GtQ[a, 0]
 

rule 1396

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_)*((d_) + (e_.)*(x_)^(n_))^(q_.), x 
_Symbol] :> Simp[(a + c*x^(2*n))^FracPart[p]/((d + e*x^n)^FracPart[p]*(a/d 
+ c*(x^n/e))^FracPart[p])   Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, 
x], x] /; FreeQ[{a, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a* 
e^2, 0] &&  !IntegerQ[p] &&  !(EqQ[q, 1] && EqQ[n, 2])
 

Maple [A] (verified)

Time = 6.33 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.25

method	result	size
elliptic	\(-\frac {2 \left (-e^{2} x^{2}-d e \right ) x}{e \sqrt {\left (x^{2}-\frac {d}{e}\right ) \left (-e^{2} x^{2}-d e \right )}}-\frac {d \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{\sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}+\frac {3 d \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{\sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\)	\(184\)
default	\(d^{3} \left (\frac {x}{2 d^{2} \sqrt {-\left (x^{4}-\frac {d^{2}}{e^{2}}\right ) e^{2}}}+\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{2 d^{2} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\right )+e^{3} \left (\frac {x^{3}}{2 e^{2} \sqrt {-\left (x^{4}-\frac {d^{2}}{e^{2}}\right ) e^{2}}}+\frac {3 d \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{2 e^{3} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\right )+3 d \,e^{2} \left (\frac {x}{2 e^{2} \sqrt {-\left (x^{4}-\frac {d^{2}}{e^{2}}\right ) e^{2}}}-\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{2 e^{2} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\right )+3 d^{2} e \left (\frac {x^{3}}{2 d^{2} \sqrt {-\left (x^{4}-\frac {d^{2}}{e^{2}}\right ) e^{2}}}+\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{2 d \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}\, e}\right )\)	\(416\)

Input:

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

Output:

-2*(-e^2*x^2-d*e)*x/e/((x^2-d/e)*(-e^2*x^2-d*e))^(1/2)-d/(e/d)^(1/2)*(1-e* 
x^2/d)^(1/2)*(1+e*x^2/d)^(1/2)/(-e^2*x^4+d^2)^(1/2)*EllipticF(x*(e/d)^(1/2 
),I)+3*d/(e/d)^(1/2)*(1-e*x^2/d)^(1/2)*(1+e*x^2/d)^(1/2)/(-e^2*x^4+d^2)^(1 
/2)*(EllipticF(x*(e/d)^(1/2),I)-EllipticE(x*(e/d)^(1/2),I))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.98 \[ \int \frac {\left (d+e x^2\right )^3}{\left (d^2-e^2 x^4\right )^{3/2}} \, dx=\frac {3 \, {\left (d e x^{3} - d^{2} x\right )} \sqrt {-e^{2}} \sqrt {\frac {d}{e}} E(\arcsin \left (\frac {\sqrt {\frac {d}{e}}}{x}\right )\,|\,-1) - {\left ({\left (3 \, d e + e^{2}\right )} x^{3} - {\left (3 \, d^{2} + d e\right )} x\right )} \sqrt {-e^{2}} \sqrt {\frac {d}{e}} F(\arcsin \left (\frac {\sqrt {\frac {d}{e}}}{x}\right )\,|\,-1) + \sqrt {-e^{2} x^{4} + d^{2}} {\left (e^{2} x^{2} - 3 \, d e\right )}}{e^{3} x^{3} - d e^{2} x} \] Input:

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

Output:

(3*(d*e*x^3 - d^2*x)*sqrt(-e^2)*sqrt(d/e)*elliptic_e(arcsin(sqrt(d/e)/x), 
-1) - ((3*d*e + e^2)*x^3 - (3*d^2 + d*e)*x)*sqrt(-e^2)*sqrt(d/e)*elliptic_ 
f(arcsin(sqrt(d/e)/x), -1) + sqrt(-e^2*x^4 + d^2)*(e^2*x^2 - 3*d*e))/(e^3* 
x^3 - d*e^2*x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (d+e x^2\right )^3}{\left (d^2-e^2 x^4\right )^{3/2}} \, dx=\frac {\sqrt {-e^{2} x^{4}+d^{2}}\, x +2 \left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}\, x^{4}}{e^{3} x^{6}-d \,e^{2} x^{4}-d^{2} e \,x^{2}+d^{3}}d x \right ) d \,e^{2}-2 \left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}\, x^{4}}{e^{3} x^{6}-d \,e^{2} x^{4}-d^{2} e \,x^{2}+d^{3}}d x \right ) e^{3} x^{2}}{-e \,x^{2}+d} \] Input:

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

Output:

(sqrt(d**2 - e**2*x**4)*x + 2*int((sqrt(d**2 - e**2*x**4)*x**4)/(d**3 - d* 
*2*e*x**2 - d*e**2*x**4 + e**3*x**6),x)*d*e**2 - 2*int((sqrt(d**2 - e**2*x 
**4)*x**4)/(d**3 - d**2*e*x**2 - d*e**2*x**4 + e**3*x**6),x)*e**3*x**2)/(d 
 - e*x**2)