\(\int \sqrt {d+e x^2} (d^2-e^2 x^4)^{3/2} \, dx\) [139]

Optimal result

Integrand size = 28, antiderivative size = 196 \[ \int \sqrt {d+e x^2} \left (d^2-e^2 x^4\right )^{3/2} \, dx=\frac {61 d^3 x \sqrt {d^2-e^2 x^4}}{128 \sqrt {d+e x^2}}+\frac {61 d^2 e x^3 \sqrt {d^2-e^2 x^4}}{192 \sqrt {d+e x^2}}-\frac {7 d e^2 x^5 \sqrt {d^2-e^2 x^4}}{48 \sqrt {d+e x^2}}-\frac {e^3 x^7 \sqrt {d^2-e^2 x^4}}{8 \sqrt {d+e x^2}}+\frac {67 d^4 \arctan \left (\frac {\sqrt {e} x \sqrt {d+e x^2}}{\sqrt {d^2-e^2 x^4}}\right )}{128 \sqrt {e}} \] Output:

61/128*d^3*x*(-e^2*x^4+d^2)^(1/2)/(e*x^2+d)^(1/2)+61/192*d^2*e*x^3*(-e^2*x 
^4+d^2)^(1/2)/(e*x^2+d)^(1/2)-7/48*d*e^2*x^5*(-e^2*x^4+d^2)^(1/2)/(e*x^2+d 
)^(1/2)-1/8*e^3*x^7*(-e^2*x^4+d^2)^(1/2)/(e*x^2+d)^(1/2)+67/128*d^4*arctan 
(e^(1/2)*x*(e*x^2+d)^(1/2)/(-e^2*x^4+d^2)^(1/2))/e^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 3.01 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.61 \[ \int \sqrt {d+e x^2} \left (d^2-e^2 x^4\right )^{3/2} \, dx=\frac {1}{384} \left (\frac {x \sqrt {d^2-e^2 x^4} \left (183 d^3+122 d^2 e x^2-56 d e^2 x^4-48 e^3 x^6\right )}{\sqrt {d+e x^2}}+\frac {201 i d^4 \log \left (-2 i \sqrt {e} x+\frac {2 \sqrt {d^2-e^2 x^4}}{\sqrt {d+e x^2}}\right )}{\sqrt {e}}\right ) \] Input:

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

Output:

((x*Sqrt[d^2 - e^2*x^4]*(183*d^3 + 122*d^2*e*x^2 - 56*d*e^2*x^4 - 48*e^3*x 
^6))/Sqrt[d + e*x^2] + ((201*I)*d^4*Log[(-2*I)*Sqrt[e]*x + (2*Sqrt[d^2 - e 
^2*x^4])/Sqrt[d + e*x^2]])/Sqrt[e])/384
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.84, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {1396, 318, 25, 27, 299, 211, 211, 224, 216}

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.

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

\(\Big \downarrow \) 1396

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

\(\Big \downarrow \) 318

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (-\frac {\int -d e \left (d-e x^2\right )^{3/2} \left (13 e x^2+9 d\right )dx}{8 e}-\frac {1}{8} x \left (d+e x^2\right ) \left (d-e x^2\right )^{5/2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {\int d e \left (d-e x^2\right )^{3/2} \left (13 e x^2+9 d\right )dx}{8 e}-\frac {1}{8} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{8} d \int \left (d-e x^2\right )^{3/2} \left (13 e x^2+9 d\right )dx-\frac {1}{8} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 299

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{8} d \left (\frac {67}{6} d \int \left (d-e x^2\right )^{3/2}dx-\frac {13}{6} x \left (d-e x^2\right )^{5/2}\right )-\frac {1}{8} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{8} d \left (\frac {67}{6} d \left (\frac {3}{4} d \int \sqrt {d-e x^2}dx+\frac {1}{4} x \left (d-e x^2\right )^{3/2}\right )-\frac {13}{6} x \left (d-e x^2\right )^{5/2}\right )-\frac {1}{8} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{8} d \left (\frac {67}{6} d \left (\frac {3}{4} d \left (\frac {1}{2} d \int \frac {1}{\sqrt {d-e x^2}}dx+\frac {1}{2} x \sqrt {d-e x^2}\right )+\frac {1}{4} x \left (d-e x^2\right )^{3/2}\right )-\frac {13}{6} x \left (d-e x^2\right )^{5/2}\right )-\frac {1}{8} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{8} d \left (\frac {67}{6} d \left (\frac {3}{4} d \left (\frac {1}{2} d \int \frac {1}{\frac {e x^2}{d-e x^2}+1}d\frac {x}{\sqrt {d-e x^2}}+\frac {1}{2} x \sqrt {d-e x^2}\right )+\frac {1}{4} x \left (d-e x^2\right )^{3/2}\right )-\frac {13}{6} x \left (d-e x^2\right )^{5/2}\right )-\frac {1}{8} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{8} d \left (\frac {67}{6} d \left (\frac {3}{4} d \left (\frac {d \arctan \left (\frac {\sqrt {e} x}{\sqrt {d-e x^2}}\right )}{2 \sqrt {e}}+\frac {1}{2} x \sqrt {d-e x^2}\right )+\frac {1}{4} x \left (d-e x^2\right )^{3/2}\right )-\frac {13}{6} x \left (d-e x^2\right )^{5/2}\right )-\frac {1}{8} x \left (d-e x^2\right )^{5/2} \left (d+e x^2\right )\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

Input:

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

Output:

(Sqrt[d^2 - e^2*x^4]*(-1/8*(x*(d - e*x^2)^(5/2)*(d + e*x^2)) + (d*((-13*x* 
(d - e*x^2)^(5/2))/6 + (67*d*((x*(d - e*x^2)^(3/2))/4 + (3*d*((x*Sqrt[d - 
e*x^2])/2 + (d*ArcTan[(Sqrt[e]*x)/Sqrt[d - e*x^2]])/(2*Sqrt[e])))/4))/6))/ 
8))/(Sqrt[d - e*x^2]*Sqrt[d + e*x^2])
 

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 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

rule 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

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 299

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x 
*((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 
*p + 3))   Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && NeQ[2*p + 3, 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 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 = 0.20 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.70

method	result	size
default	\(\frac {\sqrt {-e^{2} x^{4}+d^{2}}\, \left (-48 e^{\frac {7}{2}} x^{7} \sqrt {-e \,x^{2}+d}-56 d \,e^{\frac {5}{2}} x^{5} \sqrt {-e \,x^{2}+d}+122 d^{2} e^{\frac {3}{2}} x^{3} \sqrt {-e \,x^{2}+d}+183 \sqrt {-e \,x^{2}+d}\, \sqrt {e}\, d^{3} x +201 \arctan \left (\frac {\sqrt {e}\, x}{\sqrt {-e \,x^{2}+d}}\right ) d^{4}\right )}{384 \sqrt {e \,x^{2}+d}\, \sqrt {-e \,x^{2}+d}\, \sqrt {e}}\)	\(138\)
risch	\(\frac {x \left (-48 e^{3} x^{6}-56 d \,e^{2} x^{4}+122 d^{2} e \,x^{2}+183 d^{3}\right ) \sqrt {-e \,x^{2}+d}\, \sqrt {\frac {-e^{2} x^{4}+d^{2}}{e \,x^{2}+d}}\, \sqrt {e \,x^{2}+d}}{384 \sqrt {-e^{2} x^{4}+d^{2}}}+\frac {67 d^{4} \arctan \left (\frac {\sqrt {e}\, x}{\sqrt {-e \,x^{2}+d}}\right ) \sqrt {\frac {-e^{2} x^{4}+d^{2}}{e \,x^{2}+d}}\, \sqrt {e \,x^{2}+d}}{128 \sqrt {e}\, \sqrt {-e^{2} x^{4}+d^{2}}}\)	\(165\)

Input:

int((e*x^2+d)^(1/2)*(-e^2*x^4+d^2)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

1/384*(-e^2*x^4+d^2)^(1/2)*(-48*e^(7/2)*x^7*(-e*x^2+d)^(1/2)-56*d*e^(5/2)* 
x^5*(-e*x^2+d)^(1/2)+122*d^2*e^(3/2)*x^3*(-e*x^2+d)^(1/2)+183*(-e*x^2+d)^( 
1/2)*e^(1/2)*d^3*x+201*arctan(e^(1/2)*x/(-e*x^2+d)^(1/2))*d^4)/(e*x^2+d)^( 
1/2)/(-e*x^2+d)^(1/2)/e^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.52 \[ \int \sqrt {d+e x^2} \left (d^2-e^2 x^4\right )^{3/2} \, dx=\left [-\frac {201 \, {\left (d^{4} e x^{2} + d^{5}\right )} \sqrt {-e} \log \left (-\frac {2 \, e^{2} x^{4} + d e x^{2} - 2 \, \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {e x^{2} + d} \sqrt {-e} x - d^{2}}{e x^{2} + d}\right ) + 2 \, {\left (48 \, e^{4} x^{7} + 56 \, d e^{3} x^{5} - 122 \, d^{2} e^{2} x^{3} - 183 \, d^{3} e x\right )} \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {e x^{2} + d}}{768 \, {\left (e^{2} x^{2} + d e\right )}}, -\frac {201 \, {\left (d^{4} e x^{2} + d^{5}\right )} \sqrt {e} \arctan \left (\frac {\sqrt {-e^{2} x^{4} + d^{2}} \sqrt {e x^{2} + d} \sqrt {e} x}{e^{2} x^{4} - d^{2}}\right ) + {\left (48 \, e^{4} x^{7} + 56 \, d e^{3} x^{5} - 122 \, d^{2} e^{2} x^{3} - 183 \, d^{3} e x\right )} \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {e x^{2} + d}}{384 \, {\left (e^{2} x^{2} + d e\right )}}\right ] \] Input:

integrate((e*x^2+d)^(1/2)*(-e^2*x^4+d^2)^(3/2),x, algorithm="fricas")
 

Output:

[-1/768*(201*(d^4*e*x^2 + d^5)*sqrt(-e)*log(-(2*e^2*x^4 + d*e*x^2 - 2*sqrt 
(-e^2*x^4 + d^2)*sqrt(e*x^2 + d)*sqrt(-e)*x - d^2)/(e*x^2 + d)) + 2*(48*e^ 
4*x^7 + 56*d*e^3*x^5 - 122*d^2*e^2*x^3 - 183*d^3*e*x)*sqrt(-e^2*x^4 + d^2) 
*sqrt(e*x^2 + d))/(e^2*x^2 + d*e), -1/384*(201*(d^4*e*x^2 + d^5)*sqrt(e)*a 
rctan(sqrt(-e^2*x^4 + d^2)*sqrt(e*x^2 + d)*sqrt(e)*x/(e^2*x^4 - d^2)) + (4 
8*e^4*x^7 + 56*d*e^3*x^5 - 122*d^2*e^2*x^3 - 183*d^3*e*x)*sqrt(-e^2*x^4 + 
d^2)*sqrt(e*x^2 + d))/(e^2*x^2 + d*e)]
 

Sympy [F]

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

integrate((e*x**2+d)**(1/2)*(-e**2*x**4+d**2)**(3/2),x)
 

Output:

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

Maxima [F]

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

integrate((e*x^2+d)^(1/2)*(-e^2*x^4+d^2)^(3/2),x, algorithm="maxima")
 

Output:

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

Giac [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.42 \[ \int \sqrt {d+e x^2} \left (d^2-e^2 x^4\right )^{3/2} \, dx=-\frac {67 \, d^{4} \log \left ({\left | -\sqrt {-e} x + \sqrt {-e x^{2} + d} \right |}\right )}{128 \, \sqrt {-e}} + \frac {1}{384} \, {\left (183 \, d^{3} + 2 \, {\left (61 \, d^{2} e - 4 \, {\left (6 \, e^{3} x^{2} + 7 \, d e^{2}\right )} x^{2}\right )} x^{2}\right )} \sqrt {-e x^{2} + d} x \] Input:

integrate((e*x^2+d)^(1/2)*(-e^2*x^4+d^2)^(3/2),x, algorithm="giac")
 

Output:

-67/128*d^4*log(abs(-sqrt(-e)*x + sqrt(-e*x^2 + d)))/sqrt(-e) + 1/384*(183 
*d^3 + 2*(61*d^2*e - 4*(6*e^3*x^2 + 7*d*e^2)*x^2)*x^2)*sqrt(-e*x^2 + d)*x
 

Mupad [F(-1)]

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

int((d^2 - e^2*x^4)^(3/2)*(d + e*x^2)^(1/2),x)
 

Output:

int((d^2 - e^2*x^4)^(3/2)*(d + e*x^2)^(1/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.47 \[ \int \sqrt {d+e x^2} \left (d^2-e^2 x^4\right )^{3/2} \, dx=\frac {201 \sqrt {e}\, \mathit {asin} \left (\frac {\sqrt {e}\, x}{\sqrt {d}}\right ) d^{4}+183 \sqrt {-e \,x^{2}+d}\, d^{3} e x +122 \sqrt {-e \,x^{2}+d}\, d^{2} e^{2} x^{3}-56 \sqrt {-e \,x^{2}+d}\, d \,e^{3} x^{5}-48 \sqrt {-e \,x^{2}+d}\, e^{4} x^{7}}{384 e} \] Input:

int((e*x^2+d)^(1/2)*(-e^2*x^4+d^2)^(3/2),x)
 

Output:

(201*sqrt(e)*asin((sqrt(e)*x)/sqrt(d))*d**4 + 183*sqrt(d - e*x**2)*d**3*e* 
x + 122*sqrt(d - e*x**2)*d**2*e**2*x**3 - 56*sqrt(d - e*x**2)*d*e**3*x**5 
- 48*sqrt(d - e*x**2)*e**4*x**7)/(384*e)