\(\int \frac {1}{(d+e x^3)^{3/2} (a+c x^6)} \, dx\) [6]

Optimal result

Integrand size = 21, antiderivative size = 146 \[ \int \frac {1}{\left (d+e x^3\right )^{3/2} \left (a+c x^6\right )} \, dx=\frac {x \sqrt {1+\frac {e x^3}{d}} \operatorname {AppellF1}\left (\frac {1}{3},\frac {3}{2},1,\frac {4}{3},-\frac {e x^3}{d},-\frac {\sqrt {c} x^3}{\sqrt {-a}}\right )}{2 a d \sqrt {d+e x^3}}+\frac {x \sqrt {1+\frac {e x^3}{d}} \operatorname {AppellF1}\left (\frac {1}{3},\frac {3}{2},1,\frac {4}{3},-\frac {e x^3}{d},\frac {\sqrt {c} x^3}{\sqrt {-a}}\right )}{2 a d \sqrt {d+e x^3}} \] Output:

1/2*x*(1+e*x^3/d)^(1/2)*AppellF1(1/3,1,3/2,4/3,-c^(1/2)*x^3/(-a)^(1/2),-e* 
x^3/d)/a/d/(e*x^3+d)^(1/2)+1/2*x*(1+e*x^3/d)^(1/2)*AppellF1(1/3,1,3/2,4/3, 
c^(1/2)*x^3/(-a)^(1/2),-e*x^3/d)/a/d/(e*x^3+d)^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 21.28 (sec) , antiderivative size = 5979, normalized size of antiderivative = 40.95 \[ \int \frac {1}{\left (d+e x^3\right )^{3/2} \left (a+c x^6\right )} \, dx=\text {Result too large to show} \] Input:

Integrate[1/((d + e*x^3)^(3/2)*(a + c*x^6)),x]
 

Output:

Result too large to show
 

Rubi [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(438\) vs. \(2(146)=292\).

Time = 0.98 (sec) , antiderivative size = 438, normalized size of antiderivative = 3.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {1757, 749, 759, 7276, 2009}

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.

\(\displaystyle \int \frac {1}{\left (a+c x^6\right ) \left (d+e x^3\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 1757

\(\displaystyle \frac {e^2 \int \frac {1}{\left (e x^3+d\right )^{3/2}}dx}{a e^2+c d^2}+\frac {c \int \frac {d-e x^3}{\sqrt {e x^3+d} \left (c x^6+a\right )}dx}{a e^2+c d^2}\)

\(\Big \downarrow \) 749

\(\displaystyle \frac {e^2 \left (\frac {\int \frac {1}{\sqrt {e x^3+d}}dx}{3 d}+\frac {2 x}{3 d \sqrt {d+e x^3}}\right )}{a e^2+c d^2}+\frac {c \int \frac {d-e x^3}{\sqrt {e x^3+d} \left (c x^6+a\right )}dx}{a e^2+c d^2}\)

\(\Big \downarrow \) 759

\(\displaystyle \frac {c \int \frac {d-e x^3}{\sqrt {e x^3+d} \left (c x^6+a\right )}dx}{a e^2+c d^2}+\frac {e^2 \left (\frac {2 \sqrt {2+\sqrt {3}} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \sqrt {\frac {d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{e} x+\left (1-\sqrt {3}\right ) \sqrt [3]{d}}{\sqrt [3]{e} x+\left (1+\sqrt {3}\right ) \sqrt [3]{d}}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} d \sqrt [3]{e} \sqrt {\frac {\sqrt [3]{d} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \sqrt {d+e x^3}}+\frac {2 x}{3 d \sqrt {d+e x^3}}\right )}{a e^2+c d^2}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {c \int \left (\frac {-\sqrt {c} d-\sqrt {-a} e}{2 \sqrt {-a} \sqrt {c} \left (\sqrt {c} x^3+\sqrt {-a}\right ) \sqrt {e x^3+d}}-\frac {\sqrt {c} d-\sqrt {-a} e}{2 \sqrt {-a} \sqrt {c} \left (\sqrt {-a}-\sqrt {c} x^3\right ) \sqrt {e x^3+d}}\right )dx}{a e^2+c d^2}+\frac {e^2 \left (\frac {2 \sqrt {2+\sqrt {3}} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \sqrt {\frac {d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{e} x+\left (1-\sqrt {3}\right ) \sqrt [3]{d}}{\sqrt [3]{e} x+\left (1+\sqrt {3}\right ) \sqrt [3]{d}}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} d \sqrt [3]{e} \sqrt {\frac {\sqrt [3]{d} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \sqrt {d+e x^3}}+\frac {2 x}{3 d \sqrt {d+e x^3}}\right )}{a e^2+c d^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {c \left (\frac {x \sqrt {\frac {e x^3}{d}+1} \left (\frac {\sqrt {-a} e}{\sqrt {c}}+d\right ) \operatorname {AppellF1}\left (\frac {1}{3},1,\frac {1}{2},\frac {4}{3},-\frac {\sqrt {c} x^3}{\sqrt {-a}},-\frac {e x^3}{d}\right )}{2 a \sqrt {d+e x^3}}+\frac {x \sqrt {\frac {e x^3}{d}+1} \left (d-\frac {\sqrt {-a} e}{\sqrt {c}}\right ) \operatorname {AppellF1}\left (\frac {1}{3},1,\frac {1}{2},\frac {4}{3},\frac {\sqrt {c} x^3}{\sqrt {-a}},-\frac {e x^3}{d}\right )}{2 a \sqrt {d+e x^3}}\right )}{a e^2+c d^2}+\frac {e^2 \left (\frac {2 \sqrt {2+\sqrt {3}} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \sqrt {\frac {d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{e} x+\left (1-\sqrt {3}\right ) \sqrt [3]{d}}{\sqrt [3]{e} x+\left (1+\sqrt {3}\right ) \sqrt [3]{d}}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} d \sqrt [3]{e} \sqrt {\frac {\sqrt [3]{d} \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{d}+\sqrt [3]{e} x\right )^2}} \sqrt {d+e x^3}}+\frac {2 x}{3 d \sqrt {d+e x^3}}\right )}{a e^2+c d^2}\)

Input:

Int[1/((d + e*x^3)^(3/2)*(a + c*x^6)),x]
 

Output:

(c*(((d + (Sqrt[-a]*e)/Sqrt[c])*x*Sqrt[1 + (e*x^3)/d]*AppellF1[1/3, 1, 1/2 
, 4/3, -((Sqrt[c]*x^3)/Sqrt[-a]), -((e*x^3)/d)])/(2*a*Sqrt[d + e*x^3]) + ( 
(d - (Sqrt[-a]*e)/Sqrt[c])*x*Sqrt[1 + (e*x^3)/d]*AppellF1[1/3, 1, 1/2, 4/3 
, (Sqrt[c]*x^3)/Sqrt[-a], -((e*x^3)/d)])/(2*a*Sqrt[d + e*x^3])))/(c*d^2 + 
a*e^2) + (e^2*((2*x)/(3*d*Sqrt[d + e*x^3]) + (2*Sqrt[2 + Sqrt[3]]*(d^(1/3) 
 + e^(1/3)*x)*Sqrt[(d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^2)/((1 + Sqrt[ 
3])*d^(1/3) + e^(1/3)*x)^2]*EllipticF[ArcSin[((1 - Sqrt[3])*d^(1/3) + e^(1 
/3)*x)/((1 + Sqrt[3])*d^(1/3) + e^(1/3)*x)], -7 - 4*Sqrt[3]])/(3*3^(1/4)*d 
*e^(1/3)*Sqrt[(d^(1/3)*(d^(1/3) + e^(1/3)*x))/((1 + Sqrt[3])*d^(1/3) + e^( 
1/3)*x)^2]*Sqrt[d + e*x^3])))/(c*d^2 + a*e^2)
 

Defintions of rubi rules used

rule 749

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 
 1)/(a*n*(p + 1))), x] + Simp[(n*(p + 1) + 1)/(a*n*(p + 1))   Int[(a + b*x^ 
n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (Inte 
gerQ[2*p] || Denominator[p + 1/n] < Denominator[p])
 

rule 759

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], 
s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 + Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s 
*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[s* 
((s + r*x)/((1 + Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 - Sqrt[3])*s 
+ r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x] & 
& PosQ[a]
 

rule 1757

Int[((d_) + (e_.)*(x_)^(n_))^(q_)/((a_) + (c_.)*(x_)^(n2_)), x_Symbol] :> S 
imp[e^2/(c*d^2 + a*e^2)   Int[(d + e*x^n)^q, x], x] + Simp[c/(c*d^2 + a*e^2 
)   Int[(d + e*x^n)^(q + 1)*((d - e*x^n)/(a + c*x^(2*n))), x], x] /; FreeQ[ 
{a, c, d, e, n}, x] && EqQ[n2, 2*n] && NeQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[ 
q] && LtQ[q, -1]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7276

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE 
xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ 
[n, 0]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 6.

Time = 0.36 (sec) , antiderivative size = 937, normalized size of antiderivative = 6.42

method	result	size
default	\(\text {Expression too large to display}\)	\(937\)
elliptic	\(\text {Expression too large to display}\)	\(937\)

Input:

int(1/(e*x^3+d)^(3/2)/(c*x^6+a),x,method=_RETURNVERBOSE)
 

Output:

2/3*e^2*x/d/(a*e^2+c*d^2)/((x^3+d/e)*e)^(1/2)-2/9*I/d*e/(a*e^2+c*d^2)*3^(1 
/2)*(-d*e^2)^(1/3)*(I*(x+1/2/e*(-d*e^2)^(1/3)-1/2*I*3^(1/2)/e*(-d*e^2)^(1/ 
3))*3^(1/2)*e/(-d*e^2)^(1/3))^(1/2)*((x-1/e*(-d*e^2)^(1/3))/(-3/2/e*(-d*e^ 
2)^(1/3)+1/2*I*3^(1/2)/e*(-d*e^2)^(1/3)))^(1/2)*(-I*(x+1/2/e*(-d*e^2)^(1/3 
)+1/2*I*3^(1/2)/e*(-d*e^2)^(1/3))*3^(1/2)*e/(-d*e^2)^(1/3))^(1/2)/(e*x^3+d 
)^(1/2)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/e*(-d*e^2)^(1/3)-1/2*I*3^(1/2)/e*( 
-d*e^2)^(1/3))*3^(1/2)*e/(-d*e^2)^(1/3))^(1/2),(I*3^(1/2)/e*(-d*e^2)^(1/3) 
/(-3/2/e*(-d*e^2)^(1/3)+1/2*I*3^(1/2)/e*(-d*e^2)^(1/3)))^(1/2))-1/6*I*c/e^ 
2*2^(1/2)*sum((_alpha^3*e-d)/(a*e^2+c*d^2)^2/_alpha^5*(-d*e^2)^(1/3)*(1/2* 
I*e*(2*x+1/e*(-I*3^(1/2)*(-d*e^2)^(1/3)+(-d*e^2)^(1/3)))/(-d*e^2)^(1/3))^( 
1/2)*(e*(x-1/e*(-d*e^2)^(1/3))/(-3*(-d*e^2)^(1/3)+I*3^(1/2)*(-d*e^2)^(1/3) 
))^(1/2)*(-1/2*I*e*(2*x+1/e*(I*3^(1/2)*(-d*e^2)^(1/3)+(-d*e^2)^(1/3)))/(-d 
*e^2)^(1/3))^(1/2)/(e*x^3+d)^(1/2)*(2*e^2*(-_alpha^5*e+_alpha^2*d)-I*(-d*e 
^2)^(1/3)*_alpha^4*3^(1/2)*e^2+I*_alpha^3*3^(1/2)*(-d*e^2)^(2/3)*e+(-d*e^2 
)^(1/3)*_alpha^4*e^2+I*(-d*e^2)^(1/3)*d*_alpha*3^(1/2)*e+_alpha^3*(-d*e^2) 
^(2/3)*e-I*3^(1/2)*(-d*e^2)^(2/3)*d-(-d*e^2)^(1/3)*d*_alpha*e-(-d*e^2)^(2/ 
3)*d)*EllipticPi(1/3*3^(1/2)*(I*(x+1/2/e*(-d*e^2)^(1/3)-1/2*I*3^(1/2)/e*(- 
d*e^2)^(1/3))*3^(1/2)*e/(-d*e^2)^(1/3))^(1/2),-1/2*c/e*(-2*I*3^(1/2)*(-d*e 
^2)^(1/3)*_alpha^5*e^2+I*3^(1/2)*(-d*e^2)^(2/3)*_alpha^4*e-I*3^(1/2)*_alph 
a^3*d*e^2+2*I*3^(1/2)*(-d*e^2)^(1/3)*_alpha^2*d*e+3*(-d*e^2)^(2/3)*_alp...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{\left (d+e x^3\right )^{3/2} \left (a+c x^6\right )} \, dx=\text {Timed out} \] Input:

integrate(1/(e*x^3+d)^(3/2)/(c*x^6+a),x, algorithm="fricas")
 

Output:

Timed out
                                                                                    
                                                                                    
 

Sympy [F]

\[ \int \frac {1}{\left (d+e x^3\right )^{3/2} \left (a+c x^6\right )} \, dx=\int \frac {1}{\left (a + c x^{6}\right ) \left (d + e x^{3}\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(1/(e*x**3+d)**(3/2)/(c*x**6+a),x)
 

Output:

Integral(1/((a + c*x**6)*(d + e*x**3)**(3/2)), x)
 

Maxima [F]

\[ \int \frac {1}{\left (d+e x^3\right )^{3/2} \left (a+c x^6\right )} \, dx=\int { \frac {1}{{\left (c x^{6} + a\right )} {\left (e x^{3} + d\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(e*x^3+d)^(3/2)/(c*x^6+a),x, algorithm="maxima")
 

Output:

integrate(1/((c*x^6 + a)*(e*x^3 + d)^(3/2)), x)
 

Giac [F]

\[ \int \frac {1}{\left (d+e x^3\right )^{3/2} \left (a+c x^6\right )} \, dx=\int { \frac {1}{{\left (c x^{6} + a\right )} {\left (e x^{3} + d\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(e*x^3+d)^(3/2)/(c*x^6+a),x, algorithm="giac")
 

Output:

integrate(1/((c*x^6 + a)*(e*x^3 + d)^(3/2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (d+e x^3\right )^{3/2} \left (a+c x^6\right )} \, dx=\int \frac {1}{\left (c\,x^6+a\right )\,{\left (e\,x^3+d\right )}^{3/2}} \,d x \] Input:

int(1/((a + c*x^6)*(d + e*x^3)^(3/2)),x)
 

Output:

int(1/((a + c*x^6)*(d + e*x^3)^(3/2)), x)
 

Reduce [F]

\[ \int \frac {1}{\left (d+e x^3\right )^{3/2} \left (a+c x^6\right )} \, dx=\int \frac {\sqrt {e \,x^{3}+d}}{c \,e^{2} x^{12}+2 c d e \,x^{9}+a \,e^{2} x^{6}+c \,d^{2} x^{6}+2 a d e \,x^{3}+a \,d^{2}}d x \] Input:

int(1/(e*x^3+d)^(3/2)/(c*x^6+a),x)
 

Output:

int(sqrt(d + e*x**3)/(a*d**2 + 2*a*d*e*x**3 + a*e**2*x**6 + c*d**2*x**6 + 
2*c*d*e*x**9 + c*e**2*x**12),x)