\(\int \frac {\sqrt {d+e x^3}}{a+c x^6} \, dx\) [4]

Optimal result

Integrand size = 21, antiderivative size = 140 \[ \int \frac {\sqrt {d+e x^3}}{a+c x^6} \, dx=\frac {x \sqrt {d+e x^3} \operatorname {AppellF1}\left (\frac {1}{3},-\frac {1}{2},1,\frac {4}{3},-\frac {e x^3}{d},-\frac {\sqrt {c} x^3}{\sqrt {-a}}\right )}{2 a \sqrt {1+\frac {e x^3}{d}}}+\frac {x \sqrt {d+e x^3} \operatorname {AppellF1}\left (\frac {1}{3},-\frac {1}{2},1,\frac {4}{3},-\frac {e x^3}{d},\frac {\sqrt {c} x^3}{\sqrt {-a}}\right )}{2 a \sqrt {1+\frac {e x^3}{d}}} \] Output:

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

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 21.19 (sec) , antiderivative size = 5647, normalized size of antiderivative = 40.34 \[ \int \frac {\sqrt {d+e x^3}}{a+c x^6} \, dx=\text {Result too large to show} \] Input:

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

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {1759, 27, 937, 936}

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.

\(\displaystyle \int \frac {\sqrt {d+e x^3}}{a+c x^6} \, dx\)

\(\Big \downarrow \) 1759

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 937

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

\(\Big \downarrow \) 936

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

Input:

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

Output:

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

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 936

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] 
:> Simp[a^p*c^q*x*AppellF1[1/n, -p, -q, 1 + 1/n, (-b)*(x^n/a), (-d)*(x^n/c) 
], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] 
 && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
 

rule 937

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] 
:> Simp[a^IntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p]) 
  Int[(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, p, q 
}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] &&  !(IntegerQ[p] || GtQ[a, 0])
 

rule 1759

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

Maple [C] (warning: unable to verify)

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

Time = 0.35 (sec) , antiderivative size = 605, normalized size of antiderivative = 4.32

method	result	size
default	\(-\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{6} c +a \right )}{\sum }\frac {\left (-e \,\underline {\hspace {1.25 ex}}\alpha ^{3}-d \right ) \left (-d \,e^{2}\right )^{\frac {1}{3}} \sqrt {2}\, \sqrt {\frac {i e \left (2 x +\frac {-i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}+\left (-d \,e^{2}\right )^{\frac {1}{3}}}{e}\right )}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {e \left (x -\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{e}\right )}{-3 \left (-d \,e^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i e \left (2 x +\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}+\left (-d \,e^{2}\right )^{\frac {1}{3}}}{e}\right )}{2 \left (-d \,e^{2}\right )^{\frac {1}{3}}}}\, \left (2 e^{2} \left (-e \,\underline {\hspace {1.25 ex}}\alpha ^{5}+d \,\underline {\hspace {1.25 ex}}\alpha ^{2}\right )-i \left (-d \,e^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{4} \sqrt {3}\, e^{2}+i \underline {\hspace {1.25 ex}}\alpha ^{3} \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {2}{3}} e +\left (-d \,e^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{4} e^{2}+i \left (-d \,e^{2}\right )^{\frac {1}{3}} d \underline {\hspace {1.25 ex}}\alpha \sqrt {3}\, e +\underline {\hspace {1.25 ex}}\alpha ^{3} \left (-d \,e^{2}\right )^{\frac {2}{3}} e -i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {2}{3}} d -\left (-d \,e^{2}\right )^{\frac {1}{3}} d \underline {\hspace {1.25 ex}}\alpha e -\left (-d \,e^{2}\right )^{\frac {2}{3}} d \right ) \operatorname {EllipticPi}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}-\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}}{3}, -\frac {c \left (-2 i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{5} e^{2}+i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha ^{4} e -i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{3} d \,e^{2}+2 i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} d e +3 \left (-d \,e^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha ^{4} e -i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha d +3 \underline {\hspace {1.25 ex}}\alpha ^{3} d \,e^{2}+i \sqrt {3}\, d^{2} e -3 \left (-d \,e^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha d -3 d^{2} e \right )}{2 e \left (a \,e^{2}+c \,d^{2}\right )}, \sqrt {\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{e \left (-\frac {3 \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right )}}\right )}{2 \underline {\hspace {1.25 ex}}\alpha ^{5} \left (a \,e^{2}+c \,d^{2}\right ) \sqrt {e \,x^{3}+d}}\right )}{6 e^{2}}\)	\(605\)
elliptic	\(-\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{6} c +a \right )}{\sum }\frac {\left (-e \,\underline {\hspace {1.25 ex}}\alpha ^{3}-d \right ) \left (-d \,e^{2}\right )^{\frac {1}{3}} \sqrt {2}\, \sqrt {\frac {i e \left (2 x +\frac {-i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}+\left (-d \,e^{2}\right )^{\frac {1}{3}}}{e}\right )}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {e \left (x -\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{e}\right )}{-3 \left (-d \,e^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i e \left (2 x +\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}+\left (-d \,e^{2}\right )^{\frac {1}{3}}}{e}\right )}{2 \left (-d \,e^{2}\right )^{\frac {1}{3}}}}\, \left (2 e^{2} \left (-e \,\underline {\hspace {1.25 ex}}\alpha ^{5}+d \,\underline {\hspace {1.25 ex}}\alpha ^{2}\right )-i \left (-d \,e^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{4} \sqrt {3}\, e^{2}+i \underline {\hspace {1.25 ex}}\alpha ^{3} \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {2}{3}} e +\left (-d \,e^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{4} e^{2}+i \left (-d \,e^{2}\right )^{\frac {1}{3}} d \underline {\hspace {1.25 ex}}\alpha \sqrt {3}\, e +\underline {\hspace {1.25 ex}}\alpha ^{3} \left (-d \,e^{2}\right )^{\frac {2}{3}} e -i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {2}{3}} d -\left (-d \,e^{2}\right )^{\frac {1}{3}} d \underline {\hspace {1.25 ex}}\alpha e -\left (-d \,e^{2}\right )^{\frac {2}{3}} d \right ) \operatorname {EllipticPi}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}-\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right ) \sqrt {3}\, e}{\left (-d \,e^{2}\right )^{\frac {1}{3}}}}}{3}, -\frac {c \left (-2 i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{5} e^{2}+i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha ^{4} e -i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{3} d \,e^{2}+2 i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} d e +3 \left (-d \,e^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha ^{4} e -i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha d +3 \underline {\hspace {1.25 ex}}\alpha ^{3} d \,e^{2}+i \sqrt {3}\, d^{2} e -3 \left (-d \,e^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha d -3 d^{2} e \right )}{2 e \left (a \,e^{2}+c \,d^{2}\right )}, \sqrt {\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{e \left (-\frac {3 \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}+\frac {i \sqrt {3}\, \left (-d \,e^{2}\right )^{\frac {1}{3}}}{2 e}\right )}}\right )}{2 \underline {\hspace {1.25 ex}}\alpha ^{5} \left (a \,e^{2}+c \,d^{2}\right ) \sqrt {e \,x^{3}+d}}\right )}{6 e^{2}}\)	\(605\)

Input:

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

Output:

-1/6*I/e^2*2^(1/2)*sum((-_alpha^3*e-d)/_alpha^5/(a*e^2+c*d^2)*(-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)*_alpha^3*d*e^2+2*I*3^(1/2)*(-d*e^2)^(1/3)*_alpha^2*d*e+3*(-d*e^2)^(2/3) 
*_alpha^4*e-I*3^(1/2)*(-d*e^2)^(2/3)*_alpha*d+3*_alpha^3*d*e^2+I*3^(1/2)*d 
^2*e-3*(-d*e^2)^(2/3)*_alpha*d-3*d^2*e)/(a*e^2+c*d^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)),_al 
pha=RootOf(_Z^6*c+a))
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\sqrt {d+e x^3}}{a+c x^6} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Sympy [F]

\[ \int \frac {\sqrt {d+e x^3}}{a+c x^6} \, dx=\int \frac {\sqrt {d + e x^{3}}}{a + c x^{6}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {\sqrt {d+e x^3}}{a+c x^6} \, dx=\int { \frac {\sqrt {e x^{3} + d}}{c x^{6} + a} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {\sqrt {d+e x^3}}{a+c x^6} \, dx=\int { \frac {\sqrt {e x^{3} + d}}{c x^{6} + a} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {d+e x^3}}{a+c x^6} \, dx=\int \frac {\sqrt {e\,x^3+d}}{c\,x^6+a} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\sqrt {d+e x^3}}{a+c x^6} \, dx=\int \frac {\sqrt {e \,x^{3}+d}}{c \,x^{6}+a}d x \] Input:

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

Output:

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