Integrand size = 21, antiderivative size = 140 \[ \int \frac {1}{\sqrt {d+e x^3} \left (a+c x^6\right )} \, dx=\frac {x \sqrt {1+\frac {e x^3}{d}} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},1,\frac {4}{3},-\frac {e x^3}{d},-\frac {\sqrt {c} x^3}{\sqrt {-a}}\right )}{2 a \sqrt {d+e x^3}}+\frac {x \sqrt {1+\frac {e x^3}{d}} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},1,\frac {4}{3},-\frac {e x^3}{d},\frac {\sqrt {c} x^3}{\sqrt {-a}}\right )}{2 a \sqrt {d+e x^3}} \] Output:
1/2*x*(1+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/(e*x^3+d)^(1/2)+1/2*x*(1+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/(e*x^3+d)^(1/2)
Result contains complex when optimal does not.
Time = 19.89 (sec) , antiderivative size = 932, normalized size of antiderivative = 6.66 \[ \int \frac {1}{\sqrt {d+e x^3} \left (a+c x^6\right )} \, dx =\text {Too large to display} \] Input:
Integrate[1/(Sqrt[d + e*x^3]*(a + c*x^6)),x]
Output:
(2*(-1)^(2/3)*d^(1/3)*Sqrt[(d^(1/3) + e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3) )]*Sqrt[1 - (e^(1/3)*x)/d^(1/3) + (e^(2/3)*x^2)/d^(2/3)]*(((-1)^(2/3)*Elli pticPi[((-1)^(1/6)*(1 + (-1)^(1/3))*c^(1/6)*d^(1/3))/((-1)^(1/6)*c^(1/6)*d ^(1/3) - a^(1/6)*e^(1/3)), ArcSin[Sqrt[(d^(1/3) + (-1)^(2/3)*e^(1/3)*x)/(( 1 + (-1)^(1/3))*d^(1/3))]], (-1)^(1/3)])/(-2*a^(2/3)*c^(1/6)*d^(1/3) + (-I + Sqrt[3])*a^(5/6)*e^(1/3)) - EllipticPi[((1 + (-1)^(1/3))*c^(1/6)*d^(1/3 ))/(c^(1/6)*d^(1/3) - I*a^(1/6)*e^(1/3)), ArcSin[Sqrt[(d^(1/3) + (-1)^(2/3 )*e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))]], (-1)^(1/3)]/((I + Sqrt[3])*a^(2 /3)*(I*c^(1/6)*d^(1/3) + a^(1/6)*e^(1/3))) + (3*EllipticPi[(I*Sqrt[3]*c^(1 /6)*d^(1/3))/((-1)^(1/3)*c^(1/6)*d^(1/3) - I*a^(1/6)*e^(1/3)), ArcSin[Sqrt [(d^(1/3) + (-1)^(2/3)*e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))]], (-1)^(1/3) ])/(6*a^(2/3)*c^(1/6)*d^(1/3) - 3*(I + Sqrt[3])*a^(5/6)*e^(1/3)) - Ellipti cPi[((1 + (-1)^(1/3))*c^(1/6)*d^(1/3))/(c^(1/6)*d^(1/3) + I*a^(1/6)*e^(1/3 )), ArcSin[Sqrt[(d^(1/3) + (-1)^(2/3)*e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3) )]], (-1)^(1/3)]/((I + Sqrt[3])*a^(2/3)*((-I)*c^(1/6)*d^(1/3) + a^(1/6)*e^ (1/3))) - EllipticPi[(I*Sqrt[3]*c^(1/6)*d^(1/3))/((-1)^(1/3)*c^(1/6)*d^(1/ 3) + I*a^(1/6)*e^(1/3)), ArcSin[Sqrt[(d^(1/3) + (-1)^(2/3)*e^(1/3)*x)/((1 + (-1)^(1/3))*d^(1/3))]], (-1)^(1/3)]/(2*a^(2/3)*c^(1/6)*d^(1/3) + (I + Sq rt[3])*a^(5/6)*e^(1/3)) + ((-1)^(2/3)*EllipticPi[((-1)^(1/6)*(1 + (-1)^(1/ 3))*c^(1/6)*d^(1/3))/((-1)^(1/6)*c^(1/6)*d^(1/3) + a^(1/6)*e^(1/3)), Ar...
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.
| \(\displaystyle \int \frac {1}{\left (a+c x^6\right ) \sqrt {d+e x^3}} \, dx\) |
| \(\Big \downarrow \) 1759 |
| \(\displaystyle -\frac {\sqrt {c} \int \frac {1}{\sqrt {c} \left (\sqrt {-a}-\sqrt {c} x^3\right ) \sqrt {e x^3+d}}dx}{2 \sqrt {-a}}-\frac {\sqrt {c} \int \frac {1}{\sqrt {c} \left (\sqrt {c} x^3+\sqrt {-a}\right ) \sqrt {e x^3+d}}dx}{2 \sqrt {-a}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {1}{\left (\sqrt {-a}-\sqrt {c} x^3\right ) \sqrt {e x^3+d}}dx}{2 \sqrt {-a}}-\frac {\int \frac {1}{\left (\sqrt {c} x^3+\sqrt {-a}\right ) \sqrt {e x^3+d}}dx}{2 \sqrt {-a}}\) |
| \(\Big \downarrow \) 937 |
| \(\displaystyle -\frac {\sqrt {\frac {e x^3}{d}+1} \int \frac {1}{\left (\sqrt {-a}-\sqrt {c} x^3\right ) \sqrt {\frac {e x^3}{d}+1}}dx}{2 \sqrt {-a} \sqrt {d+e x^3}}-\frac {\sqrt {\frac {e x^3}{d}+1} \int \frac {1}{\left (\sqrt {c} x^3+\sqrt {-a}\right ) \sqrt {\frac {e x^3}{d}+1}}dx}{2 \sqrt {-a} \sqrt {d+e x^3}}\) |
| \(\Big \downarrow \) 936 |
| \(\displaystyle \frac {x \sqrt {\frac {e x^3}{d}+1} \operatorname {AppellF1}\left (\frac {1}{3},1,\frac {1}{2},\frac {4}{3},-\frac {\sqrt {c} x^3}{\sqrt {-a}},-\frac {e x^3}{d}\right )}{2 a \sqrt {d+e x^3}}+\frac {x \sqrt {\frac {e x^3}{d}+1} \operatorname {AppellF1}\left (\frac {1}{3},1,\frac {1}{2},\frac {4}{3},\frac {\sqrt {c} x^3}{\sqrt {-a}},-\frac {e x^3}{d}\right )}{2 a \sqrt {d+e x^3}}\) |
Input:
Int[1/(Sqrt[d + e*x^3]*(a + c*x^6)),x]
Output:
(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]) + (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])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
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])
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]
Result contains higher order function than in optimal. Order 9 vs. order 6.
Time = 0.32 (sec) , antiderivative size = 595, normalized size of antiderivative = 4.25
| 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 (-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}}\) | \(595\) |
| 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 (-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}}\) | \(595\) |
Input:
int(1/(e*x^3+d)^(1/2)/(c*x^6+a),x,method=_RETURNVERBOSE)
Output:
1/6*I/e^2*2^(1/2)*sum(1/_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)*E llipticPi(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)),_alpha=RootOf(_Z^6 *c+a))
Timed out. \[ \int \frac {1}{\sqrt {d+e x^3} \left (a+c x^6\right )} \, dx=\text {Timed out} \] Input:
integrate(1/(e*x^3+d)^(1/2)/(c*x^6+a),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {1}{\sqrt {d+e x^3} \left (a+c x^6\right )} \, dx=\int \frac {1}{\left (a + c x^{6}\right ) \sqrt {d + e x^{3}}}\, dx \] Input:
integrate(1/(e*x**3+d)**(1/2)/(c*x**6+a),x)
Output:
Integral(1/((a + c*x**6)*sqrt(d + e*x**3)), x)
\[ \int \frac {1}{\sqrt {d+e x^3} \left (a+c x^6\right )} \, dx=\int { \frac {1}{{\left (c x^{6} + a\right )} \sqrt {e x^{3} + d}} \,d x } \] Input:
integrate(1/(e*x^3+d)^(1/2)/(c*x^6+a),x, algorithm="maxima")
Output:
integrate(1/((c*x^6 + a)*sqrt(e*x^3 + d)), x)
\[ \int \frac {1}{\sqrt {d+e x^3} \left (a+c x^6\right )} \, dx=\int { \frac {1}{{\left (c x^{6} + a\right )} \sqrt {e x^{3} + d}} \,d x } \] Input:
integrate(1/(e*x^3+d)^(1/2)/(c*x^6+a),x, algorithm="giac")
Output:
integrate(1/((c*x^6 + a)*sqrt(e*x^3 + d)), x)
Timed out. \[ \int \frac {1}{\sqrt {d+e x^3} \left (a+c x^6\right )} \, dx=\int \frac {1}{\left (c\,x^6+a\right )\,\sqrt {e\,x^3+d}} \,d x \] Input:
int(1/((a + c*x^6)*(d + e*x^3)^(1/2)),x)
Output:
int(1/((a + c*x^6)*(d + e*x^3)^(1/2)), x)
\[ \int \frac {1}{\sqrt {d+e x^3} \left (a+c x^6\right )} \, dx=\int \frac {\sqrt {e \,x^{3}+d}}{c e \,x^{9}+c d \,x^{6}+a e \,x^{3}+a d}d x \] Input:
int(1/(e*x^3+d)^(1/2)/(c*x^6+a),x)
Output:
int(sqrt(d + e*x**3)/(a*d + a*e*x**3 + c*d*x**6 + c*e*x**9),x)