Integrand size = 28, antiderivative size = 327 \[ \int \frac {\sqrt {c+d x^3} \left (e+f x^3\right )}{a+b x^3} \, dx=\frac {2 f x \sqrt {c+d x^3}}{5 b}+\frac {(b c-a d) (b e-a f) x \sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (\frac {1}{3},1,\frac {1}{2},\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{a b^2 \sqrt {c+d x^3}}+\frac {2 \sqrt {2+\sqrt {3}} (5 b d e+3 b c f-5 a d f) \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x}\right ),-7-4 \sqrt {3}\right )}{5 \sqrt [4]{3} b^2 \sqrt [3]{d} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}} \] Output:
2/5*f*x*(d*x^3+c)^(1/2)/b+(-a*d+b*c)*(-a*f+b*e)*x*(1+d*x^3/c)^(1/2)*Appell F1(1/3,1,1/2,4/3,-b*x^3/a,-d*x^3/c)/a/b^2/(d*x^3+c)^(1/2)+2/15*(1/2*6^(1/2 )+1/2*2^(1/2))*(-5*a*d*f+3*b*c*f+5*b*d*e)*(c^(1/3)+d^(1/3)*x)*((c^(2/3)-c^ (1/3)*d^(1/3)*x+d^(2/3)*x^2)/((1+3^(1/2))*c^(1/3)+d^(1/3)*x)^2)^(1/2)*Elli pticF(((1-3^(1/2))*c^(1/3)+d^(1/3)*x)/((1+3^(1/2))*c^(1/3)+d^(1/3)*x),I*3^ (1/2)+2*I)*3^(3/4)/b^2/d^(1/3)/(c^(1/3)*(c^(1/3)+d^(1/3)*x)/((1+3^(1/2))*c ^(1/3)+d^(1/3)*x)^2)^(1/2)/(d*x^3+c)^(1/2)
Time = 6.74 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt {c+d x^3} \left (e+f x^3\right )}{a+b x^3} \, dx=\frac {x \left (\frac {(5 b d e+3 b c f-5 a d f) x^3 \sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},1,\frac {7}{3},-\frac {d x^3}{c},-\frac {b x^3}{a}\right )}{a}+\frac {8 \left (-4 a c \left (2 a d f x^3+b \left (5 c e+2 c f x^3+2 d f x^6\right )\right ) \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},1,\frac {4}{3},-\frac {d x^3}{c},-\frac {b x^3}{a}\right )+3 f x^3 \left (a+b x^3\right ) \left (c+d x^3\right ) \left (2 b c \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},2,\frac {7}{3},-\frac {d x^3}{c},-\frac {b x^3}{a}\right )+a d \operatorname {AppellF1}\left (\frac {4}{3},\frac {3}{2},1,\frac {7}{3},-\frac {d x^3}{c},-\frac {b x^3}{a}\right )\right )\right )}{\left (a+b x^3\right ) \left (-8 a c \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},1,\frac {4}{3},-\frac {d x^3}{c},-\frac {b x^3}{a}\right )+3 x^3 \left (2 b c \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},2,\frac {7}{3},-\frac {d x^3}{c},-\frac {b x^3}{a}\right )+a d \operatorname {AppellF1}\left (\frac {4}{3},\frac {3}{2},1,\frac {7}{3},-\frac {d x^3}{c},-\frac {b x^3}{a}\right )\right )\right )}\right )}{20 b \sqrt {c+d x^3}} \] Input:
Integrate[(Sqrt[c + d*x^3]*(e + f*x^3))/(a + b*x^3),x]
Output:
(x*(((5*b*d*e + 3*b*c*f - 5*a*d*f)*x^3*Sqrt[1 + (d*x^3)/c]*AppellF1[4/3, 1 /2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)])/a + (8*(-4*a*c*(2*a*d*f*x^3 + b*( 5*c*e + 2*c*f*x^3 + 2*d*f*x^6))*AppellF1[1/3, 1/2, 1, 4/3, -((d*x^3)/c), - ((b*x^3)/a)] + 3*f*x^3*(a + b*x^3)*(c + d*x^3)*(2*b*c*AppellF1[4/3, 1/2, 2 , 7/3, -((d*x^3)/c), -((b*x^3)/a)] + a*d*AppellF1[4/3, 3/2, 1, 7/3, -((d*x ^3)/c), -((b*x^3)/a)])))/((a + b*x^3)*(-8*a*c*AppellF1[1/3, 1/2, 1, 4/3, - ((d*x^3)/c), -((b*x^3)/a)] + 3*x^3*(2*b*c*AppellF1[4/3, 1/2, 2, 7/3, -((d* x^3)/c), -((b*x^3)/a)] + a*d*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), -((b *x^3)/a)])))))/(20*b*Sqrt[c + d*x^3])
Time = 0.77 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.02, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {1025, 27, 1021, 759, 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.
| \(\displaystyle \int \frac {\sqrt {c+d x^3} \left (e+f x^3\right )}{a+b x^3} \, dx\) |
| \(\Big \downarrow \) 1025 |
| \(\displaystyle \frac {2 \int \frac {(5 b d e+3 b c f-5 a d f) x^3+c (5 b e-2 a f)}{2 \left (b x^3+a\right ) \sqrt {d x^3+c}}dx}{5 b}+\frac {2 f x \sqrt {c+d x^3}}{5 b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(5 b d e+3 b c f-5 a d f) x^3+c (5 b e-2 a f)}{\left (b x^3+a\right ) \sqrt {d x^3+c}}dx}{5 b}+\frac {2 f x \sqrt {c+d x^3}}{5 b}\) |
| \(\Big \downarrow \) 1021 |
| \(\displaystyle \frac {\frac {(-5 a d f+3 b c f+5 b d e) \int \frac {1}{\sqrt {d x^3+c}}dx}{b}+\frac {5 (b c-a d) (b e-a f) \int \frac {1}{\left (b x^3+a\right ) \sqrt {d x^3+c}}dx}{b}}{5 b}+\frac {2 f x \sqrt {c+d x^3}}{5 b}\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle \frac {\frac {5 (b c-a d) (b e-a f) \int \frac {1}{\left (b x^3+a\right ) \sqrt {d x^3+c}}dx}{b}+\frac {2 \sqrt {2+\sqrt {3}} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right ),-7-4 \sqrt {3}\right ) (-5 a d f+3 b c f+5 b d e)}{\sqrt [4]{3} b \sqrt [3]{d} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}}{5 b}+\frac {2 f x \sqrt {c+d x^3}}{5 b}\) |
| \(\Big \downarrow \) 937 |
| \(\displaystyle \frac {\frac {5 \sqrt {\frac {d x^3}{c}+1} (b c-a d) (b e-a f) \int \frac {1}{\left (b x^3+a\right ) \sqrt {\frac {d x^3}{c}+1}}dx}{b \sqrt {c+d x^3}}+\frac {2 \sqrt {2+\sqrt {3}} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right ),-7-4 \sqrt {3}\right ) (-5 a d f+3 b c f+5 b d e)}{\sqrt [4]{3} b \sqrt [3]{d} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}}{5 b}+\frac {2 f x \sqrt {c+d x^3}}{5 b}\) |
| \(\Big \downarrow \) 936 |
| \(\displaystyle \frac {\frac {5 x \sqrt {\frac {d x^3}{c}+1} (b c-a d) (b e-a f) \operatorname {AppellF1}\left (\frac {1}{3},1,\frac {1}{2},\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{a b \sqrt {c+d x^3}}+\frac {2 \sqrt {2+\sqrt {3}} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right ) \sqrt {\frac {c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{d} x+\left (1-\sqrt {3}\right ) \sqrt [3]{c}}{\sqrt [3]{d} x+\left (1+\sqrt {3}\right ) \sqrt [3]{c}}\right ),-7-4 \sqrt {3}\right ) (-5 a d f+3 b c f+5 b d e)}{\sqrt [4]{3} b \sqrt [3]{d} \sqrt {\frac {\sqrt [3]{c} \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{c}+\sqrt [3]{d} x\right )^2}} \sqrt {c+d x^3}}}{5 b}+\frac {2 f x \sqrt {c+d x^3}}{5 b}\) |
Input:
Int[(Sqrt[c + d*x^3]*(e + f*x^3))/(a + b*x^3),x]
Output:
(2*f*x*Sqrt[c + d*x^3])/(5*b) + ((5*(b*c - a*d)*(b*e - a*f)*x*Sqrt[1 + (d* x^3)/c]*AppellF1[1/3, 1, 1/2, 4/3, -((b*x^3)/a), -((d*x^3)/c)])/(a*b*Sqrt[ c + d*x^3]) + (2*Sqrt[2 + Sqrt[3]]*(5*b*d*e + 3*b*c*f - 5*a*d*f)*(c^(1/3) + d^(1/3)*x)*Sqrt[(c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2)/((1 + Sqrt[3 ])*c^(1/3) + d^(1/3)*x)^2]*EllipticF[ArcSin[((1 - Sqrt[3])*c^(1/3) + d^(1/ 3)*x)/((1 + Sqrt[3])*c^(1/3) + d^(1/3)*x)], -7 - 4*Sqrt[3]])/(3^(1/4)*b*d^ (1/3)*Sqrt[(c^(1/3)*(c^(1/3) + d^(1/3)*x))/((1 + Sqrt[3])*c^(1/3) + d^(1/3 )*x)^2]*Sqrt[c + d*x^3]))/(5*b)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*Sqrt[(c_) + (d_.)*(x _)^(n_)]), x_Symbol] :> Simp[f/b Int[1/Sqrt[c + d*x^n], x], x] + Simp[(b* e - a*f)/b Int[1/((a + b*x^n)*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x]
Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + ( f_.)*(x_)^(n_)), x_Symbol] :> Simp[f*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/( b*(n*(p + q + 1) + 1))), x] + Simp[1/(b*(n*(p + q + 1) + 1)) Int[(a + b*x ^n)^p*(c + d*x^n)^(q - 1)*Simp[c*(b*e - a*f + b*e*n*(p + q + 1)) + (d*(b*e - a*f) + f*n*q*(b*c - a*d) + b*d*e*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[ {a, b, c, d, e, f, n, p}, x] && GtQ[q, 0] && NeQ[n*(p + q + 1) + 1, 0]
Result contains higher order function than in optimal. Order 9 vs. order 6.
Time = 1.60 (sec) , antiderivative size = 781, normalized size of antiderivative = 2.39
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(781\) |
| elliptic | \(\text {Expression too large to display}\) | \(784\) |
| default | \(\text {Expression too large to display}\) | \(1020\) |
Input:
int((d*x^3+c)^(1/2)*(f*x^3+e)/(b*x^3+a),x,method=_RETURNVERBOSE)
Output:
2/5*f*x*(d*x^3+c)^(1/2)/b-1/5/b*(-2/3*I*(5*a*d*f-3*b*c*f-5*b*d*e)/b*3^(1/2 )/d*(-c*d^2)^(1/3)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/ 3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2)*((x-1/d*(-c*d^2)^(1/3))/(-3/2/d*(-c*d^ 2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)))^(1/2)*(-I*(x+1/2/d*(-c*d^2)^(1/3 )+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2)/(d*x^3+c )^(1/2)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*( -c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2),(I*3^(1/2)/d*(-c*d^2)^(1/3) /(-3/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)))^(1/2))+5/3*I*(a^2 *d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b/d^2*2^(1/2)*sum(1/_alpha^2/(a*d-b*c)*(-c*d ^2)^(1/3)*(1/2*I*d*(2*x+1/d*(-I*3^(1/2)*(-c*d^2)^(1/3)+(-c*d^2)^(1/3)))/(- c*d^2)^(1/3))^(1/2)*(d*(x-1/d*(-c*d^2)^(1/3))/(I*3^(1/2)*(-c*d^2)^(1/3)-3* (-c*d^2)^(1/3)))^(1/2)*(-1/2*I*d*(2*x+1/d*(I*3^(1/2)*(-c*d^2)^(1/3)+(-c*d^ 2)^(1/3)))/(-c*d^2)^(1/3))^(1/2)/(d*x^3+c)^(1/2)*(I*(-c*d^2)^(1/3)*_alpha* 3^(1/2)*d-I*(-c*d^2)^(2/3)*3^(1/2)+2*_alpha^2*d^2-(-c*d^2)^(1/3)*_alpha*d- (-c*d^2)^(2/3))*EllipticPi(1/3*3^(1/2)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^ (1/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2),1/2*b/d*(2*I*3^(1/ 2)*(-c*d^2)^(1/3)*_alpha^2*d-I*3^(1/2)*(-c*d^2)^(2/3)*_alpha+I*3^(1/2)*c*d -3*(-c*d^2)^(2/3)*_alpha-3*c*d)/(a*d-b*c),(I*3^(1/2)/d*(-c*d^2)^(1/3)/(-3/ 2/d*(-c*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)))^(1/2)),_alpha=RootOf(_ Z^3*b+a)))
Timed out. \[ \int \frac {\sqrt {c+d x^3} \left (e+f x^3\right )}{a+b x^3} \, dx=\text {Timed out} \] Input:
integrate((d*x^3+c)^(1/2)*(f*x^3+e)/(b*x^3+a),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {\sqrt {c+d x^3} \left (e+f x^3\right )}{a+b x^3} \, dx=\int \frac {\sqrt {c + d x^{3}} \left (e + f x^{3}\right )}{a + b x^{3}}\, dx \] Input:
integrate((d*x**3+c)**(1/2)*(f*x**3+e)/(b*x**3+a),x)
Output:
Integral(sqrt(c + d*x**3)*(e + f*x**3)/(a + b*x**3), x)
\[ \int \frac {\sqrt {c+d x^3} \left (e+f x^3\right )}{a+b x^3} \, dx=\int { \frac {\sqrt {d x^{3} + c} {\left (f x^{3} + e\right )}}{b x^{3} + a} \,d x } \] Input:
integrate((d*x^3+c)^(1/2)*(f*x^3+e)/(b*x^3+a),x, algorithm="maxima")
Output:
integrate(sqrt(d*x^3 + c)*(f*x^3 + e)/(b*x^3 + a), x)
\[ \int \frac {\sqrt {c+d x^3} \left (e+f x^3\right )}{a+b x^3} \, dx=\int { \frac {\sqrt {d x^{3} + c} {\left (f x^{3} + e\right )}}{b x^{3} + a} \,d x } \] Input:
integrate((d*x^3+c)^(1/2)*(f*x^3+e)/(b*x^3+a),x, algorithm="giac")
Output:
integrate(sqrt(d*x^3 + c)*(f*x^3 + e)/(b*x^3 + a), x)
Timed out. \[ \int \frac {\sqrt {c+d x^3} \left (e+f x^3\right )}{a+b x^3} \, dx=\int \frac {\sqrt {d\,x^3+c}\,\left (f\,x^3+e\right )}{b\,x^3+a} \,d x \] Input:
int(((c + d*x^3)^(1/2)*(e + f*x^3))/(a + b*x^3),x)
Output:
int(((c + d*x^3)^(1/2)*(e + f*x^3))/(a + b*x^3), x)
\[ \int \frac {\sqrt {c+d x^3} \left (e+f x^3\right )}{a+b x^3} \, dx=\frac {2 \sqrt {d \,x^{3}+c}\, f x -2 \left (\int \frac {\sqrt {d \,x^{3}+c}}{b d \,x^{6}+a d \,x^{3}+b c \,x^{3}+a c}d x \right ) a c f +5 \left (\int \frac {\sqrt {d \,x^{3}+c}}{b d \,x^{6}+a d \,x^{3}+b c \,x^{3}+a c}d x \right ) b c e -5 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{3}}{b d \,x^{6}+a d \,x^{3}+b c \,x^{3}+a c}d x \right ) a d f +3 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{3}}{b d \,x^{6}+a d \,x^{3}+b c \,x^{3}+a c}d x \right ) b c f +5 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{3}}{b d \,x^{6}+a d \,x^{3}+b c \,x^{3}+a c}d x \right ) b d e}{5 b} \] Input:
int((d*x^3+c)^(1/2)*(f*x^3+e)/(b*x^3+a),x)
Output:
(2*sqrt(c + d*x**3)*f*x - 2*int(sqrt(c + d*x**3)/(a*c + a*d*x**3 + b*c*x** 3 + b*d*x**6),x)*a*c*f + 5*int(sqrt(c + d*x**3)/(a*c + a*d*x**3 + b*c*x**3 + b*d*x**6),x)*b*c*e - 5*int((sqrt(c + d*x**3)*x**3)/(a*c + a*d*x**3 + b* c*x**3 + b*d*x**6),x)*a*d*f + 3*int((sqrt(c + d*x**3)*x**3)/(a*c + a*d*x** 3 + b*c*x**3 + b*d*x**6),x)*b*c*f + 5*int((sqrt(c + d*x**3)*x**3)/(a*c + a *d*x**3 + b*c*x**3 + b*d*x**6),x)*b*d*e)/(5*b)