Integrand size = 24, antiderivative size = 62 \[ \int \frac {1}{x^2 \left (a+b x^3\right )^2 \sqrt {c+d x^3}} \, dx=-\frac {\sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (-\frac {1}{3},2,\frac {1}{2},\frac {2}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{a^2 x \sqrt {c+d x^3}} \] Output:
-(1+d*x^3/c)^(1/2)*AppellF1(-1/3,2,1/2,2/3,-b*x^3/a,-d*x^3/c)/a^2/x/(d*x^3 +c)^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(226\) vs. \(2(62)=124\).
Time = 10.20 (sec) , antiderivative size = 226, normalized size of antiderivative = 3.65 \[ \int \frac {1}{x^2 \left (a+b x^3\right )^2 \sqrt {c+d x^3}} \, dx=\frac {20 a \left (c+d x^3\right ) \left (3 a^2 d-4 b^2 c x^3-3 a b \left (c-d x^3\right )\right )-5 \left (8 b^2 c^2-15 a b c d+3 a^2 d^2\right ) x^3 \left (a+b x^3\right ) \sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},1,\frac {5}{3},-\frac {d x^3}{c},-\frac {b x^3}{a}\right )+2 b d (4 b c-3 a d) x^6 \left (a+b x^3\right ) \sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (\frac {5}{3},\frac {1}{2},1,\frac {8}{3},-\frac {d x^3}{c},-\frac {b x^3}{a}\right )}{60 a^3 c (b c-a d) x \left (a+b x^3\right ) \sqrt {c+d x^3}} \] Input:
Integrate[1/(x^2*(a + b*x^3)^2*Sqrt[c + d*x^3]),x]
Output:
(20*a*(c + d*x^3)*(3*a^2*d - 4*b^2*c*x^3 - 3*a*b*(c - d*x^3)) - 5*(8*b^2*c ^2 - 15*a*b*c*d + 3*a^2*d^2)*x^3*(a + b*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1[ 2/3, 1/2, 1, 5/3, -((d*x^3)/c), -((b*x^3)/a)] + 2*b*d*(4*b*c - 3*a*d)*x^6* (a + b*x^3)*Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), - ((b*x^3)/a)])/(60*a^3*c*(b*c - a*d)*x*(a + b*x^3)*Sqrt[c + d*x^3])
Time = 0.35 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1013, 1012}
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.
| \(\displaystyle \int \frac {1}{x^2 \left (a+b x^3\right )^2 \sqrt {c+d x^3}} \, dx\) |
| \(\Big \downarrow \) 1013 |
| \(\displaystyle \frac {\sqrt {\frac {d x^3}{c}+1} \int \frac {1}{x^2 \left (b x^3+a\right )^2 \sqrt {\frac {d x^3}{c}+1}}dx}{\sqrt {c+d x^3}}\) |
| \(\Big \downarrow \) 1012 |
| \(\displaystyle -\frac {\sqrt {\frac {d x^3}{c}+1} \operatorname {AppellF1}\left (-\frac {1}{3},2,\frac {1}{2},\frac {2}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{a^2 x \sqrt {c+d x^3}}\) |
Input:
Int[1/(x^2*(a + b*x^3)^2*Sqrt[c + d*x^3]),x]
Output:
-((Sqrt[1 + (d*x^3)/c]*AppellF1[-1/3, 2, 1/2, 2/3, -((b*x^3)/a), -((d*x^3) /c)])/(a^2*x*Sqrt[c + d*x^3]))
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[a^p*c^q*((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
Int[((e_.)*(x_))^(m_.)*((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[(e*x)^m*(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] & & NeQ[m, n - 1] && !(IntegerQ[p] || GtQ[a, 0])
Result contains higher order function than in optimal. Order 9 vs. order 6.
Time = 4.49 (sec) , antiderivative size = 963, normalized size of antiderivative = 15.53
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(963\) |
| default | \(\text {Expression too large to display}\) | \(1818\) |
| risch | \(\text {Expression too large to display}\) | \(1819\) |
Input:
int(1/x^2/(b*x^3+a)^2/(d*x^3+c)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/3/(a*d-b*c)/a^2*b^2*x^2*(d*x^3+c)^(1/2)/(b*x^3+a)-1/c/a^2*(d*x^3+c)^(1/2 )/x-2/3*I*(-1/6*d*b/(a*d-b*c)/a^2+1/2*d/c/a^2)*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)*((-3/2/d*(-c* d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*EllipticE(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))+1/d*(-c*d^2)^(1/3)*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)))+1/18*I/a^2/d^2*b*2^(1/2)*sum((11*a*d-8*b*c)/(a* d-b*c)^2/_alpha*(-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))/(-3*(-c *d^2)^(1/3)+I*3^(1/2)*(-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*3^(1/2)*(-c*d^2)^(2/3)+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))...
Timed out. \[ \int \frac {1}{x^2 \left (a+b x^3\right )^2 \sqrt {c+d x^3}} \, dx=\text {Timed out} \] Input:
integrate(1/x^2/(b*x^3+a)^2/(d*x^3+c)^(1/2),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {1}{x^2 \left (a+b x^3\right )^2 \sqrt {c+d x^3}} \, dx=\int \frac {1}{x^{2} \left (a + b x^{3}\right )^{2} \sqrt {c + d x^{3}}}\, dx \] Input:
integrate(1/x**2/(b*x**3+a)**2/(d*x**3+c)**(1/2),x)
Output:
Integral(1/(x**2*(a + b*x**3)**2*sqrt(c + d*x**3)), x)
\[ \int \frac {1}{x^2 \left (a+b x^3\right )^2 \sqrt {c+d x^3}} \, dx=\int { \frac {1}{{\left (b x^{3} + a\right )}^{2} \sqrt {d x^{3} + c} x^{2}} \,d x } \] Input:
integrate(1/x^2/(b*x^3+a)^2/(d*x^3+c)^(1/2),x, algorithm="maxima")
Output:
integrate(1/((b*x^3 + a)^2*sqrt(d*x^3 + c)*x^2), x)
\[ \int \frac {1}{x^2 \left (a+b x^3\right )^2 \sqrt {c+d x^3}} \, dx=\int { \frac {1}{{\left (b x^{3} + a\right )}^{2} \sqrt {d x^{3} + c} x^{2}} \,d x } \] Input:
integrate(1/x^2/(b*x^3+a)^2/(d*x^3+c)^(1/2),x, algorithm="giac")
Output:
integrate(1/((b*x^3 + a)^2*sqrt(d*x^3 + c)*x^2), x)
Timed out. \[ \int \frac {1}{x^2 \left (a+b x^3\right )^2 \sqrt {c+d x^3}} \, dx=\int \frac {1}{x^2\,{\left (b\,x^3+a\right )}^2\,\sqrt {d\,x^3+c}} \,d x \] Input:
int(1/(x^2*(a + b*x^3)^2*(c + d*x^3)^(1/2)),x)
Output:
int(1/(x^2*(a + b*x^3)^2*(c + d*x^3)^(1/2)), x)
\[ \int \frac {1}{x^2 \left (a+b x^3\right )^2 \sqrt {c+d x^3}} \, dx=\frac {-2 \sqrt {d \,x^{3}+c}-5 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{4}}{b^{2} d \,x^{9}+2 a b d \,x^{6}+b^{2} c \,x^{6}+a^{2} d \,x^{3}+2 a b c \,x^{3}+a^{2} c}d x \right ) a b d x -5 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{4}}{b^{2} d \,x^{9}+2 a b d \,x^{6}+b^{2} c \,x^{6}+a^{2} d \,x^{3}+2 a b c \,x^{3}+a^{2} c}d x \right ) b^{2} d \,x^{4}+\left (\int \frac {\sqrt {d \,x^{3}+c}\, x}{b^{2} d \,x^{9}+2 a b d \,x^{6}+b^{2} c \,x^{6}+a^{2} d \,x^{3}+2 a b c \,x^{3}+a^{2} c}d x \right ) a^{2} d x -8 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x}{b^{2} d \,x^{9}+2 a b d \,x^{6}+b^{2} c \,x^{6}+a^{2} d \,x^{3}+2 a b c \,x^{3}+a^{2} c}d x \right ) a b c x +\left (\int \frac {\sqrt {d \,x^{3}+c}\, x}{b^{2} d \,x^{9}+2 a b d \,x^{6}+b^{2} c \,x^{6}+a^{2} d \,x^{3}+2 a b c \,x^{3}+a^{2} c}d x \right ) a b d \,x^{4}-8 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x}{b^{2} d \,x^{9}+2 a b d \,x^{6}+b^{2} c \,x^{6}+a^{2} d \,x^{3}+2 a b c \,x^{3}+a^{2} c}d x \right ) b^{2} c \,x^{4}}{2 a c x \left (b \,x^{3}+a \right )} \] Input:
int(1/x^2/(b*x^3+a)^2/(d*x^3+c)^(1/2),x)
Output:
( - 2*sqrt(c + d*x**3) - 5*int((sqrt(c + d*x**3)*x**4)/(a**2*c + a**2*d*x* *3 + 2*a*b*c*x**3 + 2*a*b*d*x**6 + b**2*c*x**6 + b**2*d*x**9),x)*a*b*d*x - 5*int((sqrt(c + d*x**3)*x**4)/(a**2*c + a**2*d*x**3 + 2*a*b*c*x**3 + 2*a* b*d*x**6 + b**2*c*x**6 + b**2*d*x**9),x)*b**2*d*x**4 + int((sqrt(c + d*x** 3)*x)/(a**2*c + a**2*d*x**3 + 2*a*b*c*x**3 + 2*a*b*d*x**6 + b**2*c*x**6 + b**2*d*x**9),x)*a**2*d*x - 8*int((sqrt(c + d*x**3)*x)/(a**2*c + a**2*d*x** 3 + 2*a*b*c*x**3 + 2*a*b*d*x**6 + b**2*c*x**6 + b**2*d*x**9),x)*a*b*c*x + int((sqrt(c + d*x**3)*x)/(a**2*c + a**2*d*x**3 + 2*a*b*c*x**3 + 2*a*b*d*x* *6 + b**2*c*x**6 + b**2*d*x**9),x)*a*b*d*x**4 - 8*int((sqrt(c + d*x**3)*x) /(a**2*c + a**2*d*x**3 + 2*a*b*c*x**3 + 2*a*b*d*x**6 + b**2*c*x**6 + b**2* d*x**9),x)*b**2*c*x**4)/(2*a*c*x*(a + b*x**3))