Integrand size = 22, antiderivative size = 65 \[ \int \frac {x \left (c+d x^3\right )^{3/2}}{\left (a+b x^3\right )^2} \, dx=\frac {c x^2 \sqrt {c+d x^3} \operatorname {AppellF1}\left (\frac {2}{3},2,-\frac {3}{2},\frac {5}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{2 a^2 \sqrt {1+\frac {d x^3}{c}}} \] Output:
1/2*c*x^2*(d*x^3+c)^(1/2)*AppellF1(2/3,2,-3/2,5/3,-b*x^3/a,-d*x^3/c)/a^2/( 1+d*x^3/c)^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(177\) vs. \(2(65)=130\).
Time = 10.14 (sec) , antiderivative size = 177, normalized size of antiderivative = 2.72 \[ \int \frac {x \left (c+d x^3\right )^{3/2}}{\left (a+b x^3\right )^2} \, dx=\frac {x^2 \left (-10 a (-b c+a d) \left (c+d x^3\right )+5 c (b c+2 a d) \left (a+b x^3\right ) \sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},1,\frac {5}{3},-\frac {d x^3}{c},-\frac {b x^3}{a}\right )-d (b c-7 a d) x^3 \left (a+b x^3\right ) \sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (\frac {5}{3},\frac {1}{2},1,\frac {8}{3},-\frac {d x^3}{c},-\frac {b x^3}{a}\right )\right )}{30 a^2 b \left (a+b x^3\right ) \sqrt {c+d x^3}} \] Input:
Integrate[(x*(c + d*x^3)^(3/2))/(a + b*x^3)^2,x]
Output:
(x^2*(-10*a*(-(b*c) + a*d)*(c + d*x^3) + 5*c*(b*c + 2*a*d)*(a + b*x^3)*Sqr t[1 + (d*x^3)/c]*AppellF1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), -((b*x^3)/a)] - d*(b*c - 7*a*d)*x^3*(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)]))/(30*a^2*b*(a + b*x^3)*Sqrt[c + d*x^3])
Time = 0.33 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, 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.
\(\displaystyle \int \frac {x \left (c+d x^3\right )^{3/2}}{\left (a+b x^3\right )^2} \, dx\) |
\(\Big \downarrow \) 1013 |
\(\displaystyle \frac {c \sqrt {c+d x^3} \int \frac {x \left (\frac {d x^3}{c}+1\right )^{3/2}}{\left (b x^3+a\right )^2}dx}{\sqrt {\frac {d x^3}{c}+1}}\) |
\(\Big \downarrow \) 1012 |
\(\displaystyle \frac {c x^2 \sqrt {c+d x^3} \operatorname {AppellF1}\left (\frac {2}{3},2,-\frac {3}{2},\frac {5}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{2 a^2 \sqrt {\frac {d x^3}{c}+1}}\) |
Input:
Int[(x*(c + d*x^3)^(3/2))/(a + b*x^3)^2,x]
Output:
(c*x^2*Sqrt[c + d*x^3]*AppellF1[2/3, 2, -3/2, 5/3, -((b*x^3)/a), -((d*x^3) /c)])/(2*a^2*Sqrt[1 + (d*x^3)/c])
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 = 1.23 (sec) , antiderivative size = 955, normalized size of antiderivative = 14.69
method | result | size |
default | \(\text {Expression too large to display}\) | \(955\) |
elliptic | \(\text {Expression too large to display}\) | \(955\) |
Input:
int(x*(d*x^3+c)^(3/2)/(b*x^3+a)^2,x,method=_RETURNVERBOSE)
Output:
-1/3*(a*d-b*c)/b/a*x^2*(d*x^3+c)^(1/2)/(b*x^3+a)-2/3*I*(d^2/b^2+1/6/b^2*d* (a*d-b*c)/a)*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/b^2/d^2*2^(1/2)*sum((7*a^2*d^2-5*a*b*c*d-2*b^2*c^2)/_alpha/(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))/(-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)*_alp ha*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))^(1/2),1/2*b/d*(2*I...
Timed out. \[ \int \frac {x \left (c+d x^3\right )^{3/2}}{\left (a+b x^3\right )^2} \, dx=\text {Timed out} \] Input:
integrate(x*(d*x^3+c)^(3/2)/(b*x^3+a)^2,x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {x \left (c+d x^3\right )^{3/2}}{\left (a+b x^3\right )^2} \, dx=\int \frac {x \left (c + d x^{3}\right )^{\frac {3}{2}}}{\left (a + b x^{3}\right )^{2}}\, dx \] Input:
integrate(x*(d*x**3+c)**(3/2)/(b*x**3+a)**2,x)
Output:
Integral(x*(c + d*x**3)**(3/2)/(a + b*x**3)**2, x)
\[ \int \frac {x \left (c+d x^3\right )^{3/2}}{\left (a+b x^3\right )^2} \, dx=\int { \frac {{\left (d x^{3} + c\right )}^{\frac {3}{2}} x}{{\left (b x^{3} + a\right )}^{2}} \,d x } \] Input:
integrate(x*(d*x^3+c)^(3/2)/(b*x^3+a)^2,x, algorithm="maxima")
Output:
integrate((d*x^3 + c)^(3/2)*x/(b*x^3 + a)^2, x)
\[ \int \frac {x \left (c+d x^3\right )^{3/2}}{\left (a+b x^3\right )^2} \, dx=\int { \frac {{\left (d x^{3} + c\right )}^{\frac {3}{2}} x}{{\left (b x^{3} + a\right )}^{2}} \,d x } \] Input:
integrate(x*(d*x^3+c)^(3/2)/(b*x^3+a)^2,x, algorithm="giac")
Output:
integrate((d*x^3 + c)^(3/2)*x/(b*x^3 + a)^2, x)
Timed out. \[ \int \frac {x \left (c+d x^3\right )^{3/2}}{\left (a+b x^3\right )^2} \, dx=\int \frac {x\,{\left (d\,x^3+c\right )}^{3/2}}{{\left (b\,x^3+a\right )}^2} \,d x \] Input:
int((x*(c + d*x^3)^(3/2))/(a + b*x^3)^2,x)
Output:
int((x*(c + d*x^3)^(3/2))/(a + b*x^3)^2, x)
\[ \int \frac {x \left (c+d x^3\right )^{3/2}}{\left (a+b x^3\right )^2} \, dx =\text {Too large to display} \] Input:
int(x*(d*x^3+c)^(3/2)/(b*x^3+a)^2,x)
Output:
(4*sqrt(c + d*x**3)*c*d*x**2 + 49*int((sqrt(c + d*x**3)*x**7)/(7*a**3*c*d + 7*a**3*d**2*x**3 - 2*a**2*b*c**2 + 12*a**2*b*c*d*x**3 + 14*a**2*b*d**2*x **6 - 4*a*b**2*c**2*x**3 + 3*a*b**2*c*d*x**6 + 7*a*b**2*d**2*x**9 - 2*b**3 *c**2*x**6 - 2*b**3*c*d*x**9),x)*a**3*d**4 - 42*int((sqrt(c + d*x**3)*x**7 )/(7*a**3*c*d + 7*a**3*d**2*x**3 - 2*a**2*b*c**2 + 12*a**2*b*c*d*x**3 + 14 *a**2*b*d**2*x**6 - 4*a*b**2*c**2*x**3 + 3*a*b**2*c*d*x**6 + 7*a*b**2*d**2 *x**9 - 2*b**3*c**2*x**6 - 2*b**3*c*d*x**9),x)*a**2*b*c*d**3 + 49*int((sqr t(c + d*x**3)*x**7)/(7*a**3*c*d + 7*a**3*d**2*x**3 - 2*a**2*b*c**2 + 12*a* *2*b*c*d*x**3 + 14*a**2*b*d**2*x**6 - 4*a*b**2*c**2*x**3 + 3*a*b**2*c*d*x* *6 + 7*a*b**2*d**2*x**9 - 2*b**3*c**2*x**6 - 2*b**3*c*d*x**9),x)*a**2*b*d* *4*x**3 + 8*int((sqrt(c + d*x**3)*x**7)/(7*a**3*c*d + 7*a**3*d**2*x**3 - 2 *a**2*b*c**2 + 12*a**2*b*c*d*x**3 + 14*a**2*b*d**2*x**6 - 4*a*b**2*c**2*x* *3 + 3*a*b**2*c*d*x**6 + 7*a*b**2*d**2*x**9 - 2*b**3*c**2*x**6 - 2*b**3*c* d*x**9),x)*a*b**2*c**2*d**2 - 42*int((sqrt(c + d*x**3)*x**7)/(7*a**3*c*d + 7*a**3*d**2*x**3 - 2*a**2*b*c**2 + 12*a**2*b*c*d*x**3 + 14*a**2*b*d**2*x* *6 - 4*a*b**2*c**2*x**3 + 3*a*b**2*c*d*x**6 + 7*a*b**2*d**2*x**9 - 2*b**3* c**2*x**6 - 2*b**3*c*d*x**9),x)*a*b**2*c*d**3*x**3 + 8*int((sqrt(c + d*x** 3)*x**7)/(7*a**3*c*d + 7*a**3*d**2*x**3 - 2*a**2*b*c**2 + 12*a**2*b*c*d*x* *3 + 14*a**2*b*d**2*x**6 - 4*a*b**2*c**2*x**3 + 3*a*b**2*c*d*x**6 + 7*a*b* *2*d**2*x**9 - 2*b**3*c**2*x**6 - 2*b**3*c*d*x**9),x)*b**3*c**2*d**2*x*...