Integrand size = 22, antiderivative size = 65 \[ \int \frac {x \left (c+d x^3\right )^{3/2}}{a+b x^3} \, dx=\frac {c x^2 \sqrt {c+d x^3} \operatorname {AppellF1}\left (\frac {2}{3},1,-\frac {3}{2},\frac {5}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{2 a \sqrt {1+\frac {d x^3}{c}}} \] Output:
1/2*c*x^2*(d*x^3+c)^(1/2)*AppellF1(2/3,1,-3/2,5/3,-b*x^3/a,-d*x^3/c)/a/(1+ d*x^3/c)^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(149\) vs. \(2(65)=130\).
Time = 10.17 (sec) , antiderivative size = 149, normalized size of antiderivative = 2.29 \[ \int \frac {x \left (c+d x^3\right )^{3/2}}{a+b x^3} \, dx=\frac {x^2 \left (20 a d \left (c+d x^3\right )+5 c (7 b c-4 a d) \sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{2},1,\frac {5}{3},-\frac {d x^3}{c},-\frac {b x^3}{a}\right )+2 d (10 b c-7 a d) x^3 \sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (\frac {5}{3},\frac {1}{2},1,\frac {8}{3},-\frac {d x^3}{c},-\frac {b x^3}{a}\right )\right )}{70 a b \sqrt {c+d x^3}} \] Input:
Integrate[(x*(c + d*x^3)^(3/2))/(a + b*x^3),x]
Output:
(x^2*(20*a*d*(c + d*x^3) + 5*c*(7*b*c - 4*a*d)*Sqrt[1 + (d*x^3)/c]*AppellF 1[2/3, 1/2, 1, 5/3, -((d*x^3)/c), -((b*x^3)/a)] + 2*d*(10*b*c - 7*a*d)*x^3 *Sqrt[1 + (d*x^3)/c]*AppellF1[5/3, 1/2, 1, 8/3, -((d*x^3)/c), -((b*x^3)/a) ]))/(70*a*b*Sqrt[c + d*x^3])
Time = 0.32 (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.
\(\displaystyle \int \frac {x \left (c+d x^3\right )^{3/2}}{a+b x^3} \, dx\) |
\(\Big \downarrow \) 1013 |
\(\displaystyle \frac {c \sqrt {c+d x^3} \int \frac {x \left (\frac {d x^3}{c}+1\right )^{3/2}}{b x^3+a}dx}{\sqrt {\frac {d x^3}{c}+1}}\) |
\(\Big \downarrow \) 1012 |
\(\displaystyle \frac {c x^2 \sqrt {c+d x^3} \operatorname {AppellF1}\left (\frac {2}{3},1,-\frac {3}{2},\frac {5}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{2 a \sqrt {\frac {d x^3}{c}+1}}\) |
Input:
Int[(x*(c + d*x^3)^(3/2))/(a + b*x^3),x]
Output:
(c*x^2*Sqrt[c + d*x^3]*AppellF1[2/3, 1, -3/2, 5/3, -((b*x^3)/a), -((d*x^3) /c)])/(2*a*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 = 2.55 (sec) , antiderivative size = 921, normalized size of antiderivative = 14.17
method | result | size |
risch | \(\text {Expression too large to display}\) | \(921\) |
default | \(\text {Expression too large to display}\) | \(930\) |
elliptic | \(\text {Expression too large to display}\) | \(930\) |
Input:
int(x*(d*x^3+c)^(3/2)/(b*x^3+a),x,method=_RETURNVERBOSE)
Output:
2/7*d/b*x^2*(d*x^3+c)^(1/2)-1/7/b*(-2/3*I*(7*a*d-10*b*c)/b*3^(1/2)*(-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)))+7/3*I*(a^2*d^2-2*a*b*c*d+b^2*c^2)/b/d ^2*2^(1/2)*sum(1/_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)*_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))^(1/2),1/2*b/d*(2*I*(-c*d^2)^(1/3)*_alpha^2*3^(1/2)*d-I*...
Timed out. \[ \int \frac {x \left (c+d x^3\right )^{3/2}}{a+b x^3} \, dx=\text {Timed out} \] Input:
integrate(x*(d*x^3+c)^(3/2)/(b*x^3+a),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {x \left (c+d x^3\right )^{3/2}}{a+b x^3} \, dx=\int \frac {x \left (c + d x^{3}\right )^{\frac {3}{2}}}{a + b x^{3}}\, dx \] Input:
integrate(x*(d*x**3+c)**(3/2)/(b*x**3+a),x)
Output:
Integral(x*(c + d*x**3)**(3/2)/(a + b*x**3), x)
\[ \int \frac {x \left (c+d x^3\right )^{3/2}}{a+b x^3} \, dx=\int { \frac {{\left (d x^{3} + c\right )}^{\frac {3}{2}} x}{b x^{3} + a} \,d x } \] Input:
integrate(x*(d*x^3+c)^(3/2)/(b*x^3+a),x, algorithm="maxima")
Output:
integrate((d*x^3 + c)^(3/2)*x/(b*x^3 + a), x)
\[ \int \frac {x \left (c+d x^3\right )^{3/2}}{a+b x^3} \, dx=\int { \frac {{\left (d x^{3} + c\right )}^{\frac {3}{2}} x}{b x^{3} + a} \,d x } \] Input:
integrate(x*(d*x^3+c)^(3/2)/(b*x^3+a),x, algorithm="giac")
Output:
integrate((d*x^3 + c)^(3/2)*x/(b*x^3 + a), x)
Timed out. \[ \int \frac {x \left (c+d x^3\right )^{3/2}}{a+b x^3} \, dx=\int \frac {x\,{\left (d\,x^3+c\right )}^{3/2}}{b\,x^3+a} \,d x \] Input:
int((x*(c + d*x^3)^(3/2))/(a + b*x^3),x)
Output:
int((x*(c + d*x^3)^(3/2))/(a + b*x^3), x)
\[ \int \frac {x \left (c+d x^3\right )^{3/2}}{a+b x^3} \, dx=\frac {2 \sqrt {d \,x^{3}+c}\, d \,x^{2}-7 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{4}}{b d \,x^{6}+a d \,x^{3}+b c \,x^{3}+a c}d x \right ) a \,d^{2}+10 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{4}}{b d \,x^{6}+a d \,x^{3}+b c \,x^{3}+a c}d x \right ) b c d -4 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x}{b d \,x^{6}+a d \,x^{3}+b c \,x^{3}+a c}d x \right ) a c d +7 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x}{b d \,x^{6}+a d \,x^{3}+b c \,x^{3}+a c}d x \right ) b \,c^{2}}{7 b} \] Input:
int(x*(d*x^3+c)^(3/2)/(b*x^3+a),x)
Output:
(2*sqrt(c + d*x**3)*d*x**2 - 7*int((sqrt(c + d*x**3)*x**4)/(a*c + a*d*x**3 + b*c*x**3 + b*d*x**6),x)*a*d**2 + 10*int((sqrt(c + d*x**3)*x**4)/(a*c + a*d*x**3 + b*c*x**3 + b*d*x**6),x)*b*c*d - 4*int((sqrt(c + d*x**3)*x)/(a*c + a*d*x**3 + b*c*x**3 + b*d*x**6),x)*a*c*d + 7*int((sqrt(c + d*x**3)*x)/( a*c + a*d*x**3 + b*c*x**3 + b*d*x**6),x)*b*c**2)/(7*b)