Integrand size = 24, antiderivative size = 65 \[ \int \frac {x^3 \left (c+d x^3\right )^{3/2}}{a+b x^3} \, dx=\frac {c x^4 \sqrt {c+d x^3} \operatorname {AppellF1}\left (\frac {4}{3},1,-\frac {3}{2},\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{4 a \sqrt {1+\frac {d x^3}{c}}} \] Output:
1/4*c*x^4*(d*x^3+c)^(1/2)*AppellF1(4/3,1,-3/2,7/3,-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. \(280\) vs. \(2(65)=130\).
Time = 8.94 (sec) , antiderivative size = 280, normalized size of antiderivative = 4.31 \[ \int \frac {x^3 \left (c+d x^3\right )^{3/2}}{a+b x^3} \, dx=\frac {x \left (8 \left (c+d x^3\right ) \left (14 b c-11 a d+5 b d x^3\right )+\frac {\left (27 b^2 c^2-88 a b c d+55 a^2 d^2\right ) x^3 \sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},1,\frac {7}{3},-\frac {d x^3}{c},-\frac {b x^3}{a}\right )}{a}-\frac {64 a^2 c^2 (-14 b c+11 a d) \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},1,\frac {4}{3},-\frac {d x^3}{c},-\frac {b x^3}{a}\right )}{\left (a+b x^3\right ) \left (-8 a c \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},1,\frac {4}{3},-\frac {d x^3}{c},-\frac {b x^3}{a}\right )+3 x^3 \left (2 b c \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},2,\frac {7}{3},-\frac {d x^3}{c},-\frac {b x^3}{a}\right )+a d \operatorname {AppellF1}\left (\frac {4}{3},\frac {3}{2},1,\frac {7}{3},-\frac {d x^3}{c},-\frac {b x^3}{a}\right )\right )\right )}\right )}{220 b^2 \sqrt {c+d x^3}} \] Input:
Integrate[(x^3*(c + d*x^3)^(3/2))/(a + b*x^3),x]
Output:
(x*(8*(c + d*x^3)*(14*b*c - 11*a*d + 5*b*d*x^3) + ((27*b^2*c^2 - 88*a*b*c* d + 55*a^2*d^2)*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 - (64*a^2*c^2*(-14*b*c + 11*a*d)*AppellF1[1/3, 1/2 , 1, 4/3, -((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)])))))/(220*b^2*Sqrt[c + d*x^3])
Time = 0.34 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, 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.
| \(\displaystyle \int \frac {x^3 \left (c+d x^3\right )^{3/2}}{a+b x^3} \, dx\) |
| \(\Big \downarrow \) 1013 |
| \(\displaystyle \frac {c \sqrt {c+d x^3} \int \frac {x^3 \left (\frac {d x^3}{c}+1\right )^{3/2}}{b x^3+a}dx}{\sqrt {\frac {d x^3}{c}+1}}\) |
| \(\Big \downarrow \) 1012 |
| \(\displaystyle \frac {c x^4 \sqrt {c+d x^3} \operatorname {AppellF1}\left (\frac {4}{3},1,-\frac {3}{2},\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{4 a \sqrt {\frac {d x^3}{c}+1}}\) |
Input:
Int[(x^3*(c + d*x^3)^(3/2))/(a + b*x^3),x]
Output:
(c*x^4*Sqrt[c + d*x^3]*AppellF1[4/3, 1, -3/2, 7/3, -((b*x^3)/a), -((d*x^3) /c)])/(4*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 = 2.38 (sec) , antiderivative size = 800, normalized size of antiderivative = 12.31
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(800\) |
| elliptic | \(\text {Expression too large to display}\) | \(846\) |
| default | \(\text {Expression too large to display}\) | \(1101\) |
Input:
int(x^3*(d*x^3+c)^(3/2)/(b*x^3+a),x,method=_RETURNVERBOSE)
Output:
-2/55*x*(-5*b*d*x^3+11*a*d-14*b*c)*(d*x^3+c)^(1/2)/b^2+1/55/b^2*(-2/3*I*(5 5*a^2*d^2-88*a*b*c*d+27*b^2*c^2)/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))+55/3*I*a*(a^2*d^2-2*a*b*c*d+b^2*c^2)/b/d^2* 2^(1/2)*sum(1/(a*d-b*c)/_alpha^2*(-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*(-c* d^2)^(2/3)*_alpha*3^(1/2)+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. \[ \int \frac {x^3 \left (c+d x^3\right )^{3/2}}{a+b x^3} \, dx=\text {Timed out} \] Input:
integrate(x^3*(d*x^3+c)^(3/2)/(b*x^3+a),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {x^3 \left (c+d x^3\right )^{3/2}}{a+b x^3} \, dx=\int \frac {x^{3} \left (c + d x^{3}\right )^{\frac {3}{2}}}{a + b x^{3}}\, dx \] Input:
integrate(x**3*(d*x**3+c)**(3/2)/(b*x**3+a),x)
Output:
Integral(x**3*(c + d*x**3)**(3/2)/(a + b*x**3), x)
\[ \int \frac {x^3 \left (c+d x^3\right )^{3/2}}{a+b x^3} \, dx=\int { \frac {{\left (d x^{3} + c\right )}^{\frac {3}{2}} x^{3}}{b x^{3} + a} \,d x } \] Input:
integrate(x^3*(d*x^3+c)^(3/2)/(b*x^3+a),x, algorithm="maxima")
Output:
integrate((d*x^3 + c)^(3/2)*x^3/(b*x^3 + a), x)
\[ \int \frac {x^3 \left (c+d x^3\right )^{3/2}}{a+b x^3} \, dx=\int { \frac {{\left (d x^{3} + c\right )}^{\frac {3}{2}} x^{3}}{b x^{3} + a} \,d x } \] Input:
integrate(x^3*(d*x^3+c)^(3/2)/(b*x^3+a),x, algorithm="giac")
Output:
integrate((d*x^3 + c)^(3/2)*x^3/(b*x^3 + a), x)
Timed out. \[ \int \frac {x^3 \left (c+d x^3\right )^{3/2}}{a+b x^3} \, dx=\int \frac {x^3\,{\left (d\,x^3+c\right )}^{3/2}}{b\,x^3+a} \,d x \] Input:
int((x^3*(c + d*x^3)^(3/2))/(a + b*x^3),x)
Output:
int((x^3*(c + d*x^3)^(3/2))/(a + b*x^3), x)
\[ \int \frac {x^3 \left (c+d x^3\right )^{3/2}}{a+b x^3} \, dx=\frac {-22 \sqrt {d \,x^{3}+c}\, a d x +28 \sqrt {d \,x^{3}+c}\, b c x +10 \sqrt {d \,x^{3}+c}\, b d \,x^{4}+22 \left (\int \frac {\sqrt {d \,x^{3}+c}}{b d \,x^{6}+a d \,x^{3}+b c \,x^{3}+a c}d x \right ) a^{2} c d -28 \left (\int \frac {\sqrt {d \,x^{3}+c}}{b d \,x^{6}+a d \,x^{3}+b c \,x^{3}+a c}d x \right ) a b \,c^{2}+55 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{3}}{b d \,x^{6}+a d \,x^{3}+b c \,x^{3}+a c}d x \right ) a^{2} d^{2}-88 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{3}}{b d \,x^{6}+a d \,x^{3}+b c \,x^{3}+a c}d x \right ) a b c d +27 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{3}}{b d \,x^{6}+a d \,x^{3}+b c \,x^{3}+a c}d x \right ) b^{2} c^{2}}{55 b^{2}} \] Input:
int(x^3*(d*x^3+c)^(3/2)/(b*x^3+a),x)
Output:
( - 22*sqrt(c + d*x**3)*a*d*x + 28*sqrt(c + d*x**3)*b*c*x + 10*sqrt(c + d* x**3)*b*d*x**4 + 22*int(sqrt(c + d*x**3)/(a*c + a*d*x**3 + b*c*x**3 + b*d* x**6),x)*a**2*c*d - 28*int(sqrt(c + d*x**3)/(a*c + a*d*x**3 + b*c*x**3 + b *d*x**6),x)*a*b*c**2 + 55*int((sqrt(c + d*x**3)*x**3)/(a*c + a*d*x**3 + b* c*x**3 + b*d*x**6),x)*a**2*d**2 - 88*int((sqrt(c + d*x**3)*x**3)/(a*c + a* d*x**3 + b*c*x**3 + b*d*x**6),x)*a*b*c*d + 27*int((sqrt(c + d*x**3)*x**3)/ (a*c + a*d*x**3 + b*c*x**3 + b*d*x**6),x)*b**2*c**2)/(55*b**2)