Integrand size = 23, antiderivative size = 87 \[ \int \frac {\left (c+d x^3\right )^{3/2}}{\left (a+b x^3\right )^{3/2}} \, dx=\frac {c x \sqrt {1+\frac {b x^3}{a}} \sqrt {c+d x^3} \operatorname {AppellF1}\left (\frac {1}{3},\frac {3}{2},-\frac {3}{2},\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{a \sqrt {a+b x^3} \sqrt {1+\frac {d x^3}{c}}} \] Output:
c*x*(1+b*x^3/a)^(1/2)*(d*x^3+c)^(1/2)*AppellF1(1/3,3/2,-3/2,4/3,-b*x^3/a,- d*x^3/c)/a/(b*x^3+a)^(1/2)/(1+d*x^3/c)^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(369\) vs. \(2(87)=174\).
Time = 5.16 (sec) , antiderivative size = 369, normalized size of antiderivative = 4.24 \[ \int \frac {\left (c+d x^3\right )^{3/2}}{\left (a+b x^3\right )^{3/2}} \, dx=\frac {x \left (d (-2 b c+5 a d) x^3 \sqrt {1+\frac {b x^3}{a}} \sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},\frac {1}{2},\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+\frac {8 \left (4 a c \left (-2 a d^2 x^3+b c \left (3 c+2 d x^3\right )\right ) \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},\frac {1}{2},\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+3 (-b c+a d) x^3 \left (c+d x^3\right ) \left (a d \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},\frac {3}{2},\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+b c \operatorname {AppellF1}\left (\frac {4}{3},\frac {3}{2},\frac {1}{2},\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right )\right )}{8 a c \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},\frac {1}{2},\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )-3 x^3 \left (a d \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},\frac {3}{2},\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+b c \operatorname {AppellF1}\left (\frac {4}{3},\frac {3}{2},\frac {1}{2},\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right )}\right )}{12 a b \sqrt {a+b x^3} \sqrt {c+d x^3}} \] Input:
Integrate[(c + d*x^3)^(3/2)/(a + b*x^3)^(3/2),x]
Output:
(x*(d*(-2*b*c + 5*a*d)*x^3*Sqrt[1 + (b*x^3)/a]*Sqrt[1 + (d*x^3)/c]*AppellF 1[4/3, 1/2, 1/2, 7/3, -((b*x^3)/a), -((d*x^3)/c)] + (8*(4*a*c*(-2*a*d^2*x^ 3 + b*c*(3*c + 2*d*x^3))*AppellF1[1/3, 1/2, 1/2, 4/3, -((b*x^3)/a), -((d*x ^3)/c)] + 3*(-(b*c) + a*d)*x^3*(c + d*x^3)*(a*d*AppellF1[4/3, 1/2, 3/2, 7/ 3, -((b*x^3)/a), -((d*x^3)/c)] + b*c*AppellF1[4/3, 3/2, 1/2, 7/3, -((b*x^3 )/a), -((d*x^3)/c)])))/(8*a*c*AppellF1[1/3, 1/2, 1/2, 4/3, -((b*x^3)/a), - ((d*x^3)/c)] - 3*x^3*(a*d*AppellF1[4/3, 1/2, 3/2, 7/3, -((b*x^3)/a), -((d* x^3)/c)] + b*c*AppellF1[4/3, 3/2, 1/2, 7/3, -((b*x^3)/a), -((d*x^3)/c)]))) )/(12*a*b*Sqrt[a + b*x^3]*Sqrt[c + d*x^3])
Time = 0.35 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {937, 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.
| \(\displaystyle \int \frac {\left (c+d x^3\right )^{3/2}}{\left (a+b x^3\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 937 |
| \(\displaystyle \frac {\sqrt {\frac {b x^3}{a}+1} \int \frac {\left (d x^3+c\right )^{3/2}}{\left (\frac {b x^3}{a}+1\right )^{3/2}}dx}{a \sqrt {a+b x^3}}\) |
| \(\Big \downarrow \) 937 |
| \(\displaystyle \frac {c \sqrt {\frac {b x^3}{a}+1} \sqrt {c+d x^3} \int \frac {\left (\frac {d x^3}{c}+1\right )^{3/2}}{\left (\frac {b x^3}{a}+1\right )^{3/2}}dx}{a \sqrt {a+b x^3} \sqrt {\frac {d x^3}{c}+1}}\) |
| \(\Big \downarrow \) 936 |
| \(\displaystyle \frac {c x \sqrt {\frac {b x^3}{a}+1} \sqrt {c+d x^3} \operatorname {AppellF1}\left (\frac {1}{3},\frac {3}{2},-\frac {3}{2},\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{a \sqrt {a+b x^3} \sqrt {\frac {d x^3}{c}+1}}\) |
Input:
Int[(c + d*x^3)^(3/2)/(a + b*x^3)^(3/2),x]
Output:
(c*x*Sqrt[1 + (b*x^3)/a]*Sqrt[c + d*x^3]*AppellF1[1/3, 3/2, -3/2, 4/3, -(( b*x^3)/a), -((d*x^3)/c)])/(a*Sqrt[a + b*x^3]*Sqrt[1 + (d*x^3)/c])
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])
\[\int \frac {\left (d \,x^{3}+c \right )^{\frac {3}{2}}}{\left (b \,x^{3}+a \right )^{\frac {3}{2}}}d x\]
Input:
int((d*x^3+c)^(3/2)/(b*x^3+a)^(3/2),x)
Output:
int((d*x^3+c)^(3/2)/(b*x^3+a)^(3/2),x)
\[ \int \frac {\left (c+d x^3\right )^{3/2}}{\left (a+b x^3\right )^{3/2}} \, dx=\int { \frac {{\left (d x^{3} + c\right )}^{\frac {3}{2}}}{{\left (b x^{3} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((d*x^3+c)^(3/2)/(b*x^3+a)^(3/2),x, algorithm="fricas")
Output:
integral(sqrt(b*x^3 + a)*(d*x^3 + c)^(3/2)/(b^2*x^6 + 2*a*b*x^3 + a^2), x)
\[ \int \frac {\left (c+d x^3\right )^{3/2}}{\left (a+b x^3\right )^{3/2}} \, dx=\int \frac {\left (c + d x^{3}\right )^{\frac {3}{2}}}{\left (a + b x^{3}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((d*x**3+c)**(3/2)/(b*x**3+a)**(3/2),x)
Output:
Integral((c + d*x**3)**(3/2)/(a + b*x**3)**(3/2), x)
\[ \int \frac {\left (c+d x^3\right )^{3/2}}{\left (a+b x^3\right )^{3/2}} \, dx=\int { \frac {{\left (d x^{3} + c\right )}^{\frac {3}{2}}}{{\left (b x^{3} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((d*x^3+c)^(3/2)/(b*x^3+a)^(3/2),x, algorithm="maxima")
Output:
integrate((d*x^3 + c)^(3/2)/(b*x^3 + a)^(3/2), x)
\[ \int \frac {\left (c+d x^3\right )^{3/2}}{\left (a+b x^3\right )^{3/2}} \, dx=\int { \frac {{\left (d x^{3} + c\right )}^{\frac {3}{2}}}{{\left (b x^{3} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((d*x^3+c)^(3/2)/(b*x^3+a)^(3/2),x, algorithm="giac")
Output:
integrate((d*x^3 + c)^(3/2)/(b*x^3 + a)^(3/2), x)
Timed out. \[ \int \frac {\left (c+d x^3\right )^{3/2}}{\left (a+b x^3\right )^{3/2}} \, dx=\int \frac {{\left (d\,x^3+c\right )}^{3/2}}{{\left (b\,x^3+a\right )}^{3/2}} \,d x \] Input:
int((c + d*x^3)^(3/2)/(a + b*x^3)^(3/2),x)
Output:
int((c + d*x^3)^(3/2)/(a + b*x^3)^(3/2), x)
\[ \int \frac {\left (c+d x^3\right )^{3/2}}{\left (a+b x^3\right )^{3/2}} \, dx=\text {too large to display} \] Input:
int((d*x^3+c)^(3/2)/(b*x^3+a)^(3/2),x)
Output:
(4*sqrt(c + d*x**3)*sqrt(a + b*x**3)*c*d*x + 25*int((sqrt(c + d*x**3)*sqrt (a + b*x**3)*x**6)/(5*a**3*c*d + 5*a**3*d**2*x**3 - a**2*b*c**2 + 9*a**2*b *c*d*x**3 + 10*a**2*b*d**2*x**6 - 2*a*b**2*c**2*x**3 + 3*a*b**2*c*d*x**6 + 5*a*b**2*d**2*x**9 - b**3*c**2*x**6 - b**3*c*d*x**9),x)*a**3*d**4 - 30*in t((sqrt(c + d*x**3)*sqrt(a + b*x**3)*x**6)/(5*a**3*c*d + 5*a**3*d**2*x**3 - a**2*b*c**2 + 9*a**2*b*c*d*x**3 + 10*a**2*b*d**2*x**6 - 2*a*b**2*c**2*x* *3 + 3*a*b**2*c*d*x**6 + 5*a*b**2*d**2*x**9 - b**3*c**2*x**6 - b**3*c*d*x* *9),x)*a**2*b*c*d**3 + 25*int((sqrt(c + d*x**3)*sqrt(a + b*x**3)*x**6)/(5* a**3*c*d + 5*a**3*d**2*x**3 - a**2*b*c**2 + 9*a**2*b*c*d*x**3 + 10*a**2*b* d**2*x**6 - 2*a*b**2*c**2*x**3 + 3*a*b**2*c*d*x**6 + 5*a*b**2*d**2*x**9 - b**3*c**2*x**6 - b**3*c*d*x**9),x)*a**2*b*d**4*x**3 + 5*int((sqrt(c + d*x* *3)*sqrt(a + b*x**3)*x**6)/(5*a**3*c*d + 5*a**3*d**2*x**3 - a**2*b*c**2 + 9*a**2*b*c*d*x**3 + 10*a**2*b*d**2*x**6 - 2*a*b**2*c**2*x**3 + 3*a*b**2*c* d*x**6 + 5*a*b**2*d**2*x**9 - b**3*c**2*x**6 - b**3*c*d*x**9),x)*a*b**2*c* *2*d**2 - 30*int((sqrt(c + d*x**3)*sqrt(a + b*x**3)*x**6)/(5*a**3*c*d + 5* a**3*d**2*x**3 - a**2*b*c**2 + 9*a**2*b*c*d*x**3 + 10*a**2*b*d**2*x**6 - 2 *a*b**2*c**2*x**3 + 3*a*b**2*c*d*x**6 + 5*a*b**2*d**2*x**9 - b**3*c**2*x** 6 - b**3*c*d*x**9),x)*a*b**2*c*d**3*x**3 + 5*int((sqrt(c + d*x**3)*sqrt(a + b*x**3)*x**6)/(5*a**3*c*d + 5*a**3*d**2*x**3 - a**2*b*c**2 + 9*a**2*b*c* d*x**3 + 10*a**2*b*d**2*x**6 - 2*a*b**2*c**2*x**3 + 3*a*b**2*c*d*x**6 +...