Integrand size = 23, antiderivative size = 83 \[ \int \frac {\sqrt {c+d x^3}}{\sqrt {a+b x^3}} \, dx=\frac {x \sqrt {1+\frac {b x^3}{a}} \sqrt {c+d x^3} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},-\frac {1}{2},\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{\sqrt {a+b x^3} \sqrt {1+\frac {d x^3}{c}}} \] Output:
x*(1+b*x^3/a)^(1/2)*(d*x^3+c)^(1/2)*AppellF1(1/3,1/2,-1/2,4/3,-b*x^3/a,-d* x^3/c)/(b*x^3+a)^(1/2)/(1+d*x^3/c)^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(171\) vs. \(2(83)=166\).
Time = 1.81 (sec) , antiderivative size = 171, normalized size of antiderivative = 2.06 \[ \int \frac {\sqrt {c+d x^3}}{\sqrt {a+b x^3}} \, dx=\frac {8 a c x \sqrt {c+d x^3} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},-\frac {1}{2},\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{\sqrt {a+b x^3} \left (8 a c \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},-\frac {1}{2},\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+3 x^3 \left (a d \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},\frac {1}{2},\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )-b c \operatorname {AppellF1}\left (\frac {4}{3},\frac {3}{2},-\frac {1}{2},\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right )\right )} \] Input:
Integrate[Sqrt[c + d*x^3]/Sqrt[a + b*x^3],x]
Output:
(8*a*c*x*Sqrt[c + d*x^3]*AppellF1[1/3, 1/2, -1/2, 4/3, -((b*x^3)/a), -((d* x^3)/c)])/(Sqrt[a + b*x^3]*(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, 1/2, 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)])))
Time = 0.35 (sec) , antiderivative size = 83, 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.
\(\displaystyle \int \frac {\sqrt {c+d x^3}}{\sqrt {a+b x^3}} \, dx\) |
\(\Big \downarrow \) 937 |
\(\displaystyle \frac {\sqrt {\frac {b x^3}{a}+1} \int \frac {\sqrt {d x^3+c}}{\sqrt {\frac {b x^3}{a}+1}}dx}{\sqrt {a+b x^3}}\) |
\(\Big \downarrow \) 937 |
\(\displaystyle \frac {\sqrt {\frac {b x^3}{a}+1} \sqrt {c+d x^3} \int \frac {\sqrt {\frac {d x^3}{c}+1}}{\sqrt {\frac {b x^3}{a}+1}}dx}{\sqrt {a+b x^3} \sqrt {\frac {d x^3}{c}+1}}\) |
\(\Big \downarrow \) 936 |
\(\displaystyle \frac {x \sqrt {\frac {b x^3}{a}+1} \sqrt {c+d x^3} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},-\frac {1}{2},\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{\sqrt {a+b x^3} \sqrt {\frac {d x^3}{c}+1}}\) |
Input:
Int[Sqrt[c + d*x^3]/Sqrt[a + b*x^3],x]
Output:
(x*Sqrt[1 + (b*x^3)/a]*Sqrt[c + d*x^3]*AppellF1[1/3, 1/2, -1/2, 4/3, -((b* x^3)/a), -((d*x^3)/c)])/(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])
\[\int \frac {\sqrt {d \,x^{3}+c}}{\sqrt {b \,x^{3}+a}}d x\]
Input:
int((d*x^3+c)^(1/2)/(b*x^3+a)^(1/2),x)
Output:
int((d*x^3+c)^(1/2)/(b*x^3+a)^(1/2),x)
\[ \int \frac {\sqrt {c+d x^3}}{\sqrt {a+b x^3}} \, dx=\int { \frac {\sqrt {d x^{3} + c}}{\sqrt {b x^{3} + a}} \,d x } \] Input:
integrate((d*x^3+c)^(1/2)/(b*x^3+a)^(1/2),x, algorithm="fricas")
Output:
integral(sqrt(d*x^3 + c)/sqrt(b*x^3 + a), x)
\[ \int \frac {\sqrt {c+d x^3}}{\sqrt {a+b x^3}} \, dx=\int \frac {\sqrt {c + d x^{3}}}{\sqrt {a + b x^{3}}}\, dx \] Input:
integrate((d*x**3+c)**(1/2)/(b*x**3+a)**(1/2),x)
Output:
Integral(sqrt(c + d*x**3)/sqrt(a + b*x**3), x)
\[ \int \frac {\sqrt {c+d x^3}}{\sqrt {a+b x^3}} \, dx=\int { \frac {\sqrt {d x^{3} + c}}{\sqrt {b x^{3} + a}} \,d x } \] Input:
integrate((d*x^3+c)^(1/2)/(b*x^3+a)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(d*x^3 + c)/sqrt(b*x^3 + a), x)
\[ \int \frac {\sqrt {c+d x^3}}{\sqrt {a+b x^3}} \, dx=\int { \frac {\sqrt {d x^{3} + c}}{\sqrt {b x^{3} + a}} \,d x } \] Input:
integrate((d*x^3+c)^(1/2)/(b*x^3+a)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(d*x^3 + c)/sqrt(b*x^3 + a), x)
Timed out. \[ \int \frac {\sqrt {c+d x^3}}{\sqrt {a+b x^3}} \, dx=\int \frac {\sqrt {d\,x^3+c}}{\sqrt {b\,x^3+a}} \,d x \] Input:
int((c + d*x^3)^(1/2)/(a + b*x^3)^(1/2),x)
Output:
int((c + d*x^3)^(1/2)/(a + b*x^3)^(1/2), x)
\[ \int \frac {\sqrt {c+d x^3}}{\sqrt {a+b x^3}} \, dx=\int \frac {\sqrt {d \,x^{3}+c}\, \sqrt {b \,x^{3}+a}}{b \,x^{3}+a}d x \] Input:
int((d*x^3+c)^(1/2)/(b*x^3+a)^(1/2),x)
Output:
int((sqrt(c + d*x**3)*sqrt(a + b*x**3))/(a + b*x**3),x)