Integrand size = 23, antiderivative size = 83 \[ \int \frac {1}{\sqrt {a+b x^3} \sqrt {c+d x^3}} \, dx=\frac {x \sqrt {1+\frac {b x^3}{a}} \sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},\frac {1}{2},\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{\sqrt {a+b x^3} \sqrt {c+d x^3}} \] Output:
x*(1+b*x^3/a)^(1/2)*(1+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)/(d*x^3+c)^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(170\) vs. \(2(83)=166\).
Time = 0.04 (sec) , antiderivative size = 170, normalized size of antiderivative = 2.05 \[ \int \frac {1}{\sqrt {a+b x^3} \sqrt {c+d x^3}} \, dx=-\frac {8 a c x \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},\frac {1}{2},\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{\sqrt {a+b x^3} \sqrt {c+d x^3} \left (-8 a c \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},\frac {1}{2},\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+3 x^3 \left (a d \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},\frac {3}{2},\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+b c \operatorname {AppellF1}\left (\frac {4}{3},\frac {3}{2},\frac {1}{2},\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right )\right )} \] Input:
Integrate[1/(Sqrt[a + b*x^3]*Sqrt[c + d*x^3]),x]
Output:
(-8*a*c*x*AppellF1[1/3, 1/2, 1/2, 4/3, -((b*x^3)/a), -((d*x^3)/c)])/(Sqrt[ a + b*x^3]*Sqrt[c + d*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, 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 )])))
Time = 0.37 (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.
\(\displaystyle \int \frac {1}{\sqrt {a+b x^3} \sqrt {c+d x^3}} \, dx\) |
\(\Big \downarrow \) 937 |
\(\displaystyle \frac {\sqrt {\frac {b x^3}{a}+1} \int \frac {1}{\sqrt {\frac {b x^3}{a}+1} \sqrt {d x^3+c}}dx}{\sqrt {a+b x^3}}\) |
\(\Big \downarrow \) 937 |
\(\displaystyle \frac {\sqrt {\frac {b x^3}{a}+1} \sqrt {\frac {d x^3}{c}+1} \int \frac {1}{\sqrt {\frac {b x^3}{a}+1} \sqrt {\frac {d x^3}{c}+1}}dx}{\sqrt {a+b x^3} \sqrt {c+d x^3}}\) |
\(\Big \downarrow \) 936 |
\(\displaystyle \frac {x \sqrt {\frac {b x^3}{a}+1} \sqrt {\frac {d x^3}{c}+1} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},\frac {1}{2},\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{\sqrt {a+b x^3} \sqrt {c+d x^3}}\) |
Input:
Int[1/(Sqrt[a + b*x^3]*Sqrt[c + d*x^3]),x]
Output:
(x*Sqrt[1 + (b*x^3)/a]*Sqrt[1 + (d*x^3)/c]*AppellF1[1/3, 1/2, 1/2, 4/3, -( (b*x^3)/a), -((d*x^3)/c)])/(Sqrt[a + b*x^3]*Sqrt[c + d*x^3])
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])
\[\int \frac {1}{\sqrt {b \,x^{3}+a}\, \sqrt {d \,x^{3}+c}}d x\]
Input:
int(1/(b*x^3+a)^(1/2)/(d*x^3+c)^(1/2),x)
Output:
int(1/(b*x^3+a)^(1/2)/(d*x^3+c)^(1/2),x)
\[ \int \frac {1}{\sqrt {a+b x^3} \sqrt {c+d x^3}} \, dx=\int { \frac {1}{\sqrt {b x^{3} + a} \sqrt {d x^{3} + c}} \,d x } \] Input:
integrate(1/(b*x^3+a)^(1/2)/(d*x^3+c)^(1/2),x, algorithm="fricas")
Output:
integral(sqrt(b*x^3 + a)*sqrt(d*x^3 + c)/(b*d*x^6 + (b*c + a*d)*x^3 + a*c) , x)
\[ \int \frac {1}{\sqrt {a+b x^3} \sqrt {c+d x^3}} \, dx=\int \frac {1}{\sqrt {a + b x^{3}} \sqrt {c + d x^{3}}}\, dx \] Input:
integrate(1/(b*x**3+a)**(1/2)/(d*x**3+c)**(1/2),x)
Output:
Integral(1/(sqrt(a + b*x**3)*sqrt(c + d*x**3)), x)
\[ \int \frac {1}{\sqrt {a+b x^3} \sqrt {c+d x^3}} \, dx=\int { \frac {1}{\sqrt {b x^{3} + a} \sqrt {d x^{3} + c}} \,d x } \] Input:
integrate(1/(b*x^3+a)^(1/2)/(d*x^3+c)^(1/2),x, algorithm="maxima")
Output:
integrate(1/(sqrt(b*x^3 + a)*sqrt(d*x^3 + c)), x)
\[ \int \frac {1}{\sqrt {a+b x^3} \sqrt {c+d x^3}} \, dx=\int { \frac {1}{\sqrt {b x^{3} + a} \sqrt {d x^{3} + c}} \,d x } \] Input:
integrate(1/(b*x^3+a)^(1/2)/(d*x^3+c)^(1/2),x, algorithm="giac")
Output:
integrate(1/(sqrt(b*x^3 + a)*sqrt(d*x^3 + c)), x)
Timed out. \[ \int \frac {1}{\sqrt {a+b x^3} \sqrt {c+d x^3}} \, dx=\int \frac {1}{\sqrt {b\,x^3+a}\,\sqrt {d\,x^3+c}} \,d x \] Input:
int(1/((a + b*x^3)^(1/2)*(c + d*x^3)^(1/2)),x)
Output:
int(1/((a + b*x^3)^(1/2)*(c + d*x^3)^(1/2)), x)
\[ \int \frac {1}{\sqrt {a+b x^3} \sqrt {c+d x^3}} \, dx=\int \frac {\sqrt {d \,x^{3}+c}\, \sqrt {b \,x^{3}+a}}{b d \,x^{6}+a d \,x^{3}+b c \,x^{3}+a c}d x \] Input:
int(1/(b*x^3+a)^(1/2)/(d*x^3+c)^(1/2),x)
Output:
int((sqrt(c + d*x**3)*sqrt(a + b*x**3))/(a*c + a*d*x**3 + b*c*x**3 + b*d*x **6),x)