\(\int \frac {1}{x^3 \sqrt {a+b x^3} \sqrt {c+d x^3}} \, dx\) [881]

Optimal result

Integrand size = 26, antiderivative size = 88 \[ \int \frac {1}{x^3 \sqrt {a+b x^3} \sqrt {c+d x^3}} \, dx=-\frac {\sqrt {1+\frac {b x^3}{a}} \sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (-\frac {2}{3},\frac {1}{2},\frac {1}{2},\frac {1}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{2 x^2 \sqrt {a+b x^3} \sqrt {c+d x^3}} \] Output:

-1/2*(1+b*x^3/a)^(1/2)*(1+d*x^3/c)^(1/2)*AppellF1(-2/3,1/2,1/2,1/3,-b*x^3/ 
a,-d*x^3/c)/x^2/(b*x^3+a)^(1/2)/(d*x^3+c)^(1/2)
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(365\) vs. \(2(88)=176\).

Time = 2.55 (sec) , antiderivative size = 365, normalized size of antiderivative = 4.15 \[ \int \frac {1}{x^3 \sqrt {a+b x^3} \sqrt {c+d x^3}} \, dx=\frac {b d x^6 \sqrt {1+\frac {b x^3}{a}} \sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},\frac {1}{2},\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+\frac {4 \left (-4 a c \left (2 a c+3 b c x^3+3 a d x^3+2 b d x^6\right ) \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},\frac {1}{2},\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+3 x^3 \left (a+b x^3\right ) \left (c+d x^3\right ) \left (a d \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},\frac {3}{2},\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+b c \operatorname {AppellF1}\left (\frac {4}{3},\frac {3}{2},\frac {1}{2},\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right )\right )}{8 a c \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},\frac {1}{2},\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )-3 x^3 \left (a d \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},\frac {3}{2},\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+b c \operatorname {AppellF1}\left (\frac {4}{3},\frac {3}{2},\frac {1}{2},\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right )}}{8 a c x^2 \sqrt {a+b x^3} \sqrt {c+d x^3}} \] Input:

Integrate[1/(x^3*Sqrt[a + b*x^3]*Sqrt[c + d*x^3]),x]
 

Output:

(b*d*x^6*Sqrt[1 + (b*x^3)/a]*Sqrt[1 + (d*x^3)/c]*AppellF1[4/3, 1/2, 1/2, 7 
/3, -((b*x^3)/a), -((d*x^3)/c)] + (4*(-4*a*c*(2*a*c + 3*b*c*x^3 + 3*a*d*x^ 
3 + 2*b*d*x^6)*AppellF1[1/3, 1/2, 1/2, 4/3, -((b*x^3)/a), -((d*x^3)/c)] + 
3*x^3*(a + b*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)])))/(8*a*c*x^2*S 
qrt[a + b*x^3]*Sqrt[c + d*x^3])
 

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1013, 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.

\(\displaystyle \int \frac {1}{x^3 \sqrt {a+b x^3} \sqrt {c+d x^3}} \, dx\)

\(\Big \downarrow \) 1013

\(\displaystyle \frac {\sqrt {\frac {b x^3}{a}+1} \int \frac {1}{x^3 \sqrt {\frac {b x^3}{a}+1} \sqrt {d x^3+c}}dx}{\sqrt {a+b x^3}}\)

\(\Big \downarrow \) 1013

\(\displaystyle \frac {\sqrt {\frac {b x^3}{a}+1} \sqrt {\frac {d x^3}{c}+1} \int \frac {1}{x^3 \sqrt {\frac {b x^3}{a}+1} \sqrt {\frac {d x^3}{c}+1}}dx}{\sqrt {a+b x^3} \sqrt {c+d x^3}}\)

\(\Big \downarrow \) 1012

\(\displaystyle -\frac {\sqrt {\frac {b x^3}{a}+1} \sqrt {\frac {d x^3}{c}+1} \operatorname {AppellF1}\left (-\frac {2}{3},\frac {1}{2},\frac {1}{2},\frac {1}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{2 x^2 \sqrt {a+b x^3} \sqrt {c+d x^3}}\)

Input:

Int[1/(x^3*Sqrt[a + b*x^3]*Sqrt[c + d*x^3]),x]
 

Output:

-1/2*(Sqrt[1 + (b*x^3)/a]*Sqrt[1 + (d*x^3)/c]*AppellF1[-2/3, 1/2, 1/2, 1/3 
, -((b*x^3)/a), -((d*x^3)/c)])/(x^2*Sqrt[a + b*x^3]*Sqrt[c + d*x^3])
 

Defintions of rubi rules used

rule 1012

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])
 

rule 1013

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])
 

Maple [F]

\[\int \frac {1}{x^{3} \sqrt {b \,x^{3}+a}\, \sqrt {d \,x^{3}+c}}d x\]

Input:

int(1/x^3/(b*x^3+a)^(1/2)/(d*x^3+c)^(1/2),x)
 

Output:

int(1/x^3/(b*x^3+a)^(1/2)/(d*x^3+c)^(1/2),x)
 

Fricas [F]

\[ \int \frac {1}{x^3 \sqrt {a+b x^3} \sqrt {c+d x^3}} \, dx=\int { \frac {1}{\sqrt {b x^{3} + a} \sqrt {d x^{3} + c} x^{3}} \,d x } \] Input:

integrate(1/x^3/(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^9 + (b*c + a*d)*x^6 + a*c* 
x^3), x)
 

Sympy [F]

\[ \int \frac {1}{x^3 \sqrt {a+b x^3} \sqrt {c+d x^3}} \, dx=\int \frac {1}{x^{3} \sqrt {a + b x^{3}} \sqrt {c + d x^{3}}}\, dx \] Input:

integrate(1/x**3/(b*x**3+a)**(1/2)/(d*x**3+c)**(1/2),x)
 

Output:

Integral(1/(x**3*sqrt(a + b*x**3)*sqrt(c + d*x**3)), x)
 

Maxima [F]

\[ \int \frac {1}{x^3 \sqrt {a+b x^3} \sqrt {c+d x^3}} \, dx=\int { \frac {1}{\sqrt {b x^{3} + a} \sqrt {d x^{3} + c} x^{3}} \,d x } \] Input:

integrate(1/x^3/(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^3), x)
 

Giac [F]

\[ \int \frac {1}{x^3 \sqrt {a+b x^3} \sqrt {c+d x^3}} \, dx=\int { \frac {1}{\sqrt {b x^{3} + a} \sqrt {d x^{3} + c} x^{3}} \,d x } \] Input:

integrate(1/x^3/(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^3), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^3 \sqrt {a+b x^3} \sqrt {c+d x^3}} \, dx=\int \frac {1}{x^3\,\sqrt {b\,x^3+a}\,\sqrt {d\,x^3+c}} \,d x \] Input:

int(1/(x^3*(a + b*x^3)^(1/2)*(c + d*x^3)^(1/2)),x)
 

Output:

int(1/(x^3*(a + b*x^3)^(1/2)*(c + d*x^3)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {1}{x^3 \sqrt {a+b x^3} \sqrt {c+d x^3}} \, dx=\int \frac {\sqrt {d \,x^{3}+c}\, \sqrt {b \,x^{3}+a}}{b d \,x^{9}+a d \,x^{6}+b c \,x^{6}+a c \,x^{3}}d x \] Input:

int(1/x^3/(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*x**3 + a*d*x**6 + b*c*x**6 + 
b*d*x**9),x)