\(\int \frac {1}{\sqrt {a+b x^3} (c+d x^3)^{3/2}} \, dx\) [61]

Optimal result

Integrand size = 23, antiderivative size = 86 \[ \int \frac {1}{\sqrt {a+b x^3} \left (c+d x^3\right )^{3/2}} \, dx=\frac {x \sqrt {1+\frac {b x^3}{a}} \sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},\frac {3}{2},\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{c \sqrt {a+b x^3} \sqrt {c+d x^3}} \] Output:

x*(1+b*x^3/a)^(1/2)*(1+d*x^3/c)^(1/2)*AppellF1(1/3,1/2,3/2,4/3,-b*x^3/a,-d 
*x^3/c)/c/(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. \(419\) vs. \(2(86)=172\).

Time = 3.48 (sec) , antiderivative size = 419, normalized size of antiderivative = 4.87 \[ \int \frac {1}{\sqrt {a+b x^3} \left (c+d x^3\right )^{3/2}} \, dx=\frac {-8 a c x \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},\frac {1}{2},\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right ) \left (6 b c-6 a d-4 b d x^3+b d x^3 \sqrt {1+\frac {b x^3}{a}} \sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},\frac {1}{2},\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right )-3 d x^4 \left (4 \left (a+b x^3\right )-b x^3 \sqrt {1+\frac {b x^3}{a}} \sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},\frac {1}{2},\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right ) \left (a d \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},\frac {3}{2},\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+b c \operatorname {AppellF1}\left (\frac {4}{3},\frac {3}{2},\frac {1}{2},\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right )}{6 c (b c-a d) \sqrt {a+b x^3} \sqrt {c+d x^3} \left (-8 a c \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},\frac {1}{2},\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+3 x^3 \left (a d \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},\frac {3}{2},\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+b c \operatorname {AppellF1}\left (\frac {4}{3},\frac {3}{2},\frac {1}{2},\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right )\right )} \] Input:

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

Output:

(-8*a*c*x*AppellF1[1/3, 1/2, 1/2, 4/3, -((b*x^3)/a), -((d*x^3)/c)]*(6*b*c 
- 6*a*d - 4*b*d*x^3 + b*d*x^3*Sqrt[1 + (b*x^3)/a]*Sqrt[1 + (d*x^3)/c]*Appe 
llF1[4/3, 1/2, 1/2, 7/3, -((b*x^3)/a), -((d*x^3)/c)]) - 3*d*x^4*(4*(a + b* 
x^3) - b*x^3*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)])*(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)]))/(6*c*(b*c - a*d)*Sqrt[a + b*x^3]*Sqrt[c + d*x^3]*(-8*a*c*A 
ppellF1[1/3, 1/2, 1/2, 4/3, -((b*x^3)/a), -((d*x^3)/c)] + 3*x^3*(a*d*Appel 
lF1[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)])))
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 86, 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.

\(\displaystyle \int \frac {1}{\sqrt {a+b x^3} \left (c+d x^3\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 937

\(\displaystyle \frac {\sqrt {\frac {b x^3}{a}+1} \int \frac {1}{\sqrt {\frac {b x^3}{a}+1} \left (d x^3+c\right )^{3/2}}dx}{\sqrt {a+b x^3}}\)

\(\Big \downarrow \) 937

\(\displaystyle \frac {\sqrt {\frac {b x^3}{a}+1} \sqrt {\frac {d x^3}{c}+1} \int \frac {1}{\sqrt {\frac {b x^3}{a}+1} \left (\frac {d x^3}{c}+1\right )^{3/2}}dx}{c \sqrt {a+b x^3} \sqrt {c+d x^3}}\)

\(\Big \downarrow \) 936

\(\displaystyle \frac {x \sqrt {\frac {b x^3}{a}+1} \sqrt {\frac {d x^3}{c}+1} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},\frac {3}{2},\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{c \sqrt {a+b x^3} \sqrt {c+d x^3}}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 936

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])
 

rule 937

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])
 

Maple [F]

\[\int \frac {1}{\sqrt {b \,x^{3}+a}\, \left (d \,x^{3}+c \right )^{\frac {3}{2}}}d x\]

Input:

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

Output:

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

Fricas [F]

\[ \int \frac {1}{\sqrt {a+b x^3} \left (c+d x^3\right )^{3/2}} \, dx=\int { \frac {1}{\sqrt {b x^{3} + a} {\left (d x^{3} + c\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(b*x^3+a)^(1/2)/(d*x^3+c)^(3/2),x, algorithm="fricas")
 

Output:

integral(sqrt(b*x^3 + a)*sqrt(d*x^3 + c)/(b*d^2*x^9 + (2*b*c*d + a*d^2)*x^ 
6 + (b*c^2 + 2*a*c*d)*x^3 + a*c^2), x)
 

Sympy [F]

\[ \int \frac {1}{\sqrt {a+b x^3} \left (c+d x^3\right )^{3/2}} \, dx=\int \frac {1}{\sqrt {a + b x^{3}} \left (c + d x^{3}\right )^{\frac {3}{2}}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {1}{\sqrt {a+b x^3} \left (c+d x^3\right )^{3/2}} \, dx=\int { \frac {1}{\sqrt {b x^{3} + a} {\left (d x^{3} + c\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(b*x^3+a)^(1/2)/(d*x^3+c)^(3/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int \frac {1}{\sqrt {a+b x^3} \left (c+d x^3\right )^{3/2}} \, dx=\int { \frac {1}{\sqrt {b x^{3} + a} {\left (d x^{3} + c\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(b*x^3+a)^(1/2)/(d*x^3+c)^(3/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {a+b x^3} \left (c+d x^3\right )^{3/2}} \, dx=\int \frac {1}{\sqrt {b\,x^3+a}\,{\left (d\,x^3+c\right )}^{3/2}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {1}{\sqrt {a+b x^3} \left (c+d x^3\right )^{3/2}} \, dx=\int \frac {\sqrt {d \,x^{3}+c}\, \sqrt {b \,x^{3}+a}}{b \,d^{2} x^{9}+a \,d^{2} x^{6}+2 b c d \,x^{6}+2 a c d \,x^{3}+b \,c^{2} x^{3}+a \,c^{2}}d x \] Input:

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

Output:

int((sqrt(c + d*x**3)*sqrt(a + b*x**3))/(a*c**2 + 2*a*c*d*x**3 + a*d**2*x* 
*6 + b*c**2*x**3 + 2*b*c*d*x**6 + b*d**2*x**9),x)