\(\int \frac {\sqrt {c+d x^3}}{(a+b x^3)^{3/2}} \, dx\) [64]

Optimal result

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

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

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(256\) vs. \(2(86)=172\).

Time = 3.23 (sec) , antiderivative size = 256, normalized size of antiderivative = 2.98 \[ \int \frac {\sqrt {c+d x^3}}{\left (a+b x^3\right )^{3/2}} \, dx=\frac {x \left (\frac {4 \left (c+d x^3\right )}{a}-\frac {d x^3 \sqrt {1+\frac {b x^3}{a}} \sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},\frac {1}{2},\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{a}+\frac {16 c^2 \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},\frac {1}{2},\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{8 a c \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},\frac {1}{2},\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )-3 x^3 \left (a d \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},\frac {3}{2},\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+b c \operatorname {AppellF1}\left (\frac {4}{3},\frac {3}{2},\frac {1}{2},\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right )}\right )}{6 \sqrt {a+b x^3} \sqrt {c+d x^3}} \] Input:

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

Output:

(x*((4*(c + d*x^3))/a - (d*x^3*Sqrt[1 + (b*x^3)/a]*Sqrt[1 + (d*x^3)/c]*App 
ellF1[4/3, 1/2, 1/2, 7/3, -((b*x^3)/a), -((d*x^3)/c)])/a + (16*c^2*AppellF 
1[1/3, 1/2, 1/2, 4/3, -((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)]))))/(6*Sqrt[a + b*x^3]*Sqrt[c + d*x^3])
 

Rubi [A] (verified)

Time = 0.35 (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.

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

\(\Big \downarrow \) 937

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

\(\Big \downarrow \) 937

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

\(\Big \downarrow \) 936

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

Input:

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

Output:

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

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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