\(\int \frac {(c+d x^3)^{3/2}}{\sqrt {a+b x^3}} \, dx\) [58]

Optimal result

Integrand size = 23, antiderivative size = 84 \[ \int \frac {\left (c+d x^3\right )^{3/2}}{\sqrt {a+b x^3}} \, dx=\frac {c x \sqrt {1+\frac {b x^3}{a}} \sqrt {c+d x^3} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},-\frac {3}{2},\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{\sqrt {a+b x^3} \sqrt {1+\frac {d x^3}{c}}} \] Output:

c*x*(1+b*x^3/a)^(1/2)*(d*x^3+c)^(1/2)*AppellF1(1/3,1/2,-3/2,4/3,-b*x^3/a,- 
d*x^3/c)/(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. \(374\) vs. \(2(84)=168\).

Time = 3.32 (sec) , antiderivative size = 374, normalized size of antiderivative = 4.45 \[ \int \frac {\left (c+d x^3\right )^{3/2}}{\sqrt {a+b x^3}} \, dx=\frac {x \left (-d (-11 b c+5 a d) x^3 \sqrt {1+\frac {b x^3}{a}} \sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},\frac {1}{2},\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+\frac {8 \left (-8 a c \left (a d^2 x^3+b \left (4 c^2+c d x^3+d^2 x^6\right )\right ) \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},\frac {1}{2},\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+3 d x^3 \left (a+b x^3\right ) \left (c+d x^3\right ) \left (a d \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},\frac {3}{2},\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+b c \operatorname {AppellF1}\left (\frac {4}{3},\frac {3}{2},\frac {1}{2},\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right )\right )}{-8 a c \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},\frac {1}{2},\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+3 x^3 \left (a d \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},\frac {3}{2},\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+b c \operatorname {AppellF1}\left (\frac {4}{3},\frac {3}{2},\frac {1}{2},\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right )}\right )}{32 b \sqrt {a+b x^3} \sqrt {c+d x^3}} \] Input:

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

Output:

(x*(-(d*(-11*b*c + 5*a*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)]) + (8*(-8*a*c*(a*d^2* 
x^3 + b*(4*c^2 + c*d*x^3 + d^2*x^6))*AppellF1[1/3, 1/2, 1/2, 4/3, -((b*x^3 
)/a), -((d*x^3)/c)] + 3*d*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)]))))/(32*b*Sqrt[a + b*x^3]*Sqrt[c + d*x^3])
 

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 84, 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.

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

\(\Big \downarrow \) 937

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

\(\Big \downarrow \) 937

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

\(\Big \downarrow \) 936

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

Input:

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

Output:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (c+d x^3\right )^{3/2}}{\sqrt {a+b x^3}} \, dx=\frac {2 \sqrt {d \,x^{3}+c}\, \sqrt {b \,x^{3}+a}\, d x -5 \left (\int \frac {\sqrt {d \,x^{3}+c}\, \sqrt {b \,x^{3}+a}\, x^{3}}{b d \,x^{6}+a d \,x^{3}+b c \,x^{3}+a c}d x \right ) a \,d^{2}+11 \left (\int \frac {\sqrt {d \,x^{3}+c}\, \sqrt {b \,x^{3}+a}\, x^{3}}{b d \,x^{6}+a d \,x^{3}+b c \,x^{3}+a c}d x \right ) b c d -2 \left (\int \frac {\sqrt {d \,x^{3}+c}\, \sqrt {b \,x^{3}+a}}{b d \,x^{6}+a d \,x^{3}+b c \,x^{3}+a c}d x \right ) a c d +8 \left (\int \frac {\sqrt {d \,x^{3}+c}\, \sqrt {b \,x^{3}+a}}{b d \,x^{6}+a d \,x^{3}+b c \,x^{3}+a c}d x \right ) b \,c^{2}}{8 b} \] Input:

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

Output:

(2*sqrt(c + d*x**3)*sqrt(a + b*x**3)*d*x - 5*int((sqrt(c + d*x**3)*sqrt(a 
+ b*x**3)*x**3)/(a*c + a*d*x**3 + b*c*x**3 + b*d*x**6),x)*a*d**2 + 11*int( 
(sqrt(c + d*x**3)*sqrt(a + b*x**3)*x**3)/(a*c + a*d*x**3 + b*c*x**3 + b*d* 
x**6),x)*b*c*d - 2*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)*a*c*d + 8*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)*b*c**2)/(8*b)