\(\int \frac {(c+d x^3)^{5/2}}{\sqrt {a+b x^3}} \, dx\) [57]

Optimal result

Integrand size = 23, antiderivative size = 86 \[ \int \frac {\left (c+d x^3\right )^{5/2}}{\sqrt {a+b x^3}} \, dx=\frac {c^2 x \sqrt {1+\frac {b x^3}{a}} \sqrt {c+d x^3} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},-\frac {5}{2},\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{\sqrt {a+b x^3} \sqrt {1+\frac {d x^3}{c}}} \] Output:

c^2*x*(1+b*x^3/a)^(1/2)*(d*x^3+c)^(1/2)*AppellF1(1/3,1/2,-5/2,4/3,-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. \(323\) vs. \(2(86)=172\).

Time = 4.21 (sec) , antiderivative size = 323, normalized size of antiderivative = 3.76 \[ \int \frac {\left (c+d x^3\right )^{5/2}}{\sqrt {a+b x^3}} \, dx=\frac {x \left (8 d \left (a+b x^3\right ) \left (c+d x^3\right ) \left (31 b c-11 a d+8 b d x^3\right )+d \left (181 b^2 c^2-164 a b c d+55 a^2 d^2\right ) x^3 \sqrt {1+\frac {b x^3}{a}} \sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},\frac {1}{2},\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+\frac {64 a c^2 \left (56 b^2 c^2-31 a b c d+11 a^2 d^2\right ) \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},\frac {1}{2},\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{8 a c \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},\frac {1}{2},\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )-3 x^3 \left (a d \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},\frac {3}{2},\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+b c \operatorname {AppellF1}\left (\frac {4}{3},\frac {3}{2},\frac {1}{2},\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right )}\right )}{448 b^2 \sqrt {a+b x^3} \sqrt {c+d x^3}} \] Input:

Integrate[(c + d*x^3)^(5/2)/Sqrt[a + b*x^3],x]
 

Output:

(x*(8*d*(a + b*x^3)*(c + d*x^3)*(31*b*c - 11*a*d + 8*b*d*x^3) + d*(181*b^2 
*c^2 - 164*a*b*c*d + 55*a^2*d^2)*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)] + (64*a*c^2*(5 
6*b^2*c^2 - 31*a*b*c*d + 11*a^2*d^2)*AppellF1[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)]))))/ 
(448*b^2*Sqrt[a + b*x^3]*Sqrt[c + d*x^3])
 

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.

\(\displaystyle \int \frac {\left (c+d x^3\right )^{5/2}}{\sqrt {a+b x^3}} \, dx\)

\(\Big \downarrow \) 937

\(\displaystyle \frac {\sqrt {\frac {b x^3}{a}+1} \int \frac {\left (d x^3+c\right )^{5/2}}{\sqrt {\frac {b x^3}{a}+1}}dx}{\sqrt {a+b x^3}}\)

\(\Big \downarrow \) 937

\(\displaystyle \frac {c^2 \sqrt {\frac {b x^3}{a}+1} \sqrt {c+d x^3} \int \frac {\left (\frac {d x^3}{c}+1\right )^{5/2}}{\sqrt {\frac {b x^3}{a}+1}}dx}{\sqrt {a+b x^3} \sqrt {\frac {d x^3}{c}+1}}\)

\(\Big \downarrow \) 936

\(\displaystyle \frac {c^2 x \sqrt {\frac {b x^3}{a}+1} \sqrt {c+d x^3} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},-\frac {5}{2},\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{\sqrt {a+b x^3} \sqrt {\frac {d x^3}{c}+1}}\)

Input:

Int[(c + d*x^3)^(5/2)/Sqrt[a + b*x^3],x]
 

Output:

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

\[\int \frac {\left (d \,x^{3}+c \right )^{\frac {5}{2}}}{\sqrt {b \,x^{3}+a}}d x\]

Input:

int((d*x^3+c)^(5/2)/(b*x^3+a)^(1/2),x)
 

Output:

int((d*x^3+c)^(5/2)/(b*x^3+a)^(1/2),x)
 

Fricas [F]

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

integrate((d*x^3+c)^(5/2)/(b*x^3+a)^(1/2),x, algorithm="fricas")
 

Output:

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

Sympy [F]

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

integrate((d*x**3+c)**(5/2)/(b*x**3+a)**(1/2),x)
 

Output:

Integral((c + d*x**3)**(5/2)/sqrt(a + b*x**3), x)
 

Maxima [F]

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

integrate((d*x^3+c)^(5/2)/(b*x^3+a)^(1/2),x, algorithm="maxima")
 

Output:

integrate((d*x^3 + c)^(5/2)/sqrt(b*x^3 + a), x)
 

Giac [F]

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

integrate((d*x^3+c)^(5/2)/(b*x^3+a)^(1/2),x, algorithm="giac")
 

Output:

integrate((d*x^3 + c)^(5/2)/sqrt(b*x^3 + a), x)
 

Mupad [F(-1)]

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

int((c + d*x^3)^(5/2)/(a + b*x^3)^(1/2),x)
 

Output:

int((c + d*x^3)^(5/2)/(a + b*x^3)^(1/2), x)
 

Reduce [F]

\[ \int \frac {\left (c+d x^3\right )^{5/2}}{\sqrt {a+b x^3}} \, dx=\frac {-22 \sqrt {d \,x^{3}+c}\, \sqrt {b \,x^{3}+a}\, a \,d^{2} x +62 \sqrt {d \,x^{3}+c}\, \sqrt {b \,x^{3}+a}\, b c d x +16 \sqrt {d \,x^{3}+c}\, \sqrt {b \,x^{3}+a}\, b \,d^{2} x^{4}+55 \left (\int \frac {\sqrt {d \,x^{3}+c}\, \sqrt {b \,x^{3}+a}\, x^{3}}{b d \,x^{6}+a d \,x^{3}+b c \,x^{3}+a c}d x \right ) a^{2} d^{3}-164 \left (\int \frac {\sqrt {d \,x^{3}+c}\, \sqrt {b \,x^{3}+a}\, x^{3}}{b d \,x^{6}+a d \,x^{3}+b c \,x^{3}+a c}d x \right ) a b c \,d^{2}+181 \left (\int \frac {\sqrt {d \,x^{3}+c}\, \sqrt {b \,x^{3}+a}\, x^{3}}{b d \,x^{6}+a d \,x^{3}+b c \,x^{3}+a c}d x \right ) b^{2} c^{2} d +22 \left (\int \frac {\sqrt {d \,x^{3}+c}\, \sqrt {b \,x^{3}+a}}{b d \,x^{6}+a d \,x^{3}+b c \,x^{3}+a c}d x \right ) a^{2} c \,d^{2}-62 \left (\int \frac {\sqrt {d \,x^{3}+c}\, \sqrt {b \,x^{3}+a}}{b d \,x^{6}+a d \,x^{3}+b c \,x^{3}+a c}d x \right ) a b \,c^{2} d +112 \left (\int \frac {\sqrt {d \,x^{3}+c}\, \sqrt {b \,x^{3}+a}}{b d \,x^{6}+a d \,x^{3}+b c \,x^{3}+a c}d x \right ) b^{2} c^{3}}{112 b^{2}} \] Input:

int((d*x^3+c)^(5/2)/(b*x^3+a)^(1/2),x)
 

Output:

( - 22*sqrt(c + d*x**3)*sqrt(a + b*x**3)*a*d**2*x + 62*sqrt(c + d*x**3)*sq 
rt(a + b*x**3)*b*c*d*x + 16*sqrt(c + d*x**3)*sqrt(a + b*x**3)*b*d**2*x**4 
+ 55*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**2*d**3 - 164*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*b*c*d**2 + 181*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**2*c**2*d + 22*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**2*c*d**2 - 62*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*b*c**2*d + 112*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**2*c**3)/(112*b**2)