\(\int \frac {(c+d x^3)^{5/2}}{(a+b x^3)^{3/2}} \, dx\) [62]

Optimal result

Integrand size = 23, antiderivative size = 89 \[ \int \frac {\left (c+d x^3\right )^{5/2}}{\left (a+b x^3\right )^{3/2}} \, dx=\frac {c^2 x \sqrt {1+\frac {b x^3}{a}} \sqrt {c+d x^3} \operatorname {AppellF1}\left (\frac {1}{3},\frac {3}{2},-\frac {5}{2},\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{a \sqrt {a+b x^3} \sqrt {1+\frac {d x^3}{c}}} \] Output:

c^2*x*(1+b*x^3/a)^(1/2)*(d*x^3+c)^(1/2)*AppellF1(1/3,3/2,-5/2,4/3,-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. \(432\) vs. \(2(89)=178\).

Time = 5.81 (sec) , antiderivative size = 432, normalized size of antiderivative = 4.85 \[ \int \frac {\left (c+d x^3\right )^{5/2}}{\left (a+b x^3\right )^{3/2}} \, dx=\frac {-d \left (16 b^2 c^2-89 a b c d+55 a^2 d^2\right ) x^4 \sqrt {1+\frac {b x^3}{a}} \sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},\frac {1}{2},\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+\frac {64 a c x \left (11 a^2 d^3 x^3+4 b^2 c^2 \left (3 c+2 d x^3\right )+a b d^2 x^3 \left (-13 c+3 d x^3\right )\right ) \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},\frac {1}{2},\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )-24 x^4 \left (c+d x^3\right ) \left (8 b^2 c^2+11 a^2 d^2+a b d \left (-16 c+3 d x^3\right )\right ) \left (a d \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},\frac {3}{2},\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+b c \operatorname {AppellF1}\left (\frac {4}{3},\frac {3}{2},\frac {1}{2},\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right )}{8 a c \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},\frac {1}{2},\frac {4}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )-3 x^3 \left (a d \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},\frac {3}{2},\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+b c \operatorname {AppellF1}\left (\frac {4}{3},\frac {3}{2},\frac {1}{2},\frac {7}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right )}}{96 a b^2 \sqrt {a+b x^3} \sqrt {c+d x^3}} \] Input:

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

Output:

(-(d*(16*b^2*c^2 - 89*a*b*c*d + 55*a^2*d^2)*x^4*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*x*(11*a^2*d^3*x^3 + 4*b^2*c^2*(3*c + 2*d*x^3) + a*b*d^2*x^3*(-13*c 
 + 3*d*x^3))*AppellF1[1/3, 1/2, 1/2, 4/3, -((b*x^3)/a), -((d*x^3)/c)] - 24 
*x^4*(c + d*x^3)*(8*b^2*c^2 + 11*a^2*d^2 + a*b*d*(-16*c + 3*d*x^3))*(a*d*A 
ppellF1[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)])))/(96*a*b^2*Sqrt[a + b*x^3]*Sqrt[c + d*x^3])
 

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 89, 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.

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

\(\Big \downarrow \) 937

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

\(\Big \downarrow \) 937

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

\(\Big \downarrow \) 936

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

Input:

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

Output:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 16*sqrt(c + d*x**3)*sqrt(a + b*x**3)*a*c*d**2*x + 10*sqrt(c + d*x**3)* 
sqrt(a + b*x**3)*a*d**3*x**4 + 48*sqrt(c + d*x**3)*sqrt(a + b*x**3)*b*c**2 
*d*x - 2*sqrt(c + d*x**3)*sqrt(a + b*x**3)*b*c*d**2*x**4 - 275*int((sqrt(c 
 + d*x**3)*sqrt(a + b*x**3)*x**6)/(5*a**3*c*d + 5*a**3*d**2*x**3 - a**2*b* 
c**2 + 9*a**2*b*c*d*x**3 + 10*a**2*b*d**2*x**6 - 2*a*b**2*c**2*x**3 + 3*a* 
b**2*c*d*x**6 + 5*a*b**2*d**2*x**9 - b**3*c**2*x**6 - b**3*c*d*x**9),x)*a* 
*4*d**5 + 665*int((sqrt(c + d*x**3)*sqrt(a + b*x**3)*x**6)/(5*a**3*c*d + 5 
*a**3*d**2*x**3 - a**2*b*c**2 + 9*a**2*b*c*d*x**3 + 10*a**2*b*d**2*x**6 - 
2*a*b**2*c**2*x**3 + 3*a*b**2*c*d*x**6 + 5*a*b**2*d**2*x**9 - b**3*c**2*x* 
*6 - b**3*c*d*x**9),x)*a**3*b*c*d**4 - 275*int((sqrt(c + d*x**3)*sqrt(a + 
b*x**3)*x**6)/(5*a**3*c*d + 5*a**3*d**2*x**3 - a**2*b*c**2 + 9*a**2*b*c*d* 
x**3 + 10*a**2*b*d**2*x**6 - 2*a*b**2*c**2*x**3 + 3*a*b**2*c*d*x**6 + 5*a* 
b**2*d**2*x**9 - b**3*c**2*x**6 - b**3*c*d*x**9),x)*a**3*b*d**5*x**3 - 457 
*int((sqrt(c + d*x**3)*sqrt(a + b*x**3)*x**6)/(5*a**3*c*d + 5*a**3*d**2*x* 
*3 - a**2*b*c**2 + 9*a**2*b*c*d*x**3 + 10*a**2*b*d**2*x**6 - 2*a*b**2*c**2 
*x**3 + 3*a*b**2*c*d*x**6 + 5*a*b**2*d**2*x**9 - b**3*c**2*x**6 - b**3*c*d 
*x**9),x)*a**2*b**2*c**2*d**3 + 665*int((sqrt(c + d*x**3)*sqrt(a + b*x**3) 
*x**6)/(5*a**3*c*d + 5*a**3*d**2*x**3 - a**2*b*c**2 + 9*a**2*b*c*d*x**3 + 
10*a**2*b*d**2*x**6 - 2*a*b**2*c**2*x**3 + 3*a*b**2*c*d*x**6 + 5*a*b**2*d* 
*2*x**9 - b**3*c**2*x**6 - b**3*c*d*x**9),x)*a**2*b**2*c*d**4*x**3 + 67...