\(\int \frac {1}{x^3 (a+b x^3)^2 (c+d x^3)^{3/2}} \, dx\) [672]

Optimal result

Integrand size = 24, antiderivative size = 67 \[ \int \frac {1}{x^3 \left (a+b x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx=-\frac {\sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (-\frac {2}{3},2,\frac {3}{2},\frac {1}{3},-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{2 a^2 c x^2 \sqrt {c+d x^3}} \] Output:

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

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(515\) vs. \(2(67)=134\).

Time = 10.72 (sec) , antiderivative size = 515, normalized size of antiderivative = 7.69 \[ \int \frac {1}{x^3 \left (a+b x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx=\frac {-b d \left (5 b^2 c^2-6 a b c d+7 a^2 d^2\right ) x^6 \sqrt {1+\frac {d x^3}{c}} \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},1,\frac {7}{3},-\frac {d x^3}{c},-\frac {b x^3}{a}\right )+\frac {a \left (32 a c \left (10 b^3 c^2 x^3 \left (3 c+d x^3\right )+3 a^3 d^2 \left (2 c+7 d x^3\right )+3 a b^2 c \left (2 c^2-13 c d x^3-4 d^2 x^6\right )+2 a^2 b d \left (-6 c^2-6 c d x^3+7 d^2 x^6\right )\right ) \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},1,\frac {4}{3},-\frac {d x^3}{c},-\frac {b x^3}{a}\right )-24 x^3 \left (5 b^3 c^2 x^3 \left (c+d x^3\right )+a^3 d^2 \left (3 c+7 d x^3\right )+3 a b^2 c \left (c^2-c d x^3-2 d^2 x^6\right )+a^2 b d \left (-6 c^2-3 c d x^3+7 d^2 x^6\right )\right ) \left (2 b c \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},2,\frac {7}{3},-\frac {d x^3}{c},-\frac {b x^3}{a}\right )+a d \operatorname {AppellF1}\left (\frac {4}{3},\frac {3}{2},1,\frac {7}{3},-\frac {d x^3}{c},-\frac {b x^3}{a}\right )\right )\right )}{\left (a+b x^3\right ) \left (-8 a c \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},1,\frac {4}{3},-\frac {d x^3}{c},-\frac {b x^3}{a}\right )+3 x^3 \left (2 b c \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},2,\frac {7}{3},-\frac {d x^3}{c},-\frac {b x^3}{a}\right )+a d \operatorname {AppellF1}\left (\frac {4}{3},\frac {3}{2},1,\frac {7}{3},-\frac {d x^3}{c},-\frac {b x^3}{a}\right )\right )\right )}}{48 a^3 c^2 (b c-a d)^2 x^2 \sqrt {c+d x^3}} \] Input:

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

Output:

(-(b*d*(5*b^2*c^2 - 6*a*b*c*d + 7*a^2*d^2)*x^6*Sqrt[1 + (d*x^3)/c]*AppellF 
1[4/3, 1/2, 1, 7/3, -((d*x^3)/c), -((b*x^3)/a)]) + (a*(32*a*c*(10*b^3*c^2* 
x^3*(3*c + d*x^3) + 3*a^3*d^2*(2*c + 7*d*x^3) + 3*a*b^2*c*(2*c^2 - 13*c*d* 
x^3 - 4*d^2*x^6) + 2*a^2*b*d*(-6*c^2 - 6*c*d*x^3 + 7*d^2*x^6))*AppellF1[1/ 
3, 1/2, 1, 4/3, -((d*x^3)/c), -((b*x^3)/a)] - 24*x^3*(5*b^3*c^2*x^3*(c + d 
*x^3) + a^3*d^2*(3*c + 7*d*x^3) + 3*a*b^2*c*(c^2 - c*d*x^3 - 2*d^2*x^6) + 
a^2*b*d*(-6*c^2 - 3*c*d*x^3 + 7*d^2*x^6))*(2*b*c*AppellF1[4/3, 1/2, 2, 7/3 
, -((d*x^3)/c), -((b*x^3)/a)] + a*d*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c 
), -((b*x^3)/a)])))/((a + b*x^3)*(-8*a*c*AppellF1[1/3, 1/2, 1, 4/3, -((d*x 
^3)/c), -((b*x^3)/a)] + 3*x^3*(2*b*c*AppellF1[4/3, 1/2, 2, 7/3, -((d*x^3)/ 
c), -((b*x^3)/a)] + a*d*AppellF1[4/3, 3/2, 1, 7/3, -((d*x^3)/c), -((b*x^3) 
/a)]))))/(48*a^3*c^2*(b*c - a*d)^2*x^2*Sqrt[c + d*x^3])
 

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1013, 1012}

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.

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

\(\Big \downarrow \) 1013

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

\(\Big \downarrow \) 1012

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 1012

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ 
))^(q_), x_Symbol] :> Simp[a^p*c^q*((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m 
+ 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; FreeQ[{a, 
 b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n 
 - 1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
 

rule 1013

Int[((e_.)*(x_))^(m_.)*((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[(e*x)^m*(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x] /; 
 FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] & 
& NeQ[m, n - 1] &&  !(IntegerQ[p] || GtQ[a, 0])
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 6.

Time = 6.11 (sec) , antiderivative size = 863, normalized size of antiderivative = 12.88

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(863\)
risch	\(\text {Expression too large to display}\)	\(1874\)
default	\(\text {Expression too large to display}\)	\(1919\)

Input:

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

Output:

-1/2/c^2/a^2*(d*x^3+c)^(1/2)/x^2-2/3*d^3*x/c^2/(a*d-b*c)^2/((x^3+c/d)*d)^( 
1/2)-1/3/(a*d-b*c)^2/a^2*b^3*x*(d*x^3+c)^(1/2)/(b*x^3+a)-2/3*I*(-1/4*d/c^2 
/a^2-1/3/c^2*d^3/(a*d-b*c)^2-1/6*b^2*d/a^2/(a*d-b*c)^2)*3^(1/2)/d*(-c*d^2) 
^(1/3)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3))*3^(1/2)* 
d/(-c*d^2)^(1/3))^(1/2)*((x-1/d*(-c*d^2)^(1/3))/(-3/2/d*(-c*d^2)^(1/3)+1/2 
*I*3^(1/2)/d*(-c*d^2)^(1/3)))^(1/2)*(-I*(x+1/2/d*(-c*d^2)^(1/3)+1/2*I*3^(1 
/2)/d*(-c*d^2)^(1/3))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2)/(d*x^3+c)^(1/2)*Elli 
pticF(1/3*3^(1/2)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3 
))*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2),(I*3^(1/2)/d*(-c*d^2)^(1/3)/(-3/2/d*(-c 
*d^2)^(1/3)+1/2*I*3^(1/2)/d*(-c*d^2)^(1/3)))^(1/2))-1/18*I*b^2/a^2/d^2*2^( 
1/2)*sum((19*a*d-10*b*c)/(a*d-b*c)^3/_alpha^2*(-c*d^2)^(1/3)*(1/2*I*d*(2*x 
+1/d*(-I*3^(1/2)*(-c*d^2)^(1/3)+(-c*d^2)^(1/3)))/(-c*d^2)^(1/3))^(1/2)*(d* 
(x-1/d*(-c*d^2)^(1/3))/(-3*(-c*d^2)^(1/3)+I*3^(1/2)*(-c*d^2)^(1/3)))^(1/2) 
*(-1/2*I*d*(2*x+1/d*(I*3^(1/2)*(-c*d^2)^(1/3)+(-c*d^2)^(1/3)))/(-c*d^2)^(1 
/3))^(1/2)/(d*x^3+c)^(1/2)*(I*(-c*d^2)^(1/3)*_alpha*3^(1/2)*d-I*3^(1/2)*(- 
c*d^2)^(2/3)+2*_alpha^2*d^2-(-c*d^2)^(1/3)*_alpha*d-(-c*d^2)^(2/3))*Ellipt 
icPi(1/3*3^(1/2)*(I*(x+1/2/d*(-c*d^2)^(1/3)-1/2*I*3^(1/2)/d*(-c*d^2)^(1/3) 
)*3^(1/2)*d/(-c*d^2)^(1/3))^(1/2),1/2*b/d*(2*I*(-c*d^2)^(1/3)*_alpha^2*3^( 
1/2)*d-I*(-c*d^2)^(2/3)*_alpha*3^(1/2)+I*3^(1/2)*c*d-3*(-c*d^2)^(2/3)*_alp 
ha-3*c*d)/(a*d-b*c),(I*3^(1/2)/d*(-c*d^2)^(1/3)/(-3/2/d*(-c*d^2)^(1/3)+...
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 2*sqrt(c + d*x**3) - 7*int(sqrt(c + d*x**3)/(a**2*c**2 + 2*a**2*c*d*x* 
*3 + a**2*d**2*x**6 + 2*a*b*c**2*x**3 + 4*a*b*c*d*x**6 + 2*a*b*d**2*x**9 + 
 b**2*c**2*x**6 + 2*b**2*c*d*x**9 + b**2*d**2*x**12),x)*a**2*c*d*x**2 - 7* 
int(sqrt(c + d*x**3)/(a**2*c**2 + 2*a**2*c*d*x**3 + a**2*d**2*x**6 + 2*a*b 
*c**2*x**3 + 4*a*b*c*d*x**6 + 2*a*b*d**2*x**9 + b**2*c**2*x**6 + 2*b**2*c* 
d*x**9 + b**2*d**2*x**12),x)*a**2*d**2*x**5 - 10*int(sqrt(c + d*x**3)/(a** 
2*c**2 + 2*a**2*c*d*x**3 + a**2*d**2*x**6 + 2*a*b*c**2*x**3 + 4*a*b*c*d*x* 
*6 + 2*a*b*d**2*x**9 + b**2*c**2*x**6 + 2*b**2*c*d*x**9 + b**2*d**2*x**12) 
,x)*a*b*c**2*x**2 - 17*int(sqrt(c + d*x**3)/(a**2*c**2 + 2*a**2*c*d*x**3 + 
 a**2*d**2*x**6 + 2*a*b*c**2*x**3 + 4*a*b*c*d*x**6 + 2*a*b*d**2*x**9 + b** 
2*c**2*x**6 + 2*b**2*c*d*x**9 + b**2*d**2*x**12),x)*a*b*c*d*x**5 - 7*int(s 
qrt(c + d*x**3)/(a**2*c**2 + 2*a**2*c*d*x**3 + a**2*d**2*x**6 + 2*a*b*c**2 
*x**3 + 4*a*b*c*d*x**6 + 2*a*b*d**2*x**9 + b**2*c**2*x**6 + 2*b**2*c*d*x** 
9 + b**2*d**2*x**12),x)*a*b*d**2*x**8 - 10*int(sqrt(c + d*x**3)/(a**2*c**2 
 + 2*a**2*c*d*x**3 + a**2*d**2*x**6 + 2*a*b*c**2*x**3 + 4*a*b*c*d*x**6 + 2 
*a*b*d**2*x**9 + b**2*c**2*x**6 + 2*b**2*c*d*x**9 + b**2*d**2*x**12),x)*b* 
*2*c**2*x**5 - 10*int(sqrt(c + d*x**3)/(a**2*c**2 + 2*a**2*c*d*x**3 + a**2 
*d**2*x**6 + 2*a*b*c**2*x**3 + 4*a*b*c*d*x**6 + 2*a*b*d**2*x**9 + b**2*c** 
2*x**6 + 2*b**2*c*d*x**9 + b**2*d**2*x**12),x)*b**2*c*d*x**8 - 13*int((sqr 
t(c + d*x**3)*x**3)/(a**2*c**2 + 2*a**2*c*d*x**3 + a**2*d**2*x**6 + 2*a...