\(\int \frac {1}{(a+b x^3)^{7/3} (c+d x^3)^3} \, dx\) [168]

Optimal result

Integrand size = 21, antiderivative size = 442 \[ \int \frac {1}{\left (a+b x^3\right )^{7/3} \left (c+d x^3\right )^3} \, dx=\frac {b \left (9 b^2 c^2+36 a b c d-10 a^2 d^2\right ) x}{36 a c^2 (b c-a d)^3 \left (a+b x^3\right )^{4/3}}+\frac {b \left (27 b^3 c^3-135 a b^2 c^2 d-42 a^2 b c d^2+10 a^3 d^3\right ) x}{36 a^2 c^2 (b c-a d)^4 \sqrt [3]{a+b x^3}}-\frac {d x}{6 c (b c-a d) \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )^2}-\frac {5 d (3 b c-a d) x}{18 c^2 (b c-a d)^2 \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )}+\frac {d^2 \left (54 b^2 c^2-24 a b c d+5 a^2 d^2\right ) \arctan \left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{9 \sqrt {3} c^{8/3} (b c-a d)^{13/3}}+\frac {d^2 \left (54 b^2 c^2-24 a b c d+5 a^2 d^2\right ) \log \left (c+d x^3\right )}{54 c^{8/3} (b c-a d)^{13/3}}-\frac {d^2 \left (54 b^2 c^2-24 a b c d+5 a^2 d^2\right ) \log \left (\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{18 c^{8/3} (b c-a d)^{13/3}} \] Output:

1/36*b*(-10*a^2*d^2+36*a*b*c*d+9*b^2*c^2)*x/a/c^2/(-a*d+b*c)^3/(b*x^3+a)^( 
4/3)+1/36*b*(10*a^3*d^3-42*a^2*b*c*d^2-135*a*b^2*c^2*d+27*b^3*c^3)*x/a^2/c 
^2/(-a*d+b*c)^4/(b*x^3+a)^(1/3)-1/6*d*x/c/(-a*d+b*c)/(b*x^3+a)^(4/3)/(d*x^ 
3+c)^2-5/18*d*(-a*d+3*b*c)*x/c^2/(-a*d+b*c)^2/(b*x^3+a)^(4/3)/(d*x^3+c)+1/ 
27*d^2*(5*a^2*d^2-24*a*b*c*d+54*b^2*c^2)*arctan(1/3*(1+2*(-a*d+b*c)^(1/3)* 
x/c^(1/3)/(b*x^3+a)^(1/3))*3^(1/2))*3^(1/2)/c^(8/3)/(-a*d+b*c)^(13/3)+1/54 
*d^2*(5*a^2*d^2-24*a*b*c*d+54*b^2*c^2)*ln(d*x^3+c)/c^(8/3)/(-a*d+b*c)^(13/ 
3)-1/18*d^2*(5*a^2*d^2-24*a*b*c*d+54*b^2*c^2)*ln((-a*d+b*c)^(1/3)*x/c^(1/3 
)-(b*x^3+a)^(1/3))/c^(8/3)/(-a*d+b*c)^(13/3)
 

Mathematica [A] (warning: unable to verify)

Time = 15.98 (sec) , antiderivative size = 337, normalized size of antiderivative = 0.76 \[ \int \frac {1}{\left (a+b x^3\right )^{7/3} \left (c+d x^3\right )^3} \, dx=\frac {1}{36} x \left (a+b x^3\right )^{2/3} \left (-\frac {9 b^3}{a (-b c+a d)^3 \left (a+b x^3\right )^2}+\frac {27 b^3 (b c-5 a d)}{a^2 (b c-a d)^4 \left (a+b x^3\right )}-\frac {6 d^3}{c (b c-a d)^3 \left (c+d x^3\right )^2}+\frac {2 d^3 (-21 b c+5 a d)}{c^2 (b c-a d)^4 \left (c+d x^3\right )}\right )+\frac {d^2 \left (54 b^2 c^2-24 a b c d+5 a^2 d^2\right ) \left (2 \sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{b+a x^3}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{c}-\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{b+a x^3}}\right )+\log \left (c^{2/3}+\frac {(b c-a d)^{2/3} x^2}{\left (b+a x^3\right )^{2/3}}+\frac {\sqrt [3]{c} \sqrt [3]{b c-a d} x}{\sqrt [3]{b+a x^3}}\right )\right )}{54 c^{8/3} (b c-a d)^{13/3}} \] Input:

Integrate[1/((a + b*x^3)^(7/3)*(c + d*x^3)^3),x]
 

Output:

(x*(a + b*x^3)^(2/3)*((-9*b^3)/(a*(-(b*c) + a*d)^3*(a + b*x^3)^2) + (27*b^ 
3*(b*c - 5*a*d))/(a^2*(b*c - a*d)^4*(a + b*x^3)) - (6*d^3)/(c*(b*c - a*d)^ 
3*(c + d*x^3)^2) + (2*d^3*(-21*b*c + 5*a*d))/(c^2*(b*c - a*d)^4*(c + d*x^3 
))))/36 + (d^2*(54*b^2*c^2 - 24*a*b*c*d + 5*a^2*d^2)*(2*Sqrt[3]*ArcTan[(1 
+ (2*(b*c - a*d)^(1/3)*x)/(c^(1/3)*(b + a*x^3)^(1/3)))/Sqrt[3]] - 2*Log[c^ 
(1/3) - ((b*c - a*d)^(1/3)*x)/(b + a*x^3)^(1/3)] + Log[c^(2/3) + ((b*c - a 
*d)^(2/3)*x^2)/(b + a*x^3)^(2/3) + (c^(1/3)*(b*c - a*d)^(1/3)*x)/(b + a*x^ 
3)^(1/3)]))/(54*c^(8/3)*(b*c - a*d)^(13/3))
 

Rubi [A] (verified)

Time = 1.08 (sec) , antiderivative size = 466, normalized size of antiderivative = 1.05, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {931, 1024, 27, 1024, 25, 27, 1024, 27, 901}

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.

\(\displaystyle \int \frac {1}{\left (a+b x^3\right )^{7/3} \left (c+d x^3\right )^3} \, dx\)

\(\Big \downarrow \) 931

\(\displaystyle \frac {\int \frac {-9 b d x^3+6 b c-5 a d}{\left (b x^3+a\right )^{7/3} \left (d x^3+c\right )^2}dx}{6 c (b c-a d)}-\frac {d x}{6 c \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )^2 (b c-a d)}\)

\(\Big \downarrow \) 1024

\(\displaystyle \frac {\frac {b x (2 a d+3 b c)}{2 a \left (a+b x^3\right )^{4/3} \left (c+d x^3\right ) (b c-a d)}-\frac {\int -\frac {2 \left (6 b d (3 b c+2 a d) x^3+9 b^2 c^2+10 a^2 d^2-24 a b c d\right )}{\left (b x^3+a\right )^{4/3} \left (d x^3+c\right )^2}dx}{4 a (b c-a d)}}{6 c (b c-a d)}-\frac {d x}{6 c \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )^2 (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {6 b d (3 b c+2 a d) x^3+9 b^2 c^2+10 a^2 d^2-24 a b c d}{\left (b x^3+a\right )^{4/3} \left (d x^3+c\right )^2}dx}{2 a (b c-a d)}+\frac {b x (2 a d+3 b c)}{2 a \left (a+b x^3\right )^{4/3} \left (c+d x^3\right ) (b c-a d)}}{6 c (b c-a d)}-\frac {d x}{6 c \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )^2 (b c-a d)}\)

\(\Big \downarrow \) 1024

\(\displaystyle \frac {\frac {\frac {b x \left (-2 a^2 d^2-42 a b c d+9 b^2 c^2\right )}{a \sqrt [3]{a+b x^3} \left (c+d x^3\right ) (b c-a d)}-\frac {\int -\frac {d \left (3 b \left (9 b^2 c^2-42 a b d c-2 a^2 d^2\right ) x^3+a \left (9 b^2 c^2+36 a b d c-10 a^2 d^2\right )\right )}{\sqrt [3]{b x^3+a} \left (d x^3+c\right )^2}dx}{a (b c-a d)}}{2 a (b c-a d)}+\frac {b x (2 a d+3 b c)}{2 a \left (a+b x^3\right )^{4/3} \left (c+d x^3\right ) (b c-a d)}}{6 c (b c-a d)}-\frac {d x}{6 c \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )^2 (b c-a d)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\frac {\int \frac {d \left (3 b \left (9 b^2 c^2-42 a b d c-2 a^2 d^2\right ) x^3+a \left (9 b^2 c^2+36 a b d c-10 a^2 d^2\right )\right )}{\sqrt [3]{b x^3+a} \left (d x^3+c\right )^2}dx}{a (b c-a d)}+\frac {b x \left (-2 a^2 d^2-42 a b c d+9 b^2 c^2\right )}{a \sqrt [3]{a+b x^3} \left (c+d x^3\right ) (b c-a d)}}{2 a (b c-a d)}+\frac {b x (2 a d+3 b c)}{2 a \left (a+b x^3\right )^{4/3} \left (c+d x^3\right ) (b c-a d)}}{6 c (b c-a d)}-\frac {d x}{6 c \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )^2 (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {d \int \frac {3 b \left (9 b^2 c^2-42 a b d c-2 a^2 d^2\right ) x^3+a \left (9 b^2 c^2+36 a b d c-10 a^2 d^2\right )}{\sqrt [3]{b x^3+a} \left (d x^3+c\right )^2}dx}{a (b c-a d)}+\frac {b x \left (-2 a^2 d^2-42 a b c d+9 b^2 c^2\right )}{a \sqrt [3]{a+b x^3} \left (c+d x^3\right ) (b c-a d)}}{2 a (b c-a d)}+\frac {b x (2 a d+3 b c)}{2 a \left (a+b x^3\right )^{4/3} \left (c+d x^3\right ) (b c-a d)}}{6 c (b c-a d)}-\frac {d x}{6 c \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )^2 (b c-a d)}\)

\(\Big \downarrow \) 1024

\(\displaystyle \frac {\frac {\frac {d \left (\frac {\int \frac {4 a^2 d \left (54 b^2 c^2-24 a b d c+5 a^2 d^2\right )}{\sqrt [3]{b x^3+a} \left (d x^3+c\right )}dx}{3 c (b c-a d)}+\frac {x \left (a+b x^3\right )^{2/3} \left (10 a^3 d^3-42 a^2 b c d^2-135 a b^2 c^2 d+27 b^3 c^3\right )}{3 c \left (c+d x^3\right ) (b c-a d)}\right )}{a (b c-a d)}+\frac {b x \left (-2 a^2 d^2-42 a b c d+9 b^2 c^2\right )}{a \sqrt [3]{a+b x^3} \left (c+d x^3\right ) (b c-a d)}}{2 a (b c-a d)}+\frac {b x (2 a d+3 b c)}{2 a \left (a+b x^3\right )^{4/3} \left (c+d x^3\right ) (b c-a d)}}{6 c (b c-a d)}-\frac {d x}{6 c \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )^2 (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {d \left (\frac {4 a^2 d \left (5 a^2 d^2-24 a b c d+54 b^2 c^2\right ) \int \frac {1}{\sqrt [3]{b x^3+a} \left (d x^3+c\right )}dx}{3 c (b c-a d)}+\frac {x \left (a+b x^3\right )^{2/3} \left (10 a^3 d^3-42 a^2 b c d^2-135 a b^2 c^2 d+27 b^3 c^3\right )}{3 c \left (c+d x^3\right ) (b c-a d)}\right )}{a (b c-a d)}+\frac {b x \left (-2 a^2 d^2-42 a b c d+9 b^2 c^2\right )}{a \sqrt [3]{a+b x^3} \left (c+d x^3\right ) (b c-a d)}}{2 a (b c-a d)}+\frac {b x (2 a d+3 b c)}{2 a \left (a+b x^3\right )^{4/3} \left (c+d x^3\right ) (b c-a d)}}{6 c (b c-a d)}-\frac {d x}{6 c \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )^2 (b c-a d)}\)

\(\Big \downarrow \) 901

\(\displaystyle \frac {\frac {\frac {b x \left (-2 a^2 d^2-42 a b c d+9 b^2 c^2\right )}{a \sqrt [3]{a+b x^3} \left (c+d x^3\right ) (b c-a d)}+\frac {d \left (\frac {4 a^2 d \left (5 a^2 d^2-24 a b c d+54 b^2 c^2\right ) \left (\frac {\arctan \left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} c^{2/3} \sqrt [3]{b c-a d}}+\frac {\log \left (c+d x^3\right )}{6 c^{2/3} \sqrt [3]{b c-a d}}-\frac {\log \left (\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 c^{2/3} \sqrt [3]{b c-a d}}\right )}{3 c (b c-a d)}+\frac {x \left (a+b x^3\right )^{2/3} \left (10 a^3 d^3-42 a^2 b c d^2-135 a b^2 c^2 d+27 b^3 c^3\right )}{3 c \left (c+d x^3\right ) (b c-a d)}\right )}{a (b c-a d)}}{2 a (b c-a d)}+\frac {b x (2 a d+3 b c)}{2 a \left (a+b x^3\right )^{4/3} \left (c+d x^3\right ) (b c-a d)}}{6 c (b c-a d)}-\frac {d x}{6 c \left (a+b x^3\right )^{4/3} \left (c+d x^3\right )^2 (b c-a d)}\)

Input:

Int[1/((a + b*x^3)^(7/3)*(c + d*x^3)^3),x]
 

Output:

-1/6*(d*x)/(c*(b*c - a*d)*(a + b*x^3)^(4/3)*(c + d*x^3)^2) + ((b*(3*b*c + 
2*a*d)*x)/(2*a*(b*c - a*d)*(a + b*x^3)^(4/3)*(c + d*x^3)) + ((b*(9*b^2*c^2 
 - 42*a*b*c*d - 2*a^2*d^2)*x)/(a*(b*c - a*d)*(a + b*x^3)^(1/3)*(c + d*x^3) 
) + (d*(((27*b^3*c^3 - 135*a*b^2*c^2*d - 42*a^2*b*c*d^2 + 10*a^3*d^3)*x*(a 
 + b*x^3)^(2/3))/(3*c*(b*c - a*d)*(c + d*x^3)) + (4*a^2*d*(54*b^2*c^2 - 24 
*a*b*c*d + 5*a^2*d^2)*(ArcTan[(1 + (2*(b*c - a*d)^(1/3)*x)/(c^(1/3)*(a + b 
*x^3)^(1/3)))/Sqrt[3]]/(Sqrt[3]*c^(2/3)*(b*c - a*d)^(1/3)) + Log[c + d*x^3 
]/(6*c^(2/3)*(b*c - a*d)^(1/3)) - Log[((b*c - a*d)^(1/3)*x)/c^(1/3) - (a + 
 b*x^3)^(1/3)]/(2*c^(2/3)*(b*c - a*d)^(1/3))))/(3*c*(b*c - a*d))))/(a*(b*c 
 - a*d)))/(2*a*(b*c - a*d)))/(6*c*(b*c - a*d))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 901

Int[1/(((a_) + (b_.)*(x_)^3)^(1/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> Wit 
h[{q = Rt[(b*c - a*d)/c, 3]}, Simp[ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3))/S 
qrt[3]]/(Sqrt[3]*c*q), x] + (-Simp[Log[q*x - (a + b*x^3)^(1/3)]/(2*c*q), x] 
 + Simp[Log[c + d*x^3]/(6*c*q), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0]
 

rule 931

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] 
:> Simp[(-b)*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - 
 a*d))), x] + Simp[1/(a*n*(p + 1)*(b*c - a*d))   Int[(a + b*x^n)^(p + 1)*(c 
 + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, 
 x], x], x] /; FreeQ[{a, b, c, d, n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, 
-1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomialQ[a, b, 
 c, d, n, p, q, x]
 

rule 1024

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f 
_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^n)^(p + 1)*((c 
+ d*x^n)^(q + 1)/(a*n*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*n*(b*c - a*d)*( 
p + 1))   Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b 
*c - a*d)*(p + 1) + d*(b*e - a*f)*(n*(p + q + 2) + 1)*x^n, x], x], x] /; Fr 
eeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]
 

Maple [A] (verified)

Time = 1.55 (sec) , antiderivative size = 424, normalized size of antiderivative = 0.96

method	result	size
pseudoelliptic	\(-\frac {-24 \left (d^{4} \left (c +\frac {5 d \,x^{3}}{8}\right ) a^{5}-3 d^{3} \left (-\frac {5}{12} d^{2} x^{6}+\frac {5}{24} c d \,x^{3}+c^{2}\right ) b \,a^{4}-6 \left (\frac {5 d \,x^{3}}{6}+c \right ) \left (-\frac {d \,x^{3}}{8}+c \right ) d^{3} b^{2} x^{3} a^{3}-9 \left (\frac {7}{12} d^{2} x^{6}+\frac {3}{2} c d \,x^{3}+c^{2}\right ) \left (\frac {d \,x^{3}}{2}+c \right ) c d \,b^{3} a^{2}+\frac {9 c^{2} \left (d \,x^{3}+c \right )^{2} \left (-\frac {15 d \,x^{3}}{4}+c \right ) b^{4} a}{4}+\frac {27 b^{5} c^{3} x^{3} \left (d \,x^{3}+c \right )^{2}}{16}\right ) c x \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}}+a^{2} d^{2} \left (b \,x^{3}+a \right )^{\frac {4}{3}} \left (d \,x^{3}+c \right )^{2} \left (5 a^{2} d^{2}-24 a b c d +54 b^{2} c^{2}\right ) \left (-2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x -2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right )}{3 \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x}\right )+\ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {2}{3}} x^{2}-\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right )-2 \ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right )\right )}{54 \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {4}{3}} \left (d \,x^{3}+c \right )^{2} c^{3} \left (a d -b c \right )^{4} a^{2}}\)	\(424\)

Input:

int(1/(b*x^3+a)^(7/3)/(d*x^3+c)^3,x,method=_RETURNVERBOSE)
 

Output:

-1/54/((a*d-b*c)/c)^(1/3)*(-24*(d^4*(c+5/8*d*x^3)*a^5-3*d^3*(-5/12*d^2*x^6 
+5/24*c*d*x^3+c^2)*b*a^4-6*(5/6*d*x^3+c)*(-1/8*d*x^3+c)*d^3*b^2*x^3*a^3-9* 
(7/12*d^2*x^6+3/2*c*d*x^3+c^2)*(1/2*d*x^3+c)*c*d*b^3*a^2+9/4*c^2*(d*x^3+c) 
^2*(-15/4*d*x^3+c)*b^4*a+27/16*b^5*c^3*x^3*(d*x^3+c)^2)*c*x*((a*d-b*c)/c)^ 
(1/3)+a^2*d^2*(b*x^3+a)^(4/3)*(d*x^3+c)^2*(5*a^2*d^2-24*a*b*c*d+54*b^2*c^2 
)*(-2*3^(1/2)*arctan(1/3*3^(1/2)*(((a*d-b*c)/c)^(1/3)*x-2*(b*x^3+a)^(1/3)) 
/((a*d-b*c)/c)^(1/3)/x)+ln((((a*d-b*c)/c)^(2/3)*x^2-((a*d-b*c)/c)^(1/3)*(b 
*x^3+a)^(1/3)*x+(b*x^3+a)^(2/3))/x^2)-2*ln((((a*d-b*c)/c)^(1/3)*x+(b*x^3+a 
)^(1/3))/x)))/(b*x^3+a)^(4/3)/(d*x^3+c)^2/c^3/(a*d-b*c)^4/a^2
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b x^3\right )^{7/3} \left (c+d x^3\right )^3} \, dx=\text {Timed out} \] Input:

integrate(1/(b*x^3+a)^(7/3)/(d*x^3+c)^3,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {1}{\left (a+b x^3\right )^{7/3} \left (c+d x^3\right )^3} \, dx=\int \frac {1}{\left (a + b x^{3}\right )^{\frac {7}{3}} \left (c + d x^{3}\right )^{3}}\, dx \] Input:

integrate(1/(b*x**3+a)**(7/3)/(d*x**3+c)**3,x)
 

Output:

Integral(1/((a + b*x**3)**(7/3)*(c + d*x**3)**3), x)
 

Maxima [F]

\[ \int \frac {1}{\left (a+b x^3\right )^{7/3} \left (c+d x^3\right )^3} \, dx=\int { \frac {1}{{\left (b x^{3} + a\right )}^{\frac {7}{3}} {\left (d x^{3} + c\right )}^{3}} \,d x } \] Input:

integrate(1/(b*x^3+a)^(7/3)/(d*x^3+c)^3,x, algorithm="maxima")
 

Output:

integrate(1/((b*x^3 + a)^(7/3)*(d*x^3 + c)^3), x)
 

Giac [F]

\[ \int \frac {1}{\left (a+b x^3\right )^{7/3} \left (c+d x^3\right )^3} \, dx=\int { \frac {1}{{\left (b x^{3} + a\right )}^{\frac {7}{3}} {\left (d x^{3} + c\right )}^{3}} \,d x } \] Input:

integrate(1/(b*x^3+a)^(7/3)/(d*x^3+c)^3,x, algorithm="giac")
 

Output:

integrate(1/((b*x^3 + a)^(7/3)*(d*x^3 + c)^3), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b x^3\right )^{7/3} \left (c+d x^3\right )^3} \, dx=\int \frac {1}{{\left (b\,x^3+a\right )}^{7/3}\,{\left (d\,x^3+c\right )}^3} \,d x \] Input:

int(1/((a + b*x^3)^(7/3)*(c + d*x^3)^3),x)
 

Output:

int(1/((a + b*x^3)^(7/3)*(c + d*x^3)^3), x)
 

Reduce [F]

\[ \int \frac {1}{\left (a+b x^3\right )^{7/3} \left (c+d x^3\right )^3} \, dx=\int \frac {1}{\left (b \,x^{3}+a \right )^{\frac {1}{3}} a^{2} c^{3}+3 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a^{2} c^{2} d \,x^{3}+3 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a^{2} c \,d^{2} x^{6}+\left (b \,x^{3}+a \right )^{\frac {1}{3}} a^{2} d^{3} x^{9}+2 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a b \,c^{3} x^{3}+6 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a b \,c^{2} d \,x^{6}+6 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a b c \,d^{2} x^{9}+2 \left (b \,x^{3}+a \right )^{\frac {1}{3}} a b \,d^{3} x^{12}+\left (b \,x^{3}+a \right )^{\frac {1}{3}} b^{2} c^{3} x^{6}+3 \left (b \,x^{3}+a \right )^{\frac {1}{3}} b^{2} c^{2} d \,x^{9}+3 \left (b \,x^{3}+a \right )^{\frac {1}{3}} b^{2} c \,d^{2} x^{12}+\left (b \,x^{3}+a \right )^{\frac {1}{3}} b^{2} d^{3} x^{15}}d x \] Input:

int(1/(b*x^3+a)^(7/3)/(d*x^3+c)^3,x)
 

Output:

int(1/((a + b*x**3)**(1/3)*a**2*c**3 + 3*(a + b*x**3)**(1/3)*a**2*c**2*d*x 
**3 + 3*(a + b*x**3)**(1/3)*a**2*c*d**2*x**6 + (a + b*x**3)**(1/3)*a**2*d* 
*3*x**9 + 2*(a + b*x**3)**(1/3)*a*b*c**3*x**3 + 6*(a + b*x**3)**(1/3)*a*b* 
c**2*d*x**6 + 6*(a + b*x**3)**(1/3)*a*b*c*d**2*x**9 + 2*(a + b*x**3)**(1/3 
)*a*b*d**3*x**12 + (a + b*x**3)**(1/3)*b**2*c**3*x**6 + 3*(a + b*x**3)**(1 
/3)*b**2*c**2*d*x**9 + 3*(a + b*x**3)**(1/3)*b**2*c*d**2*x**12 + (a + b*x* 
*3)**(1/3)*b**2*d**3*x**15),x)