\(\int (a+b x^n)^3 (c+d x^n)^{-4-\frac {1}{n}} \, dx\) [321]

Optimal result

Integrand size = 25, antiderivative size = 178 \[ \int \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-4-\frac {1}{n}} \, dx=\frac {x \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-3-\frac {1}{n}}}{c (1+3 n)}+\frac {3 a n x \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-2-\frac {1}{n}}}{c^2 \left (1+5 n+6 n^2\right )}+\frac {6 a^2 n^2 x \left (a+b x^n\right ) \left (c+d x^n\right )^{-1-\frac {1}{n}}}{c^3 (1+n) (1+2 n) (1+3 n)}+\frac {6 a^3 n^3 x \left (c+d x^n\right )^{-1/n}}{c^4 (1+n) (1+2 n) (1+3 n)} \]

[Out]

x*(a+b*x^n)^3*(c+d*x^n)^(-3-1/n)/c/(1+3*n)+3*a*n*x*(a+b*x^n)^2*(c+d*x^n)^(-2-1/n)/c^2/(6*n^2+5*n+1)+6*a^2*n^2*
x*(a+b*x^n)*(c+d*x^n)^(-1-1/n)/c^3/(6*n^3+11*n^2+6*n+1)+6*a^3*n^3*x/c^4/(6*n^3+11*n^2+6*n+1)/((c+d*x^n)^(1/n))

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {386, 197} \[ \int \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-4-\frac {1}{n}} \, dx=\frac {6 a^3 n^3 x \left (c+d x^n\right )^{-1/n}}{c^4 (n+1) (2 n+1) (3 n+1)}+\frac {6 a^2 n^2 x \left (a+b x^n\right ) \left (c+d x^n\right )^{-\frac {1}{n}-1}}{c^3 (n+1) (2 n+1) (3 n+1)}+\frac {3 a n x \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-\frac {1}{n}-2}}{c^2 \left (6 n^2+5 n+1\right )}+\frac {x \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-\frac {1}{n}-3}}{c (3 n+1)} \]

[In]

Int[(a + b*x^n)^3*(c + d*x^n)^(-4 - n^(-1)),x]

[Out]

(x*(a + b*x^n)^3*(c + d*x^n)^(-3 - n^(-1)))/(c*(1 + 3*n)) + (3*a*n*x*(a + b*x^n)^2*(c + d*x^n)^(-2 - n^(-1)))/
(c^2*(1 + 5*n + 6*n^2)) + (6*a^2*n^2*x*(a + b*x^n)*(c + d*x^n)^(-1 - n^(-1)))/(c^3*(1 + n)*(1 + 2*n)*(1 + 3*n)
) + (6*a^3*n^3*x)/(c^4*(1 + n)*(1 + 2*n)*(1 + 3*n)*(c + d*x^n)^n^(-1))

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 386

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

Rubi steps \begin{align*} \text {integral}& = \frac {x \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-3-\frac {1}{n}}}{c (1+3 n)}+\frac {(3 a n) \int \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-3-\frac {1}{n}} \, dx}{c (1+3 n)} \\ & = \frac {x \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-3-\frac {1}{n}}}{c (1+3 n)}+\frac {3 a n x \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-2-\frac {1}{n}}}{c^2 \left (1+5 n+6 n^2\right )}+\frac {\left (6 a^2 n^2\right ) \int \left (a+b x^n\right ) \left (c+d x^n\right )^{-2-\frac {1}{n}} \, dx}{c^2 \left (1+5 n+6 n^2\right )} \\ & = \frac {x \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-3-\frac {1}{n}}}{c (1+3 n)}+\frac {3 a n x \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-2-\frac {1}{n}}}{c^2 \left (1+5 n+6 n^2\right )}+\frac {6 a^2 n^2 x \left (a+b x^n\right ) \left (c+d x^n\right )^{-1-\frac {1}{n}}}{c^3 (1+n) \left (1+5 n+6 n^2\right )}+\frac {\left (6 a^3 n^3\right ) \int \left (c+d x^n\right )^{-1-\frac {1}{n}} \, dx}{c^3 (1+n) \left (1+5 n+6 n^2\right )} \\ & = \frac {x \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-3-\frac {1}{n}}}{c (1+3 n)}+\frac {3 a n x \left (a+b x^n\right )^2 \left (c+d x^n\right )^{-2-\frac {1}{n}}}{c^2 \left (1+5 n+6 n^2\right )}+\frac {6 a^2 n^2 x \left (a+b x^n\right ) \left (c+d x^n\right )^{-1-\frac {1}{n}}}{c^3 (1+n) \left (1+5 n+6 n^2\right )}+\frac {6 a^3 n^3 x \left (c+d x^n\right )^{-1/n}}{c^4 (1+n) \left (1+5 n+6 n^2\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.22 \[ \int \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-4-\frac {1}{n}} \, dx=\frac {x \left (c+d x^n\right )^{-3-\frac {1}{n}} \left (b^3 c^3 \left (1+3 n+2 n^2\right ) x^{3 n}+3 a b^2 c^2 (1+n) x^{2 n} \left (c+3 c n+d n x^n\right )+3 a^2 b c x^n \left (c^2 \left (1+5 n+6 n^2\right )+2 c d n (1+3 n) x^n+2 d^2 n^2 x^{2 n}\right )+a^3 \left (c^3 \left (1+6 n+11 n^2+6 n^3\right )+3 c^2 d n \left (1+5 n+6 n^2\right ) x^n+6 c d^2 n^2 (1+3 n) x^{2 n}+6 d^3 n^3 x^{3 n}\right )\right )}{c^4 (1+n) (1+2 n) (1+3 n)} \]

[In]

Integrate[(a + b*x^n)^3*(c + d*x^n)^(-4 - n^(-1)),x]

[Out]

(x*(c + d*x^n)^(-3 - n^(-1))*(b^3*c^3*(1 + 3*n + 2*n^2)*x^(3*n) + 3*a*b^2*c^2*(1 + n)*x^(2*n)*(c + 3*c*n + d*n
*x^n) + 3*a^2*b*c*x^n*(c^2*(1 + 5*n + 6*n^2) + 2*c*d*n*(1 + 3*n)*x^n + 2*d^2*n^2*x^(2*n)) + a^3*(c^3*(1 + 6*n
+ 11*n^2 + 6*n^3) + 3*c^2*d*n*(1 + 5*n + 6*n^2)*x^n + 6*c*d^2*n^2*(1 + 3*n)*x^(2*n) + 6*d^3*n^3*x^(3*n))))/(c^
4*(1 + n)*(1 + 2*n)*(1 + 3*n))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1287\) vs. \(2(174)=348\).

Time = 5.02 (sec) , antiderivative size = 1288, normalized size of antiderivative = 7.24

method	result	size
parallelrisch	\(\text {Expression too large to display}\)	\(1288\)

[In]

int((a+b*x^n)^3*(c+d*x^n)^(-4-1/n),x,method=_RETURNVERBOSE)

[Out]

(6*x*(x^n)^4*(c+d*x^n)^(-(1+4*n)/n)*a^3*d^4*n^3+2*x*(x^n)^3*(c+d*x^n)^(-(1+4*n)/n)*b^3*c^4*n^2+x*(x^n)^4*(c+d*
x^n)^(-(1+4*n)/n)*b^3*c^3*d+3*x*(x^n)^3*(c+d*x^n)^(-(1+4*n)/n)*b^3*c^4*n+3*x*(x^n)^2*(c+d*x^n)^(-(1+4*n)/n)*a*
b^2*c^4+x*x^n*(c+d*x^n)^(-(1+4*n)/n)*a^3*c^3*d+3*x*x^n*(c+d*x^n)^(-(1+4*n)/n)*a^2*b*c^4+2*x*(x^n)^4*(c+d*x^n)^
(-(1+4*n)/n)*b^3*c^3*d*n^2+24*x*(x^n)^3*(c+d*x^n)^(-(1+4*n)/n)*a^3*c*d^3*n^3+3*x*(x^n)^4*(c+d*x^n)^(-(1+4*n)/n
)*b^3*c^3*d*n+6*x*(x^n)^3*(c+d*x^n)^(-(1+4*n)/n)*a^3*c*d^3*n^2+36*x*(x^n)^2*(c+d*x^n)^(-(1+4*n)/n)*a^3*c^2*d^2
*n^3+21*x*(x^n)^2*(c+d*x^n)^(-(1+4*n)/n)*a^3*c^2*d^2*n^2+9*x*(x^n)^2*(c+d*x^n)^(-(1+4*n)/n)*a*b^2*c^4*n^2+24*x
*x^n*(c+d*x^n)^(-(1+4*n)/n)*a^3*c^3*d*n^3+3*x*(x^n)^3*(c+d*x^n)^(-(1+4*n)/n)*a*b^2*c^3*d+3*x*(x^n)^2*(c+d*x^n)
^(-(1+4*n)/n)*a^3*c^2*d^2*n+12*x*(x^n)^2*(c+d*x^n)^(-(1+4*n)/n)*a*b^2*c^4*n+26*x*x^n*(c+d*x^n)^(-(1+4*n)/n)*a^
3*c^3*d*n^2+18*x*x^n*(c+d*x^n)^(-(1+4*n)/n)*a^2*b*c^4*n^2+3*x*(x^n)^2*(c+d*x^n)^(-(1+4*n)/n)*a^2*b*c^3*d+9*x*x
^n*(c+d*x^n)^(-(1+4*n)/n)*a^3*c^3*d*n+6*x*(x^n)^4*(c+d*x^n)^(-(1+4*n)/n)*a^2*b*c*d^3*n^2+3*x*(x^n)^4*(c+d*x^n)
^(-(1+4*n)/n)*a*b^2*c^2*d^2*n^2+3*x*(x^n)^4*(c+d*x^n)^(-(1+4*n)/n)*a*b^2*c^2*d^2*n+24*x*(x^n)^3*(c+d*x^n)^(-(1
+4*n)/n)*a^2*b*c^2*d^2*n^2+12*x*(x^n)^3*(c+d*x^n)^(-(1+4*n)/n)*a*b^2*c^3*d*n^2+6*x*(x^n)^3*(c+d*x^n)^(-(1+4*n)
/n)*a^2*b*c^2*d^2*n+15*x*(x^n)^3*(c+d*x^n)^(-(1+4*n)/n)*a*b^2*c^3*d*n+36*x*(x^n)^2*(c+d*x^n)^(-(1+4*n)/n)*a^2*
b*c^3*d*n^2+21*x*(x^n)^2*(c+d*x^n)^(-(1+4*n)/n)*a^2*b*c^3*d*n+15*x*x^n*(c+d*x^n)^(-(1+4*n)/n)*a^2*b*c^4*n+x*(x
^n)^3*(c+d*x^n)^(-(1+4*n)/n)*b^3*c^4+6*x*(c+d*x^n)^(-(1+4*n)/n)*a^3*c^4*n^3+11*x*(c+d*x^n)^(-(1+4*n)/n)*a^3*c^
4*n^2+x*(c+d*x^n)^(-(1+4*n)/n)*a^3*c^4+6*x*(c+d*x^n)^(-(1+4*n)/n)*a^3*c^4*n)/c^4/(6*n^3+11*n^2+6*n+1)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 478 vs. \(2 (174) = 348\).

Time = 0.27 (sec) , antiderivative size = 478, normalized size of antiderivative = 2.69 \[ \int \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-4-\frac {1}{n}} \, dx=\frac {{\left (6 \, a^{3} d^{4} n^{3} + b^{3} c^{3} d + {\left (2 \, b^{3} c^{3} d + 3 \, a b^{2} c^{2} d^{2} + 6 \, a^{2} b c d^{3}\right )} n^{2} + 3 \, {\left (b^{3} c^{3} d + a b^{2} c^{2} d^{2}\right )} n\right )} x x^{4 \, n} + {\left (24 \, a^{3} c d^{3} n^{3} + b^{3} c^{4} + 3 \, a b^{2} c^{3} d + 2 \, {\left (b^{3} c^{4} + 6 \, a b^{2} c^{3} d + 12 \, a^{2} b c^{2} d^{2} + 3 \, a^{3} c d^{3}\right )} n^{2} + 3 \, {\left (b^{3} c^{4} + 5 \, a b^{2} c^{3} d + 2 \, a^{2} b c^{2} d^{2}\right )} n\right )} x x^{3 \, n} + 3 \, {\left (12 \, a^{3} c^{2} d^{2} n^{3} + a b^{2} c^{4} + a^{2} b c^{3} d + {\left (3 \, a b^{2} c^{4} + 12 \, a^{2} b c^{3} d + 7 \, a^{3} c^{2} d^{2}\right )} n^{2} + {\left (4 \, a b^{2} c^{4} + 7 \, a^{2} b c^{3} d + a^{3} c^{2} d^{2}\right )} n\right )} x x^{2 \, n} + {\left (24 \, a^{3} c^{3} d n^{3} + 3 \, a^{2} b c^{4} + a^{3} c^{3} d + 2 \, {\left (9 \, a^{2} b c^{4} + 13 \, a^{3} c^{3} d\right )} n^{2} + 3 \, {\left (5 \, a^{2} b c^{4} + 3 \, a^{3} c^{3} d\right )} n\right )} x x^{n} + {\left (6 \, a^{3} c^{4} n^{3} + 11 \, a^{3} c^{4} n^{2} + 6 \, a^{3} c^{4} n + a^{3} c^{4}\right )} x}{{\left (6 \, c^{4} n^{3} + 11 \, c^{4} n^{2} + 6 \, c^{4} n + c^{4}\right )} {\left (d x^{n} + c\right )}^{\frac {4 \, n + 1}{n}}} \]

[In]

integrate((a+b*x^n)^3*(c+d*x^n)^(-4-1/n),x, algorithm="fricas")

[Out]

((6*a^3*d^4*n^3 + b^3*c^3*d + (2*b^3*c^3*d + 3*a*b^2*c^2*d^2 + 6*a^2*b*c*d^3)*n^2 + 3*(b^3*c^3*d + a*b^2*c^2*d
^2)*n)*x*x^(4*n) + (24*a^3*c*d^3*n^3 + b^3*c^4 + 3*a*b^2*c^3*d + 2*(b^3*c^4 + 6*a*b^2*c^3*d + 12*a^2*b*c^2*d^2
+ 3*a^3*c*d^3)*n^2 + 3*(b^3*c^4 + 5*a*b^2*c^3*d + 2*a^2*b*c^2*d^2)*n)*x*x^(3*n) + 3*(12*a^3*c^2*d^2*n^3 + a*b
^2*c^4 + a^2*b*c^3*d + (3*a*b^2*c^4 + 12*a^2*b*c^3*d + 7*a^3*c^2*d^2)*n^2 + (4*a*b^2*c^4 + 7*a^2*b*c^3*d + a^3
*c^2*d^2)*n)*x*x^(2*n) + (24*a^3*c^3*d*n^3 + 3*a^2*b*c^4 + a^3*c^3*d + 2*(9*a^2*b*c^4 + 13*a^3*c^3*d)*n^2 + 3*
(5*a^2*b*c^4 + 3*a^3*c^3*d)*n)*x*x^n + (6*a^3*c^4*n^3 + 11*a^3*c^4*n^2 + 6*a^3*c^4*n + a^3*c^4)*x)/((6*c^4*n^3
+ 11*c^4*n^2 + 6*c^4*n + c^4)*(d*x^n + c)^((4*n + 1)/n))

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2822 vs. \(2 (160) = 320\).

Time = 29.87 (sec) , antiderivative size = 2822, normalized size of antiderivative = 15.85 \[ \int \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-4-\frac {1}{n}} \, dx=\text {Too large to display} \]

[In]

integrate((a+b*x**n)**3*(c+d*x**n)**(-4-1/n),x)

[Out]

6*a**3*c**3*c**(1/n)*c**(-4 - 1/n)*n**3*gamma(1/n)/(c**3*d**(1/n)*n**4*(c/(d*x**n) + 1)**(1/n)*gamma(4 + 1/n)
+ 3*c**2*d*d**(1/n)*n**4*x**n*(c/(d*x**n) + 1)**(1/n)*gamma(4 + 1/n) + 3*c*d**2*d**(1/n)*n**4*x**(2*n)*(c/(d*x
**n) + 1)**(1/n)*gamma(4 + 1/n) + d**3*d**(1/n)*n**4*x**(3*n)*(c/(d*x**n) + 1)**(1/n)*gamma(4 + 1/n)) + 11*a**
3*c**3*c**(1/n)*c**(-4 - 1/n)*n**2*gamma(1/n)/(c**3*d**(1/n)*n**4*(c/(d*x**n) + 1)**(1/n)*gamma(4 + 1/n) + 3*c
**2*d*d**(1/n)*n**4*x**n*(c/(d*x**n) + 1)**(1/n)*gamma(4 + 1/n) + 3*c*d**2*d**(1/n)*n**4*x**(2*n)*(c/(d*x**n)
+ 1)**(1/n)*gamma(4 + 1/n) + d**3*d**(1/n)*n**4*x**(3*n)*(c/(d*x**n) + 1)**(1/n)*gamma(4 + 1/n)) + 6*a**3*c**3
*c**(1/n)*c**(-4 - 1/n)*n*gamma(1/n)/(c**3*d**(1/n)*n**4*(c/(d*x**n) + 1)**(1/n)*gamma(4 + 1/n) + 3*c**2*d*d**
(1/n)*n**4*x**n*(c/(d*x**n) + 1)**(1/n)*gamma(4 + 1/n) + 3*c*d**2*d**(1/n)*n**4*x**(2*n)*(c/(d*x**n) + 1)**(1/
n)*gamma(4 + 1/n) + d**3*d**(1/n)*n**4*x**(3*n)*(c/(d*x**n) + 1)**(1/n)*gamma(4 + 1/n)) + a**3*c**3*c**(1/n)*c
**(-4 - 1/n)*gamma(1/n)/(c**3*d**(1/n)*n**4*(c/(d*x**n) + 1)**(1/n)*gamma(4 + 1/n) + 3*c**2*d*d**(1/n)*n**4*x*
*n*(c/(d*x**n) + 1)**(1/n)*gamma(4 + 1/n) + 3*c*d**2*d**(1/n)*n**4*x**(2*n)*(c/(d*x**n) + 1)**(1/n)*gamma(4 +
1/n) + d**3*d**(1/n)*n**4*x**(3*n)*(c/(d*x**n) + 1)**(1/n)*gamma(4 + 1/n)) + 18*a**3*c**2*c**(1/n)*c**(-4 - 1/
n)*d*n**3*x**n*gamma(1/n)/(c**3*d**(1/n)*n**4*(c/(d*x**n) + 1)**(1/n)*gamma(4 + 1/n) + 3*c**2*d*d**(1/n)*n**4*
x**n*(c/(d*x**n) + 1)**(1/n)*gamma(4 + 1/n) + 3*c*d**2*d**(1/n)*n**4*x**(2*n)*(c/(d*x**n) + 1)**(1/n)*gamma(4
+ 1/n) + d**3*d**(1/n)*n**4*x**(3*n)*(c/(d*x**n) + 1)**(1/n)*gamma(4 + 1/n)) + 15*a**3*c**2*c**(1/n)*c**(-4 -
1/n)*d*n**2*x**n*gamma(1/n)/(c**3*d**(1/n)*n**4*(c/(d*x**n) + 1)**(1/n)*gamma(4 + 1/n) + 3*c**2*d*d**(1/n)*n**
4*x**n*(c/(d*x**n) + 1)**(1/n)*gamma(4 + 1/n) + 3*c*d**2*d**(1/n)*n**4*x**(2*n)*(c/(d*x**n) + 1)**(1/n)*gamma(
4 + 1/n) + d**3*d**(1/n)*n**4*x**(3*n)*(c/(d*x**n) + 1)**(1/n)*gamma(4 + 1/n)) + 3*a**3*c**2*c**(1/n)*c**(-4 -
1/n)*d*n*x**n*gamma(1/n)/(c**3*d**(1/n)*n**4*(c/(d*x**n) + 1)**(1/n)*gamma(4 + 1/n) + 3*c**2*d*d**(1/n)*n**4*
x**n*(c/(d*x**n) + 1)**(1/n)*gamma(4 + 1/n) + 3*c*d**2*d**(1/n)*n**4*x**(2*n)*(c/(d*x**n) + 1)**(1/n)*gamma(4
+ 1/n) + d**3*d**(1/n)*n**4*x**(3*n)*(c/(d*x**n) + 1)**(1/n)*gamma(4 + 1/n)) + 18*a**3*c*c**(1/n)*c**(-4 - 1/n
)*d**2*n**3*x**(2*n)*gamma(1/n)/(c**3*d**(1/n)*n**4*(c/(d*x**n) + 1)**(1/n)*gamma(4 + 1/n) + 3*c**2*d*d**(1/n)
*n**4*x**n*(c/(d*x**n) + 1)**(1/n)*gamma(4 + 1/n) + 3*c*d**2*d**(1/n)*n**4*x**(2*n)*(c/(d*x**n) + 1)**(1/n)*ga
mma(4 + 1/n) + d**3*d**(1/n)*n**4*x**(3*n)*(c/(d*x**n) + 1)**(1/n)*gamma(4 + 1/n)) + 6*a**3*c*c**(1/n)*c**(-4
- 1/n)*d**2*n**2*x**(2*n)*gamma(1/n)/(c**3*d**(1/n)*n**4*(c/(d*x**n) + 1)**(1/n)*gamma(4 + 1/n) + 3*c**2*d*d**
(1/n)*n**4*x**n*(c/(d*x**n) + 1)**(1/n)*gamma(4 + 1/n) + 3*c*d**2*d**(1/n)*n**4*x**(2*n)*(c/(d*x**n) + 1)**(1/
n)*gamma(4 + 1/n) + d**3*d**(1/n)*n**4*x**(3*n)*(c/(d*x**n) + 1)**(1/n)*gamma(4 + 1/n)) + 6*a**3*c**(1/n)*c**(
-4 - 1/n)*d**3*n**3*x**(3*n)*gamma(1/n)/(c**3*d**(1/n)*n**4*(c/(d*x**n) + 1)**(1/n)*gamma(4 + 1/n) + 3*c**2*d*
d**(1/n)*n**4*x**n*(c/(d*x**n) + 1)**(1/n)*gamma(4 + 1/n) + 3*c*d**2*d**(1/n)*n**4*x**(2*n)*(c/(d*x**n) + 1)**
(1/n)*gamma(4 + 1/n) + d**3*d**(1/n)*n**4*x**(3*n)*(c/(d*x**n) + 1)**(1/n)*gamma(4 + 1/n)) + 18*a**2*b*c**2*c*
*(-4 - 1/n)*c**(1 + 1/n)*n**2*(c/(d*x**n) + 1)**(-1 - 1/n)*gamma(1 + 1/n)/(c**2*d**(1 + 1/n)*n**3*gamma(4 + 1/
n) + 2*c*d*d**(1 + 1/n)*n**3*x**n*gamma(4 + 1/n) + d**2*d**(1 + 1/n)*n**3*x**(2*n)*gamma(4 + 1/n)) + 15*a**2*b
*c**2*c**(-4 - 1/n)*c**(1 + 1/n)*n*(c/(d*x**n) + 1)**(-1 - 1/n)*gamma(1 + 1/n)/(c**2*d**(1 + 1/n)*n**3*gamma(4
+ 1/n) + 2*c*d*d**(1 + 1/n)*n**3*x**n*gamma(4 + 1/n) + d**2*d**(1 + 1/n)*n**3*x**(2*n)*gamma(4 + 1/n)) + 3*a*
*2*b*c**2*c**(-4 - 1/n)*c**(1 + 1/n)*(c/(d*x**n) + 1)**(-1 - 1/n)*gamma(1 + 1/n)/(c**2*d**(1 + 1/n)*n**3*gamma
(4 + 1/n) + 2*c*d*d**(1 + 1/n)*n**3*x**n*gamma(4 + 1/n) + d**2*d**(1 + 1/n)*n**3*x**(2*n)*gamma(4 + 1/n)) + 18
*a**2*b*c*c**(-4 - 1/n)*c**(1 + 1/n)*d*n**2*x**n*(c/(d*x**n) + 1)**(-1 - 1/n)*gamma(1 + 1/n)/(c**2*d**(1 + 1/n
)*n**3*gamma(4 + 1/n) + 2*c*d*d**(1 + 1/n)*n**3*x**n*gamma(4 + 1/n) + d**2*d**(1 + 1/n)*n**3*x**(2*n)*gamma(4
+ 1/n)) + 6*a**2*b*c*c**(-4 - 1/n)*c**(1 + 1/n)*d*n*x**n*(c/(d*x**n) + 1)**(-1 - 1/n)*gamma(1 + 1/n)/(c**2*d**
(1 + 1/n)*n**3*gamma(4 + 1/n) + 2*c*d*d**(1 + 1/n)*n**3*x**n*gamma(4 + 1/n) + d**2*d**(1 + 1/n)*n**3*x**(2*n)*
gamma(4 + 1/n)) + 6*a**2*b*c**(-4 - 1/n)*c**(1 + 1/n)*d**2*n**2*x**(2*n)*(c/(d*x**n) + 1)**(-1 - 1/n)*gamma(1
+ 1/n)/(c**2*d**(1 + 1/n)*n**3*gamma(4 + 1/n) + 2*c*d*d**(1 + 1/n)*n**3*x**n*gamma(4 + 1/n) + d**2*d**(1 + 1/n
)*n**3*x**(2*n)*gamma(4 + 1/n)) + 9*a*b**2*c*c**(-4 - 1/n)*c**(2 + 1/n)*n*(c/(d*x**n) + 1)**(-2 - 1/n)*gamma(2
+ 1/n)/(c*d**(2 + 1/n)*n**2*gamma(4 + 1/n) + d*d**(2 + 1/n)*n**2*x**n*gamma(4 + 1/n)) + 3*a*b**2*c*c**(-4 - 1
/n)*c**(2 + 1/n)*(c/(d*x**n) + 1)**(-2 - 1/n)*gamma(2 + 1/n)/(c*d**(2 + 1/n)*n**2*gamma(4 + 1/n) + d*d**(2 + 1
/n)*n**2*x**n*gamma(4 + 1/n)) + 3*a*b**2*c**(-4 - 1/n)*c**(2 + 1/n)*d*n*x**n*(c/(d*x**n) + 1)**(-2 - 1/n)*gamm
a(2 + 1/n)/(c*d**(2 + 1/n)*n**2*gamma(4 + 1/n) + d*d**(2 + 1/n)*n**2*x**n*gamma(4 + 1/n)) + b**3*c**(-4 - 1/n)
*c**(3 + 1/n)*d**(-3 - 1/n)*(c/(d*x**n) + 1)**(-3 - 1/n)*gamma(3 + 1/n)/(n*gamma(4 + 1/n))

Maxima [F]

\[ \int \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-4-\frac {1}{n}} \, dx=\int { {\left (b x^{n} + a\right )}^{3} {\left (d x^{n} + c\right )}^{-\frac {1}{n} - 4} \,d x } \]

[In]

integrate((a+b*x^n)^3*(c+d*x^n)^(-4-1/n),x, algorithm="maxima")

[Out]

integrate((b*x^n + a)^3*(d*x^n + c)^(-1/n - 4), x)

Giac [F(-2)]

Exception generated. \[ \int \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-4-\frac {1}{n}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((a+b*x^n)^3*(c+d*x^n)^(-4-1/n),x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Unable to divide, perhaps due to rounding error%%%{81,[2,0,6,4,2,4,3,0]%%%}+%%%{108,[2,0,6,3,2,4,3,0]%%%}+%
%%{54,[2,0,

Mupad [F(-1)]

Timed out. \[ \int \left (a+b x^n\right )^3 \left (c+d x^n\right )^{-4-\frac {1}{n}} \, dx=\int \frac {{\left (a+b\,x^n\right )}^3}{{\left (c+d\,x^n\right )}^{\frac {1}{n}+4}} \,d x \]

[In]

int((a + b*x^n)^3/(c + d*x^n)^(1/n + 4),x)

[Out]

int((a + b*x^n)^3/(c + d*x^n)^(1/n + 4), x)