\(\int \frac {x^{-1+2 n} (a+b x^n)^3}{c+d x^n} \, dx\) [452]

Optimal result

Integrand size = 26, antiderivative size = 130 \[ \int \frac {x^{-1+2 n} \left (a+b x^n\right )^3}{c+d x^n} \, dx=-\frac {(b c-a d)^3 x^n}{d^4 n}+\frac {b \left (b^2 c^2-3 a b c d+3 a^2 d^2\right ) x^{2 n}}{2 d^3 n}-\frac {b^2 (b c-3 a d) x^{3 n}}{3 d^2 n}+\frac {b^3 x^{4 n}}{4 d n}+\frac {c (b c-a d)^3 \log \left (c+d x^n\right )}{d^5 n} \] Output:

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

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.03 \[ \int \frac {x^{-1+2 n} \left (a+b x^n\right )^3}{c+d x^n} \, dx=\frac {d x^n \left (12 a^3 d^3+18 a^2 b d^2 \left (-2 c+d x^n\right )+6 a b^2 d \left (6 c^2-3 c d x^n+2 d^2 x^{2 n}\right )+b^3 \left (-12 c^3+6 c^2 d x^n-4 c d^2 x^{2 n}+3 d^3 x^{3 n}\right )\right )+12 c (b c-a d)^3 \log \left (c+d x^n\right )}{12 d^5 n} \] Input:

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

Output:

(d*x^n*(12*a^3*d^3 + 18*a^2*b*d^2*(-2*c + d*x^n) + 6*a*b^2*d*(6*c^2 - 3*c* 
d*x^n + 2*d^2*x^(2*n)) + b^3*(-12*c^3 + 6*c^2*d*x^n - 4*c*d^2*x^(2*n) + 3* 
d^3*x^(3*n))) + 12*c*(b*c - a*d)^3*Log[c + d*x^n])/(12*d^5*n)
 

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.92, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {948, 86, 2009}

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.

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

\(\Big \downarrow \) 948

\(\displaystyle \frac {\int \frac {x^n \left (b x^n+a\right )^3}{d x^n+c}dx^n}{n}\)

\(\Big \downarrow \) 86

\(\displaystyle \frac {\int \left (\frac {b \left (b^2 c^2-3 a b d c+3 a^2 d^2\right ) x^n}{d^3}-\frac {b^2 (b c-3 a d) x^{2 n}}{d^2}+\frac {b^3 x^{3 n}}{d}+\frac {(a d-b c)^3}{d^4}+\frac {c (b c-a d)^3}{d^4 \left (d x^n+c\right )}\right )dx^n}{n}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {b x^{2 n} \left (3 a^2 d^2-3 a b c d+b^2 c^2\right )}{2 d^3}-\frac {b^2 x^{3 n} (b c-3 a d)}{3 d^2}+\frac {c (b c-a d)^3 \log \left (c+d x^n\right )}{d^5}-\frac {x^n (b c-a d)^3}{d^4}+\frac {b^3 x^{4 n}}{4 d}}{n}\)

Input:

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

Output:

(-(((b*c - a*d)^3*x^n)/d^4) + (b*(b^2*c^2 - 3*a*b*c*d + 3*a^2*d^2)*x^(2*n) 
)/(2*d^3) - (b^2*(b*c - 3*a*d)*x^(3*n))/(3*d^2) + (b^3*x^(4*n))/(4*d) + (c 
*(b*c - a*d)^3*Log[c + d*x^n])/d^5)/n
 

Defintions of rubi rules used

rule 86

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ 
.), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; 
 FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 
] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p 
+ 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
 

rule 948

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

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 0.98 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.45

method	result	size
norman	\(\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) {\mathrm e}^{n \ln \left (x \right )}}{d^{4} n}+\frac {b^{3} {\mathrm e}^{4 n \ln \left (x \right )}}{4 d n}+\frac {b \left (3 a^{2} d^{2}-3 a b c d +b^{2} c^{2}\right ) {\mathrm e}^{2 n \ln \left (x \right )}}{2 d^{3} n}+\frac {b^{2} \left (3 a d -c b \right ) {\mathrm e}^{3 n \ln \left (x \right )}}{3 d^{2} n}-\frac {c \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (c +d \,{\mathrm e}^{n \ln \left (x \right )}\right )}{d^{5} n}\)	\(188\)
risch	\(\frac {b^{3} x^{4 n}}{4 d n}+\frac {b^{2} x^{3 n} a}{d n}-\frac {b^{3} x^{3 n} c}{3 d^{2} n}+\frac {3 b \,x^{2 n} a^{2}}{2 d n}-\frac {3 b^{2} x^{2 n} a c}{2 d^{2} n}+\frac {b^{3} x^{2 n} c^{2}}{2 d^{3} n}+\frac {x^{n} a^{3}}{d n}-\frac {3 x^{n} a^{2} b c}{d^{2} n}+\frac {3 x^{n} a \,b^{2} c^{2}}{d^{3} n}-\frac {x^{n} b^{3} c^{3}}{d^{4} n}-\frac {c \ln \left (x^{n}+\frac {c}{d}\right ) a^{3}}{d^{2} n}+\frac {3 c^{2} \ln \left (x^{n}+\frac {c}{d}\right ) a^{2} b}{d^{3} n}-\frac {3 c^{3} \ln \left (x^{n}+\frac {c}{d}\right ) a \,b^{2}}{d^{4} n}+\frac {c^{4} \ln \left (x^{n}+\frac {c}{d}\right ) b^{3}}{d^{5} n}\)	\(264\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.36 \[ \int \frac {x^{-1+2 n} \left (a+b x^n\right )^3}{c+d x^n} \, dx=\frac {3 \, b^{3} d^{4} x^{4 \, n} - 4 \, {\left (b^{3} c d^{3} - 3 \, a b^{2} d^{4}\right )} x^{3 \, n} + 6 \, {\left (b^{3} c^{2} d^{2} - 3 \, a b^{2} c d^{3} + 3 \, a^{2} b d^{4}\right )} x^{2 \, n} - 12 \, {\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - a^{3} d^{4}\right )} x^{n} + 12 \, {\left (b^{3} c^{4} - 3 \, a b^{2} c^{3} d + 3 \, a^{2} b c^{2} d^{2} - a^{3} c d^{3}\right )} \log \left (d x^{n} + c\right )}{12 \, d^{5} n} \] Input:

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

Output:

1/12*(3*b^3*d^4*x^(4*n) - 4*(b^3*c*d^3 - 3*a*b^2*d^4)*x^(3*n) + 6*(b^3*c^2 
*d^2 - 3*a*b^2*c*d^3 + 3*a^2*b*d^4)*x^(2*n) - 12*(b^3*c^3*d - 3*a*b^2*c^2* 
d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*x^n + 12*(b^3*c^4 - 3*a*b^2*c^3*d + 3*a^2*b 
*c^2*d^2 - a^3*c*d^3)*log(d*x^n + c))/(d^5*n)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 347 vs. \(2 (114) = 228\).

Time = 5.13 (sec) , antiderivative size = 347, normalized size of antiderivative = 2.67 \[ \int \frac {x^{-1+2 n} \left (a+b x^n\right )^3}{c+d x^n} \, dx=\begin {cases} \frac {\left (a + b\right )^{3} \log {\left (x \right )}}{c} & \text {for}\: d = 0 \wedge n = 0 \\\frac {\frac {a^{3} x x^{2 n - 1}}{2 n} + \frac {a^{2} b x x^{n} x^{2 n - 1}}{n} + \frac {3 a b^{2} x x^{2 n} x^{2 n - 1}}{4 n} + \frac {b^{3} x x^{3 n} x^{2 n - 1}}{5 n}}{c} & \text {for}\: d = 0 \\\frac {\left (a + b\right )^{3} \log {\left (x \right )}}{c + d} & \text {for}\: n = 0 \\- \frac {a^{3} c \log {\left (\frac {c}{d} + x^{n} \right )}}{d^{2} n} + \frac {a^{3} x^{n}}{d n} + \frac {3 a^{2} b c^{2} \log {\left (\frac {c}{d} + x^{n} \right )}}{d^{3} n} - \frac {3 a^{2} b c x^{n}}{d^{2} n} + \frac {3 a^{2} b x^{2 n}}{2 d n} - \frac {3 a b^{2} c^{3} \log {\left (\frac {c}{d} + x^{n} \right )}}{d^{4} n} + \frac {3 a b^{2} c^{2} x^{n}}{d^{3} n} - \frac {3 a b^{2} c x^{2 n}}{2 d^{2} n} + \frac {a b^{2} x^{3 n}}{d n} + \frac {b^{3} c^{4} \log {\left (\frac {c}{d} + x^{n} \right )}}{d^{5} n} - \frac {b^{3} c^{3} x^{n}}{d^{4} n} + \frac {b^{3} c^{2} x^{2 n}}{2 d^{3} n} - \frac {b^{3} c x^{3 n}}{3 d^{2} n} + \frac {b^{3} x^{4 n}}{4 d n} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise(((a + b)**3*log(x)/c, Eq(d, 0) & Eq(n, 0)), ((a**3*x*x**(2*n - 1 
)/(2*n) + a**2*b*x*x**n*x**(2*n - 1)/n + 3*a*b**2*x*x**(2*n)*x**(2*n - 1)/ 
(4*n) + b**3*x*x**(3*n)*x**(2*n - 1)/(5*n))/c, Eq(d, 0)), ((a + b)**3*log( 
x)/(c + d), Eq(n, 0)), (-a**3*c*log(c/d + x**n)/(d**2*n) + a**3*x**n/(d*n) 
 + 3*a**2*b*c**2*log(c/d + x**n)/(d**3*n) - 3*a**2*b*c*x**n/(d**2*n) + 3*a 
**2*b*x**(2*n)/(2*d*n) - 3*a*b**2*c**3*log(c/d + x**n)/(d**4*n) + 3*a*b**2 
*c**2*x**n/(d**3*n) - 3*a*b**2*c*x**(2*n)/(2*d**2*n) + a*b**2*x**(3*n)/(d* 
n) + b**3*c**4*log(c/d + x**n)/(d**5*n) - b**3*c**3*x**n/(d**4*n) + b**3*c 
**2*x**(2*n)/(2*d**3*n) - b**3*c*x**(3*n)/(3*d**2*n) + b**3*x**(4*n)/(4*d* 
n), True))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.78 \[ \int \frac {x^{-1+2 n} \left (a+b x^n\right )^3}{c+d x^n} \, dx=a^{3} {\left (\frac {x^{n}}{d n} - \frac {c \log \left (\frac {d x^{n} + c}{d}\right )}{d^{2} n}\right )} + \frac {1}{12} \, b^{3} {\left (\frac {12 \, c^{4} \log \left (\frac {d x^{n} + c}{d}\right )}{d^{5} n} + \frac {3 \, d^{3} x^{4 \, n} - 4 \, c d^{2} x^{3 \, n} + 6 \, c^{2} d x^{2 \, n} - 12 \, c^{3} x^{n}}{d^{4} n}\right )} - \frac {1}{2} \, a b^{2} {\left (\frac {6 \, c^{3} \log \left (\frac {d x^{n} + c}{d}\right )}{d^{4} n} - \frac {2 \, d^{2} x^{3 \, n} - 3 \, c d x^{2 \, n} + 6 \, c^{2} x^{n}}{d^{3} n}\right )} + \frac {3}{2} \, a^{2} b {\left (\frac {2 \, c^{2} \log \left (\frac {d x^{n} + c}{d}\right )}{d^{3} n} + \frac {d x^{2 \, n} - 2 \, c x^{n}}{d^{2} n}\right )} \] Input:

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

Output:

a^3*(x^n/(d*n) - c*log((d*x^n + c)/d)/(d^2*n)) + 1/12*b^3*(12*c^4*log((d*x 
^n + c)/d)/(d^5*n) + (3*d^3*x^(4*n) - 4*c*d^2*x^(3*n) + 6*c^2*d*x^(2*n) - 
12*c^3*x^n)/(d^4*n)) - 1/2*a*b^2*(6*c^3*log((d*x^n + c)/d)/(d^4*n) - (2*d^ 
2*x^(3*n) - 3*c*d*x^(2*n) + 6*c^2*x^n)/(d^3*n)) + 3/2*a^2*b*(2*c^2*log((d* 
x^n + c)/d)/(d^3*n) + (d*x^(2*n) - 2*c*x^n)/(d^2*n))
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.67 \[ \int \frac {x^{-1+2 n} \left (a+b x^n\right )^3}{c+d x^n} \, dx=\frac {3 x^{4 n} b^{3} d^{4}+12 x^{3 n} a \,b^{2} d^{4}-4 x^{3 n} b^{3} c \,d^{3}+18 x^{2 n} a^{2} b \,d^{4}-18 x^{2 n} a \,b^{2} c \,d^{3}+6 x^{2 n} b^{3} c^{2} d^{2}+12 x^{n} a^{3} d^{4}-36 x^{n} a^{2} b c \,d^{3}+36 x^{n} a \,b^{2} c^{2} d^{2}-12 x^{n} b^{3} c^{3} d -12 \,\mathrm {log}\left (x^{n} d +c \right ) a^{3} c \,d^{3}+36 \,\mathrm {log}\left (x^{n} d +c \right ) a^{2} b \,c^{2} d^{2}-36 \,\mathrm {log}\left (x^{n} d +c \right ) a \,b^{2} c^{3} d +12 \,\mathrm {log}\left (x^{n} d +c \right ) b^{3} c^{4}}{12 d^{5} n} \] Input:

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

Output:

(3*x**(4*n)*b**3*d**4 + 12*x**(3*n)*a*b**2*d**4 - 4*x**(3*n)*b**3*c*d**3 + 
 18*x**(2*n)*a**2*b*d**4 - 18*x**(2*n)*a*b**2*c*d**3 + 6*x**(2*n)*b**3*c** 
2*d**2 + 12*x**n*a**3*d**4 - 36*x**n*a**2*b*c*d**3 + 36*x**n*a*b**2*c**2*d 
**2 - 12*x**n*b**3*c**3*d - 12*log(x**n*d + c)*a**3*c*d**3 + 36*log(x**n*d 
 + c)*a**2*b*c**2*d**2 - 36*log(x**n*d + c)*a*b**2*c**3*d + 12*log(x**n*d 
+ c)*b**3*c**4)/(12*d**5*n)