3.3.98 \(\int \frac {(c+d x^n)^4}{a+b x^n} \, dx\) [298]

3.3.98.1 Optimal result

Integrand size = 19, antiderivative size = 310 \[ \int \frac {\left (c+d x^n\right )^4}{a+b x^n} \, dx=-\frac {d \left (a^3 d^3 \left (1+6 n+11 n^2+6 n^3\right )-b^3 c^3 \left (1+7 n+18 n^2+24 n^3\right )-a^2 b c d^2 \left (3+19 n+38 n^2+24 n^3\right )+a b^2 c^2 d \left (3+20 n+45 n^2+36 n^3\right )\right ) x}{b^4 (1+n) (1+2 n) (1+3 n)}-\frac {d \left (2 a b c d (1+3 n)^2-a^2 d^2 \left (1+5 n+6 n^2\right )-b^2 c^2 \left (1+7 n+18 n^2\right )\right ) x \left (c+d x^n\right )}{b^3 (1+n) (1+2 n) (1+3 n)}-\frac {d (a d (1+3 n)-b (c+6 c n)) x \left (c+d x^n\right )^2}{b^2 \left (1+5 n+6 n^2\right )}+\frac {d x \left (c+d x^n\right )^3}{b (1+3 n)}+\frac {(b c-a d)^4 x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )}{a b^4} \]

output

-d*(a^3*d^3*(6*n^3+11*n^2+6*n+1)-b^3*c^3*(24*n^3+18*n^2+7*n+1)-a^2*b*c*d^2 
*(24*n^3+38*n^2+19*n+3)+a*b^2*c^2*d*(36*n^3+45*n^2+20*n+3))*x/b^4/(6*n^3+1 
1*n^2+6*n+1)-d*(2*a*b*c*d*(1+3*n)^2-a^2*d^2*(6*n^2+5*n+1)-b^2*c^2*(18*n^2+ 
7*n+1))*x*(c+d*x^n)/b^3/(6*n^3+11*n^2+6*n+1)-d*(a*d*(1+3*n)-b*(6*c*n+c))*x 
*(c+d*x^n)^2/b^2/(6*n^2+5*n+1)+d*x*(c+d*x^n)^3/b/(1+3*n)+(-a*d+b*c)^4*x*hy 
pergeom([1, 1/n],[1+1/n],-b*x^n/a)/a/b^4
 

3.3.98.2 Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 5 in optimal.

Time = 3.53 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.43 \[ \int \frac {\left (c+d x^n\right )^4}{a+b x^n} \, dx=\frac {x \left (4 c^3 d x^n \Phi \left (-\frac {b x^n}{a},1,1+\frac {1}{n}\right )+6 c^2 d^2 x^{2 n} \Phi \left (-\frac {b x^n}{a},1,2+\frac {1}{n}\right )+4 c d^3 x^{3 n} \Phi \left (-\frac {b x^n}{a},1,3+\frac {1}{n}\right )+d^4 x^{4 n} \Phi \left (-\frac {b x^n}{a},1,4+\frac {1}{n}\right )+c^4 \Phi \left (-\frac {b x^n}{a},1,\frac {1}{n}\right )\right )}{a n} \]

input

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

output

(x*(4*c^3*d*x^n*HurwitzLerchPhi[-((b*x^n)/a), 1, 1 + n^(-1)] + 6*c^2*d^2*x 
^(2*n)*HurwitzLerchPhi[-((b*x^n)/a), 1, 2 + n^(-1)] + 4*c*d^3*x^(3*n)*Hurw 
itzLerchPhi[-((b*x^n)/a), 1, 3 + n^(-1)] + d^4*x^(4*n)*HurwitzLerchPhi[-(( 
b*x^n)/a), 1, 4 + n^(-1)] + c^4*HurwitzLerchPhi[-((b*x^n)/a), 1, n^(-1)])) 
/(a*n)
 

3.3.98.3 Rubi [A] (verified)

Time = 0.77 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.04, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {933, 25, 1025, 25, 1025, 25, 913, 778}

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.

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

\(\Big \downarrow \) 933

\(\displaystyle \frac {\int -\frac {\left (d x^n+c\right )^2 \left (d (a d (3 n+1)-b (6 n c+c)) x^n+c (a d-b (3 n c+c))\right )}{b x^n+a}dx}{b (3 n+1)}+\frac {d x \left (c+d x^n\right )^3}{b (3 n+1)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {d x \left (c+d x^n\right )^3}{b (3 n+1)}-\frac {\int \frac {\left (d x^n+c\right )^2 \left (d (a d (3 n+1)-b (6 n c+c)) x^n+c (a d-b (3 n c+c))\right )}{b x^n+a}dx}{b (3 n+1)}\)

\(\Big \downarrow \) 1025

\(\displaystyle \frac {d x \left (c+d x^n\right )^3}{b (3 n+1)}-\frac {\frac {\int -\frac {\left (d x^n+c\right ) \left (c \left (b^2 \left (6 n^2+5 n+1\right ) c^2-2 a b d (4 n+1) c+a^2 d^2 (3 n+1)\right )-d \left (-b^2 \left (18 n^2+7 n+1\right ) c^2+2 a b d (3 n+1)^2 c-a^2 d^2 \left (6 n^2+5 n+1\right )\right ) x^n\right )}{b x^n+a}dx}{b (2 n+1)}+\frac {d x \left (c+d x^n\right )^2 (a d (3 n+1)-b (6 c n+c))}{b (2 n+1)}}{b (3 n+1)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {d x \left (c+d x^n\right )^3}{b (3 n+1)}-\frac {\frac {d x \left (c+d x^n\right )^2 (a d (3 n+1)-b (6 c n+c))}{b (2 n+1)}-\frac {\int \frac {\left (d x^n+c\right ) \left (c \left (b^2 \left (6 n^2+5 n+1\right ) c^2-2 a b d (4 n+1) c+a^2 d^2 (3 n+1)\right )-d \left (-b^2 \left (18 n^2+7 n+1\right ) c^2+2 a b d (3 n+1)^2 c-a^2 d^2 \left (6 n^2+5 n+1\right )\right ) x^n\right )}{b x^n+a}dx}{b (2 n+1)}}{b (3 n+1)}\)

\(\Big \downarrow \) 1025

\(\displaystyle \frac {d x \left (c+d x^n\right )^3}{b (3 n+1)}-\frac {\frac {d x \left (c+d x^n\right )^2 (a d (3 n+1)-b (6 c n+c))}{b (2 n+1)}-\frac {\frac {\int -\frac {d \left (-b^3 \left (24 n^3+18 n^2+7 n+1\right ) c^3+a b^2 d \left (36 n^3+45 n^2+20 n+3\right ) c^2-a^2 b d^2 \left (24 n^3+38 n^2+19 n+3\right ) c+a^3 d^3 \left (6 n^3+11 n^2+6 n+1\right )\right ) x^n+c \left (-b^3 \left (6 n^3+11 n^2+6 n+1\right ) c^3+a b^2 d \left (26 n^2+17 n+3\right ) c^2-a^2 b d^2 \left (21 n^2+16 n+3\right ) c+a^3 d^3 \left (6 n^2+5 n+1\right )\right )}{b x^n+a}dx}{b (n+1)}-\frac {d x \left (c+d x^n\right ) \left (-a^2 d^2 \left (6 n^2+5 n+1\right )+2 a b c d (3 n+1)^2-b^2 c^2 \left (18 n^2+7 n+1\right )\right )}{b (n+1)}}{b (2 n+1)}}{b (3 n+1)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {d x \left (c+d x^n\right )^3}{b (3 n+1)}-\frac {\frac {d x \left (c+d x^n\right )^2 (a d (3 n+1)-b (6 c n+c))}{b (2 n+1)}-\frac {-\frac {\int \frac {d \left (-b^3 \left (24 n^3+18 n^2+7 n+1\right ) c^3+a b^2 d \left (36 n^3+45 n^2+20 n+3\right ) c^2-a^2 b d^2 \left (24 n^3+38 n^2+19 n+3\right ) c+a^3 d^3 \left (6 n^3+11 n^2+6 n+1\right )\right ) x^n+c \left (-b^3 \left (6 n^3+11 n^2+6 n+1\right ) c^3+a b^2 d \left (26 n^2+17 n+3\right ) c^2-a^2 b d^2 \left (21 n^2+16 n+3\right ) c+a^3 d^3 \left (6 n^2+5 n+1\right )\right )}{b x^n+a}dx}{b (n+1)}-\frac {d x \left (c+d x^n\right ) \left (-a^2 d^2 \left (6 n^2+5 n+1\right )+2 a b c d (3 n+1)^2-b^2 c^2 \left (18 n^2+7 n+1\right )\right )}{b (n+1)}}{b (2 n+1)}}{b (3 n+1)}\)

\(\Big \downarrow \) 913

\(\displaystyle \frac {d x \left (c+d x^n\right )^3}{b (3 n+1)}-\frac {\frac {d x \left (c+d x^n\right )^2 (a d (3 n+1)-b (6 c n+c))}{b (2 n+1)}-\frac {-\frac {\frac {d x \left (a^3 d^3 \left (6 n^3+11 n^2+6 n+1\right )-a^2 b c d^2 \left (24 n^3+38 n^2+19 n+3\right )+a b^2 c^2 d \left (36 n^3+45 n^2+20 n+3\right )-b^3 c^3 \left (24 n^3+18 n^2+7 n+1\right )\right )}{b}-\frac {(n+1) (2 n+1) (3 n+1) (b c-a d)^4 \int \frac {1}{b x^n+a}dx}{b}}{b (n+1)}-\frac {d x \left (c+d x^n\right ) \left (-a^2 d^2 \left (6 n^2+5 n+1\right )+2 a b c d (3 n+1)^2-b^2 c^2 \left (18 n^2+7 n+1\right )\right )}{b (n+1)}}{b (2 n+1)}}{b (3 n+1)}\)

\(\Big \downarrow \) 778

\(\displaystyle \frac {d x \left (c+d x^n\right )^3}{b (3 n+1)}-\frac {\frac {d x \left (c+d x^n\right )^2 (a d (3 n+1)-b (6 c n+c))}{b (2 n+1)}-\frac {-\frac {d x \left (c+d x^n\right ) \left (-a^2 d^2 \left (6 n^2+5 n+1\right )+2 a b c d (3 n+1)^2-b^2 c^2 \left (18 n^2+7 n+1\right )\right )}{b (n+1)}-\frac {\frac {d x \left (a^3 d^3 \left (6 n^3+11 n^2+6 n+1\right )-a^2 b c d^2 \left (24 n^3+38 n^2+19 n+3\right )+a b^2 c^2 d \left (36 n^3+45 n^2+20 n+3\right )-b^3 c^3 \left (24 n^3+18 n^2+7 n+1\right )\right )}{b}-\frac {(n+1) (2 n+1) (3 n+1) x (b c-a d)^4 \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )}{a b}}{b (n+1)}}{b (2 n+1)}}{b (3 n+1)}\)

input

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

output

(d*x*(c + d*x^n)^3)/(b*(1 + 3*n)) - ((d*(a*d*(1 + 3*n) - b*(c + 6*c*n))*x* 
(c + d*x^n)^2)/(b*(1 + 2*n)) - (-((d*(2*a*b*c*d*(1 + 3*n)^2 - a^2*d^2*(1 + 
 5*n + 6*n^2) - b^2*c^2*(1 + 7*n + 18*n^2))*x*(c + d*x^n))/(b*(1 + n))) - 
((d*(a^3*d^3*(1 + 6*n + 11*n^2 + 6*n^3) - b^3*c^3*(1 + 7*n + 18*n^2 + 24*n 
^3) - a^2*b*c*d^2*(3 + 19*n + 38*n^2 + 24*n^3) + a*b^2*c^2*d*(3 + 20*n + 4 
5*n^2 + 36*n^3))*x)/b - ((b*c - a*d)^4*(1 + n)*(1 + 2*n)*(1 + 3*n)*x*Hyper 
geometric2F1[1, n^(-1), 1 + n^(-1), -((b*x^n)/a)])/(a*b))/(b*(1 + n)))/(b* 
(1 + 2*n)))/(b*(1 + 3*n))
 

3.3.98.3.1 Defintions of rubi rules used

rule 25

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

rule 778

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F 
1[-p, 1/n, 1/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p 
, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p] || 
GtQ[a, 0])
 

rule 913

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

rule 933

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

rule 1025

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

3.3.98.4 Maple [F]

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

input

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

output

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

3.3.98.5 Fricas [F]

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

input

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

output

integral((d^4*x^(4*n) + 4*c*d^3*x^(3*n) + 6*c^2*d^2*x^(2*n) + 4*c^3*d*x^n 
+ c^4)/(b*x^n + a), x)
 

3.3.98.6 Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 3.21 (sec) , antiderivative size = 488, normalized size of antiderivative = 1.57 \[ \int \frac {\left (c+d x^n\right )^4}{a+b x^n} \, dx=\frac {a^{\frac {1}{n}} a^{-1 - \frac {1}{n}} c^{4} x \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {1}{n}\right ) \Gamma \left (\frac {1}{n}\right )}{n^{2} \Gamma \left (1 + \frac {1}{n}\right )} + \frac {4 a^{-5 - \frac {1}{n}} a^{4 + \frac {1}{n}} d^{4} x^{4 n + 1} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, 4 + \frac {1}{n}\right ) \Gamma \left (4 + \frac {1}{n}\right )}{n \Gamma \left (5 + \frac {1}{n}\right )} + \frac {a^{-5 - \frac {1}{n}} a^{4 + \frac {1}{n}} d^{4} x^{4 n + 1} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, 4 + \frac {1}{n}\right ) \Gamma \left (4 + \frac {1}{n}\right )}{n^{2} \Gamma \left (5 + \frac {1}{n}\right )} + \frac {12 a^{-4 - \frac {1}{n}} a^{3 + \frac {1}{n}} c d^{3} x^{3 n + 1} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, 3 + \frac {1}{n}\right ) \Gamma \left (3 + \frac {1}{n}\right )}{n \Gamma \left (4 + \frac {1}{n}\right )} + \frac {4 a^{-4 - \frac {1}{n}} a^{3 + \frac {1}{n}} c d^{3} x^{3 n + 1} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, 3 + \frac {1}{n}\right ) \Gamma \left (3 + \frac {1}{n}\right )}{n^{2} \Gamma \left (4 + \frac {1}{n}\right )} + \frac {12 a^{-3 - \frac {1}{n}} a^{2 + \frac {1}{n}} c^{2} d^{2} x^{2 n + 1} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, 2 + \frac {1}{n}\right ) \Gamma \left (2 + \frac {1}{n}\right )}{n \Gamma \left (3 + \frac {1}{n}\right )} + \frac {6 a^{-3 - \frac {1}{n}} a^{2 + \frac {1}{n}} c^{2} d^{2} x^{2 n + 1} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, 2 + \frac {1}{n}\right ) \Gamma \left (2 + \frac {1}{n}\right )}{n^{2} \Gamma \left (3 + \frac {1}{n}\right )} - \frac {4 a^{- \frac {1}{n}} a^{1 + \frac {1}{n}} b^{\frac {1}{n}} b^{-1 - \frac {1}{n}} c^{3} d x \Phi \left (\frac {a x^{- n} e^{i \pi }}{b}, 1, \frac {e^{i \pi }}{n}\right ) \Gamma \left (\frac {1}{n}\right )}{a n^{2} \Gamma \left (1 + \frac {1}{n}\right )} \]

input

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

output

a**(1/n)*a**(-1 - 1/n)*c**4*x*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 1/n)*g 
amma(1/n)/(n**2*gamma(1 + 1/n)) + 4*a**(-5 - 1/n)*a**(4 + 1/n)*d**4*x**(4* 
n + 1)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 4 + 1/n)*gamma(4 + 1/n)/(n*ga 
mma(5 + 1/n)) + a**(-5 - 1/n)*a**(4 + 1/n)*d**4*x**(4*n + 1)*lerchphi(b*x* 
*n*exp_polar(I*pi)/a, 1, 4 + 1/n)*gamma(4 + 1/n)/(n**2*gamma(5 + 1/n)) + 1 
2*a**(-4 - 1/n)*a**(3 + 1/n)*c*d**3*x**(3*n + 1)*lerchphi(b*x**n*exp_polar 
(I*pi)/a, 1, 3 + 1/n)*gamma(3 + 1/n)/(n*gamma(4 + 1/n)) + 4*a**(-4 - 1/n)* 
a**(3 + 1/n)*c*d**3*x**(3*n + 1)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 3 + 
 1/n)*gamma(3 + 1/n)/(n**2*gamma(4 + 1/n)) + 12*a**(-3 - 1/n)*a**(2 + 1/n) 
*c**2*d**2*x**(2*n + 1)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 2 + 1/n)*gam 
ma(2 + 1/n)/(n*gamma(3 + 1/n)) + 6*a**(-3 - 1/n)*a**(2 + 1/n)*c**2*d**2*x* 
*(2*n + 1)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 2 + 1/n)*gamma(2 + 1/n)/( 
n**2*gamma(3 + 1/n)) - 4*a**(1 + 1/n)*b**(1/n)*b**(-1 - 1/n)*c**3*d*x*lerc 
hphi(a*exp_polar(I*pi)/(b*x**n), 1, exp_polar(I*pi)/n)*gamma(1/n)/(a*a**(1 
/n)*n**2*gamma(1 + 1/n))
 

3.3.98.7 Maxima [F]

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

input

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

output

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

3.3.98.8 Giac [F]

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

input

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

output

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

3.3.98.9 Mupad [F(-1)]

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

input

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

output

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