\(\int \frac {(c x)^m (d+e x^n+f x^{2 n}+g x^{3 n})}{a+b x^n} \, dx\) [28]

Optimal result

Integrand size = 36, antiderivative size = 168 \[ \int \frac {(c x)^m \left (d+e x^n+f x^{2 n}+g x^{3 n}\right )}{a+b x^n} \, dx=\frac {\left (b^2 e-a b f+a^2 g\right ) (c x)^{1+m}}{b^3 c (1+m)}+\frac {(b f-a g) x^n (c x)^{1+m}}{b^2 c (1+m+n)}+\frac {g x^{2 n} (c x)^{1+m}}{b c (1+m+2 n)}+\frac {\left (b^3 d-a b^2 e+a^2 b f-a^3 g\right ) (c x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {b x^n}{a}\right )}{a b^3 c (1+m)} \] Output:

(a^2*g-a*b*f+b^2*e)*(c*x)^(1+m)/b^3/c/(1+m)+(-a*g+b*f)*x^n*(c*x)^(1+m)/b^2 
/c/(1+m+n)+g*x^(2*n)*(c*x)^(1+m)/b/c/(1+m+2*n)+(-a^3*g+a^2*b*f-a*b^2*e+b^3 
*d)*(c*x)^(1+m)*hypergeom([1, (1+m)/n],[(1+m+n)/n],-b*x^n/a)/a/b^3/c/(1+m)
 

Mathematica [A] (verified)

Time = 0.40 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.77 \[ \int \frac {(c x)^m \left (d+e x^n+f x^{2 n}+g x^{3 n}\right )}{a+b x^n} \, dx=\frac {x (c x)^m \left (\frac {b^2 e-a b f+a^2 g}{1+m}+\frac {b (b f-a g) x^n}{1+m+n}+\frac {b^2 g x^{2 n}}{1+m+2 n}+\frac {\left (b^3 d-a b^2 e+a^2 b f-a^3 g\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {b x^n}{a}\right )}{a (1+m)}\right )}{b^3} \] Input:

Integrate[((c*x)^m*(d + e*x^n + f*x^(2*n) + g*x^(3*n)))/(a + b*x^n),x]
 

Output:

(x*(c*x)^m*((b^2*e - a*b*f + a^2*g)/(1 + m) + (b*(b*f - a*g)*x^n)/(1 + m + 
 n) + (b^2*g*x^(2*n))/(1 + m + 2*n) + ((b^3*d - a*b^2*e + a^2*b*f - a^3*g) 
*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)])/(a*(1 + m)) 
))/b^3
 

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.96, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {2383, 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.

\(\displaystyle \int \frac {(c x)^m \left (d+e x^n+f x^{2 n}+g x^{3 n}\right )}{a+b x^n} \, dx\)

\(\Big \downarrow \) 2383

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {(c x)^{m+1} \left (a^2 g-a b f+b^2 e\right )}{b^3 c (m+1)}+\frac {(c x)^{m+1} \left (a^3 (-g)+a^2 b f-a b^2 e+b^3 d\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{n},\frac {m+n+1}{n},-\frac {b x^n}{a}\right )}{a b^3 c (m+1)}+\frac {x^{n+1} (c x)^m (b f-a g)}{b^2 (m+n+1)}+\frac {g x^{2 n+1} (c x)^m}{b (m+2 n+1)}\)

Input:

Int[((c*x)^m*(d + e*x^n + f*x^(2*n) + g*x^(3*n)))/(a + b*x^n),x]
 

Output:

((b*f - a*g)*x^(1 + n)*(c*x)^m)/(b^2*(1 + m + n)) + (g*x^(1 + 2*n)*(c*x)^m 
)/(b*(1 + m + 2*n)) + ((b^2*e - a*b*f + a^2*g)*(c*x)^(1 + m))/(b^3*c*(1 + 
m)) + ((b^3*d - a*b^2*e + a^2*b*f - a^3*g)*(c*x)^(1 + m)*Hypergeometric2F1 
[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)])/(a*b^3*c*(1 + m))
 

Defintions of rubi rules used

rule 2009

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

rule 2383

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> I 
nt[ExpandIntegrand[(c*x)^m*Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n 
, p}, x] && (PolyQ[Pq, x] || PolyQ[Pq, x^n]) &&  !IGtQ[m, 0]
 

Maple [F]

\[\int \frac {\left (c x \right )^{m} \left (d +e \,x^{n}+f \,x^{2 n}+g \,x^{3 n}\right )}{a +b \,x^{n}}d x\]

Input:

int((c*x)^m*(d+e*x^n+f*x^(2*n)+g*x^(3*n))/(a+b*x^n),x)
 

Output:

int((c*x)^m*(d+e*x^n+f*x^(2*n)+g*x^(3*n))/(a+b*x^n),x)
 

Fricas [F]

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

integrate((c*x)^m*(d+e*x^n+f*x^(2*n)+g*x^(3*n))/(a+b*x^n),x, algorithm="fr 
icas")
 

Output:

integral((g*x^(3*n) + f*x^(2*n) + e*x^n + d)*(c*x)^m/(b*x^n + a), x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 27.66 (sec) , antiderivative size = 860, normalized size of antiderivative = 5.12 \[ \int \frac {(c x)^m \left (d+e x^n+f x^{2 n}+g x^{3 n}\right )}{a+b x^n} \, dx=\text {Too large to display} \] Input:

integrate((c*x)**m*(d+e*x**n+f*x**(2*n)+g*x**(3*n))/(a+b*x**n),x)
 

Output:

a**(m/n + 1/n)*a**(-m/n - 1 - 1/n)*c**m*d*m*x**(m + 1)*lerchphi(b*x**n*exp 
_polar(I*pi)/a, 1, m/n + 1/n)*gamma(m/n + 1/n)/(n**2*gamma(m/n + 1 + 1/n)) 
 + a**(m/n + 1/n)*a**(-m/n - 1 - 1/n)*c**m*d*x**(m + 1)*lerchphi(b*x**n*ex 
p_polar(I*pi)/a, 1, m/n + 1/n)*gamma(m/n + 1/n)/(n**2*gamma(m/n + 1 + 1/n) 
) + a**(-m/n - 4 - 1/n)*a**(m/n + 3 + 1/n)*c**m*g*m*x**(m + 3*n + 1)*lerch 
phi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 3 + 1/n)*gamma(m/n + 3 + 1/n)/(n**2 
*gamma(m/n + 4 + 1/n)) + 3*a**(-m/n - 4 - 1/n)*a**(m/n + 3 + 1/n)*c**m*g*x 
**(m + 3*n + 1)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 3 + 1/n)*gamma 
(m/n + 3 + 1/n)/(n*gamma(m/n + 4 + 1/n)) + a**(-m/n - 4 - 1/n)*a**(m/n + 3 
 + 1/n)*c**m*g*x**(m + 3*n + 1)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n 
+ 3 + 1/n)*gamma(m/n + 3 + 1/n)/(n**2*gamma(m/n + 4 + 1/n)) + a**(-m/n - 3 
 - 1/n)*a**(m/n + 2 + 1/n)*c**m*f*m*x**(m + 2*n + 1)*lerchphi(b*x**n*exp_p 
olar(I*pi)/a, 1, m/n + 2 + 1/n)*gamma(m/n + 2 + 1/n)/(n**2*gamma(m/n + 3 + 
 1/n)) + 2*a**(-m/n - 3 - 1/n)*a**(m/n + 2 + 1/n)*c**m*f*x**(m + 2*n + 1)* 
lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 2 + 1/n)*gamma(m/n + 2 + 1/n)/ 
(n*gamma(m/n + 3 + 1/n)) + a**(-m/n - 3 - 1/n)*a**(m/n + 2 + 1/n)*c**m*f*x 
**(m + 2*n + 1)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 2 + 1/n)*gamma 
(m/n + 2 + 1/n)/(n**2*gamma(m/n + 3 + 1/n)) + a**(-m/n - 2 - 1/n)*a**(m/n 
+ 1 + 1/n)*c**m*e*m*x**(m + n + 1)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m 
/n + 1 + 1/n)*gamma(m/n + 1 + 1/n)/(n**2*gamma(m/n + 2 + 1/n)) + a**(-m...
 

Maxima [F]

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

integrate((c*x)^m*(d+e*x^n+f*x^(2*n)+g*x^(3*n))/(a+b*x^n),x, algorithm="ma 
xima")
 

Output:

(b^3*c^m*d - a*b^2*c^m*e + a^2*b*c^m*f - a^3*c^m*g)*integrate(x^m/(b^4*x^n 
 + a*b^3), x) + ((m^2 + m*(n + 2) + n + 1)*b^2*c^m*g*x*e^(m*log(x) + 2*n*l 
og(x)) + ((m^2 + m*(3*n + 2) + 2*n^2 + 3*n + 1)*b^2*c^m*e - (m^2 + m*(3*n 
+ 2) + 2*n^2 + 3*n + 1)*a*b*c^m*f + (m^2 + m*(3*n + 2) + 2*n^2 + 3*n + 1)* 
a^2*c^m*g)*x*x^m + ((m^2 + 2*m*(n + 1) + 2*n + 1)*b^2*c^m*f - (m^2 + 2*m*( 
n + 1) + 2*n + 1)*a*b*c^m*g)*x*e^(m*log(x) + n*log(x)))/((m^3 + 3*m^2*(n + 
 1) + (2*n^2 + 6*n + 3)*m + 2*n^2 + 3*n + 1)*b^3)
 

Giac [F]

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

integrate((c*x)^m*(d+e*x^n+f*x^(2*n)+g*x^(3*n))/(a+b*x^n),x, algorithm="gi 
ac")
 

Output:

integrate((g*x^(3*n) + f*x^(2*n) + e*x^n + d)*(c*x)^m/(b*x^n + a), x)
 

Mupad [F(-1)]

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

int(((c*x)^m*(d + e*x^n + f*x^(2*n) + g*x^(3*n)))/(a + b*x^n),x)
 

Output:

int(((c*x)^m*(d + e*x^n + f*x^(2*n) + g*x^(3*n)))/(a + b*x^n), x)
 

Reduce [F]

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

int((c*x)^m*(d+e*x^n+f*x^(2*n)+g*x^(3*n))/(a+b*x^n),x)
 

Output:

(c**m*(x**(m + 2*n)*b**2*g*m**2*x + x**(m + 2*n)*b**2*g*m*n*x + 2*x**(m + 
2*n)*b**2*g*m*x + x**(m + 2*n)*b**2*g*n*x + x**(m + 2*n)*b**2*g*x - x**(m 
+ n)*a*b*g*m**2*x - 2*x**(m + n)*a*b*g*m*n*x - 2*x**(m + n)*a*b*g*m*x - 2* 
x**(m + n)*a*b*g*n*x - x**(m + n)*a*b*g*x + x**(m + n)*b**2*f*m**2*x + 2*x 
**(m + n)*b**2*f*m*n*x + 2*x**(m + n)*b**2*f*m*x + 2*x**(m + n)*b**2*f*n*x 
 + x**(m + n)*b**2*f*x + x**m*a**2*g*m**2*x + 3*x**m*a**2*g*m*n*x + 2*x**m 
*a**2*g*m*x + 2*x**m*a**2*g*n**2*x + 3*x**m*a**2*g*n*x + x**m*a**2*g*x - x 
**m*a*b*f*m**2*x - 3*x**m*a*b*f*m*n*x - 2*x**m*a*b*f*m*x - 2*x**m*a*b*f*n* 
*2*x - 3*x**m*a*b*f*n*x - x**m*a*b*f*x + x**m*b**2*e*m**2*x + 3*x**m*b**2* 
e*m*n*x + 2*x**m*b**2*e*m*x + 2*x**m*b**2*e*n**2*x + 3*x**m*b**2*e*n*x + x 
**m*b**2*e*x - int(x**m/(x**n*b + a),x)*a**3*g*m**3 - 3*int(x**m/(x**n*b + 
 a),x)*a**3*g*m**2*n - 3*int(x**m/(x**n*b + a),x)*a**3*g*m**2 - 2*int(x**m 
/(x**n*b + a),x)*a**3*g*m*n**2 - 6*int(x**m/(x**n*b + a),x)*a**3*g*m*n - 3 
*int(x**m/(x**n*b + a),x)*a**3*g*m - 2*int(x**m/(x**n*b + a),x)*a**3*g*n** 
2 - 3*int(x**m/(x**n*b + a),x)*a**3*g*n - int(x**m/(x**n*b + a),x)*a**3*g 
+ int(x**m/(x**n*b + a),x)*a**2*b*f*m**3 + 3*int(x**m/(x**n*b + a),x)*a**2 
*b*f*m**2*n + 3*int(x**m/(x**n*b + a),x)*a**2*b*f*m**2 + 2*int(x**m/(x**n* 
b + a),x)*a**2*b*f*m*n**2 + 6*int(x**m/(x**n*b + a),x)*a**2*b*f*m*n + 3*in 
t(x**m/(x**n*b + a),x)*a**2*b*f*m + 2*int(x**m/(x**n*b + a),x)*a**2*b*f*n* 
*2 + 3*int(x**m/(x**n*b + a),x)*a**2*b*f*n + int(x**m/(x**n*b + a),x)*a...