\(\int \frac {x (e+f x)^n}{a+b x^3} \, dx\) [43]

Optimal result

Integrand size = 18, antiderivative size = 288 \[ \int \frac {x (e+f x)^n}{a+b x^3} \, dx=\frac {(e+f x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt [3]{b} (e+f x)}{\sqrt [3]{b} e-\sqrt [3]{a} f}\right )}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{b} e-\sqrt [3]{a} f\right ) (1+n)}-\frac {\sqrt [3]{-1} (e+f x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {(-1)^{2/3} \sqrt [3]{b} (e+f x)}{(-1)^{2/3} \sqrt [3]{b} e-\sqrt [3]{a} f}\right )}{3 \sqrt [3]{a} \sqrt [3]{b} \left ((-1)^{2/3} \sqrt [3]{b} e-\sqrt [3]{a} f\right ) (1+n)}-\frac {(-1)^{2/3} (e+f x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt [3]{-1} \sqrt [3]{b} (e+f x)}{\sqrt [3]{-1} \sqrt [3]{b} e+\sqrt [3]{a} f}\right )}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{-1} \sqrt [3]{b} e+\sqrt [3]{a} f\right ) (1+n)} \] Output:

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

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.82 \[ \int \frac {x (e+f x)^n}{a+b x^3} \, dx=\frac {(e+f x)^{1+n} \left (\frac {\operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt [3]{b} (e+f x)}{\sqrt [3]{b} e-\sqrt [3]{a} f}\right )}{\sqrt [3]{b} e-\sqrt [3]{a} f}-\frac {\sqrt [3]{-1} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {(-1)^{2/3} \sqrt [3]{b} (e+f x)}{(-1)^{2/3} \sqrt [3]{b} e-\sqrt [3]{a} f}\right )}{(-1)^{2/3} \sqrt [3]{b} e-\sqrt [3]{a} f}-\frac {(-1)^{2/3} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt [3]{-1} \sqrt [3]{b} (e+f x)}{\sqrt [3]{-1} \sqrt [3]{b} e+\sqrt [3]{a} f}\right )}{\sqrt [3]{-1} \sqrt [3]{b} e+\sqrt [3]{a} f}\right )}{3 \sqrt [3]{a} \sqrt [3]{b} (1+n)} \] Input:

Integrate[(x*(e + f*x)^n)/(a + b*x^3),x]
 

Output:

((e + f*x)^(1 + n)*(Hypergeometric2F1[1, 1 + n, 2 + n, (b^(1/3)*(e + f*x)) 
/(b^(1/3)*e - a^(1/3)*f)]/(b^(1/3)*e - a^(1/3)*f) - ((-1)^(1/3)*Hypergeome 
tric2F1[1, 1 + n, 2 + n, ((-1)^(2/3)*b^(1/3)*(e + f*x))/((-1)^(2/3)*b^(1/3 
)*e - a^(1/3)*f)])/((-1)^(2/3)*b^(1/3)*e - a^(1/3)*f) - ((-1)^(2/3)*Hyperg 
eometric2F1[1, 1 + n, 2 + n, ((-1)^(1/3)*b^(1/3)*(e + f*x))/((-1)^(1/3)*b^ 
(1/3)*e + a^(1/3)*f)])/((-1)^(1/3)*b^(1/3)*e + a^(1/3)*f)))/(3*a^(1/3)*b^( 
1/3)*(1 + n))
 

Rubi [A] (verified)

Time = 0.86 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {7276, 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.

\(\displaystyle \int \frac {x (e+f x)^n}{a+b x^3} \, dx\)

\(\Big \downarrow \) 7276

\(\displaystyle \int \left (-\frac {(e+f x)^n}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {(-1)^{2/3} (e+f x)^n}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}+\frac {\sqrt [3]{-1} (e+f x)^n}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {(e+f x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {\sqrt [3]{b} (e+f x)}{\sqrt [3]{b} e-\sqrt [3]{a} f}\right )}{3 \sqrt [3]{a} \sqrt [3]{b} (n+1) \left (\sqrt [3]{b} e-\sqrt [3]{a} f\right )}-\frac {\sqrt [3]{-1} (e+f x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {(-1)^{2/3} \sqrt [3]{b} (e+f x)}{(-1)^{2/3} \sqrt [3]{b} e-\sqrt [3]{a} f}\right )}{3 \sqrt [3]{a} \sqrt [3]{b} (n+1) \left ((-1)^{2/3} \sqrt [3]{b} e-\sqrt [3]{a} f\right )}-\frac {(-1)^{2/3} (e+f x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {\sqrt [3]{-1} \sqrt [3]{b} (e+f x)}{\sqrt [3]{-1} \sqrt [3]{b} e+\sqrt [3]{a} f}\right )}{3 \sqrt [3]{a} \sqrt [3]{b} (n+1) \left (\sqrt [3]{a} f+\sqrt [3]{-1} \sqrt [3]{b} e\right )}\)

Input:

Int[(x*(e + f*x)^n)/(a + b*x^3),x]
 

Output:

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

Defintions of rubi rules used

rule 2009

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

rule 7276

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE 
xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ 
[n, 0]
 

Maple [F]

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

Input:

int(x*(f*x+e)^n/(b*x^3+a),x)
 

Output:

int(x*(f*x+e)^n/(b*x^3+a),x)
 

Fricas [F]

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

integrate(x*(f*x+e)^n/(b*x^3+a),x, algorithm="fricas")
 

Output:

integral((f*x + e)^n*x/(b*x^3 + a), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x (e+f x)^n}{a+b x^3} \, dx=\text {Timed out} \] Input:

integrate(x*(f*x+e)**n/(b*x**3+a),x)
 

Output:

Timed out
 

Maxima [F]

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

integrate(x*(f*x+e)^n/(b*x^3+a),x, algorithm="maxima")
 

Output:

integrate((f*x + e)^n*x/(b*x^3 + a), x)
 

Giac [F]

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

integrate(x*(f*x+e)^n/(b*x^3+a),x, algorithm="giac")
 

Output:

integrate((f*x + e)^n*x/(b*x^3 + a), x)
 

Mupad [F(-1)]

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

int((x*(e + f*x)^n)/(a + b*x^3),x)
 

Output:

int((x*(e + f*x)^n)/(a + b*x^3), x)
 

Reduce [F]

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

int(x*(f*x+e)^n/(b*x^3+a),x)
 

Output:

int(((e + f*x)**n*x)/(a + b*x**3),x)