\(\int \frac {x^2 (e+f x)^n}{a+b x^3} \, dx\) [42]

Optimal result

Integrand size = 20, antiderivative size = 253 \[ \int \frac {x^2 (e+f x)^n}{a+b x^3} \, dx=-\frac {(e+f x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt [3]{b} (e+f x)}{\sqrt [3]{b} e-\sqrt [3]{a} f}\right )}{3 b^{2/3} \left (\sqrt [3]{b} e-\sqrt [3]{a} f\right ) (1+n)}-\frac {(e+f x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt [3]{b} (e+f x)}{\sqrt [3]{b} e+\sqrt [3]{-1} \sqrt [3]{a} f}\right )}{3 b^{2/3} \left (\sqrt [3]{b} e+\sqrt [3]{-1} \sqrt [3]{a} f\right ) (1+n)}-\frac {(e+f x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt [3]{b} (e+f x)}{\sqrt [3]{b} e-(-1)^{2/3} \sqrt [3]{a} f}\right )}{3 b^{2/3} \left (\sqrt [3]{b} e-(-1)^{2/3} \sqrt [3]{a} f\right ) (1+n)} \] 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))/b^(2/3)/(b^(1/3)*e-a^(1/3)*f)/(1+n)-1/3*(f*x+e)^(1+n)*hypergeom([ 
1, 1+n],[2+n],b^(1/3)*(f*x+e)/(b^(1/3)*e+(-1)^(1/3)*a^(1/3)*f))/b^(2/3)/(b 
^(1/3)*e+(-1)^(1/3)*a^(1/3)*f)/(1+n)-1/3*(f*x+e)^(1+n)*hypergeom([1, 1+n], 
[2+n],b^(1/3)*(f*x+e)/(b^(1/3)*e-(-1)^(2/3)*a^(1/3)*f))/b^(2/3)/(b^(1/3)*e 
-(-1)^(2/3)*a^(1/3)*f)/(1+n)
 

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.84 \[ \int \frac {x^2 (e+f x)^n}{a+b x^3} \, dx=\frac {(e+f x)^{1+n} \left (-\frac {\operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt [3]{b} (e+f x)}{\sqrt [3]{b} e-\sqrt [3]{a} f}\right )}{\sqrt [3]{b} e-\sqrt [3]{a} f}-\frac {\operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt [3]{b} (e+f x)}{\sqrt [3]{b} e+\sqrt [3]{-1} \sqrt [3]{a} f}\right )}{\sqrt [3]{b} e+\sqrt [3]{-1} \sqrt [3]{a} f}-\frac {\operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt [3]{b} (e+f x)}{\sqrt [3]{b} e-(-1)^{2/3} \sqrt [3]{a} f}\right )}{\sqrt [3]{b} e-(-1)^{2/3} \sqrt [3]{a} f}\right )}{3 b^{2/3} (1+n)} \] Input:

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

Rubi [A] (verified)

Time = 0.94 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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.

\(\displaystyle \int \frac {x^2 (e+f x)^n}{a+b x^3} \, dx\)

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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

Input:

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

Output:

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

Fricas [F]

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

integrate(x^2*(f*x+e)^n/(b*x^3+a),x, algorithm="fricas")
 

Output:

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

Sympy [F(-1)]

Timed out. \[ \int \frac {x^2 (e+f x)^n}{a+b x^3} \, dx=\text {Timed out} \] Input:

integrate(x**2*(f*x+e)**n/(b*x**3+a),x)
 

Output:

Timed out
 

Maxima [F]

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

integrate(x^2*(f*x+e)^n/(b*x^3+a),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate(x^2*(f*x+e)^n/(b*x^3+a),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^2 (e+f x)^n}{a+b x^3} \, dx=\frac {\left (f x +e \right )^{n}-\left (\int \frac {\left (f x +e \right )^{n}}{b f \,x^{4}+b e \,x^{3}+a f x +a e}d x \right ) a f n +\left (\int \frac {\left (f x +e \right )^{n} x^{2}}{b f \,x^{4}+b e \,x^{3}+a f x +a e}d x \right ) b e n}{b n} \] Input:

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

Output:

((e + f*x)**n - int((e + f*x)**n/(a*e + a*f*x + b*e*x**3 + b*f*x**4),x)*a* 
f*n + int(((e + f*x)**n*x**2)/(a*e + a*f*x + b*e*x**3 + b*f*x**4),x)*b*e*n 
)/(b*n)