\(\int \frac {(e+f x)^n}{a+b x^3} \, dx\) [44]

Optimal result

Integrand size = 17, antiderivative size = 263 \[ \int \frac {(e+f x)^n}{a+b x^3} \, dx=-\frac {(e+f x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt [3]{b} (e+f x)}{\sqrt [3]{b} e-\sqrt [3]{a} f}\right )}{3 a^{2/3} \left (\sqrt [3]{b} e-\sqrt [3]{a} f\right ) (1+n)}-\frac {(e+f x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {(-1)^{2/3} \sqrt [3]{b} (e+f x)}{(-1)^{2/3} \sqrt [3]{b} e-\sqrt [3]{a} f}\right )}{3 a^{2/3} \left ((-1)^{2/3} \sqrt [3]{b} e-\sqrt [3]{a} f\right ) (1+n)}+\frac {(e+f x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt [3]{-1} \sqrt [3]{b} (e+f x)}{\sqrt [3]{-1} \sqrt [3]{b} e+\sqrt [3]{a} f}\right )}{3 a^{2/3} \left (\sqrt [3]{-1} \sqrt [3]{b} e+\sqrt [3]{a} f\right ) (1+n)} \] 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^(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],(-1)^(2/3)*b^(1/3)*(f*x+e)/((-1)^(2/3)*b^(1/3)*e-a^(1/3)*f)) 
/a^(2/3)/((-1)^(2/3)*b^(1/3)*e-a^(1/3)*f)/(1+n)+1/3*(f*x+e)^(1+n)*hypergeo 
m([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^(2/3)/((-1)^(1/3)*b^(1/3)*e+a^(1/3)*f)/(1+n)
 

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.84 \[ \int \frac {(e+f x)^n}{a+b x^3} \, dx=\frac {(e+f x)^{1+n} \left (-\frac {\operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt [3]{b} (e+f x)}{\sqrt [3]{b} e-\sqrt [3]{a} f}\right )}{\sqrt [3]{b} e-\sqrt [3]{a} f}-\frac {\operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {(-1)^{2/3} \sqrt [3]{b} (e+f x)}{(-1)^{2/3} \sqrt [3]{b} e-\sqrt [3]{a} f}\right )}{(-1)^{2/3} \sqrt [3]{b} e-\sqrt [3]{a} f}+\frac {\operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt [3]{-1} \sqrt [3]{b} (e+f x)}{\sqrt [3]{-1} \sqrt [3]{b} e+\sqrt [3]{a} f}\right )}{\sqrt [3]{-1} \sqrt [3]{b} e+\sqrt [3]{a} f}\right )}{3 a^{2/3} (1+n)} \] Input:

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

Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, 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.

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 2009

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

Input:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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