\(\int \frac {(e+f x)^n}{x (a+b x^3)} \, dx\) [45]

Optimal result

Integrand size = 20, antiderivative size = 300 \[ \int \frac {(e+f x)^n}{x \left (a+b x^3\right )} \, dx=\frac {\sqrt [3]{b} (e+f x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt [3]{b} (e+f x)}{\sqrt [3]{b} e-\sqrt [3]{a} f}\right )}{3 a \left (\sqrt [3]{b} e-\sqrt [3]{a} f\right ) (1+n)}+\frac {\sqrt [3]{b} (e+f x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt [3]{b} (e+f x)}{\sqrt [3]{b} e+\sqrt [3]{-1} \sqrt [3]{a} f}\right )}{3 a \left (\sqrt [3]{b} e+\sqrt [3]{-1} \sqrt [3]{a} f\right ) (1+n)}+\frac {\sqrt [3]{b} (e+f x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt [3]{b} (e+f x)}{\sqrt [3]{b} e-(-1)^{2/3} \sqrt [3]{a} f}\right )}{3 a \left (\sqrt [3]{b} e-(-1)^{2/3} \sqrt [3]{a} f\right ) (1+n)}-\frac {(e+f x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,1+\frac {f x}{e}\right )}{a e (1+n)} \] Output:

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

Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.81 \[ \int \frac {(e+f x)^n}{x \left (a+b x^3\right )} \, dx=\frac {(e+f x)^{1+n} \left (\frac {\sqrt [3]{b} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt [3]{b} (e+f x)}{\sqrt [3]{b} e-\sqrt [3]{a} f}\right )}{\sqrt [3]{b} e-\sqrt [3]{a} f}+\frac {\sqrt [3]{b} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt [3]{b} (e+f x)}{\sqrt [3]{b} e+\sqrt [3]{-1} \sqrt [3]{a} f}\right )}{\sqrt [3]{b} e+\sqrt [3]{-1} \sqrt [3]{a} f}+\frac {\sqrt [3]{b} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {\sqrt [3]{b} (e+f x)}{\sqrt [3]{b} e-(-1)^{2/3} \sqrt [3]{a} f}\right )}{\sqrt [3]{b} e-(-1)^{2/3} \sqrt [3]{a} f}-\frac {3 \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,1+\frac {f x}{e}\right )}{e}\right )}{3 a (1+n)} \] Input:

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

Output:

((e + f*x)^(1 + n)*((b^(1/3)*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) + (b^(1/3)*Hyp 
ergeometric2F1[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) + (b^(1/3)*Hypergeometri 
c2F1[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*Hypergeometric2F1[1, 1 + n, 2 
 + n, 1 + (f*x)/e])/e))/(3*a*(1 + n))
 

Rubi [A] (verified)

Time = 1.29 (sec) , antiderivative size = 300, 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.

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

\(\Big \downarrow \) 7276

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(b^(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)])/(3*a*(b^(1/3)*e - a^(1/3)*f)*(1 + n)) + 
(b^(1/3)*(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*a*(b^(1/3)*e + (-1)^(1/3)* 
a^(1/3)*f)*(1 + n)) + (b^(1/3)*(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*a*(b 
^(1/3)*e - (-1)^(2/3)*a^(1/3)*f)*(1 + n)) - ((e + f*x)^(1 + n)*Hypergeomet 
ric2F1[1, 1 + n, 2 + n, 1 + (f*x)/e])/(a*e*(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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((f*x + e)^n/(b*x^4 + a*x), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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