Integrand size = 20, antiderivative size = 211 \[ \int \frac {x^m (e+f x)^n}{a+b x^3} \, dx=\frac {x^{1+m} (e+f x)^n \left (1+\frac {f x}{e}\right )^{-n} \operatorname {AppellF1}\left (1+m,-n,1,2+m,-\frac {f x}{e},-\frac {\sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 a (1+m)}+\frac {x^{1+m} (e+f x)^n \left (1+\frac {f x}{e}\right )^{-n} \operatorname {AppellF1}\left (1+m,-n,1,2+m,-\frac {f x}{e},\frac {\sqrt [3]{-1} \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 a (1+m)}+\frac {x^{1+m} (e+f x)^n \left (1+\frac {f x}{e}\right )^{-n} \operatorname {AppellF1}\left (1+m,-n,1,2+m,-\frac {f x}{e},-\frac {(-1)^{2/3} \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 a (1+m)} \] Output:
1/3*x^(1+m)*(f*x+e)^n*AppellF1(1+m,1,-n,2+m,-b^(1/3)*x/a^(1/3),-f*x/e)/a/( 1+m)/((1+f*x/e)^n)+1/3*x^(1+m)*(f*x+e)^n*AppellF1(1+m,1,-n,2+m,(-1)^(1/3)* b^(1/3)*x/a^(1/3),-f*x/e)/a/(1+m)/((1+f*x/e)^n)+1/3*x^(1+m)*(f*x+e)^n*Appe llF1(1+m,-n,1,2+m,-f*x/e,-(-1)^(2/3)*b^(1/3)*x/a^(1/3))/a/(1+m)/((1+f*x/e) ^n)
\[ \int \frac {x^m (e+f x)^n}{a+b x^3} \, dx=\int \frac {x^m (e+f x)^n}{a+b x^3} \, dx \] Input:
Integrate[(x^m*(e + f*x)^n)/(a + b*x^3),x]
Output:
Integrate[(x^m*(e + f*x)^n)/(a + b*x^3), x]
Time = 1.04 (sec) , antiderivative size = 211, 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.
\(\displaystyle \int \frac {x^m (e+f x)^n}{a+b x^3} \, dx\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle \int \left (-\frac {x^m (e+f x)^n}{3 a^{2/3} \left (-\sqrt [3]{a}-\sqrt [3]{b} x\right )}-\frac {x^m (e+f x)^n}{3 a^{2/3} \left (\sqrt [3]{-1} \sqrt [3]{b} x-\sqrt [3]{a}\right )}-\frac {x^m (e+f x)^n}{3 a^{2/3} \left (-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {x^{m+1} (e+f x)^n \left (\frac {f x}{e}+1\right )^{-n} \operatorname {AppellF1}\left (m+1,-n,1,m+2,-\frac {f x}{e},-\frac {\sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 a (m+1)}+\frac {x^{m+1} (e+f x)^n \left (\frac {f x}{e}+1\right )^{-n} \operatorname {AppellF1}\left (m+1,-n,1,m+2,-\frac {f x}{e},\frac {\sqrt [3]{-1} \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 a (m+1)}+\frac {x^{m+1} (e+f x)^n \left (\frac {f x}{e}+1\right )^{-n} \operatorname {AppellF1}\left (m+1,-n,1,m+2,-\frac {f x}{e},-\frac {(-1)^{2/3} \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 a (m+1)}\) |
Input:
Int[(x^m*(e + f*x)^n)/(a + b*x^3),x]
Output:
(x^(1 + m)*(e + f*x)^n*AppellF1[1 + m, -n, 1, 2 + m, -((f*x)/e), -((b^(1/3 )*x)/a^(1/3))])/(3*a*(1 + m)*(1 + (f*x)/e)^n) + (x^(1 + m)*(e + f*x)^n*App ellF1[1 + m, -n, 1, 2 + m, -((f*x)/e), ((-1)^(1/3)*b^(1/3)*x)/a^(1/3)])/(3 *a*(1 + m)*(1 + (f*x)/e)^n) + (x^(1 + m)*(e + f*x)^n*AppellF1[1 + m, -n, 1 , 2 + m, -((f*x)/e), -(((-1)^(2/3)*b^(1/3)*x)/a^(1/3))])/(3*a*(1 + m)*(1 + (f*x)/e)^n)
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]
\[\int \frac {x^{m} \left (f x +e \right )^{n}}{b \,x^{3}+a}d x\]
Input:
int(x^m*(f*x+e)^n/(b*x^3+a),x)
Output:
int(x^m*(f*x+e)^n/(b*x^3+a),x)
\[ \int \frac {x^m (e+f x)^n}{a+b x^3} \, dx=\int { \frac {{\left (f x + e\right )}^{n} x^{m}}{b x^{3} + a} \,d x } \] Input:
integrate(x^m*(f*x+e)^n/(b*x^3+a),x, algorithm="fricas")
Output:
integral((f*x + e)^n*x^m/(b*x^3 + a), x)
Timed out. \[ \int \frac {x^m (e+f x)^n}{a+b x^3} \, dx=\text {Timed out} \] Input:
integrate(x**m*(f*x+e)**n/(b*x**3+a),x)
Output:
Timed out
\[ \int \frac {x^m (e+f x)^n}{a+b x^3} \, dx=\int { \frac {{\left (f x + e\right )}^{n} x^{m}}{b x^{3} + a} \,d x } \] Input:
integrate(x^m*(f*x+e)^n/(b*x^3+a),x, algorithm="maxima")
Output:
integrate((f*x + e)^n*x^m/(b*x^3 + a), x)
\[ \int \frac {x^m (e+f x)^n}{a+b x^3} \, dx=\int { \frac {{\left (f x + e\right )}^{n} x^{m}}{b x^{3} + a} \,d x } \] Input:
integrate(x^m*(f*x+e)^n/(b*x^3+a),x, algorithm="giac")
Output:
integrate((f*x + e)^n*x^m/(b*x^3 + a), x)
Timed out. \[ \int \frac {x^m (e+f x)^n}{a+b x^3} \, dx=\int \frac {x^m\,{\left (e+f\,x\right )}^n}{b\,x^3+a} \,d x \] Input:
int((x^m*(e + f*x)^n)/(a + b*x^3),x)
Output:
int((x^m*(e + f*x)^n)/(a + b*x^3), x)
\[ \int \frac {x^m (e+f x)^n}{a+b x^3} \, dx=\int \frac {x^{m} \left (f x +e \right )^{n}}{b \,x^{3}+a}d x \] Input:
int(x^m*(f*x+e)^n/(b*x^3+a),x)
Output:
int((x**m*(e + f*x)**n)/(a + b*x**3),x)