Integrand size = 36, antiderivative size = 162 \[ \int \frac {(c x)^m \left (d+e x^n+f x^{2 n}+g x^{3 n}\right )}{a+b x^n} \, dx=\frac {(b f-a g) x^{1+n} (c x)^m}{b^2 (1+m+n)}+\frac {g x^{1+2 n} (c x)^m}{b (1+m+2 n)}+\frac {\left (b^2 e-a b f+a^2 g\right ) (c x)^{1+m}}{b^3 c (1+m)}+\frac {\left (b^3 d-a b^2 e+a^2 b f-a^3 g\right ) (c x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {b x^n}{a}\right )}{a b^3 c (1+m)} \]
(-a*g+b*f)*x^(1+n)*(c*x)^m/b^2/(1+m+n)+g*x^(1+2*n)*(c*x)^m/b/(1+m+2*n)+(a^ 2*g-a*b*f+b^2*e)*(c*x)^(1+m)/b^3/c/(1+m)+(-a^3*g+a^2*b*f-a*b^2*e+b^3*d)*(c *x)^(1+m)*hypergeom([1, (1+m)/n],[(1+m+n)/n],-b*x^n/a)/a/b^3/c/(1+m)
Time = 0.63 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.80 \[ \int \frac {(c x)^m \left (d+e x^n+f x^{2 n}+g x^{3 n}\right )}{a+b x^n} \, dx=\frac {x (c x)^m \left (\frac {b^2 e-a b f+a^2 g}{1+m}+\frac {b (b f-a g) x^n}{1+m+n}+\frac {b^2 g x^{2 n}}{1+m+2 n}+\frac {\left (b^3 d-a b^2 e+a^2 b f-a^3 g\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {b x^n}{a}\right )}{a (1+m)}\right )}{b^3} \]
(x*(c*x)^m*((b^2*e - a*b*f + a^2*g)/(1 + m) + (b*(b*f - a*g)*x^n)/(1 + m + n) + (b^2*g*x^(2*n))/(1 + m + 2*n) + ((b^3*d - a*b^2*e + a^2*b*f - a^3*g) *Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)])/(a*(1 + m)) ))/b^3
Time = 0.41 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {2383, 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.
\(\displaystyle \int \frac {(c x)^m \left (d+e x^n+f x^{2 n}+g x^{3 n}\right )}{a+b x^n} \, dx\) |
\(\Big \downarrow \) 2383 |
\(\displaystyle \int \left (\frac {(c x)^m \left (a^2 g-a b f+b^2 e\right )}{b^3}+\frac {(c x)^m \left (a^3 (-g)+a^2 b f-a b^2 e+b^3 d\right )}{b^3 \left (a+b x^n\right )}+\frac {x^n (c x)^m (b f-a g)}{b^2}+\frac {g x^{2 n} (c x)^m}{b}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {(c x)^{m+1} \left (a^2 g-a b f+b^2 e\right )}{b^3 c (m+1)}+\frac {(c x)^{m+1} \left (a^3 (-g)+a^2 b f-a b^2 e+b^3 d\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{n},\frac {m+n+1}{n},-\frac {b x^n}{a}\right )}{a b^3 c (m+1)}+\frac {x^{n+1} (c x)^m (b f-a g)}{b^2 (m+n+1)}+\frac {g x^{2 n+1} (c x)^m}{b (m+2 n+1)}\) |
((b*f - a*g)*x^(1 + n)*(c*x)^m)/(b^2*(1 + m + n)) + (g*x^(1 + 2*n)*(c*x)^m )/(b*(1 + m + 2*n)) + ((b^2*e - a*b*f + a^2*g)*(c*x)^(1 + m))/(b^3*c*(1 + m)) + ((b^3*d - a*b^2*e + a^2*b*f - a^3*g)*(c*x)^(1 + m)*Hypergeometric2F1 [1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)])/(a*b^3*c*(1 + m))
3.6.76.3.1 Defintions of rubi rules used
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> I nt[ExpandIntegrand[(c*x)^m*Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n , p}, x] && (PolyQ[Pq, x] || PolyQ[Pq, x^n]) && !IGtQ[m, 0]
\[\int \frac {\left (c x \right )^{m} \left (d +e \,x^{n}+f \,x^{2 n}+g \,x^{3 n}\right )}{a +b \,x^{n}}d x\]
\[ \int \frac {(c x)^m \left (d+e x^n+f x^{2 n}+g x^{3 n}\right )}{a+b x^n} \, dx=\int { \frac {{\left (g x^{3 \, n} + f x^{2 \, n} + e x^{n} + d\right )} \left (c x\right )^{m}}{b x^{n} + a} \,d x } \]
Result contains complex when optimal does not.
Time = 25.50 (sec) , antiderivative size = 860, normalized size of antiderivative = 5.31 \[ \int \frac {(c x)^m \left (d+e x^n+f x^{2 n}+g x^{3 n}\right )}{a+b x^n} \, dx=\text {Too large to display} \]
a**(m/n + 1/n)*a**(-m/n - 1 - 1/n)*c**m*d*m*x**(m + 1)*lerchphi(b*x**n*exp _polar(I*pi)/a, 1, m/n + 1/n)*gamma(m/n + 1/n)/(n**2*gamma(m/n + 1 + 1/n)) + a**(m/n + 1/n)*a**(-m/n - 1 - 1/n)*c**m*d*x**(m + 1)*lerchphi(b*x**n*ex p_polar(I*pi)/a, 1, m/n + 1/n)*gamma(m/n + 1/n)/(n**2*gamma(m/n + 1 + 1/n) ) + a**(-m/n - 4 - 1/n)*a**(m/n + 3 + 1/n)*c**m*g*m*x**(m + 3*n + 1)*lerch phi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 3 + 1/n)*gamma(m/n + 3 + 1/n)/(n**2 *gamma(m/n + 4 + 1/n)) + 3*a**(-m/n - 4 - 1/n)*a**(m/n + 3 + 1/n)*c**m*g*x **(m + 3*n + 1)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 3 + 1/n)*gamma (m/n + 3 + 1/n)/(n*gamma(m/n + 4 + 1/n)) + a**(-m/n - 4 - 1/n)*a**(m/n + 3 + 1/n)*c**m*g*x**(m + 3*n + 1)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 3 + 1/n)*gamma(m/n + 3 + 1/n)/(n**2*gamma(m/n + 4 + 1/n)) + a**(-m/n - 3 - 1/n)*a**(m/n + 2 + 1/n)*c**m*f*m*x**(m + 2*n + 1)*lerchphi(b*x**n*exp_p olar(I*pi)/a, 1, m/n + 2 + 1/n)*gamma(m/n + 2 + 1/n)/(n**2*gamma(m/n + 3 + 1/n)) + 2*a**(-m/n - 3 - 1/n)*a**(m/n + 2 + 1/n)*c**m*f*x**(m + 2*n + 1)* lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 2 + 1/n)*gamma(m/n + 2 + 1/n)/ (n*gamma(m/n + 3 + 1/n)) + a**(-m/n - 3 - 1/n)*a**(m/n + 2 + 1/n)*c**m*f*x **(m + 2*n + 1)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 2 + 1/n)*gamma (m/n + 2 + 1/n)/(n**2*gamma(m/n + 3 + 1/n)) + a**(-m/n - 2 - 1/n)*a**(m/n + 1 + 1/n)*c**m*e*m*x**(m + n + 1)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m /n + 1 + 1/n)*gamma(m/n + 1 + 1/n)/(n**2*gamma(m/n + 2 + 1/n)) + a**(-m...
\[ \int \frac {(c x)^m \left (d+e x^n+f x^{2 n}+g x^{3 n}\right )}{a+b x^n} \, dx=\int { \frac {{\left (g x^{3 \, n} + f x^{2 \, n} + e x^{n} + d\right )} \left (c x\right )^{m}}{b x^{n} + a} \,d x } \]
(b^3*c^m*d - a*b^2*c^m*e + a^2*b*c^m*f - a^3*c^m*g)*integrate(x^m/(b^4*x^n + a*b^3), x) + ((m^2 + m*(n + 2) + n + 1)*b^2*c^m*g*x*e^(m*log(x) + 2*n*l og(x)) + ((m^2 + m*(3*n + 2) + 2*n^2 + 3*n + 1)*b^2*c^m*e - (m^2 + m*(3*n + 2) + 2*n^2 + 3*n + 1)*a*b*c^m*f + (m^2 + m*(3*n + 2) + 2*n^2 + 3*n + 1)* a^2*c^m*g)*x*x^m + ((m^2 + 2*m*(n + 1) + 2*n + 1)*b^2*c^m*f - (m^2 + 2*m*( n + 1) + 2*n + 1)*a*b*c^m*g)*x*e^(m*log(x) + n*log(x)))/((m^3 + 3*m^2*(n + 1) + (2*n^2 + 6*n + 3)*m + 2*n^2 + 3*n + 1)*b^3)
\[ \int \frac {(c x)^m \left (d+e x^n+f x^{2 n}+g x^{3 n}\right )}{a+b x^n} \, dx=\int { \frac {{\left (g x^{3 \, n} + f x^{2 \, n} + e x^{n} + d\right )} \left (c x\right )^{m}}{b x^{n} + a} \,d x } \]
Timed out. \[ \int \frac {(c x)^m \left (d+e x^n+f x^{2 n}+g x^{3 n}\right )}{a+b x^n} \, dx=\int \frac {{\left (c\,x\right )}^m\,\left (d+e\,x^n+f\,x^{2\,n}+g\,x^{3\,n}\right )}{a+b\,x^n} \,d x \]