Integrand size = 26, antiderivative size = 95 \[ \int \frac {c+d x^{n/2}+e x^n}{a+b x^n} \, dx=\frac {e x}{b}+\frac {2 d x^{\frac {2+n}{2}} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} \left (1+\frac {2}{n}\right ),\frac {1}{2} \left (3+\frac {2}{n}\right ),-\frac {b x^n}{a}\right )}{a (2+n)}+\frac {(b c-a e) x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )}{a b} \] Output:
e*x/b+2*d*x^(1+1/2*n)*hypergeom([1, 1/2+1/n],[3/2+1/n],-b*x^n/a)/a/(2+n)+( -a*e+b*c)*x*hypergeom([1, 1/n],[1+1/n],-b*x^n/a)/a/b
Time = 0.21 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.93 \[ \int \frac {c+d x^{n/2}+e x^n}{a+b x^n} \, dx=\frac {x \left (2 b d x^{n/2} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2}+\frac {1}{n},\frac {3}{2}+\frac {1}{n},-\frac {b x^n}{a}\right )-(2+n) \left (-a e+(-b c+a e) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )\right )\right )}{a b (2+n)} \] Input:
Integrate[(c + d*x^(n/2) + e*x^n)/(a + b*x^n),x]
Output:
(x*(2*b*d*x^(n/2)*Hypergeometric2F1[1, 1/2 + n^(-1), 3/2 + n^(-1), -((b*x^ n)/a)] - (2 + n)*(-(a*e) + (-(b*c) + a*e)*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((b*x^n)/a)])))/(a*b*(2 + n))
Time = 0.60 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {7293, 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.
\(\displaystyle \int \frac {c+d x^{n/2}+e x^n}{a+b x^n} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {-a e+b c+b d x^{n/2}}{b \left (a+b x^n\right )}+\frac {e}{b}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {x (b c-a e) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )}{a b}+\frac {2 d x^{\frac {n+2}{2}} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} \left (1+\frac {2}{n}\right ),\frac {1}{2} \left (3+\frac {2}{n}\right ),-\frac {b x^n}{a}\right )}{a (n+2)}+\frac {e x}{b}\) |
Input:
Int[(c + d*x^(n/2) + e*x^n)/(a + b*x^n),x]
Output:
(e*x)/b + (2*d*x^((2 + n)/2)*Hypergeometric2F1[1, (1 + 2/n)/2, (3 + 2/n)/2 , -((b*x^n)/a)])/(a*(2 + n)) + ((b*c - a*e)*x*Hypergeometric2F1[1, n^(-1), 1 + n^(-1), -((b*x^n)/a)])/(a*b)
\[\int \frac {c +d \,x^{\frac {n}{2}}+e \,x^{n}}{a +b \,x^{n}}d x\]
Input:
int((c+d*x^(1/2*n)+e*x^n)/(a+b*x^n),x)
Output:
int((c+d*x^(1/2*n)+e*x^n)/(a+b*x^n),x)
\[ \int \frac {c+d x^{n/2}+e x^n}{a+b x^n} \, dx=\int { \frac {d x^{\frac {1}{2} \, n} + e x^{n} + c}{b x^{n} + a} \,d x } \] Input:
integrate((c+d*x^(1/2*n)+e*x^n)/(a+b*x^n),x, algorithm="fricas")
Output:
integral((d*x^(1/2*n) + e*x^n + c)/(b*x^n + a), x)
Result contains complex when optimal does not.
Time = 3.68 (sec) , antiderivative size = 243, normalized size of antiderivative = 2.56 \[ \int \frac {c+d x^{n/2}+e x^n}{a+b x^n} \, dx=\frac {a^{\frac {1}{n}} a^{-1 - \frac {1}{n}} c x \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {1}{n}\right ) \Gamma \left (\frac {1}{n}\right )}{n^{2} \Gamma \left (1 + \frac {1}{n}\right )} + \frac {a^{- \frac {3}{2} - \frac {1}{n}} a^{\frac {1}{2} + \frac {1}{n}} d x^{\frac {n}{2} + 1} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {1}{2} + \frac {1}{n}\right ) \Gamma \left (\frac {1}{2} + \frac {1}{n}\right )}{2 n \Gamma \left (\frac {3}{2} + \frac {1}{n}\right )} + \frac {a^{- \frac {3}{2} - \frac {1}{n}} a^{\frac {1}{2} + \frac {1}{n}} d x^{\frac {n}{2} + 1} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {1}{2} + \frac {1}{n}\right ) \Gamma \left (\frac {1}{2} + \frac {1}{n}\right )}{n^{2} \Gamma \left (\frac {3}{2} + \frac {1}{n}\right )} - \frac {a^{- \frac {1}{n}} a^{1 + \frac {1}{n}} b^{\frac {1}{n}} b^{-1 - \frac {1}{n}} e x \Phi \left (\frac {a x^{- n} e^{i \pi }}{b}, 1, \frac {e^{i \pi }}{n}\right ) \Gamma \left (\frac {1}{n}\right )}{a n^{2} \Gamma \left (1 + \frac {1}{n}\right )} \] Input:
integrate((c+d*x**(1/2*n)+e*x**n)/(a+b*x**n),x)
Output:
a**(1/n)*a**(-1 - 1/n)*c*x*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 1/n)*gamm a(1/n)/(n**2*gamma(1 + 1/n)) + a**(-3/2 - 1/n)*a**(1/2 + 1/n)*d*x**(n/2 + 1)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 1/2 + 1/n)*gamma(1/2 + 1/n)/(2*n* gamma(3/2 + 1/n)) + a**(-3/2 - 1/n)*a**(1/2 + 1/n)*d*x**(n/2 + 1)*lerchphi (b*x**n*exp_polar(I*pi)/a, 1, 1/2 + 1/n)*gamma(1/2 + 1/n)/(n**2*gamma(3/2 + 1/n)) - a**(1 + 1/n)*b**(1/n)*b**(-1 - 1/n)*e*x*lerchphi(a*exp_polar(I*p i)/(b*x**n), 1, exp_polar(I*pi)/n)*gamma(1/n)/(a*a**(1/n)*n**2*gamma(1 + 1 /n))
\[ \int \frac {c+d x^{n/2}+e x^n}{a+b x^n} \, dx=\int { \frac {d x^{\frac {1}{2} \, n} + e x^{n} + c}{b x^{n} + a} \,d x } \] Input:
integrate((c+d*x^(1/2*n)+e*x^n)/(a+b*x^n),x, algorithm="maxima")
Output:
e*x/b + integrate((b*d*x^(1/2*n) + b*c - a*e)/(b^2*x^n + a*b), x)
\[ \int \frac {c+d x^{n/2}+e x^n}{a+b x^n} \, dx=\int { \frac {d x^{\frac {1}{2} \, n} + e x^{n} + c}{b x^{n} + a} \,d x } \] Input:
integrate((c+d*x^(1/2*n)+e*x^n)/(a+b*x^n),x, algorithm="giac")
Output:
integrate((d*x^(1/2*n) + e*x^n + c)/(b*x^n + a), x)
Timed out. \[ \int \frac {c+d x^{n/2}+e x^n}{a+b x^n} \, dx=\int \frac {c+e\,x^n+d\,x^{n/2}}{a+b\,x^n} \,d x \] Input:
int((c + e*x^n + d*x^(n/2))/(a + b*x^n),x)
Output:
int((c + e*x^n + d*x^(n/2))/(a + b*x^n), x)
\[ \int \frac {c+d x^{n/2}+e x^n}{a+b x^n} \, dx=\frac {\left (\int \frac {x^{\frac {n}{2}}}{x^{n} b +a}d x \right ) b d -\left (\int \frac {1}{x^{n} b +a}d x \right ) a e +\left (\int \frac {1}{x^{n} b +a}d x \right ) b c +e x}{b} \] Input:
int((c+d*x^(1/2*n)+e*x^n)/(a+b*x^n),x)
Output:
(int(x**(n/2)/(x**n*b + a),x)*b*d - int(1/(x**n*b + a),x)*a*e + int(1/(x** n*b + a),x)*b*c + e*x)/b