Integrand size = 25, antiderivative size = 54 \[ \int \frac {a x^m+b x^n}{c x^m+d x^n} \, dx=\frac {a x}{c}+\frac {(b c-a d) x \operatorname {Hypergeometric2F1}\left (1,\frac {1}{m-n},1+\frac {1}{m-n},-\frac {c x^{m-n}}{d}\right )}{c d} \] Output:
a*x/c+(-a*d+b*c)*x*hypergeom([1, 1/(m-n)],[1+1/(m-n)],-c*x^(m-n)/d)/c/d
Time = 0.09 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.96 \[ \int \frac {a x^m+b x^n}{c x^m+d x^n} \, dx=\frac {x \left (a d+(b c-a d) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{m-n},1+\frac {1}{m-n},-\frac {c x^{m-n}}{d}\right )\right )}{c d} \] Input:
Integrate[(a*x^m + b*x^n)/(c*x^m + d*x^n),x]
Output:
(x*(a*d + (b*c - a*d)*Hypergeometric2F1[1, (m - n)^(-1), 1 + (m - n)^(-1), -((c*x^(m - n))/d)]))/(c*d)
Time = 0.38 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2027, 10, 913, 778}
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.
\(\displaystyle \int \frac {a x^m+b x^n}{c x^m+d x^n} \, dx\) |
\(\Big \downarrow \) 2027 |
\(\displaystyle \int \frac {x^n \left (a x^{m-n}+b\right )}{c x^m+d x^n}dx\) |
\(\Big \downarrow \) 10 |
\(\displaystyle \int \frac {a x^{m-n}+b}{c x^{m-n}+d}dx\) |
\(\Big \downarrow \) 913 |
\(\displaystyle \frac {(b c-a d) \int \frac {1}{c x^{m-n}+d}dx}{c}+\frac {a x}{c}\) |
\(\Big \downarrow \) 778 |
\(\displaystyle \frac {x (b c-a d) \operatorname {Hypergeometric2F1}\left (1,\frac {1}{m-n},1+\frac {1}{m-n},-\frac {c x^{m-n}}{d}\right )}{c d}+\frac {a x}{c}\) |
Input:
Int[(a*x^m + b*x^n)/(c*x^m + d*x^n),x]
Output:
(a*x)/c + ((b*c - a*d)*x*Hypergeometric2F1[1, (m - n)^(-1), 1 + (m - n)^(- 1), -((c*x^(m - n))/d)])/(c*d)
Int[(u_.)*((e_.)*(x_))^(m_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.))^(p_.), x _Symbol] :> Simp[1/e^(p*r) Int[u*(e*x)^(m + p*r)*(a + b*x^(s - r))^p, x], x] /; FreeQ[{a, b, e, m, r, s}, x] && IntegerQ[p] && (IntegerQ[p*r] || GtQ [e, 0]) && PosQ[s - r]
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F 1[-p, 1/n, 1/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, n, p}, x] && !IGtQ[p , 0] && !IntegerQ[1/n] && !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p] || GtQ[a, 0])
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Si mp[d*x*((a + b*x^n)^(p + 1)/(b*(n*(p + 1) + 1))), x] - Simp[(a*d - b*c*(n*( p + 1) + 1))/(b*(n*(p + 1) + 1)) Int[(a + b*x^n)^p, x], x] /; FreeQ[{a, b , c, d, n, p}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]
Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.))^(p_.), x_Symbol] :> Int[x^ (p*r)*(a + b*x^(s - r))^p*Fx, x] /; FreeQ[{a, b, r, s}, x] && IntegerQ[p] & & PosQ[s - r] && !(EqQ[p, 1] && EqQ[u, 1])
\[\int \frac {a \,x^{m}+b \,x^{n}}{c \,x^{m}+d \,x^{n}}d x\]
Input:
int((a*x^m+b*x^n)/(c*x^m+d*x^n),x)
Output:
int((a*x^m+b*x^n)/(c*x^m+d*x^n),x)
\[ \int \frac {a x^m+b x^n}{c x^m+d x^n} \, dx=\int { \frac {a x^{m} + b x^{n}}{c x^{m} + d x^{n}} \,d x } \] Input:
integrate((a*x^m+b*x^n)/(c*x^m+d*x^n),x, algorithm="fricas")
Output:
integral((a*x^m + b*x^n)/(c*x^m + d*x^n), x)
\[ \int \frac {a x^m+b x^n}{c x^m+d x^n} \, dx=\int \frac {a x^{m} + b x^{n}}{c x^{m} + d x^{n}}\, dx \] Input:
integrate((a*x**m+b*x**n)/(c*x**m+d*x**n),x)
Output:
Integral((a*x**m + b*x**n)/(c*x**m + d*x**n), x)
\[ \int \frac {a x^m+b x^n}{c x^m+d x^n} \, dx=\int { \frac {a x^{m} + b x^{n}}{c x^{m} + d x^{n}} \,d x } \] Input:
integrate((a*x^m+b*x^n)/(c*x^m+d*x^n),x, algorithm="maxima")
Output:
-(b*c - a*d)*integrate(x^m/(c*d*x^m + d^2*x^n), x) + b*x/d
\[ \int \frac {a x^m+b x^n}{c x^m+d x^n} \, dx=\int { \frac {a x^{m} + b x^{n}}{c x^{m} + d x^{n}} \,d x } \] Input:
integrate((a*x^m+b*x^n)/(c*x^m+d*x^n),x, algorithm="giac")
Output:
integrate((a*x^m + b*x^n)/(c*x^m + d*x^n), x)
Timed out. \[ \int \frac {a x^m+b x^n}{c x^m+d x^n} \, dx=\int \frac {a\,x^m+b\,x^n}{c\,x^m+d\,x^n} \,d x \] Input:
int((a*x^m + b*x^n)/(c*x^m + d*x^n),x)
Output:
int((a*x^m + b*x^n)/(c*x^m + d*x^n), x)
\[ \int \frac {a x^m+b x^n}{c x^m+d x^n} \, dx=\frac {\left (\int \frac {x^{m}}{x^{m} c +x^{n} d}d x \right ) a d -\left (\int \frac {x^{m}}{x^{m} c +x^{n} d}d x \right ) b c +b x}{d} \] Input:
int((a*x^m+b*x^n)/(c*x^m+d*x^n),x)
Output:
(int(x**m/(x**m*c + x**n*d),x)*a*d - int(x**m/(x**m*c + x**n*d),x)*b*c + b *x)/d