Integrand size = 16, antiderivative size = 73 \[ \int \frac {x^m (A+B x)}{(a+b x)^2} \, dx=\frac {(A b-a B) x^{1+m}}{a b (a+b x)}-\frac {(A b m-a B (1+m)) x^{1+m} \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {b x}{a}\right )}{a^2 b (1+m)} \]
(A*b-B*a)*x^(1+m)/a/b/(b*x+a)-(A*b*m-a*B*(1+m))*x^(1+m)*hypergeom([1, 1+m] ,[2+m],-b*x/a)/a^2/b/(1+m)
Time = 0.06 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.86 \[ \int \frac {x^m (A+B x)}{(a+b x)^2} \, dx=\frac {x^{1+m} \left (\frac {a (A b-a B)}{a+b x}+\frac {(-A b m+a B (1+m)) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {b x}{a}\right )}{1+m}\right )}{a^2 b} \]
(x^(1 + m)*((a*(A*b - a*B))/(a + b*x) + ((-(A*b*m) + a*B*(1 + m))*Hypergeo metric2F1[1, 1 + m, 2 + m, -((b*x)/a)])/(1 + m)))/(a^2*b)
Time = 0.19 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {87, 74}
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.
\(\displaystyle \int \frac {x^m (A+B x)}{(a+b x)^2} \, dx\) |
\(\Big \downarrow \) 87 |
\(\displaystyle \frac {x^{m+1} (A b-a B)}{a b (a+b x)}-\frac {(A b m-a B (m+1)) \int \frac {x^m}{a+b x}dx}{a b}\) |
\(\Big \downarrow \) 74 |
\(\displaystyle \frac {x^{m+1} (A b-a B)}{a b (a+b x)}-\frac {x^{m+1} (A b m-a B (m+1)) \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,-\frac {b x}{a}\right )}{a^2 b (m+1)}\) |
((A*b - a*B)*x^(1 + m))/(a*b*(a + b*x)) - ((A*b*m - a*B*(1 + m))*x^(1 + m) *Hypergeometric2F1[1, 1 + m, 2 + m, -((b*x)/a)])/(a^2*b*(1 + m))
3.4.76.3.1 Defintions of rubi rules used
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^n*((b*x )^(m + 1)/(b*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] && !IntegerQ[m] && (IntegerQ[n] || (GtQ[c, 0] && !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-d/(b*c), 0])))
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
\[\int \frac {x^{m} \left (B x +A \right )}{\left (b x +a \right )^{2}}d x\]
\[ \int \frac {x^m (A+B x)}{(a+b x)^2} \, dx=\int { \frac {{\left (B x + A\right )} x^{m}}{{\left (b x + a\right )}^{2}} \,d x } \]
Result contains complex when optimal does not.
Time = 2.36 (sec) , antiderivative size = 631, normalized size of antiderivative = 8.64 \[ \int \frac {x^m (A+B x)}{(a+b x)^2} \, dx=A \left (- \frac {a m^{2} x^{m + 1} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{a^{3} \Gamma \left (m + 2\right ) + a^{2} b x \Gamma \left (m + 2\right )} - \frac {a m x^{m + 1} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{a^{3} \Gamma \left (m + 2\right ) + a^{2} b x \Gamma \left (m + 2\right )} + \frac {a m x^{m + 1} \Gamma \left (m + 1\right )}{a^{3} \Gamma \left (m + 2\right ) + a^{2} b x \Gamma \left (m + 2\right )} + \frac {a x^{m + 1} \Gamma \left (m + 1\right )}{a^{3} \Gamma \left (m + 2\right ) + a^{2} b x \Gamma \left (m + 2\right )} - \frac {b m^{2} x x^{m + 1} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{a^{3} \Gamma \left (m + 2\right ) + a^{2} b x \Gamma \left (m + 2\right )} - \frac {b m x x^{m + 1} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{a^{3} \Gamma \left (m + 2\right ) + a^{2} b x \Gamma \left (m + 2\right )}\right ) + B \left (- \frac {a m^{2} x^{m + 2} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{a^{3} \Gamma \left (m + 3\right ) + a^{2} b x \Gamma \left (m + 3\right )} - \frac {3 a m x^{m + 2} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{a^{3} \Gamma \left (m + 3\right ) + a^{2} b x \Gamma \left (m + 3\right )} + \frac {a m x^{m + 2} \Gamma \left (m + 2\right )}{a^{3} \Gamma \left (m + 3\right ) + a^{2} b x \Gamma \left (m + 3\right )} - \frac {2 a x^{m + 2} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{a^{3} \Gamma \left (m + 3\right ) + a^{2} b x \Gamma \left (m + 3\right )} + \frac {2 a x^{m + 2} \Gamma \left (m + 2\right )}{a^{3} \Gamma \left (m + 3\right ) + a^{2} b x \Gamma \left (m + 3\right )} - \frac {b m^{2} x x^{m + 2} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{a^{3} \Gamma \left (m + 3\right ) + a^{2} b x \Gamma \left (m + 3\right )} - \frac {3 b m x x^{m + 2} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{a^{3} \Gamma \left (m + 3\right ) + a^{2} b x \Gamma \left (m + 3\right )} - \frac {2 b x x^{m + 2} \Phi \left (\frac {b x e^{i \pi }}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{a^{3} \Gamma \left (m + 3\right ) + a^{2} b x \Gamma \left (m + 3\right )}\right ) \]
A*(-a*m**2*x**(m + 1)*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 1)*gamma(m + 1)/(a**3*gamma(m + 2) + a**2*b*x*gamma(m + 2)) - a*m*x**(m + 1)*lerchphi(b *x*exp_polar(I*pi)/a, 1, m + 1)*gamma(m + 1)/(a**3*gamma(m + 2) + a**2*b*x *gamma(m + 2)) + a*m*x**(m + 1)*gamma(m + 1)/(a**3*gamma(m + 2) + a**2*b*x *gamma(m + 2)) + a*x**(m + 1)*gamma(m + 1)/(a**3*gamma(m + 2) + a**2*b*x*g amma(m + 2)) - b*m**2*x*x**(m + 1)*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 1)*gamma(m + 1)/(a**3*gamma(m + 2) + a**2*b*x*gamma(m + 2)) - b*m*x*x**(m + 1)*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 1)*gamma(m + 1)/(a**3*gamma(m + 2) + a**2*b*x*gamma(m + 2))) + B*(-a*m**2*x**(m + 2)*lerchphi(b*x*exp_po lar(I*pi)/a, 1, m + 2)*gamma(m + 2)/(a**3*gamma(m + 3) + a**2*b*x*gamma(m + 3)) - 3*a*m*x**(m + 2)*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 2)*gamma(m + 2)/(a**3*gamma(m + 3) + a**2*b*x*gamma(m + 3)) + a*m*x**(m + 2)*gamma(m + 2)/(a**3*gamma(m + 3) + a**2*b*x*gamma(m + 3)) - 2*a*x**(m + 2)*lerchph i(b*x*exp_polar(I*pi)/a, 1, m + 2)*gamma(m + 2)/(a**3*gamma(m + 3) + a**2* b*x*gamma(m + 3)) + 2*a*x**(m + 2)*gamma(m + 2)/(a**3*gamma(m + 3) + a**2* b*x*gamma(m + 3)) - b*m**2*x*x**(m + 2)*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 2)*gamma(m + 2)/(a**3*gamma(m + 3) + a**2*b*x*gamma(m + 3)) - 3*b*m*x *x**(m + 2)*lerchphi(b*x*exp_polar(I*pi)/a, 1, m + 2)*gamma(m + 2)/(a**3*g amma(m + 3) + a**2*b*x*gamma(m + 3)) - 2*b*x*x**(m + 2)*lerchphi(b*x*exp_p olar(I*pi)/a, 1, m + 2)*gamma(m + 2)/(a**3*gamma(m + 3) + a**2*b*x*gamm...
\[ \int \frac {x^m (A+B x)}{(a+b x)^2} \, dx=\int { \frac {{\left (B x + A\right )} x^{m}}{{\left (b x + a\right )}^{2}} \,d x } \]
\[ \int \frac {x^m (A+B x)}{(a+b x)^2} \, dx=\int { \frac {{\left (B x + A\right )} x^{m}}{{\left (b x + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {x^m (A+B x)}{(a+b x)^2} \, dx=\int \frac {x^m\,\left (A+B\,x\right )}{{\left (a+b\,x\right )}^2} \,d x \]