Integrand size = 22, antiderivative size = 66 \[ \int \frac {(e x)^m (a+b x)}{(a c-b c x)^2} \, dx=\frac {2 (e x)^{1+m}}{c^2 e (a-b x)}-\frac {(1+2 m) (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {b x}{a}\right )}{a c^2 e (1+m)} \]
2*(e*x)^(1+m)/c^2/e/(-b*x+a)-(1+2*m)*(e*x)^(1+m)*hypergeom([1, 1+m],[2+m], b*x/a)/a/c^2/e/(1+m)
Time = 0.06 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.94 \[ \int \frac {(e x)^m (a+b x)}{(a c-b c x)^2} \, dx=-\frac {x (e x)^m \left (2 a (1+m)-(1+2 m) (a-b x) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {b x}{a}\right )\right )}{a c^2 (1+m) (-a+b x)} \]
-((x*(e*x)^m*(2*a*(1 + m) - (1 + 2*m)*(a - b*x)*Hypergeometric2F1[1, 1 + m , 2 + m, (b*x)/a]))/(a*c^2*(1 + m)*(-a + b*x)))
Time = 0.18 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {87, 27, 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.
| \(\displaystyle \int \frac {(a+b x) (e x)^m}{(a c-b c x)^2} \, dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {2 (e x)^{m+1}}{c^2 e (a-b x)}-\frac {(2 m+1) \int \frac {(e x)^m}{c (a-b x)}dx}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 (e x)^{m+1}}{c^2 e (a-b x)}-\frac {(2 m+1) \int \frac {(e x)^m}{a-b x}dx}{c^2}\) |
| \(\Big \downarrow \) 74 |
| \(\displaystyle \frac {2 (e x)^{m+1}}{c^2 e (a-b x)}-\frac {(2 m+1) (e x)^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,\frac {b x}{a}\right )}{a c^2 e (m+1)}\) |
(2*(e*x)^(1 + m))/(c^2*e*(a - b*x)) - ((1 + 2*m)*(e*x)^(1 + m)*Hypergeomet ric2F1[1, 1 + m, 2 + m, (b*x)/a])/(a*c^2*e*(1 + m))
3.1.73.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]))))
\[\int \frac {\left (e x \right )^{m} \left (b x +a \right )}{\left (-b c x +a c \right )^{2}}d x\]
\[ \int \frac {(e x)^m (a+b x)}{(a c-b c x)^2} \, dx=\int { \frac {{\left (b x + a\right )} \left (e x\right )^{m}}{{\left (b c x - a c\right )}^{2}} \,d x } \]
Result contains complex when optimal does not.
Time = 2.80 (sec) , antiderivative size = 790, normalized size of antiderivative = 11.97 \[ \int \frac {(e x)^m (a+b x)}{(a c-b c x)^2} \, dx=a \left (\frac {a e^{m} m^{2} x^{m + 1} \Phi \left (\frac {b x e^{2 i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{- a^{3} c^{2} \Gamma \left (m + 2\right ) + a^{2} b c^{2} x \Gamma \left (m + 2\right )} + \frac {a e^{m} m x^{m + 1} \Phi \left (\frac {b x e^{2 i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{- a^{3} c^{2} \Gamma \left (m + 2\right ) + a^{2} b c^{2} x \Gamma \left (m + 2\right )} - \frac {a e^{m} m x^{m + 1} \Gamma \left (m + 1\right )}{- a^{3} c^{2} \Gamma \left (m + 2\right ) + a^{2} b c^{2} x \Gamma \left (m + 2\right )} - \frac {a e^{m} x^{m + 1} \Gamma \left (m + 1\right )}{- a^{3} c^{2} \Gamma \left (m + 2\right ) + a^{2} b c^{2} x \Gamma \left (m + 2\right )} - \frac {b e^{m} m^{2} x x^{m + 1} \Phi \left (\frac {b x e^{2 i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{- a^{3} c^{2} \Gamma \left (m + 2\right ) + a^{2} b c^{2} x \Gamma \left (m + 2\right )} - \frac {b e^{m} m x x^{m + 1} \Phi \left (\frac {b x e^{2 i \pi }}{a}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{- a^{3} c^{2} \Gamma \left (m + 2\right ) + a^{2} b c^{2} x \Gamma \left (m + 2\right )}\right ) + b \left (\frac {a e^{m} m^{2} x^{m + 2} \Phi \left (\frac {b x e^{2 i \pi }}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{- a^{3} c^{2} \Gamma \left (m + 3\right ) + a^{2} b c^{2} x \Gamma \left (m + 3\right )} + \frac {3 a e^{m} m x^{m + 2} \Phi \left (\frac {b x e^{2 i \pi }}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{- a^{3} c^{2} \Gamma \left (m + 3\right ) + a^{2} b c^{2} x \Gamma \left (m + 3\right )} - \frac {a e^{m} m x^{m + 2} \Gamma \left (m + 2\right )}{- a^{3} c^{2} \Gamma \left (m + 3\right ) + a^{2} b c^{2} x \Gamma \left (m + 3\right )} + \frac {2 a e^{m} x^{m + 2} \Phi \left (\frac {b x e^{2 i \pi }}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{- a^{3} c^{2} \Gamma \left (m + 3\right ) + a^{2} b c^{2} x \Gamma \left (m + 3\right )} - \frac {2 a e^{m} x^{m + 2} \Gamma \left (m + 2\right )}{- a^{3} c^{2} \Gamma \left (m + 3\right ) + a^{2} b c^{2} x \Gamma \left (m + 3\right )} - \frac {b e^{m} m^{2} x x^{m + 2} \Phi \left (\frac {b x e^{2 i \pi }}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{- a^{3} c^{2} \Gamma \left (m + 3\right ) + a^{2} b c^{2} x \Gamma \left (m + 3\right )} - \frac {3 b e^{m} m x x^{m + 2} \Phi \left (\frac {b x e^{2 i \pi }}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{- a^{3} c^{2} \Gamma \left (m + 3\right ) + a^{2} b c^{2} x \Gamma \left (m + 3\right )} - \frac {2 b e^{m} x x^{m + 2} \Phi \left (\frac {b x e^{2 i \pi }}{a}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{- a^{3} c^{2} \Gamma \left (m + 3\right ) + a^{2} b c^{2} x \Gamma \left (m + 3\right )}\right ) \]
a*(a*e**m*m**2*x**(m + 1)*lerchphi(b*x*exp_polar(2*I*pi)/a, 1, m + 1)*gamm a(m + 1)/(-a**3*c**2*gamma(m + 2) + a**2*b*c**2*x*gamma(m + 2)) + a*e**m*m *x**(m + 1)*lerchphi(b*x*exp_polar(2*I*pi)/a, 1, m + 1)*gamma(m + 1)/(-a** 3*c**2*gamma(m + 2) + a**2*b*c**2*x*gamma(m + 2)) - a*e**m*m*x**(m + 1)*ga mma(m + 1)/(-a**3*c**2*gamma(m + 2) + a**2*b*c**2*x*gamma(m + 2)) - a*e**m *x**(m + 1)*gamma(m + 1)/(-a**3*c**2*gamma(m + 2) + a**2*b*c**2*x*gamma(m + 2)) - b*e**m*m**2*x*x**(m + 1)*lerchphi(b*x*exp_polar(2*I*pi)/a, 1, m + 1)*gamma(m + 1)/(-a**3*c**2*gamma(m + 2) + a**2*b*c**2*x*gamma(m + 2)) - b *e**m*m*x*x**(m + 1)*lerchphi(b*x*exp_polar(2*I*pi)/a, 1, m + 1)*gamma(m + 1)/(-a**3*c**2*gamma(m + 2) + a**2*b*c**2*x*gamma(m + 2))) + b*(a*e**m*m* *2*x**(m + 2)*lerchphi(b*x*exp_polar(2*I*pi)/a, 1, m + 2)*gamma(m + 2)/(-a **3*c**2*gamma(m + 3) + a**2*b*c**2*x*gamma(m + 3)) + 3*a*e**m*m*x**(m + 2 )*lerchphi(b*x*exp_polar(2*I*pi)/a, 1, m + 2)*gamma(m + 2)/(-a**3*c**2*gam ma(m + 3) + a**2*b*c**2*x*gamma(m + 3)) - a*e**m*m*x**(m + 2)*gamma(m + 2) /(-a**3*c**2*gamma(m + 3) + a**2*b*c**2*x*gamma(m + 3)) + 2*a*e**m*x**(m + 2)*lerchphi(b*x*exp_polar(2*I*pi)/a, 1, m + 2)*gamma(m + 2)/(-a**3*c**2*g amma(m + 3) + a**2*b*c**2*x*gamma(m + 3)) - 2*a*e**m*x**(m + 2)*gamma(m + 2)/(-a**3*c**2*gamma(m + 3) + a**2*b*c**2*x*gamma(m + 3)) - b*e**m*m**2*x* x**(m + 2)*lerchphi(b*x*exp_polar(2*I*pi)/a, 1, m + 2)*gamma(m + 2)/(-a**3 *c**2*gamma(m + 3) + a**2*b*c**2*x*gamma(m + 3)) - 3*b*e**m*m*x*x**(m +...
\[ \int \frac {(e x)^m (a+b x)}{(a c-b c x)^2} \, dx=\int { \frac {{\left (b x + a\right )} \left (e x\right )^{m}}{{\left (b c x - a c\right )}^{2}} \,d x } \]
\[ \int \frac {(e x)^m (a+b x)}{(a c-b c x)^2} \, dx=\int { \frac {{\left (b x + a\right )} \left (e x\right )^{m}}{{\left (b c x - a c\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {(e x)^m (a+b x)}{(a c-b c x)^2} \, dx=\int \frac {{\left (e\,x\right )}^m\,\left (a+b\,x\right )}{{\left (a\,c-b\,c\,x\right )}^2} \,d x \]