Integrand size = 24, antiderivative size = 98 \[ \int \frac {(e x)^m}{(a+b x) (a c-b c x)^2} \, dx=\frac {(e x)^{1+m} \operatorname {Hypergeometric2F1}\left (2,\frac {1+m}{2},\frac {3+m}{2},\frac {b^2 x^2}{a^2}\right )}{a^3 c^2 e (1+m)}+\frac {b (e x)^{2+m} \operatorname {Hypergeometric2F1}\left (2,\frac {2+m}{2},\frac {4+m}{2},\frac {b^2 x^2}{a^2}\right )}{a^4 c^2 e^2 (2+m)} \] Output:
(e*x)^(1+m)*hypergeom([2, 1/2+1/2*m],[3/2+1/2*m],b^2*x^2/a^2)/a^3/c^2/e/(1 +m)+b*(e*x)^(2+m)*hypergeom([2, 1+1/2*m],[2+1/2*m],b^2*x^2/a^2)/a^4/c^2/e^ 2/(2+m)
Time = 0.08 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.89 \[ \int \frac {(e x)^m}{(a+b x) (a c-b c x)^2} \, dx=\frac {x (e x)^m \left (b (1+m) x \operatorname {Hypergeometric2F1}\left (2,1+\frac {m}{2},2+\frac {m}{2},\frac {b^2 x^2}{a^2}\right )+a (2+m) \operatorname {Hypergeometric2F1}\left (2,\frac {1+m}{2},\frac {3+m}{2},\frac {b^2 x^2}{a^2}\right )\right )}{a^4 c^2 (1+m) (2+m)} \] Input:
Integrate[(e*x)^m/((a + b*x)*(a*c - b*c*x)^2),x]
Output:
(x*(e*x)^m*(b*(1 + m)*x*Hypergeometric2F1[2, 1 + m/2, 2 + m/2, (b^2*x^2)/a ^2] + a*(2 + m)*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, (b^2*x^2)/a^2]) )/(a^4*c^2*(1 + m)*(2 + m))
Time = 0.20 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {92, 27, 82, 278}
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.
\(\displaystyle \int \frac {(e x)^m}{(a+b x) (a c-b c x)^2} \, dx\) |
\(\Big \downarrow \) 92 |
\(\displaystyle a \int \frac {(e x)^m}{c^2 (a-b x)^2 (a+b x)^2}dx+\frac {b \int \frac {(e x)^{m+1}}{c^2 (a-b x)^2 (a+b x)^2}dx}{e}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {a \int \frac {(e x)^m}{(a-b x)^2 (a+b x)^2}dx}{c^2}+\frac {b \int \frac {(e x)^{m+1}}{(a-b x)^2 (a+b x)^2}dx}{c^2 e}\) |
\(\Big \downarrow \) 82 |
\(\displaystyle \frac {a \int \frac {(e x)^m}{\left (a^2-b^2 x^2\right )^2}dx}{c^2}+\frac {b \int \frac {(e x)^{m+1}}{\left (a^2-b^2 x^2\right )^2}dx}{c^2 e}\) |
\(\Big \downarrow \) 278 |
\(\displaystyle \frac {b (e x)^{m+2} \operatorname {Hypergeometric2F1}\left (2,\frac {m+2}{2},\frac {m+4}{2},\frac {b^2 x^2}{a^2}\right )}{a^4 c^2 e^2 (m+2)}+\frac {(e x)^{m+1} \operatorname {Hypergeometric2F1}\left (2,\frac {m+1}{2},\frac {m+3}{2},\frac {b^2 x^2}{a^2}\right )}{a^3 c^2 e (m+1)}\) |
Input:
Int[(e*x)^m/((a + b*x)*(a*c - b*c*x)^2),x]
Output:
((e*x)^(1 + m)*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, (b^2*x^2)/a^2])/ (a^3*c^2*e*(1 + m)) + (b*(e*x)^(2 + m)*Hypergeometric2F1[2, (2 + m)/2, (4 + m)/2, (b^2*x^2)/a^2])/(a^4*c^2*e^2*(2 + m))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_) )^(p_.), x_] :> Int[(a*c + b*d*x^2)^m*(e + f*x)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[n, m] && IntegerQ[m]
Int[((f_.)*(x_))^(p_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_] :> Simp[a Int[(a + b*x)^n*(c + d*x)^n*(f*x)^p, x], x] + Simp[b/f In t[(a + b*x)^n*(c + d*x)^n*(f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[m - n - 1, 0] && !RationalQ[p] && !IGtQ[m, 0] && NeQ[m + n + p + 2, 0]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( -b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] && !IGtQ[p, 0] && (ILtQ[p, 0 ] || GtQ[a, 0])
\[\int \frac {\left (e x \right )^{m}}{\left (b x +a \right ) \left (-b c x +a c \right )^{2}}d x\]
Input:
int((e*x)^m/(b*x+a)/(-b*c*x+a*c)^2,x)
Output:
int((e*x)^m/(b*x+a)/(-b*c*x+a*c)^2,x)
\[ \int \frac {(e x)^m}{(a+b x) (a c-b c x)^2} \, dx=\int { \frac {\left (e x\right )^{m}}{{\left (b c x - a c\right )}^{2} {\left (b x + a\right )}} \,d x } \] Input:
integrate((e*x)^m/(b*x+a)/(-b*c*x+a*c)^2,x, algorithm="fricas")
Output:
integral((e*x)^m/(b^3*c^2*x^3 - a*b^2*c^2*x^2 - a^2*b*c^2*x + a^3*c^2), x)
Result contains complex when optimal does not.
Time = 1.48 (sec) , antiderivative size = 440, normalized size of antiderivative = 4.49 \[ \int \frac {(e x)^m}{(a+b x) (a c-b c x)^2} \, dx=- \frac {2 a e^{m} m^{2} x^{m} \Phi \left (\frac {a}{b x}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{- 4 a^{3} b c^{2} \Gamma \left (1 - m\right ) + 4 a^{2} b^{2} c^{2} x \Gamma \left (1 - m\right )} + \frac {a e^{m} m x^{m} \Phi \left (\frac {a}{b x}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{- 4 a^{3} b c^{2} \Gamma \left (1 - m\right ) + 4 a^{2} b^{2} c^{2} x \Gamma \left (1 - m\right )} - \frac {a e^{m} m x^{m} \Phi \left (\frac {a e^{i \pi }}{b x}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{- 4 a^{3} b c^{2} \Gamma \left (1 - m\right ) + 4 a^{2} b^{2} c^{2} x \Gamma \left (1 - m\right )} + \frac {2 b e^{m} m^{2} x x^{m} \Phi \left (\frac {a}{b x}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{- 4 a^{3} b c^{2} \Gamma \left (1 - m\right ) + 4 a^{2} b^{2} c^{2} x \Gamma \left (1 - m\right )} - \frac {b e^{m} m x x^{m} \Phi \left (\frac {a}{b x}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{- 4 a^{3} b c^{2} \Gamma \left (1 - m\right ) + 4 a^{2} b^{2} c^{2} x \Gamma \left (1 - m\right )} + \frac {b e^{m} m x x^{m} \Phi \left (\frac {a e^{i \pi }}{b x}, 1, m e^{i \pi }\right ) \Gamma \left (- m\right )}{- 4 a^{3} b c^{2} \Gamma \left (1 - m\right ) + 4 a^{2} b^{2} c^{2} x \Gamma \left (1 - m\right )} + \frac {2 b e^{m} m x x^{m} \Gamma \left (- m\right )}{- 4 a^{3} b c^{2} \Gamma \left (1 - m\right ) + 4 a^{2} b^{2} c^{2} x \Gamma \left (1 - m\right )} \] Input:
integrate((e*x)**m/(b*x+a)/(-b*c*x+a*c)**2,x)
Output:
-2*a*e**m*m**2*x**m*lerchphi(a/(b*x), 1, m*exp_polar(I*pi))*gamma(-m)/(-4* a**3*b*c**2*gamma(1 - m) + 4*a**2*b**2*c**2*x*gamma(1 - m)) + a*e**m*m*x** m*lerchphi(a/(b*x), 1, m*exp_polar(I*pi))*gamma(-m)/(-4*a**3*b*c**2*gamma( 1 - m) + 4*a**2*b**2*c**2*x*gamma(1 - m)) - a*e**m*m*x**m*lerchphi(a*exp_p olar(I*pi)/(b*x), 1, m*exp_polar(I*pi))*gamma(-m)/(-4*a**3*b*c**2*gamma(1 - m) + 4*a**2*b**2*c**2*x*gamma(1 - m)) + 2*b*e**m*m**2*x*x**m*lerchphi(a/ (b*x), 1, m*exp_polar(I*pi))*gamma(-m)/(-4*a**3*b*c**2*gamma(1 - m) + 4*a* *2*b**2*c**2*x*gamma(1 - m)) - b*e**m*m*x*x**m*lerchphi(a/(b*x), 1, m*exp_ polar(I*pi))*gamma(-m)/(-4*a**3*b*c**2*gamma(1 - m) + 4*a**2*b**2*c**2*x*g amma(1 - m)) + b*e**m*m*x*x**m*lerchphi(a*exp_polar(I*pi)/(b*x), 1, m*exp_ polar(I*pi))*gamma(-m)/(-4*a**3*b*c**2*gamma(1 - m) + 4*a**2*b**2*c**2*x*g amma(1 - m)) + 2*b*e**m*m*x*x**m*gamma(-m)/(-4*a**3*b*c**2*gamma(1 - m) + 4*a**2*b**2*c**2*x*gamma(1 - m))
\[ \int \frac {(e x)^m}{(a+b x) (a c-b c x)^2} \, dx=\int { \frac {\left (e x\right )^{m}}{{\left (b c x - a c\right )}^{2} {\left (b x + a\right )}} \,d x } \] Input:
integrate((e*x)^m/(b*x+a)/(-b*c*x+a*c)^2,x, algorithm="maxima")
Output:
integrate((e*x)^m/((b*c*x - a*c)^2*(b*x + a)), x)
\[ \int \frac {(e x)^m}{(a+b x) (a c-b c x)^2} \, dx=\int { \frac {\left (e x\right )^{m}}{{\left (b c x - a c\right )}^{2} {\left (b x + a\right )}} \,d x } \] Input:
integrate((e*x)^m/(b*x+a)/(-b*c*x+a*c)^2,x, algorithm="giac")
Output:
integrate((e*x)^m/((b*c*x - a*c)^2*(b*x + a)), x)
Timed out. \[ \int \frac {(e x)^m}{(a+b x) (a c-b c x)^2} \, dx=\int \frac {{\left (e\,x\right )}^m}{{\left (a\,c-b\,c\,x\right )}^2\,\left (a+b\,x\right )} \,d x \] Input:
int((e*x)^m/((a*c - b*c*x)^2*(a + b*x)),x)
Output:
int((e*x)^m/((a*c - b*c*x)^2*(a + b*x)), x)
\[ \int \frac {(e x)^m}{(a+b x) (a c-b c x)^2} \, dx=\frac {e^{m} \left (\int \frac {x^{m}}{b^{3} x^{3}-a \,b^{2} x^{2}-a^{2} b x +a^{3}}d x \right )}{c^{2}} \] Input:
int((e*x)^m/(b*x+a)/(-b*c*x+a*c)^2,x)
Output:
(e**m*int(x**m/(a**3 - a**2*b*x - a*b**2*x**2 + b**3*x**3),x))/c**2