\(\int \frac {(e x)^m}{(a+b x) (a c-b c x)^3} \, dx\) [325]

Optimal result

Integrand size = 24, antiderivative size = 178 \[ \int \frac {(e x)^m}{(a+b x) (a c-b c x)^3} \, dx=\frac {(e x)^{1+m}}{4 a^2 c^3 e (a-b x)^2}+\frac {(2-m) (e x)^{1+m}}{4 a^3 c^3 e (a-b x)}+\frac {(1-m)^2 (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},\frac {b^2 x^2}{a^2}\right )}{4 a^4 c^3 e (1+m)}-\frac {b (2-m) m (e x)^{2+m} \operatorname {Hypergeometric2F1}\left (1,\frac {2+m}{2},\frac {4+m}{2},\frac {b^2 x^2}{a^2}\right )}{4 a^5 c^3 e^2 (2+m)} \] Output:

1/4*(e*x)^(1+m)/a^2/c^3/e/(-b*x+a)^2+1/4*(2-m)*(e*x)^(1+m)/a^3/c^3/e/(-b*x 
+a)+1/4*(1-m)^2*(e*x)^(1+m)*hypergeom([1, 1/2+1/2*m],[3/2+1/2*m],b^2*x^2/a 
^2)/a^4/c^3/e/(1+m)-1/4*b*(2-m)*m*(e*x)^(2+m)*hypergeom([1, 1+1/2*m],[2+1/ 
2*m],b^2*x^2/a^2)/a^5/c^3/e^2/(2+m)
 

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.60 \[ \int \frac {(e x)^m}{(a+b x) (a c-b c x)^3} \, dx=\frac {x (e x)^m \left (-2 a (1+m) (a (-3+m)-b (-2+m) x)+(a-b x)^2 \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {b x}{a}\right )+\left (1-4 m+2 m^2\right ) (a-b x)^2 \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {b x}{a}\right )\right )}{8 a^4 c^3 (1+m) (a-b x)^2} \] Input:

Integrate[(e*x)^m/((a + b*x)*(a*c - b*c*x)^3),x]
 

Output:

(x*(e*x)^m*(-2*a*(1 + m)*(a*(-3 + m) - b*(-2 + m)*x) + (a - b*x)^2*Hyperge 
ometric2F1[1, 1 + m, 2 + m, -((b*x)/a)] + (1 - 4*m + 2*m^2)*(a - b*x)^2*Hy 
pergeometric2F1[1, 1 + m, 2 + m, (b*x)/a]))/(8*a^4*c^3*(1 + m)*(a - b*x)^2 
)
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.87, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {114, 25, 27, 168, 27, 174, 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.

\(\displaystyle \int \frac {(e x)^m}{(a+b x) (a c-b c x)^3} \, dx\)

\(\Big \downarrow \) 114

\(\displaystyle \frac {(e x)^{m+1}}{4 a^2 c^3 e (a-b x)^2}-\frac {\int -\frac {b e (e x)^m (a (3-m)+b (1-m) x)}{c (a-b x)^2 (a+b x)}dx}{4 a^2 b c^2 e}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {b e (e x)^m (a (3-m)+b (1-m) x)}{c (a-b x)^2 (a+b x)}dx}{4 a^2 b c^2 e}+\frac {(e x)^{m+1}}{4 a^2 c^3 e (a-b x)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {(e x)^m (a (3-m)+b (1-m) x)}{(a-b x)^2 (a+b x)}dx}{4 a^2 c^3}+\frac {(e x)^{m+1}}{4 a^2 c^3 e (a-b x)^2}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {\frac {(2-m) (e x)^{m+1}}{a e (a-b x)}-\frac {\int -\frac {2 a b e (e x)^m \left (a (1-m)^2-b (2-m) m x\right )}{(a-b x) (a+b x)}dx}{2 a^2 b e}}{4 a^2 c^3}+\frac {(e x)^{m+1}}{4 a^2 c^3 e (a-b x)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {(e x)^m \left (a (1-m)^2-b (2-m) m x\right )}{(a-b x) (a+b x)}dx}{a}+\frac {(2-m) (e x)^{m+1}}{a e (a-b x)}}{4 a^2 c^3}+\frac {(e x)^{m+1}}{4 a^2 c^3 e (a-b x)^2}\)

\(\Big \downarrow \) 174

\(\displaystyle \frac {\frac {\frac {1}{2} \left (2 m^2-4 m+1\right ) \int \frac {(e x)^m}{a-b x}dx+\frac {1}{2} \int \frac {(e x)^m}{a+b x}dx}{a}+\frac {(2-m) (e x)^{m+1}}{a e (a-b x)}}{4 a^2 c^3}+\frac {(e x)^{m+1}}{4 a^2 c^3 e (a-b x)^2}\)

\(\Big \downarrow \) 74

\(\displaystyle \frac {\frac {\frac {\left (2 m^2-4 m+1\right ) (e x)^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,\frac {b x}{a}\right )}{2 a e (m+1)}+\frac {(e x)^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,-\frac {b x}{a}\right )}{2 a e (m+1)}}{a}+\frac {(2-m) (e x)^{m+1}}{a e (a-b x)}}{4 a^2 c^3}+\frac {(e x)^{m+1}}{4 a^2 c^3 e (a-b x)^2}\)

Input:

Int[(e*x)^m/((a + b*x)*(a*c - b*c*x)^3),x]
 

Output:

(e*x)^(1 + m)/(4*a^2*c^3*e*(a - b*x)^2) + (((2 - m)*(e*x)^(1 + m))/(a*e*(a 
 - b*x)) + (((e*x)^(1 + m)*Hypergeometric2F1[1, 1 + m, 2 + m, -((b*x)/a)]) 
/(2*a*e*(1 + m)) + ((1 - 4*m + 2*m^2)*(e*x)^(1 + m)*Hypergeometric2F1[1, 1 
 + m, 2 + m, (b*x)/a])/(2*a*e*(1 + m)))/a)/(4*a^2*c^3)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 74

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])))
 

rule 114

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 
)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e 
 - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) 
 - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || 
 IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
 

rule 168

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + 
 d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S 
imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n 
*(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
 

rule 174

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* 
((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d)   Int[(e + f*x)^ 
p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d)   Int[(e + f*x)^p/(c + d 
*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
 

Maple [F]

\[\int \frac {\left (e x \right )^{m}}{\left (b x +a \right ) \left (-b c x +a c \right )^{3}}d x\]

Input:

int((e*x)^m/(b*x+a)/(-b*c*x+a*c)^3,x)
 

Output:

int((e*x)^m/(b*x+a)/(-b*c*x+a*c)^3,x)
 

Fricas [F]

\[ \int \frac {(e x)^m}{(a+b x) (a c-b c x)^3} \, dx=\int { -\frac {\left (e x\right )^{m}}{{\left (b c x - a c\right )}^{3} {\left (b x + a\right )}} \,d x } \] Input:

integrate((e*x)^m/(b*x+a)/(-b*c*x+a*c)^3,x, algorithm="fricas")
 

Output:

integral(-(e*x)^m/(b^4*c^3*x^4 - 2*a*b^3*c^3*x^3 + 2*a^3*b*c^3*x - a^4*c^3 
), x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 2.23 (sec) , antiderivative size = 1363, normalized size of antiderivative = 7.66 \[ \int \frac {(e x)^m}{(a+b x) (a c-b c x)^3} \, dx=\text {Too large to display} \] Input:

integrate((e*x)**m/(b*x+a)/(-b*c*x+a*c)**3,x)
 

Output:

-2*a**2*e**m*m**3*x**m*lerchphi(a/(b*x), 1, m*exp_polar(I*pi))*gamma(-m)/( 
8*a**5*b*c**3*gamma(1 - m) - 16*a**4*b**2*c**3*x*gamma(1 - m) + 8*a**3*b** 
3*c**3*x**2*gamma(1 - m)) + 4*a**2*e**m*m**2*x**m*lerchphi(a/(b*x), 1, m*e 
xp_polar(I*pi))*gamma(-m)/(8*a**5*b*c**3*gamma(1 - m) - 16*a**4*b**2*c**3* 
x*gamma(1 - m) + 8*a**3*b**3*c**3*x**2*gamma(1 - m)) - a**2*e**m*m*x**m*le 
rchphi(a/(b*x), 1, m*exp_polar(I*pi))*gamma(-m)/(8*a**5*b*c**3*gamma(1 - m 
) - 16*a**4*b**2*c**3*x*gamma(1 - m) + 8*a**3*b**3*c**3*x**2*gamma(1 - m)) 
 + a**2*e**m*m*x**m*lerchphi(a*exp_polar(I*pi)/(b*x), 1, m*exp_polar(I*pi) 
)*gamma(-m)/(8*a**5*b*c**3*gamma(1 - m) - 16*a**4*b**2*c**3*x*gamma(1 - m) 
 + 8*a**3*b**3*c**3*x**2*gamma(1 - m)) + 4*a*b*e**m*m**3*x*x**m*lerchphi(a 
/(b*x), 1, m*exp_polar(I*pi))*gamma(-m)/(8*a**5*b*c**3*gamma(1 - m) - 16*a 
**4*b**2*c**3*x*gamma(1 - m) + 8*a**3*b**3*c**3*x**2*gamma(1 - m)) - 8*a*b 
*e**m*m**2*x*x**m*lerchphi(a/(b*x), 1, m*exp_polar(I*pi))*gamma(-m)/(8*a** 
5*b*c**3*gamma(1 - m) - 16*a**4*b**2*c**3*x*gamma(1 - m) + 8*a**3*b**3*c** 
3*x**2*gamma(1 - m)) + 2*a*b*e**m*m**2*x*x**m*gamma(-m)/(8*a**5*b*c**3*gam 
ma(1 - m) - 16*a**4*b**2*c**3*x*gamma(1 - m) + 8*a**3*b**3*c**3*x**2*gamma 
(1 - m)) + 2*a*b*e**m*m*x*x**m*lerchphi(a/(b*x), 1, m*exp_polar(I*pi))*gam 
ma(-m)/(8*a**5*b*c**3*gamma(1 - m) - 16*a**4*b**2*c**3*x*gamma(1 - m) + 8* 
a**3*b**3*c**3*x**2*gamma(1 - m)) - 2*a*b*e**m*m*x*x**m*lerchphi(a*exp_pol 
ar(I*pi)/(b*x), 1, m*exp_polar(I*pi))*gamma(-m)/(8*a**5*b*c**3*gamma(1 ...
 

Maxima [F]

\[ \int \frac {(e x)^m}{(a+b x) (a c-b c x)^3} \, dx=\int { -\frac {\left (e x\right )^{m}}{{\left (b c x - a c\right )}^{3} {\left (b x + a\right )}} \,d x } \] Input:

integrate((e*x)^m/(b*x+a)/(-b*c*x+a*c)^3,x, algorithm="maxima")
 

Output:

-integrate((e*x)^m/((b*c*x - a*c)^3*(b*x + a)), x)
 

Giac [F]

\[ \int \frac {(e x)^m}{(a+b x) (a c-b c x)^3} \, dx=\int { -\frac {\left (e x\right )^{m}}{{\left (b c x - a c\right )}^{3} {\left (b x + a\right )}} \,d x } \] Input:

integrate((e*x)^m/(b*x+a)/(-b*c*x+a*c)^3,x, algorithm="giac")
 

Output:

integrate(-(e*x)^m/((b*c*x - a*c)^3*(b*x + a)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(e x)^m}{(a+b x) (a c-b c x)^3} \, dx=\int \frac {{\left (e\,x\right )}^m}{{\left (a\,c-b\,c\,x\right )}^3\,\left (a+b\,x\right )} \,d x \] Input:

int((e*x)^m/((a*c - b*c*x)^3*(a + b*x)),x)
 

Output:

int((e*x)^m/((a*c - b*c*x)^3*(a + b*x)), x)
 

Reduce [F]

\[ \int \frac {(e x)^m}{(a+b x) (a c-b c x)^3} \, dx=\frac {e^{m} \left (\int \frac {x^{m}}{-b^{4} x^{4}+2 a \,b^{3} x^{3}-2 a^{3} b x +a^{4}}d x \right )}{c^{3}} \] Input:

int((e*x)^m/(b*x+a)/(-b*c*x+a*c)^3,x)
 

Output:

(e**m*int(x**m/(a**4 - 2*a**3*b*x + 2*a*b**3*x**3 - b**4*x**4),x))/c**3