Integrand size = 24, antiderivative size = 151 \[ \int \frac {(e x)^m}{(a+b x) (a c-b c x)^3} \, dx=\frac {(e x)^{1+m}}{4 a^2 c^3 e (a-b x)^2}+\frac {(2-m) (e x)^{1+m}}{4 a^3 c^3 e (a-b x)}+\frac {(e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {b x}{a}\right )}{8 a^4 c^3 e (1+m)}+\frac {\left (1-4 m+2 m^2\right ) (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {b x}{a}\right )}{8 a^4 c^3 e (1+m)} \]
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)+1/8*(e*x)^(1+m)*hypergeom([1, 1+m],[2+m],-b*x/a)/a^4/c^3/e/(1+m)+1/8*( 2*m^2-4*m+1)*(e*x)^(1+m)*hypergeom([1, 1+m],[2+m],b*x/a)/a^4/c^3/e/(1+m)
Time = 0.13 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.70 \[ \int \frac {(e x)^m}{(a+b x) (a c-b c x)^3} \, dx=\frac {x (e x)^m \left (-2 a (1+m) (a (-3+m)-b (-2+m) x)+(a-b x)^2 \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {b x}{a}\right )+\left (1-4 m+2 m^2\right ) (a-b x)^2 \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {b x}{a}\right )\right )}{8 a^4 c^3 (1+m) (a-b x)^2} \]
(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 )
Time = 0.30 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.03, 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 |
| \(\displaystyle \frac {\frac {\frac {\left (2 m^2-4 m+1\right ) (e x)^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,\frac {b x}{a}\right )}{2 a e (m+1)}+\frac {(e x)^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,-\frac {b x}{a}\right )}{2 a e (m+1)}}{a}+\frac {(2-m) (e x)^{m+1}}{a e (a-b x)}}{4 a^2 c^3}+\frac {(e x)^{m+1}}{4 a^2 c^3 e (a-b x)^2}\) |
(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)
3.1.79.3.1 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_))^(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])
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]
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]
\[\int \frac {\left (e x \right )^{m}}{\left (b x +a \right ) \left (-b c x +a c \right )^{3}}d x\]
\[ \int \frac {(e x)^m}{(a+b x) (a c-b c x)^3} \, dx=\int { -\frac {\left (e x\right )^{m}}{{\left (b c x - a c\right )}^{3} {\left (b x + a\right )}} \,d x } \]
Result contains complex when optimal does not.
Time = 2.39 (sec) , antiderivative size = 1363, normalized size of antiderivative = 9.03 \[ \int \frac {(e x)^m}{(a+b x) (a c-b c x)^3} \, dx=\text {Too large to display} \]
-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 ...
\[ \int \frac {(e x)^m}{(a+b x) (a c-b c x)^3} \, dx=\int { -\frac {\left (e x\right )^{m}}{{\left (b c x - a c\right )}^{3} {\left (b x + a\right )}} \,d x } \]
\[ \int \frac {(e x)^m}{(a+b x) (a c-b c x)^3} \, dx=\int { -\frac {\left (e x\right )^{m}}{{\left (b c x - a c\right )}^{3} {\left (b x + a\right )}} \,d x } \]
Timed out. \[ \int \frac {(e x)^m}{(a+b x) (a c-b c x)^3} \, dx=\int \frac {{\left (e\,x\right )}^m}{{\left (a\,c-b\,c\,x\right )}^3\,\left (a+b\,x\right )} \,d x \]