\(\int \frac {(e x)^m (c+d x)}{(a x^2+b x^3)^2} \, dx\) [360]

Optimal result

Integrand size = 24, antiderivative size = 85 \[ \int \frac {(e x)^m (c+d x)}{\left (a x^2+b x^3\right )^2} \, dx=-\frac {d e^3 (e x)^{-3+m}}{b (4-m) (a+b x)}-\frac {e^3 \left (\frac {c}{3-m}-\frac {a d}{4 b-b m}\right ) (e x)^{-3+m} \operatorname {Hypergeometric2F1}\left (2,-3+m,-2+m,-\frac {b x}{a}\right )}{a^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.81 \[ \int \frac {(e x)^m (c+d x)}{\left (a x^2+b x^3\right )^2} \, dx=\frac {(e x)^m \left (\frac {a (b c-a d)}{a+b x}-\frac {(b c (-4+m)-a d (-3+m)) \operatorname {Hypergeometric2F1}\left (1,-3+m,-2+m,-\frac {b x}{a}\right )}{-3+m}\right )}{a^2 b x^3} \] Input:

Integrate[((e*x)^m*(c + d*x))/(a*x^2 + b*x^3)^2,x]
 

Output:

((e*x)^m*((a*(b*c - a*d))/(a + b*x) - ((b*c*(-4 + m) - a*d*(-3 + m))*Hyper 
geometric2F1[1, -3 + m, -2 + m, -((b*x)/a)])/(-3 + m)))/(a^2*b*x^3)
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.11, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {9, 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.

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

\(\Big \downarrow \) 9

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

\(\Big \downarrow \) 87

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

\(\Big \downarrow \) 74

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

Input:

Int[((e*x)^m*(c + d*x))/(a*x^2 + b*x^3)^2,x]
 

Output:

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

Defintions of rubi rules used

rule 9

Int[(u_.)*(Px_)^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> With[{r = Expon[Px, 
x, Min]}, Simp[1/e^(p*r)   Int[u*(e*x)^(m + p*r)*ExpandToSum[Px/x^r, x]^p, 
x], x] /; IGtQ[r, 0]] /; FreeQ[{e, m}, x] && PolyQ[Px, x] && IntegerQ[p] && 
  !MonomialQ[Px, 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 87

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]))))
 

Maple [F]

\[\int \frac {\left (e x \right )^{m} \left (d x +c \right )}{\left (b \,x^{3}+a \,x^{2}\right )^{2}}d x\]

Input:

int((e*x)^m*(d*x+c)/(b*x^3+a*x^2)^2,x)
 

Output:

int((e*x)^m*(d*x+c)/(b*x^3+a*x^2)^2,x)
 

Fricas [F]

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

integrate((e*x)^m*(d*x+c)/(b*x^3+a*x^2)^2,x, algorithm="fricas")
 

Output:

integral((d*x + c)*(e*x)^m/(b^2*x^6 + 2*a*b*x^5 + a^2*x^4), x)
 

Sympy [F]

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

integrate((e*x)**m*(d*x+c)/(b*x**3+a*x**2)**2,x)
 

Output:

Integral((e*x)**m*(c + d*x)/(x**4*(a + b*x)**2), x)
 

Maxima [F]

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

integrate((e*x)^m*(d*x+c)/(b*x^3+a*x^2)^2,x, algorithm="maxima")
 

Output:

integrate((d*x + c)*(e*x)^m/(b*x^3 + a*x^2)^2, x)
 

Giac [F]

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

integrate((e*x)^m*(d*x+c)/(b*x^3+a*x^2)^2,x, algorithm="giac")
 

Output:

integrate((d*x + c)*(e*x)^m/(b*x^3 + a*x^2)^2, x)
 

Mupad [F(-1)]

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

int(((e*x)^m*(c + d*x))/(a*x^2 + b*x^3)^2,x)
 

Output:

int(((e*x)^m*(c + d*x))/(a*x^2 + b*x^3)^2, x)
 

Reduce [F]

\[ \int \frac {(e x)^m (c+d x)}{\left (a x^2+b x^3\right )^2} \, dx=\frac {e^{m} \left (x^{m} d -\left (\int \frac {x^{m}}{b^{2} m \,x^{6}+2 a b m \,x^{5}-4 b^{2} x^{6}+a^{2} m \,x^{4}-8 a b \,x^{5}-4 a^{2} x^{4}}d x \right ) a^{2} d \,m^{2} x^{3}+7 \left (\int \frac {x^{m}}{b^{2} m \,x^{6}+2 a b m \,x^{5}-4 b^{2} x^{6}+a^{2} m \,x^{4}-8 a b \,x^{5}-4 a^{2} x^{4}}d x \right ) a^{2} d m \,x^{3}-12 \left (\int \frac {x^{m}}{b^{2} m \,x^{6}+2 a b m \,x^{5}-4 b^{2} x^{6}+a^{2} m \,x^{4}-8 a b \,x^{5}-4 a^{2} x^{4}}d x \right ) a^{2} d \,x^{3}+\left (\int \frac {x^{m}}{b^{2} m \,x^{6}+2 a b m \,x^{5}-4 b^{2} x^{6}+a^{2} m \,x^{4}-8 a b \,x^{5}-4 a^{2} x^{4}}d x \right ) a b c \,m^{2} x^{3}-8 \left (\int \frac {x^{m}}{b^{2} m \,x^{6}+2 a b m \,x^{5}-4 b^{2} x^{6}+a^{2} m \,x^{4}-8 a b \,x^{5}-4 a^{2} x^{4}}d x \right ) a b c m \,x^{3}+16 \left (\int \frac {x^{m}}{b^{2} m \,x^{6}+2 a b m \,x^{5}-4 b^{2} x^{6}+a^{2} m \,x^{4}-8 a b \,x^{5}-4 a^{2} x^{4}}d x \right ) a b c \,x^{3}-\left (\int \frac {x^{m}}{b^{2} m \,x^{6}+2 a b m \,x^{5}-4 b^{2} x^{6}+a^{2} m \,x^{4}-8 a b \,x^{5}-4 a^{2} x^{4}}d x \right ) a b d \,m^{2} x^{4}+7 \left (\int \frac {x^{m}}{b^{2} m \,x^{6}+2 a b m \,x^{5}-4 b^{2} x^{6}+a^{2} m \,x^{4}-8 a b \,x^{5}-4 a^{2} x^{4}}d x \right ) a b d m \,x^{4}-12 \left (\int \frac {x^{m}}{b^{2} m \,x^{6}+2 a b m \,x^{5}-4 b^{2} x^{6}+a^{2} m \,x^{4}-8 a b \,x^{5}-4 a^{2} x^{4}}d x \right ) a b d \,x^{4}+\left (\int \frac {x^{m}}{b^{2} m \,x^{6}+2 a b m \,x^{5}-4 b^{2} x^{6}+a^{2} m \,x^{4}-8 a b \,x^{5}-4 a^{2} x^{4}}d x \right ) b^{2} c \,m^{2} x^{4}-8 \left (\int \frac {x^{m}}{b^{2} m \,x^{6}+2 a b m \,x^{5}-4 b^{2} x^{6}+a^{2} m \,x^{4}-8 a b \,x^{5}-4 a^{2} x^{4}}d x \right ) b^{2} c m \,x^{4}+16 \left (\int \frac {x^{m}}{b^{2} m \,x^{6}+2 a b m \,x^{5}-4 b^{2} x^{6}+a^{2} m \,x^{4}-8 a b \,x^{5}-4 a^{2} x^{4}}d x \right ) b^{2} c \,x^{4}\right )}{b \,x^{3} \left (b m x +a m -4 b x -4 a \right )} \] Input:

int((e*x)^m*(d*x+c)/(b*x^3+a*x^2)^2,x)
 

Output:

(e**m*(x**m*d - int(x**m/(a**2*m*x**4 - 4*a**2*x**4 + 2*a*b*m*x**5 - 8*a*b 
*x**5 + b**2*m*x**6 - 4*b**2*x**6),x)*a**2*d*m**2*x**3 + 7*int(x**m/(a**2* 
m*x**4 - 4*a**2*x**4 + 2*a*b*m*x**5 - 8*a*b*x**5 + b**2*m*x**6 - 4*b**2*x* 
*6),x)*a**2*d*m*x**3 - 12*int(x**m/(a**2*m*x**4 - 4*a**2*x**4 + 2*a*b*m*x* 
*5 - 8*a*b*x**5 + b**2*m*x**6 - 4*b**2*x**6),x)*a**2*d*x**3 + int(x**m/(a* 
*2*m*x**4 - 4*a**2*x**4 + 2*a*b*m*x**5 - 8*a*b*x**5 + b**2*m*x**6 - 4*b**2 
*x**6),x)*a*b*c*m**2*x**3 - 8*int(x**m/(a**2*m*x**4 - 4*a**2*x**4 + 2*a*b* 
m*x**5 - 8*a*b*x**5 + b**2*m*x**6 - 4*b**2*x**6),x)*a*b*c*m*x**3 + 16*int( 
x**m/(a**2*m*x**4 - 4*a**2*x**4 + 2*a*b*m*x**5 - 8*a*b*x**5 + b**2*m*x**6 
- 4*b**2*x**6),x)*a*b*c*x**3 - int(x**m/(a**2*m*x**4 - 4*a**2*x**4 + 2*a*b 
*m*x**5 - 8*a*b*x**5 + b**2*m*x**6 - 4*b**2*x**6),x)*a*b*d*m**2*x**4 + 7*i 
nt(x**m/(a**2*m*x**4 - 4*a**2*x**4 + 2*a*b*m*x**5 - 8*a*b*x**5 + b**2*m*x* 
*6 - 4*b**2*x**6),x)*a*b*d*m*x**4 - 12*int(x**m/(a**2*m*x**4 - 4*a**2*x**4 
 + 2*a*b*m*x**5 - 8*a*b*x**5 + b**2*m*x**6 - 4*b**2*x**6),x)*a*b*d*x**4 + 
int(x**m/(a**2*m*x**4 - 4*a**2*x**4 + 2*a*b*m*x**5 - 8*a*b*x**5 + b**2*m*x 
**6 - 4*b**2*x**6),x)*b**2*c*m**2*x**4 - 8*int(x**m/(a**2*m*x**4 - 4*a**2* 
x**4 + 2*a*b*m*x**5 - 8*a*b*x**5 + b**2*m*x**6 - 4*b**2*x**6),x)*b**2*c*m* 
x**4 + 16*int(x**m/(a**2*m*x**4 - 4*a**2*x**4 + 2*a*b*m*x**5 - 8*a*b*x**5 
+ b**2*m*x**6 - 4*b**2*x**6),x)*b**2*c*x**4))/(b*x**3*(a*m - 4*a + b*m*x - 
 4*b*x))