\(\int \frac {(d+e x)^{-2+m} (a+c x^2)}{(f+g x)^3} \, dx\) [155]

Optimal result

Integrand size = 24, antiderivative size = 194 \[ \int \frac {(d+e x)^{-2+m} \left (a+c x^2\right )}{(f+g x)^3} \, dx=\frac {\left (c f^2+a g^2\right ) (d+e x)^{-1+m}}{2 g^2 (e f-d g) (f+g x)^2}-\frac {c (d+e x)^{-1+m}}{e g^2 (2-m) (f+g x)}-\frac {\left (c \left (2 d^2 g^2+4 d e f g (1-m)-e^2 f^2 (1-m) m\right )+a e^2 g^2 \left (6-5 m+m^2\right )\right ) (d+e x)^{-1+m} \operatorname {Hypergeometric2F1}\left (2,-1+m,m,-\frac {g (d+e x)}{e f-d g}\right )}{2 g^2 (e f-d g)^3 (1-m) (2-m)} \] Output:

1/2*(a*g^2+c*f^2)*(e*x+d)^(-1+m)/g^2/(-d*g+e*f)/(g*x+f)^2-c*(e*x+d)^(-1+m) 
/e/g^2/(2-m)/(g*x+f)-1/2*(c*(2*d^2*g^2+4*d*e*f*g*(1-m)-e^2*f^2*(1-m)*m)+a* 
e^2*g^2*(m^2-5*m+6))*(e*x+d)^(-1+m)*hypergeom([2, -1+m],[m],-g*(e*x+d)/(-d 
*g+e*f))/g^2/(-d*g+e*f)^3/(1-m)/(2-m)
 

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.72 \[ \int \frac {(d+e x)^{-2+m} \left (a+c x^2\right )}{(f+g x)^3} \, dx=\frac {(d+e x)^{-1+m} \left (-c (e f-d g)^2 \operatorname {Hypergeometric2F1}\left (1,-1+m,m,\frac {g (d+e x)}{-e f+d g}\right )+e \left (2 c f (e f-d g) \operatorname {Hypergeometric2F1}\left (2,-1+m,m,\frac {g (d+e x)}{-e f+d g}\right )-e \left (c f^2+a g^2\right ) \operatorname {Hypergeometric2F1}\left (3,-1+m,m,\frac {g (d+e x)}{-e f+d g}\right )\right )\right )}{g^2 (-e f+d g)^3 (-1+m)} \] Input:

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

Output:

((d + e*x)^(-1 + m)*(-(c*(e*f - d*g)^2*Hypergeometric2F1[1, -1 + m, m, (g* 
(d + e*x))/(-(e*f) + d*g)]) + e*(2*c*f*(e*f - d*g)*Hypergeometric2F1[2, -1 
 + m, m, (g*(d + e*x))/(-(e*f) + d*g)] - e*(c*f^2 + a*g^2)*Hypergeometric2 
F1[3, -1 + m, m, (g*(d + e*x))/(-(e*f) + d*g)])))/(g^2*(-(e*f) + d*g)^3*(- 
1 + m))
 

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.14, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {650, 87, 78}

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.

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

\(\Big \downarrow \) 650

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

\(\Big \downarrow \) 87

\(\displaystyle \frac {\frac {\left (a e^2 g^2 \left (m^2-5 m+6\right )+c \left (2 d^2 g^2+4 d e f g (1-m)-e^2 f^2 (1-m) m\right )\right ) \int \frac {(d+e x)^{m-2}}{f+g x}dx}{g^2 (e f-d g)}+\frac {(d+e x)^{m-1} \left (a e g^2 (3-m)+c f (4 d g-e f (m+1))\right )}{g^2 (f+g x) (e f-d g)}}{2 (e f-d g)}+\frac {\left (a+\frac {c f^2}{g^2}\right ) (d+e x)^{m-1}}{2 (f+g x)^2 (e f-d g)}\)

\(\Big \downarrow \) 78

\(\displaystyle \frac {\frac {(d+e x)^{m-1} \left (a e g^2 (3-m)+c f (4 d g-e f (m+1))\right )}{g^2 (f+g x) (e f-d g)}-\frac {(d+e x)^{m-1} \left (a e^2 g^2 \left (m^2-5 m+6\right )+c \left (2 d^2 g^2+4 d e f g (1-m)-e^2 f^2 (1-m) m\right )\right ) \operatorname {Hypergeometric2F1}\left (1,m-1,m,-\frac {g (d+e x)}{e f-d g}\right )}{g^2 (1-m) (e f-d g)^2}}{2 (e f-d g)}+\frac {\left (a+\frac {c f^2}{g^2}\right ) (d+e x)^{m-1}}{2 (f+g x)^2 (e f-d g)}\)

Input:

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

Output:

((a + (c*f^2)/g^2)*(d + e*x)^(-1 + m))/(2*(e*f - d*g)*(f + g*x)^2) + (((a* 
e*g^2*(3 - m) + c*f*(4*d*g - e*f*(1 + m)))*(d + e*x)^(-1 + m))/(g^2*(e*f - 
 d*g)*(f + g*x)) - ((c*(2*d^2*g^2 + 4*d*e*f*g*(1 - m) - e^2*f^2*(1 - m)*m) 
 + a*e^2*g^2*(6 - 5*m + m^2))*(d + e*x)^(-1 + m)*Hypergeometric2F1[1, -1 + 
 m, m, -((g*(d + e*x))/(e*f - d*g))])/(g^2*(e*f - d*g)^2*(1 - m)))/(2*(e*f 
 - d*g))
 

Defintions of rubi rules used

rule 78

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(b 
*c - a*d)^n*((a + b*x)^(m + 1)/(b^(n + 1)*(m + 1)))*Hypergeometric2F1[-n, m 
 + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m}, x] 
 &&  !IntegerQ[m] && IntegerQ[n]
 

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]))))
 

rule 650

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^ 
2)^(p_.), x_Symbol] :> With[{Qx = PolynomialQuotient[(a + c*x^2)^p, d + e*x 
, x], R = PolynomialRemainder[(a + c*x^2)^p, d + e*x, x]}, Simp[R*(d + e*x) 
^(m + 1)*((f + g*x)^(n + 1)/((m + 1)*(e*f - d*g))), x] + Simp[1/((m + 1)*(e 
*f - d*g))   Int[(d + e*x)^(m + 1)*(f + g*x)^n*ExpandToSum[(m + 1)*(e*f - d 
*g)*Qx - g*R*(m + n + 2), x], x], x]] /; FreeQ[{a, c, d, e, f, g, n}, x] && 
 IGtQ[p, 0] && ILtQ[2*m, -2] &&  !IntegerQ[n]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((c*x^2 + a)*(e*x + d)^(m - 2)/(g^3*x^3 + 3*f*g^2*x^2 + 3*f^2*g*x 
+ f^3), x)
 

Sympy [F]

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

integrate((e*x+d)**(-2+m)*(c*x**2+a)/(g*x+f)**3,x)
 

Output:

Integral((a + c*x**2)*(d + e*x)**(m - 2)/(f + g*x)**3, x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(d+e x)^{-2+m} \left (a+c x^2\right )}{(f+g x)^3} \, dx=\text {too large to display} \] Input:

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

Output:

( - (d + e*x)**m*a*e*g*m + 2*(d + e*x)**m*a*e*g + (d + e*x)**m*c*d*f + 2*( 
d + e*x)**m*c*d*g*x - (d + e*x)**m*c*e*f*m*x + (d + e*x)**m*c*e*f*x + 2*in 
t(((d + e*x)**m*x)/(2*d**3*f**3*g*m - 4*d**3*f**3*g + 6*d**3*f**2*g**2*m*x 
 - 12*d**3*f**2*g**2*x + 6*d**3*f*g**3*m*x**2 - 12*d**3*f*g**3*x**2 + 2*d* 
*3*g**4*m*x**3 - 4*d**3*g**4*x**3 - d**2*e*f**4*m**2 + 3*d**2*e*f**4*m - 2 
*d**2*e*f**4 - 3*d**2*e*f**3*g*m**2*x + 13*d**2*e*f**3*g*m*x - 14*d**2*e*f 
**3*g*x - 3*d**2*e*f**2*g**2*m**2*x**2 + 21*d**2*e*f**2*g**2*m*x**2 - 30*d 
**2*e*f**2*g**2*x**2 - d**2*e*f*g**3*m**2*x**3 + 15*d**2*e*f*g**3*m*x**3 - 
 26*d**2*e*f*g**3*x**3 + 4*d**2*e*g**4*m*x**4 - 8*d**2*e*g**4*x**4 - 2*d*e 
**2*f**4*m**2*x + 6*d*e**2*f**4*m*x - 4*d*e**2*f**4*x - 6*d*e**2*f**3*g*m* 
*2*x**2 + 20*d*e**2*f**3*g*m*x**2 - 16*d*e**2*f**3*g*x**2 - 6*d*e**2*f**2* 
g**2*m**2*x**3 + 24*d*e**2*f**2*g**2*m*x**3 - 24*d*e**2*f**2*g**2*x**3 - 2 
*d*e**2*f*g**3*m**2*x**4 + 12*d*e**2*f*g**3*m*x**4 - 16*d*e**2*f*g**3*x**4 
 + 2*d*e**2*g**4*m*x**5 - 4*d*e**2*g**4*x**5 - e**3*f**4*m**2*x**2 + 3*e** 
3*f**4*m*x**2 - 2*e**3*f**4*x**2 - 3*e**3*f**3*g*m**2*x**3 + 9*e**3*f**3*g 
*m*x**3 - 6*e**3*f**3*g*x**3 - 3*e**3*f**2*g**2*m**2*x**4 + 9*e**3*f**2*g* 
*2*m*x**4 - 6*e**3*f**2*g**2*x**4 - e**3*f*g**3*m**2*x**5 + 3*e**3*f*g**3* 
m*x**5 - 2*e**3*f*g**3*x**5),x)*a*d**2*e**2*f**2*g**3*m**3 - 14*int(((d + 
e*x)**m*x)/(2*d**3*f**3*g*m - 4*d**3*f**3*g + 6*d**3*f**2*g**2*m*x - 12*d* 
*3*f**2*g**2*x + 6*d**3*f*g**3*m*x**2 - 12*d**3*f*g**3*x**2 + 2*d**3*g*...