\(\int \frac {(d+e x)^m (a+c x^2)^2}{(f+g x)^3} \, dx\) [160]

Optimal result

Integrand size = 24, antiderivative size = 302 \[ \int \frac {(d+e x)^m \left (a+c x^2\right )^2}{(f+g x)^3} \, dx=-\frac {c^2 (3 e f+d g) (d+e x)^{1+m}}{e^2 g^4 (1+m)}+\frac {c^2 (d+e x)^{2+m}}{e^2 g^3 (2+m)}+\frac {\left (c f^2+a g^2\right )^2 (d+e x)^{1+m}}{2 g^4 (e f-d g) (f+g x)^2}+\frac {2 c \left (3 c f^2+a g^2\right ) (d+e x)^{1+m}}{e g^4 m (f+g x)}+\frac {\left (a^2 e^2 g^4 (1-m) m-2 a c g^2 \left (2 d^2 g^2-4 d e f g (1+m)+e^2 f^2 \left (2+3 m+m^2\right )\right )-c^2 f^2 \left (12 d^2 g^2-8 d e f g (3+m)+e^2 f^2 \left (12+7 m+m^2\right )\right )\right ) (d+e x)^{1+m} \operatorname {Hypergeometric2F1}\left (2,1+m,2+m,-\frac {g (d+e x)}{e f-d g}\right )}{2 g^4 (e f-d g)^3 m (1+m)} \] Output:

-c^2*(d*g+3*e*f)*(e*x+d)^(1+m)/e^2/g^4/(1+m)+c^2*(e*x+d)^(2+m)/e^2/g^3/(2+ 
m)+1/2*(a*g^2+c*f^2)^2*(e*x+d)^(1+m)/g^4/(-d*g+e*f)/(g*x+f)^2+2*c*(a*g^2+3 
*c*f^2)*(e*x+d)^(1+m)/e/g^4/m/(g*x+f)+1/2*(a^2*e^2*g^4*(1-m)*m-2*a*c*g^2*( 
2*d^2*g^2-4*d*e*f*g*(1+m)+e^2*f^2*(m^2+3*m+2))-c^2*f^2*(12*d^2*g^2-8*d*e*f 
*g*(3+m)+e^2*f^2*(m^2+7*m+12)))*(e*x+d)^(1+m)*hypergeom([2, 1+m],[2+m],-g* 
(e*x+d)/(-d*g+e*f))/g^4/(-d*g+e*f)^3/m/(1+m)
 

Mathematica [A] (verified)

Time = 0.68 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.73 \[ \int \frac {(d+e x)^m \left (a+c x^2\right )^2}{(f+g x)^3} \, dx=\frac {(d+e x)^{1+m} \left (-\frac {c^2 (3 e f+d g)}{e^2 (1+m)}+\frac {c^2 g (d+e x)}{e^2 (2+m)}+\frac {2 c \left (3 c f^2+a g^2\right ) \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {g (d+e x)}{-e f+d g}\right )}{(e f-d g) (1+m)}-\frac {4 c e f \left (c f^2+a g^2\right ) \operatorname {Hypergeometric2F1}\left (2,1+m,2+m,\frac {g (d+e x)}{-e f+d g}\right )}{(e f-d g)^2 (1+m)}+\frac {e^2 \left (c f^2+a g^2\right )^2 \operatorname {Hypergeometric2F1}\left (3,1+m,2+m,\frac {g (d+e x)}{-e f+d g}\right )}{(e f-d g)^3 (1+m)}\right )}{g^4} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 2.11 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.24, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {650, 2124, 25, 1195, 2009}

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.

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

\(\Big \downarrow \) 650

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

\(\Big \downarrow \) 2124

\(\displaystyle \frac {\frac {\int -\frac {(d+e x)^m \left (a^2 (1-m) m e^2-\frac {2 c^2 (e f-d g)^2 x^2}{g^2}-\frac {2 a c \left (e^2 \left (m^2+3 m+2\right ) f^2-4 d e g (m+1) f+2 d^2 g^2\right )}{g^2}-\frac {c^2 f^2 \left (e^2 \left (m^2+7 m+6\right ) f^2-4 d e g (2 m+3) f+6 d^2 g^2\right )}{g^4}+\frac {4 c^2 f (e f-d g)^2 x}{g^3}\right )}{f+g x}dx}{e f-d g}+\frac {\left (a g^2+c f^2\right ) (d+e x)^{m+1} \left (a e g^2 (1-m)+c f (8 d g-e f (m+7))\right )}{g^4 (f+g x) (e f-d g)}}{2 (e f-d g)}+\frac {\left (a g^2+c f^2\right )^2 (d+e x)^{m+1}}{2 g^4 (f+g x)^2 (e f-d g)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\left (a g^2+c f^2\right ) (d+e x)^{m+1} \left (a e g^2 (1-m)+c f (8 d g-e f (m+7))\right )}{g^4 (f+g x) (e f-d g)}-\frac {\int \frac {(d+e x)^m \left (a^2 (1-m) m e^2-\frac {2 c^2 (e f-d g)^2 x^2}{g^2}-\frac {2 a c \left (e^2 \left (m^2+3 m+2\right ) f^2-4 d e g (m+1) f+2 d^2 g^2\right )}{g^2}-\frac {c^2 f^2 \left (e^2 \left (m^2+7 m+6\right ) f^2-4 d e g (2 m+3) f+6 d^2 g^2\right )}{g^4}+\frac {4 c^2 f (e f-d g)^2 x}{g^3}\right )}{f+g x}dx}{e f-d g}}{2 (e f-d g)}+\frac {\left (a g^2+c f^2\right )^2 (d+e x)^{m+1}}{2 g^4 (f+g x)^2 (e f-d g)}\)

\(\Big \downarrow \) 1195

\(\displaystyle \frac {\frac {\left (a g^2+c f^2\right ) (d+e x)^{m+1} \left (a e g^2 (1-m)+c f (8 d g-e f (m+7))\right )}{g^4 (f+g x) (e f-d g)}-\frac {\int \left (\frac {2 c^2 (e f-d g)^2 (3 e f+d g) (d+e x)^m}{e g^4}+\frac {\left (a^2 e^2 (1-m) m g^4-2 a c \left (e^2 \left (m^2+3 m+2\right ) f^2-4 d e g (m+1) f+2 d^2 g^2\right ) g^2-c^2 f^2 \left (e^2 \left (m^2+7 m+12\right ) f^2-8 d e g (m+3) f+12 d^2 g^2\right )\right ) (d+e x)^m}{g^4 (f+g x)}-\frac {2 c^2 (e f-d g)^2 (d+e x)^{m+1}}{e g^3}\right )dx}{e f-d g}}{2 (e f-d g)}+\frac {\left (a g^2+c f^2\right )^2 (d+e x)^{m+1}}{2 g^4 (f+g x)^2 (e f-d g)}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {\left (a g^2+c f^2\right ) (d+e x)^{m+1} \left (a e g^2 (1-m)+c f (8 d g-e f (m+7))\right )}{g^4 (f+g x) (e f-d g)}-\frac {\frac {(d+e x)^{m+1} \left (a^2 e^2 g^4 (1-m) m-2 a c g^2 \left (2 d^2 g^2-4 d e f g (m+1)+e^2 f^2 \left (m^2+3 m+2\right )\right )-c^2 f^2 \left (12 d^2 g^2-8 d e f g (m+3)+e^2 f^2 \left (m^2+7 m+12\right )\right )\right ) \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,-\frac {g (d+e x)}{e f-d g}\right )}{g^4 (m+1) (e f-d g)}+\frac {2 c^2 (e f-d g)^2 (d g+3 e f) (d+e x)^{m+1}}{e^2 g^4 (m+1)}-\frac {2 c^2 (e f-d g)^2 (d+e x)^{m+2}}{e^2 g^3 (m+2)}}{e f-d g}}{2 (e f-d g)}+\frac {\left (a g^2+c f^2\right )^2 (d+e x)^{m+1}}{2 g^4 (f+g x)^2 (e f-d g)}\)

Input:

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

Output:

((c*f^2 + a*g^2)^2*(d + e*x)^(1 + m))/(2*g^4*(e*f - d*g)*(f + g*x)^2) + (( 
(c*f^2 + a*g^2)*(a*e*g^2*(1 - m) + c*f*(8*d*g - e*f*(7 + m)))*(d + e*x)^(1 
 + m))/(g^4*(e*f - d*g)*(f + g*x)) - ((2*c^2*(e*f - d*g)^2*(3*e*f + d*g)*( 
d + e*x)^(1 + m))/(e^2*g^4*(1 + m)) - (2*c^2*(e*f - d*g)^2*(d + e*x)^(2 + 
m))/(e^2*g^3*(2 + m)) + ((a^2*e^2*g^4*(1 - m)*m - 2*a*c*g^2*(2*d^2*g^2 - 4 
*d*e*f*g*(1 + m) + e^2*f^2*(2 + 3*m + m^2)) - c^2*f^2*(12*d^2*g^2 - 8*d*e* 
f*g*(3 + m) + e^2*f^2*(12 + 7*m + m^2)))*(d + e*x)^(1 + m)*Hypergeometric2 
F1[1, 1 + m, 2 + m, -((g*(d + e*x))/(e*f - d*g))])/(g^4*(e*f - d*g)*(1 + m 
)))/(e*f - d*g))/(2*(e*f - d*g))
 

Defintions of rubi rules used

rule 25

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

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]
 

rule 1195

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x 
_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + 
 g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x 
] && IGtQ[p, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2124

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] : 
> With[{Qx = PolynomialQuotient[Px, a + b*x, x], R = PolynomialRemainder[Px 
, a + b*x, x]}, Simp[R*(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((m + 1)*(b*c - 
 a*d))), x] + Simp[1/((m + 1)*(b*c - a*d))   Int[(a + b*x)^(m + 1)*(c + d*x 
)^n*ExpandToSum[(m + 1)*(b*c - a*d)*Qx - d*R*(m + n + 2), x], x], x]] /; Fr 
eeQ[{a, b, c, d, n}, x] && PolyQ[Px, x] && LtQ[m, -1] && (IntegerQ[m] ||  ! 
ILtQ[n, -1])
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((c^2*x^4 + 2*a*c*x^2 + a^2)*(e*x + d)^m/(g^3*x^3 + 3*f*g^2*x^2 + 
3*f^2*g*x + f^3), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - (d + e*x)**m*a**2*d*e**2*g**4*m**4 - 2*(d + e*x)**m*a**2*d*e**2*g**4*m 
**3 + (d + e*x)**m*a**2*d*e**2*g**4*m**2 + 2*(d + e*x)**m*a**2*d*e**2*g**4 
*m + 2*(d + e*x)**m*a*c*d**2*e*f*g**3*m**3 + 6*(d + e*x)**m*a*c*d**2*e*f*g 
**3*m**2 + 4*(d + e*x)**m*a*c*d**2*e*f*g**3*m + 4*(d + e*x)**m*a*c*d**2*e* 
g**4*m**3*x + 12*(d + e*x)**m*a*c*d**2*e*g**4*m**2*x + 8*(d + e*x)**m*a*c* 
d**2*e*g**4*m*x - 2*(d + e*x)**m*a*c*d*e**2*f**2*g**2*m**3 - 10*(d + e*x)* 
*m*a*c*d*e**2*f**2*g**2*m**2 - 16*(d + e*x)**m*a*c*d*e**2*f**2*g**2*m - 8* 
(d + e*x)**m*a*c*d*e**2*f**2*g**2 - 2*(d + e*x)**m*a*c*d*e**2*f*g**3*m**4* 
x - 10*(d + e*x)**m*a*c*d*e**2*f*g**3*m**3*x - 24*(d + e*x)**m*a*c*d*e**2* 
f*g**3*m**2*x - 32*(d + e*x)**m*a*c*d*e**2*f*g**3*m*x - 16*(d + e*x)**m*a* 
c*d*e**2*f*g**3*x + 4*(d + e*x)**m*a*c*d*e**2*g**4*m**3*x**2 + 8*(d + e*x) 
**m*a*c*d*e**2*g**4*m**2*x**2 - 4*(d + e*x)**m*a*c*d*e**2*g**4*m*x**2 - 8* 
(d + e*x)**m*a*c*d*e**2*g**4*x**2 + 2*(d + e*x)**m*a*c*e**3*f**2*g**2*m**4 
*x + 10*(d + e*x)**m*a*c*e**3*f**2*g**2*m**3*x + 16*(d + e*x)**m*a*c*e**3* 
f**2*g**2*m**2*x + 8*(d + e*x)**m*a*c*e**3*f**2*g**2*m*x - 2*(d + e*x)**m* 
a*c*e**3*f*g**3*m**4*x**2 - 4*(d + e*x)**m*a*c*e**3*f*g**3*m**3*x**2 + 2*( 
d + e*x)**m*a*c*e**3*f*g**3*m**2*x**2 + 4*(d + e*x)**m*a*c*e**3*f*g**3*m*x 
**2 - 2*(d + e*x)**m*c**2*d**3*f**2*g**2*m**2 + 2*(d + e*x)**m*c**2*d**3*f 
**2*g**2*m - 4*(d + e*x)**m*c**2*d**3*f*g**3*m**2*x + 4*(d + e*x)**m*c**2* 
d**3*f*g**3*m*x - 2*(d + e*x)**m*c**2*d**3*g**4*m**2*x**2 + 2*(d + e*x)...