\(\int \frac {(f+g x) (a+b x+c x^2)^p}{(d+e x)^3} \, dx\) [1124]

Optimal result

Integrand size = 25, antiderivative size = 580 \[ \int \frac {(f+g x) \left (a+b x+c x^2\right )^p}{(d+e x)^3} \, dx=-\frac {(e f-d g) \left (a+b x+c x^2\right )^{1+p}}{2 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}-\frac {(e (2 a e g-b e f (1-p)-b d g (1+p))+2 c d (d g p+e (f-f p))) \left (a+b x+c x^2\right )^{1+p}}{2 \left (c d^2-b d e+a e^2\right )^2 (d+e x)}-\frac {4^{-1+p} \left (2 c^2 d^2 (d g (1+2 p)+e (f-2 f p))-b e^2 (2 a e g-b d g (1+p)-b e (f-f p))-2 c e (a e (e f-3 d g)+b d (2 d g (1+p)+e (f-2 f p)))\right ) \left (\frac {e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{c (d+e x)}\right )^{-p} \left (\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{c (d+e x)}\right )^{-p} \left (a+b x+c x^2\right )^p \operatorname {AppellF1}\left (-2 p,-p,-p,1-2 p,\frac {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}{2 c (d+e x)},\frac {2 d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{c}}{2 (d+e x)}\right )}{e^2 \left (c d^2-b d e+a e^2\right )^2}+\frac {4^{-1-p} (1+2 p) (e (2 a e g-b e f (1-p)-b d g (1+p))+2 c d (d g p+e (f-f p))) (b+2 c x) \left (a+b x+c x^2\right )^p \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{e \left (c d^2-b d e+a e^2\right )^2} \] Output:

-1/2*(-d*g+e*f)*(c*x^2+b*x+a)^(p+1)/(a*e^2-b*d*e+c*d^2)/(e*x+d)^2-1/2*(e*( 
2*a*e*g-b*e*f*(1-p)-b*d*g*(p+1))+2*c*d*(d*g*p+e*(-f*p+f)))*(c*x^2+b*x+a)^( 
p+1)/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)-4^(-1+p)*(2*c^2*d^2*(d*g*(1+2*p)+e*(-2* 
f*p+f))-b*e^2*(2*a*e*g-b*d*g*(p+1)-b*e*(-f*p+f))-2*c*e*(a*e*(-3*d*g+e*f)+b 
*d*(2*d*g*(p+1)+e*(-2*f*p+f))))*(c*x^2+b*x+a)^p*AppellF1(-2*p,-p,-p,1-2*p, 
(2*d-(b+(-4*a*c+b^2)^(1/2))*e/c)/(2*e*x+2*d),1/2*(2*c*d-(b-(-4*a*c+b^2)^(1 
/2))*e)/c/(e*x+d))/e^2/(a*e^2-b*d*e+c*d^2)^2/((e*(b-(-4*a*c+b^2)^(1/2)+2*c 
*x)/c/(e*x+d))^p)/((e*(b+(-4*a*c+b^2)^(1/2)+2*c*x)/c/(e*x+d))^p)+4^(-1-p)* 
(1+2*p)*(e*(2*a*e*g-b*e*f*(1-p)-b*d*g*(p+1))+2*c*d*(d*g*p+e*(-f*p+f)))*(2* 
c*x+b)*(c*x^2+b*x+a)^p*hypergeom([1/2, -p],[3/2],(2*c*x+b)^2/(-4*a*c+b^2)) 
/e/(a*e^2-b*d*e+c*d^2)^2/((-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^p)
 

Mathematica [A] (warning: unable to verify)

Time = 6.16 (sec) , antiderivative size = 315, normalized size of antiderivative = 0.54 \[ \int \frac {(f+g x) \left (a+b x+c x^2\right )^p}{(d+e x)^3} \, dx=\frac {2^{-1+2 p} \left (\frac {e \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{c (d+e x)}\right )^{-p} \left (\frac {e \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{c (d+e x)}\right )^{-p} (a+x (b+c x))^p \left (2 g (-1+p) (d+e x) \operatorname {AppellF1}\left (1-2 p,-p,-p,2-2 p,\frac {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c (d+e x)},\frac {2 c d-b e+\sqrt {b^2-4 a c} e}{2 c d+2 c e x}\right )+(e f-d g) (-1+2 p) \operatorname {AppellF1}\left (2-2 p,-p,-p,3-2 p,\frac {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c (d+e x)},\frac {2 c d-b e+\sqrt {b^2-4 a c} e}{2 c d+2 c e x}\right )\right )}{e^2 (-1+p) (-1+2 p) (d+e x)^2} \] Input:

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

Output:

(2^(-1 + 2*p)*(a + x*(b + c*x))^p*(2*g*(-1 + p)*(d + e*x)*AppellF1[1 - 2*p 
, -p, -p, 2 - 2*p, (2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)/(2*c*(d + e*x)), (2 
*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)/(2*c*d + 2*c*e*x)] + (e*f - d*g)*(-1 + 2 
*p)*AppellF1[2 - 2*p, -p, -p, 3 - 2*p, (2*c*d - (b + Sqrt[b^2 - 4*a*c])*e) 
/(2*c*(d + e*x)), (2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)/(2*c*d + 2*c*e*x)])) 
/(e^2*(-1 + p)*(-1 + 2*p)*((e*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x 
)))^p*((e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(c*(d + e*x)))^p*(d + e*x)^2)
 

Rubi [A] (warning: unable to verify)

Time = 2.14 (sec) , antiderivative size = 628, normalized size of antiderivative = 1.08, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {1237, 25, 1237, 25, 1269, 1096, 1178, 150}

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.

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

\(\Big \downarrow \) 1237

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1237

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1269

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

\(\Big \downarrow \) 1096

\(\displaystyle \frac {\frac {-\frac {p \left (-2 c e (a e (e f-3 d g)+b d (2 d g (p+1)+e (f-2 f p)))-b e^2 (2 a e g-b d g (p+1)-b e (f-f p))+2 c^2 d^2 (d g (2 p+1)+e (f-2 f p))\right ) \int \frac {\left (c x^2+b x+a\right )^p}{d+e x}dx}{e}-\frac {c 2^{p+1} (2 p+1) \left (a+b x+c x^2\right )^{p+1} \left (-\frac {-\sqrt {b^2-4 a c}+b+2 c x}{\sqrt {b^2-4 a c}}\right )^{-p-1} \operatorname {Hypergeometric2F1}\left (-p,p+1,p+2,\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}\right ) (e (2 a e g-b d g (p+1)-b e f (1-p))+2 c d (d g p+e (f-f p)))}{e (p+1) \sqrt {b^2-4 a c}}}{a e^2-b d e+c d^2}-\frac {\left (a+b x+c x^2\right )^{p+1} (e (2 a e g-b d g (p+1)-b e f (1-p))+2 c d (d g p+e (f-f p)))}{(d+e x) \left (a e^2-b d e+c d^2\right )}}{2 \left (a e^2-b d e+c d^2\right )}-\frac {(e f-d g) \left (a+b x+c x^2\right )^{p+1}}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1178

\(\displaystyle \frac {\frac {\frac {4^p p \left (\frac {1}{d+e x}\right )^{2 p} \left (a+b x+c x^2\right )^p \left (\frac {e \left (-\sqrt {b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{-p} \left (\frac {e \left (\sqrt {b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{-p} \left (-2 c e (a e (e f-3 d g)+b d (2 d g (p+1)+e (f-2 f p)))-b e^2 (2 a e g-b d g (p+1)-b e (f-f p))+2 c^2 d^2 (d g (2 p+1)+e (f-2 f p))\right ) \int \left (\frac {1}{d+e x}\right )^{-2 p-1} \left (1-\frac {2 d-\frac {\left (b-\sqrt {b^2-4 a c}\right ) e}{c}}{2 (d+e x)}\right )^p \left (1-\frac {2 d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{c}}{2 (d+e x)}\right )^pd\frac {1}{d+e x}}{e^2}-\frac {c 2^{p+1} (2 p+1) \left (-\frac {-\sqrt {b^2-4 a c}+b+2 c x}{\sqrt {b^2-4 a c}}\right )^{-p-1} \left (a+b x+c x^2\right )^{p+1} \operatorname {Hypergeometric2F1}\left (-p,p+1,p+2,\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}\right ) (e (2 a e g-b d g (p+1)-b e f (1-p))+2 c d (d g p+e (f-f p)))}{e (p+1) \sqrt {b^2-4 a c}}}{a e^2-b d e+c d^2}-\frac {\left (a+b x+c x^2\right )^{p+1} (e (2 a e g-b d g (p+1)-b e f (1-p))+2 c d (d g p+e (f-f p)))}{(d+e x) \left (a e^2-b d e+c d^2\right )}}{2 \left (a e^2-b d e+c d^2\right )}-\frac {(e f-d g) \left (a+b x+c x^2\right )^{p+1}}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 150

\(\displaystyle \frac {\frac {-\frac {2^{2 p-1} \left (a+b x+c x^2\right )^p \left (\frac {e \left (-\sqrt {b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{-p} \left (\frac {e \left (\sqrt {b^2-4 a c}+b+2 c x\right )}{c (d+e x)}\right )^{-p} \operatorname {AppellF1}\left (-2 p,-p,-p,1-2 p,\frac {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}{2 c (d+e x)},\frac {2 d-\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{c}}{2 (d+e x)}\right ) \left (-2 c e (a e (e f-3 d g)+b d (2 d g (p+1)+e (f-2 f p)))-b e^2 (2 a e g-b d g (p+1)-b e (f-f p))+2 c^2 d^2 (d g (2 p+1)+e (f-2 f p))\right )}{e^2}-\frac {c 2^{p+1} (2 p+1) \left (a+b x+c x^2\right )^{p+1} \left (-\frac {-\sqrt {b^2-4 a c}+b+2 c x}{\sqrt {b^2-4 a c}}\right )^{-p-1} \operatorname {Hypergeometric2F1}\left (-p,p+1,p+2,\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}\right ) (e (2 a e g-b d g (p+1)-b e f (1-p))+2 c d (d g p+e (f-f p)))}{e (p+1) \sqrt {b^2-4 a c}}}{a e^2-b d e+c d^2}-\frac {\left (a+b x+c x^2\right )^{p+1} (e (2 a e g-b d g (p+1)-b e f (1-p))+2 c d (d g p+e (f-f p)))}{(d+e x) \left (a e^2-b d e+c d^2\right )}}{2 \left (a e^2-b d e+c d^2\right )}-\frac {(e f-d g) \left (a+b x+c x^2\right )^{p+1}}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}\)

Input:

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

Output:

-1/2*((e*f - d*g)*(a + b*x + c*x^2)^(1 + p))/((c*d^2 - b*d*e + a*e^2)*(d + 
 e*x)^2) + (-(((e*(2*a*e*g - b*e*f*(1 - p) - b*d*g*(1 + p)) + 2*c*d*(d*g*p 
 + e*(f - f*p)))*(a + b*x + c*x^2)^(1 + p))/((c*d^2 - b*d*e + a*e^2)*(d + 
e*x))) + (-((2^(-1 + 2*p)*(2*c^2*d^2*(d*g*(1 + 2*p) + e*(f - 2*f*p)) - b*e 
^2*(2*a*e*g - b*d*g*(1 + p) - b*e*(f - f*p)) - 2*c*e*(a*e*(e*f - 3*d*g) + 
b*d*(2*d*g*(1 + p) + e*(f - 2*f*p))))*(a + b*x + c*x^2)^p*AppellF1[-2*p, - 
p, -p, 1 - 2*p, (2*c*d - (b - Sqrt[b^2 - 4*a*c])*e)/(2*c*(d + e*x)), (2*d 
- ((b + Sqrt[b^2 - 4*a*c])*e)/c)/(2*(d + e*x))])/(e^2*((e*(b - Sqrt[b^2 - 
4*a*c] + 2*c*x))/(c*(d + e*x)))^p*((e*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(c* 
(d + e*x)))^p)) - (2^(1 + p)*c*(1 + 2*p)*(e*(2*a*e*g - b*e*f*(1 - p) - b*d 
*g*(1 + p)) + 2*c*d*(d*g*p + e*(f - f*p)))*(-((b - Sqrt[b^2 - 4*a*c] + 2*c 
*x)/Sqrt[b^2 - 4*a*c]))^(-1 - p)*(a + b*x + c*x^2)^(1 + p)*Hypergeometric2 
F1[-p, 1 + p, 2 + p, (b + Sqrt[b^2 - 4*a*c] + 2*c*x)/(2*Sqrt[b^2 - 4*a*c]) 
])/(Sqrt[b^2 - 4*a*c]*e*(1 + p)))/(c*d^2 - b*d*e + a*e^2))/(2*(c*d^2 - b*d 
*e + a*e^2))
 

Defintions of rubi rules used

rule 25

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

rule 150

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ 
] :> Simp[c^n*e^p*((b*x)^(m + 1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2 
, (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p}, x] &&  !In 
tegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])
 

rule 1096

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Rt[b^2 
 - 4*a*c, 2]}, Simp[(-(a + b*x + c*x^2)^(p + 1)/(q*(p + 1)*((q - b - 2*c*x) 
/(2*q))^(p + 1)))*Hypergeometric2F1[-p, p + 1, p + 2, (b + q + 2*c*x)/(2*q) 
], x]] /; FreeQ[{a, b, c, p}, x] &&  !IntegerQ[4*p] &&  !IntegerQ[3*p]
 

rule 1178

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(-(1/(d + e*x))^(2*p))*((a + 
b*x + c*x^2)^p/(e*(e*((b - q + 2*c*x)/(2*c*(d + e*x))))^p*(e*((b + q + 2*c* 
x)/(2*c*(d + e*x))))^p))   Subst[Int[x^(-m - 2*(p + 1))*Simp[1 - (d - e*((b 
 - q)/(2*c)))*x, x]^p*Simp[1 - (d - e*((b + q)/(2*c)))*x, x]^p, x], x, 1/(d 
 + e*x)], x]] /; FreeQ[{a, b, c, d, e, p}, x] && ILtQ[m, 0]
 

rule 1237

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

rule 1269

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + b*x + 
 c*x^2)^p, x], x] + Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + b*x + c*x^2)^ 
p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] &&  !IGtQ[m, 0]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(2*(a + b*x + c*x**2)**p*a*e*g*p - (a + b*x + c*x**2)**p*b*d*g*p - (a + b* 
x + c*x**2)**p*b*d*g + 2*(a + b*x + c*x**2)**p*b*e*f*p - (a + b*x + c*x**2 
)**p*b*e*f + (a + b*x + c*x**2)**p*b*e*g*p*x - 2*(a + b*x + c*x**2)**p*b*e 
*g*x + 2*(a + b*x + c*x**2)**p*c*d*g*p*x + 8*int((a + b*x + c*x**2)**p/(2* 
a*b*d**3*e*p**2 - 5*a*b*d**3*e*p + 2*a*b*d**3*e + 6*a*b*d**2*e**2*p**2*x - 
 15*a*b*d**2*e**2*p*x + 6*a*b*d**2*e**2*x + 6*a*b*d*e**3*p**2*x**2 - 15*a* 
b*d*e**3*p*x**2 + 6*a*b*d*e**3*x**2 + 2*a*b*e**4*p**2*x**3 - 5*a*b*e**4*p* 
x**3 + 2*a*b*e**4*x**3 + 4*a*c*d**4*p**2 - 2*a*c*d**4*p + 12*a*c*d**3*e*p* 
*2*x - 6*a*c*d**3*e*p*x + 12*a*c*d**2*e**2*p**2*x**2 - 6*a*c*d**2*e**2*p*x 
**2 + 4*a*c*d*e**3*p**2*x**3 - 2*a*c*d*e**3*p*x**3 + 2*b**2*d**3*e*p**2*x 
- 5*b**2*d**3*e*p*x + 2*b**2*d**3*e*x + 6*b**2*d**2*e**2*p**2*x**2 - 15*b* 
*2*d**2*e**2*p*x**2 + 6*b**2*d**2*e**2*x**2 + 6*b**2*d*e**3*p**2*x**3 - 15 
*b**2*d*e**3*p*x**3 + 6*b**2*d*e**3*x**3 + 2*b**2*e**4*p**2*x**4 - 5*b**2* 
e**4*p*x**4 + 2*b**2*e**4*x**4 + 4*b*c*d**4*p**2*x - 2*b*c*d**4*p*x + 14*b 
*c*d**3*e*p**2*x**2 - 11*b*c*d**3*e*p*x**2 + 2*b*c*d**3*e*x**2 + 18*b*c*d* 
*2*e**2*p**2*x**3 - 21*b*c*d**2*e**2*p*x**3 + 6*b*c*d**2*e**2*x**3 + 10*b* 
c*d*e**3*p**2*x**4 - 17*b*c*d*e**3*p*x**4 + 6*b*c*d*e**3*x**4 + 2*b*c*e**4 
*p**2*x**5 - 5*b*c*e**4*p*x**5 + 2*b*c*e**4*x**5 + 4*c**2*d**4*p**2*x**2 - 
 2*c**2*d**4*p*x**2 + 12*c**2*d**3*e*p**2*x**3 - 6*c**2*d**3*e*p*x**3 + 12 
*c**2*d**2*e**2*p**2*x**4 - 6*c**2*d**2*e**2*p*x**4 + 4*c**2*d*e**3*p**...