\(\int (d+e x)^m (-c d^2+b d e+b e^2 x+c e^2 x^2)^p ((-c d+b e) f+(c e f-c d g+b e g) x+c e g x^2) \, dx\) [3]

Optimal result

Integrand size = 69, antiderivative size = 187 \[ \int (d+e x)^m \left (-c d^2+b d e+b e^2 x+c e^2 x^2\right )^p \left ((-c d+b e) f+(c e f-c d g+b e g) x+c e g x^2\right ) \, dx=\frac {g (d+e x)^{-1+m} \left (-d (c d-b e)+b e^2 x+c e^2 x^2\right )^{2+p}}{c e^2 (3+m+2 p)}+\frac {(b e g (1+m+p)+c (d g (1-m)-e f (3+m+2 p))) (d+e x)^{-1+m} \left (-d (c d-b e)+b e^2 x+c e^2 x^2\right )^{2+p} \operatorname {Hypergeometric2F1}\left (1,3+m+2 p,2+m+p,\frac {c (d+e x)}{2 c d-b e}\right )}{c e^2 (2 c d-b e) (1+m+p) (3+m+2 p)} \] Output:

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

Mathematica [A] (warning: unable to verify)

Time = 0.68 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.88 \[ \int (d+e x)^m \left (-c d^2+b d e+b e^2 x+c e^2 x^2\right )^p \left ((-c d+b e) f+(c e f-c d g+b e g) x+c e g x^2\right ) \, dx=\frac {(d+e x)^m (-c d+b e+c e x)^2 (-((d+e x) (-b e+c (d-e x))))^p \left (c e g (d+e x)+\frac {e (c d g (-1+m)-b e g (1+m+p)+c e f (3+m+2 p)) \left (\frac {c (d+e x)}{2 c d-b e}\right )^{-m-p} \operatorname {Hypergeometric2F1}\left (-m-p,2+p,3+p,\frac {-c d+b e+c e x}{-2 c d+b e}\right )}{2+p}\right )}{c^2 e^3 (3+m+2 p)} \] Input:

Integrate[(d + e*x)^m*(-(c*d^2) + b*d*e + b*e^2*x + c*e^2*x^2)^p*((-(c*d) 
+ b*e)*f + (c*e*f - c*d*g + b*e*g)*x + c*e*g*x^2),x]
 

Output:

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

Rubi [A] (warning: unable to verify)

Time = 0.60 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.22, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.101, Rules used = {2163, 27, 1221, 1139, 1138, 80, 79}

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.

\(\displaystyle \int (d+e x)^m \left (b d e+b e^2 x-c d^2+c e^2 x^2\right )^p \left (x (b e g-c d g+c e f)+f (b e-c d)+c e g x^2\right ) \, dx\)

\(\Big \downarrow \) 2163

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1221

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

\(\Big \downarrow \) 1139

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

\(\Big \downarrow \) 1138

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

\(\Big \downarrow \) 80

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

\(\Big \downarrow \) 79

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

Input:

Int[(d + e*x)^m*(-(c*d^2) + b*d*e + b*e^2*x + c*e^2*x^2)^p*((-(c*d) + b*e) 
*f + (c*e*f - c*d*g + b*e*g)*x + c*e*g*x^2),x]
 

Output:

(g*(d + e*x)^(-1 + m)*(-(d*(c*d - b*e)) + b*e^2*x + c*e^2*x^2)^(2 + p))/(c 
*e^2*(3 + m + 2*p)) - ((c*d*g*(1 - m) + b*e*g*(1 + m + p) - c*e*f*(3 + m + 
 2*p))*(d + e*x)^m*((c*(d + e*x))/(2*c*d - b*e))^(-m - p)*(-(d*(c*d - b*e) 
) + c*d*e*x)^2*(-(d*(c*d - b*e)) + b*e^2*x + c*e^2*x^2)^p*Hypergeometric2F 
1[-m - p, 2 + p, 3 + p, (c*d - b*e - c*e*x)/(2*c*d - b*e)])/(c^2*d^2*e^2*( 
2 + p)*(3 + m + 2*p))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 79

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( 
a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 
, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] 
&&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] 
 ||  !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
 

rule 80

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

rule 1138

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Sy 
mbol] :> Simp[d^m*((a + b*x + c*x^2)^FracPart[p]/((1 + e*(x/d))^FracPart[p] 
*(a/d + (c*x)/e)^FracPart[p]))   Int[(1 + e*(x/d))^(m + p)*(a/d + (c/e)*x)^ 
p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] 
&& (IntegerQ[m] || GtQ[d, 0]) &&  !(IGtQ[m, 0] && (IntegerQ[3*p] || Integer 
Q[4*p]))
 

rule 1139

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Sy 
mbol] :> Simp[d^IntPart[m]*((d + e*x)^FracPart[m]/(1 + e*(x/d))^FracPart[m] 
)   Int[(1 + e*(x/d))^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e 
, m}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !(IntegerQ[m] || GtQ[d, 0])
 

rule 1221

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

rule 2163

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_ 
.), x_Symbol] :> Simp[d*e   Int[(d + e*x)^(m - 1)*PolynomialQuotient[Pq, a* 
e + c*d*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m, 
 p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2 
, 0] && EqQ[PolynomialRemainder[Pq, a*e + c*d*x, x], 0]
 

Maple [F]

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

Input:

int((e*x+d)^m*(c*e^2*x^2+b*e^2*x+b*d*e-c*d^2)^p*((b*e-c*d)*f+(b*e*g-c*d*g+ 
c*e*f)*x+c*e*g*x^2),x)
 

Output:

int((e*x+d)^m*(c*e^2*x^2+b*e^2*x+b*d*e-c*d^2)^p*((b*e-c*d)*f+(b*e*g-c*d*g+ 
c*e*f)*x+c*e*g*x^2),x)
 

Fricas [F]

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

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

Output:

integral((c*e*g*x^2 - (c*d - b*e)*f + (c*e*f - (c*d - b*e)*g)*x)*(c*e^2*x^ 
2 + b*e^2*x - c*d^2 + b*d*e)^p*(e*x + d)^m, x)
 

Sympy [F(-2)]

Exception generated. \[ \int (d+e x)^m \left (-c d^2+b d e+b e^2 x+c e^2 x^2\right )^p \left ((-c d+b e) f+(c e f-c d g+b e g) x+c e g x^2\right ) \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:

integrate((e*x+d)**m*(c*e**2*x**2+b*e**2*x+b*d*e-c*d**2)**p*((b*e-c*d)*f+( 
b*e*g-c*d*g+c*e*f)*x+c*e*g*x**2),x)
 

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

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

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

Output:

integrate((c*e*g*x^2 - (c*d - b*e)*f + (c*e*f - c*d*g + b*e*g)*x)*(c*e^2*x 
^2 + b*e^2*x - c*d^2 + b*d*e)^p*(e*x + d)^m, x)
 

Giac [F]

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

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

Output:

integrate((c*e*g*x^2 - (c*d - b*e)*f + (c*e*f - c*d*g + b*e*g)*x)*(c*e^2*x 
^2 + b*e^2*x - c*d^2 + b*d*e)^p*(e*x + d)^m, x)
 

Mupad [F(-1)]

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

int((d + e*x)^m*(f*(b*e - c*d) + x*(b*e*g - c*d*g + c*e*f) + c*e*g*x^2)*(c 
*e^2*x^2 - c*d^2 + b*d*e + b*e^2*x)^p,x)
 

Output:

int((d + e*x)^m*(f*(b*e - c*d) + x*(b*e*g - c*d*g + c*e*f) + c*e*g*x^2)*(c 
*e^2*x^2 - c*d^2 + b*d*e + b*e^2*x)^p, x)
 

Reduce [F]

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

int((e*x+d)^m*(c*e^2*x^2+b*e^2*x+b*d*e-c*d^2)^p*((b*e-c*d)*f+(b*e*g-c*d*g+ 
c*e*f)*x+c*e*g*x^2),x)
 

Output:

( - (d + e*x)**m*(b*d*e + b*e**2*x - c*d**2 + c*e**2*x**2)**p*b**3*d*e**3* 
g*p**2 - (d + e*x)**m*(b*d*e + b*e**2*x - c*d**2 + c*e**2*x**2)**p*b**3*d* 
e**3*g*p + (d + e*x)**m*(b*d*e + b*e**2*x - c*d**2 + c*e**2*x**2)**p*b**3* 
e**4*g*m*p**2*x + (d + e*x)**m*(b*d*e + b*e**2*x - c*d**2 + c*e**2*x**2)** 
p*b**3*e**4*g*m*p*x + (d + e*x)**m*(b*d*e + b*e**2*x - c*d**2 + c*e**2*x** 
2)**p*b**3*e**4*g*p**3*x + (d + e*x)**m*(b*d*e + b*e**2*x - c*d**2 + c*e** 
2*x**2)**p*b**3*e**4*g*p**2*x - (d + e*x)**m*(b*d*e + b*e**2*x - c*d**2 + 
c*e**2*x**2)**p*b**2*c*d**2*e**2*g*m**2 - 5*(d + e*x)**m*(b*d*e + b*e**2*x 
 - c*d**2 + c*e**2*x**2)**p*b**2*c*d**2*e**2*g*m*p - 3*(d + e*x)**m*(b*d*e 
 + b*e**2*x - c*d**2 + c*e**2*x**2)**p*b**2*c*d**2*e**2*g*m - (d + e*x)**m 
*(b*d*e + b*e**2*x - c*d**2 + c*e**2*x**2)**p*b**2*c*d**2*e**2*g*p**2 + (d 
 + e*x)**m*(b*d*e + b*e**2*x - c*d**2 + c*e**2*x**2)**p*b**2*c*d*e**3*f*m* 
*3 + 6*(d + e*x)**m*(b*d*e + b*e**2*x - c*d**2 + c*e**2*x**2)**p*b**2*c*d* 
e**3*f*m**2*p + 5*(d + e*x)**m*(b*d*e + b*e**2*x - c*d**2 + c*e**2*x**2)** 
p*b**2*c*d*e**3*f*m**2 + 12*(d + e*x)**m*(b*d*e + b*e**2*x - c*d**2 + c*e* 
*2*x**2)**p*b**2*c*d*e**3*f*m*p**2 + 19*(d + e*x)**m*(b*d*e + b*e**2*x - c 
*d**2 + c*e**2*x**2)**p*b**2*c*d*e**3*f*m*p + 6*(d + e*x)**m*(b*d*e + b*e* 
*2*x - c*d**2 + c*e**2*x**2)**p*b**2*c*d*e**3*f*m + 8*(d + e*x)**m*(b*d*e 
+ b*e**2*x - c*d**2 + c*e**2*x**2)**p*b**2*c*d*e**3*f*p**3 + 18*(d + e*x)* 
*m*(b*d*e + b*e**2*x - c*d**2 + c*e**2*x**2)**p*b**2*c*d*e**3*f*p**2 + ...