\(\int (a+b x)^m (c+d x)^{2-m} (e+f x)^3 \, dx\) [1767]

Optimal result

Integrand size = 26, antiderivative size = 447 \[ \int (a+b x)^m (c+d x)^{2-m} (e+f x)^3 \, dx=\frac {f \left (a^2 d^2 f^2 \left (60-13 m+m^2\right )-2 a b d f \left (9 d e (8-m)-c f \left (12+4 m-m^2\right )\right )+b^2 \left (90 d^2 e^2-18 c d e f (2+m)+c^2 f^2 \left (6+5 m+m^2\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{3-m}}{120 b^3 d^3}+\frac {f^2 (18 b d e-a d f (15-m)-b c f (3+m)) (a+b x)^{2+m} (c+d x)^{3-m}}{30 b^3 d^2}+\frac {f^3 (a+b x)^{3+m} (c+d x)^{3-m}}{6 b^3 d}-\frac {\left (a^3 d^3 f^3 \left (60-47 m+12 m^2-m^3\right )-3 a^2 b d^2 f^2 \left (12-7 m+m^2\right ) (6 d e-c f (1+m))+3 a b^2 d f (3-m) \left (30 d^2 e^2-12 c d e f (1+m)+c^2 f^2 \left (2+3 m+m^2\right )\right )-b^3 \left (120 d^3 e^3-90 c d^2 e^2 f (1+m)+18 c^2 d e f^2 \left (2+3 m+m^2\right )-c^3 f^3 \left (6+11 m+6 m^2+m^3\right )\right )\right ) (a+b x)^{1+m} (c+d x)^{3-m} \operatorname {Hypergeometric2F1}\left (1,4,2+m,-\frac {d (a+b x)}{b c-a d}\right )}{120 b^3 d^3 (b c-a d) (1+m)} \] Output:

1/120*f*(a^2*d^2*f^2*(m^2-13*m+60)-2*a*b*d*f*(9*d*e*(8-m)-c*f*(-m^2+4*m+12 
))+b^2*(90*d^2*e^2-18*c*d*e*f*(2+m)+c^2*f^2*(m^2+5*m+6)))*(b*x+a)^(1+m)*(d 
*x+c)^(3-m)/b^3/d^3+1/30*f^2*(18*b*d*e-a*d*f*(15-m)-b*c*f*(3+m))*(b*x+a)^( 
2+m)*(d*x+c)^(3-m)/b^3/d^2+1/6*f^3*(b*x+a)^(3+m)*(d*x+c)^(3-m)/b^3/d-1/120 
*(a^3*d^3*f^3*(-m^3+12*m^2-47*m+60)-3*a^2*b*d^2*f^2*(m^2-7*m+12)*(6*d*e-c* 
f*(1+m))+3*a*b^2*d*f*(3-m)*(30*d^2*e^2-12*c*d*e*f*(1+m)+c^2*f^2*(m^2+3*m+2 
))-b^3*(120*d^3*e^3-90*c*d^2*e^2*f*(1+m)+18*c^2*d*e*f^2*(m^2+3*m+2)-c^3*f^ 
3*(m^3+6*m^2+11*m+6)))*(b*x+a)^(1+m)*(d*x+c)^(3-m)*hypergeom([1, 4],[2+m], 
-d*(b*x+a)/(-a*d+b*c))/b^3/d^3/(-a*d+b*c)/(1+m)
 

Mathematica [A] (warning: unable to verify)

Time = 0.59 (sec) , antiderivative size = 320, normalized size of antiderivative = 0.72 \[ \int (a+b x)^m (c+d x)^{2-m} (e+f x)^3 \, dx=\frac {(a+b x)^{1+m} (c+d x)^{-m} \left (b^5 d^2 f (1+m) (c+d x)^3 (e+f x)^2+(b c-a d)^4 f^2 (8 b d e+a d f (-5+m)-b c f (3+m)) \left (\frac {b (c+d x)}{b c-a d}\right )^m \operatorname {Hypergeometric2F1}\left (-4+m,1+m,2+m,\frac {d (a+b x)}{-b c+a d}\right )+2 b (b c-a d)^3 f (d e-c f) (7 b d e+a d f (-4+m)-b c f (3+m)) \left (\frac {b (c+d x)}{b c-a d}\right )^m \operatorname {Hypergeometric2F1}\left (-3+m,1+m,2+m,\frac {d (a+b x)}{-b c+a d}\right )+b^2 (b c-a d)^2 (d e-c f)^2 (6 b d e+a d f (-3+m)-b c f (3+m)) \left (\frac {b (c+d x)}{b c-a d}\right )^m \operatorname {Hypergeometric2F1}\left (-2+m,1+m,2+m,\frac {d (a+b x)}{-b c+a d}\right )\right )}{6 b^6 d^3 (1+m)} \] Input:

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

Output:

((a + b*x)^(1 + m)*(b^5*d^2*f*(1 + m)*(c + d*x)^3*(e + f*x)^2 + (b*c - a*d 
)^4*f^2*(8*b*d*e + a*d*f*(-5 + m) - b*c*f*(3 + m))*((b*(c + d*x))/(b*c - a 
*d))^m*Hypergeometric2F1[-4 + m, 1 + m, 2 + m, (d*(a + b*x))/(-(b*c) + a*d 
)] + 2*b*(b*c - a*d)^3*f*(d*e - c*f)*(7*b*d*e + a*d*f*(-4 + m) - b*c*f*(3 
+ m))*((b*(c + d*x))/(b*c - a*d))^m*Hypergeometric2F1[-3 + m, 1 + m, 2 + m 
, (d*(a + b*x))/(-(b*c) + a*d)] + b^2*(b*c - a*d)^2*(d*e - c*f)^2*(6*b*d*e 
 + a*d*f*(-3 + m) - b*c*f*(3 + m))*((b*(c + d*x))/(b*c - a*d))^m*Hypergeom 
etric2F1[-2 + m, 1 + m, 2 + m, (d*(a + b*x))/(-(b*c) + a*d)]))/(6*b^6*d^3* 
(1 + m)*(c + d*x)^m)
 

Rubi [A] (warning: unable to verify)

Time = 0.64 (sec) , antiderivative size = 457, normalized size of antiderivative = 1.02, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {111, 25, 164, 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.

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

\(\Big \downarrow \) 111

\(\displaystyle \frac {\int -(a+b x)^m (c+d x)^{2-m} (e+f x) (a f (2 c f+d e (3-m))-b e (6 d e-c f (m+1))-f (8 b d e-a d f (5-m)-b c f (m+3)) x)dx}{6 b d}+\frac {f (e+f x)^2 (a+b x)^{m+1} (c+d x)^{3-m}}{6 b d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {f (e+f x)^2 (a+b x)^{m+1} (c+d x)^{3-m}}{6 b d}-\frac {\int (a+b x)^m (c+d x)^{2-m} (e+f x) (a f (2 c f+d e (3-m))-b e (6 d e-c f (m+1))-f (8 b d e-a d f (5-m)-b c f (m+3)) x)dx}{6 b d}\)

\(\Big \downarrow \) 164

\(\displaystyle \frac {f (e+f x)^2 (a+b x)^{m+1} (c+d x)^{3-m}}{6 b d}-\frac {\frac {\left (a^3 d^3 f^3 \left (-m^3+12 m^2-47 m+60\right )-3 a^2 b d^2 f^2 \left (m^2-7 m+12\right ) (6 d e-c f (m+1))+3 a b^2 d f (3-m) \left (c^2 f^2 \left (m^2+3 m+2\right )-12 c d e f (m+1)+30 d^2 e^2\right )-\left (b^3 \left (-c^3 f^3 \left (m^3+6 m^2+11 m+6\right )+18 c^2 d e f^2 \left (m^2+3 m+2\right )-90 c d^2 e^2 f (m+1)+120 d^3 e^3\right )\right )\right ) \int (a+b x)^m (c+d x)^{2-m}dx}{20 b^2 d^2}-\frac {f (a+b x)^{m+1} (c+d x)^{3-m} \left (a^2 d^2 f^2 \left (m^2-9 m+20\right )-2 a b d f \left (9 d e (4-m)-c f \left (-m^2+2 m+6\right )\right )+4 b d f x (-a d f (5-m)-b c f (m+3)+8 b d e)+b^2 \left (c^2 f^2 \left (m^2+5 m+6\right )-18 c d e f (m+2)+70 d^2 e^2\right )\right )}{20 b^2 d^2}}{6 b d}\)

\(\Big \downarrow \) 80

\(\displaystyle \frac {f (e+f x)^2 (a+b x)^{m+1} (c+d x)^{3-m}}{6 b d}-\frac {\frac {(b c-a d)^2 (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m \left (a^3 d^3 f^3 \left (-m^3+12 m^2-47 m+60\right )-3 a^2 b d^2 f^2 \left (m^2-7 m+12\right ) (6 d e-c f (m+1))+3 a b^2 d f (3-m) \left (c^2 f^2 \left (m^2+3 m+2\right )-12 c d e f (m+1)+30 d^2 e^2\right )-\left (b^3 \left (-c^3 f^3 \left (m^3+6 m^2+11 m+6\right )+18 c^2 d e f^2 \left (m^2+3 m+2\right )-90 c d^2 e^2 f (m+1)+120 d^3 e^3\right )\right )\right ) \int (a+b x)^m \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^{2-m}dx}{20 b^4 d^2}-\frac {f (a+b x)^{m+1} (c+d x)^{3-m} \left (a^2 d^2 f^2 \left (m^2-9 m+20\right )-2 a b d f \left (9 d e (4-m)-c f \left (-m^2+2 m+6\right )\right )+4 b d f x (-a d f (5-m)-b c f (m+3)+8 b d e)+b^2 \left (c^2 f^2 \left (m^2+5 m+6\right )-18 c d e f (m+2)+70 d^2 e^2\right )\right )}{20 b^2 d^2}}{6 b d}\)

\(\Big \downarrow \) 79

\(\displaystyle \frac {f (e+f x)^2 (a+b x)^{m+1} (c+d x)^{3-m}}{6 b d}-\frac {\frac {(b c-a d)^2 (a+b x)^{m+1} (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m \left (a^3 d^3 f^3 \left (-m^3+12 m^2-47 m+60\right )-3 a^2 b d^2 f^2 \left (m^2-7 m+12\right ) (6 d e-c f (m+1))+3 a b^2 d f (3-m) \left (c^2 f^2 \left (m^2+3 m+2\right )-12 c d e f (m+1)+30 d^2 e^2\right )-\left (b^3 \left (-c^3 f^3 \left (m^3+6 m^2+11 m+6\right )+18 c^2 d e f^2 \left (m^2+3 m+2\right )-90 c d^2 e^2 f (m+1)+120 d^3 e^3\right )\right )\right ) \operatorname {Hypergeometric2F1}\left (m-2,m+1,m+2,-\frac {d (a+b x)}{b c-a d}\right )}{20 b^5 d^2 (m+1)}-\frac {f (a+b x)^{m+1} (c+d x)^{3-m} \left (a^2 d^2 f^2 \left (m^2-9 m+20\right )-2 a b d f \left (9 d e (4-m)-c f \left (-m^2+2 m+6\right )\right )+4 b d f x (-a d f (5-m)-b c f (m+3)+8 b d e)+b^2 \left (c^2 f^2 \left (m^2+5 m+6\right )-18 c d e f (m+2)+70 d^2 e^2\right )\right )}{20 b^2 d^2}}{6 b d}\)

Input:

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

Output:

(f*(a + b*x)^(1 + m)*(c + d*x)^(3 - m)*(e + f*x)^2)/(6*b*d) - (-1/20*(f*(a 
 + b*x)^(1 + m)*(c + d*x)^(3 - m)*(a^2*d^2*f^2*(20 - 9*m + m^2) - 2*a*b*d* 
f*(9*d*e*(4 - m) - c*f*(6 + 2*m - m^2)) + b^2*(70*d^2*e^2 - 18*c*d*e*f*(2 
+ m) + c^2*f^2*(6 + 5*m + m^2)) + 4*b*d*f*(8*b*d*e - a*d*f*(5 - m) - b*c*f 
*(3 + m))*x))/(b^2*d^2) + ((b*c - a*d)^2*(a^3*d^3*f^3*(60 - 47*m + 12*m^2 
- m^3) - 3*a^2*b*d^2*f^2*(12 - 7*m + m^2)*(6*d*e - c*f*(1 + m)) + 3*a*b^2* 
d*f*(3 - m)*(30*d^2*e^2 - 12*c*d*e*f*(1 + m) + c^2*f^2*(2 + 3*m + m^2)) - 
b^3*(120*d^3*e^3 - 90*c*d^2*e^2*f*(1 + m) + 18*c^2*d*e*f^2*(2 + 3*m + m^2) 
 - c^3*f^3*(6 + 11*m + 6*m^2 + m^3)))*(a + b*x)^(1 + m)*((b*(c + d*x))/(b* 
c - a*d))^m*Hypergeometric2F1[-2 + m, 1 + m, 2 + m, -((d*(a + b*x))/(b*c - 
 a*d))])/(20*b^5*d^2*(1 + m)*(c + d*x)^m))/(6*b*d)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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 111

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

rule 164

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_ 
))*((g_.) + (h_.)*(x_)), x_] :> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - 
 b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)^(m + 1)*(( 
c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Simp[(a^2*d^2*f*h 
*(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 
3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + 
d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3))   Int[( 
a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] 
&& NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-2)]

Exception generated. \[ \int (a+b x)^m (c+d x)^{2-m} (e+f x)^3 \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:

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

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int((a + b*x)**m/(c + d*x)**m,x)*c**2*e**3 + int(((a + b*x)**m*x**5)/(c + 
d*x)**m,x)*d**2*f**3 + 2*int(((a + b*x)**m*x**4)/(c + d*x)**m,x)*c*d*f**3 
+ 3*int(((a + b*x)**m*x**4)/(c + d*x)**m,x)*d**2*e*f**2 + int(((a + b*x)** 
m*x**3)/(c + d*x)**m,x)*c**2*f**3 + 6*int(((a + b*x)**m*x**3)/(c + d*x)**m 
,x)*c*d*e*f**2 + 3*int(((a + b*x)**m*x**3)/(c + d*x)**m,x)*d**2*e**2*f + 3 
*int(((a + b*x)**m*x**2)/(c + d*x)**m,x)*c**2*e*f**2 + 6*int(((a + b*x)**m 
*x**2)/(c + d*x)**m,x)*c*d*e**2*f + int(((a + b*x)**m*x**2)/(c + d*x)**m,x 
)*d**2*e**3 + 3*int(((a + b*x)**m*x)/(c + d*x)**m,x)*c**2*e**2*f + 2*int(( 
(a + b*x)**m*x)/(c + d*x)**m,x)*c*d*e**3