\(\int (a+b x)^m (c+d x)^{-3-m} (e+f x) (g+h x) \, dx\) [129]

Optimal result

Integrand size = 29, antiderivative size = 246 \[ \int (a+b x)^m (c+d x)^{-3-m} (e+f x) (g+h x) \, dx=-\frac {(a+b x)^{1+m} (c+d x)^{-2-m} \left (a^2 b c f h m-a^3 d f h (1+m)-b^3 c e g (2+m)+a b^2 (c (f g+e h)+d e g (1+m))-b \left (a^2 d f h (3+2 m)+b^2 (d e g+c (f g+e h) (1+m))-a b (2 c f h (1+m)+d (f g+e h) (2+m))\right ) x\right )}{b^2 (b c-a d)^2 (1+m) (2+m)}+\frac {f h (a+b x)^{3+m} (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m \operatorname {Hypergeometric2F1}\left (3+m,3+m,4+m,-\frac {d (a+b x)}{b c-a d}\right )}{(b c-a d)^3 (3+m)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {150, 72, 71} \[ \int (a+b x)^m (c+d x)^{-3-m} (e+f x) (g+h x) \, dx=\frac {f h (a+b x)^{m+3} (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m \operatorname {Hypergeometric2F1}\left (m+3,m+3,m+4,-\frac {d (a+b x)}{b c-a d}\right )}{(m+3) (b c-a d)^3}-\frac {(a+b x)^{m+1} (c+d x)^{-m-2} \left (a^3 (-d) f h (m+1)-b x \left (a^2 d f h (2 m+3)-a b (2 c f h (m+1)+d (m+2) (e h+f g))+b^2 (c (m+1) (e h+f g)+d e g)\right )+a^2 b c f h m+a b^2 (c (e h+f g)+d e g (m+1))-b^3 c e g (m+2)\right )}{b^2 (m+1) (m+2) (b c-a d)^2} \]

[In]

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

[Out]

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

Rule 71

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] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-d/(b*c - a*d), 0]))

Rule 72

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Dist[(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] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] &&
(RationalQ[m] ||  !SimplerQ[n + 1, m + 1])

Rule 150

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol] :
> Simp[((b^3*c*e*g*(m + 2) - a^3*d*f*h*(n + 2) - a^2*b*(c*f*h*m - d*(f*g + e*h)*(m + n + 3)) - a*b^2*(c*(f*g +
e*h) + d*e*g*(2*m + n + 4)) + b*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(n + 1)) + b^2*(c*(
f*g + e*h)*(m + 1) - d*e*g*(m + n + 2)))*x)/(b^2*(b*c - a*d)^2*(m + 1)*(m + 2)))*(a + b*x)^(m + 1)*(c + d*x)^(
n + 1), x] + Dist[f*(h/b^2) - (d*(m + n + 3)*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(n + 1)
) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2))))/(b^2*(b*c - a*d)^2*(m + 1)*(m + 2)), Int[(a + b*x)^(m +
2)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && (LtQ[m, -2] || (EqQ[m + n + 3, 0] &&  !L
tQ[n, -2]))

Rubi steps \begin{align*} \text {integral}& = -\frac {(a+b x)^{1+m} (c+d x)^{-2-m} \left (a^2 b c f h m-a^3 d f h (1+m)-b^3 c e g (2+m)+a b^2 (c (f g+e h)+d e g (1+m))-b \left (a^2 d f h (3+2 m)+b^2 (d e g+c (f g+e h) (1+m))-a b (2 c f h (1+m)+d (f g+e h) (2+m))\right ) x\right )}{b^2 (b c-a d)^2 (1+m) (2+m)}+\frac {(f h) \int (a+b x)^{2+m} (c+d x)^{-3-m} \, dx}{b^2} \\ & = -\frac {(a+b x)^{1+m} (c+d x)^{-2-m} \left (a^2 b c f h m-a^3 d f h (1+m)-b^3 c e g (2+m)+a b^2 (c (f g+e h)+d e g (1+m))-b \left (a^2 d f h (3+2 m)+b^2 (d e g+c (f g+e h) (1+m))-a b (2 c f h (1+m)+d (f g+e h) (2+m))\right ) x\right )}{b^2 (b c-a d)^2 (1+m) (2+m)}+\frac {\left (b f h (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m\right ) \int (a+b x)^{2+m} \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^{-3-m} \, dx}{(b c-a d)^3} \\ & = -\frac {(a+b x)^{1+m} (c+d x)^{-2-m} \left (a^2 b c f h m-a^3 d f h (1+m)-b^3 c e g (2+m)+a b^2 (c (f g+e h)+d e g (1+m))-b \left (a^2 d f h (3+2 m)+b^2 (d e g+c (f g+e h) (1+m))-a b (2 c f h (1+m)+d (f g+e h) (2+m))\right ) x\right )}{b^2 (b c-a d)^2 (1+m) (2+m)}+\frac {f h (a+b x)^{3+m} (c+d x)^{-m} \left (\frac {b (c+d x)}{b c-a d}\right )^m \, _2F_1\left (3+m,3+m;4+m;-\frac {d (a+b x)}{b c-a d}\right )}{(b c-a d)^3 (3+m)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.96 \[ \int (a+b x)^m (c+d x)^{-3-m} (e+f x) (g+h x) \, dx=-\frac {(a+b x)^m (c+d x)^{-2-m} \left (d^3 (a+b x) \left (-a^3 d f h (1+m)+a^2 b f h (c m-d (3+2 m) x)+a b^2 (c e h+d e g (1+m)+d f g (2+m) x+d e h (2+m) x+c f (g+2 h (1+m) x))-b^3 (d e g x+c (e g (2+m)+f g (1+m) x+e h (1+m) x))\right )+(b c-a d)^4 f h (1+m) \left (\frac {d (a+b x)}{-b c+a d}\right )^{-m} \operatorname {Hypergeometric2F1}\left (-2-m,-2-m,-1-m,\frac {b (c+d x)}{b c-a d}\right )\right )}{b^2 d^3 (b c-a d)^2 (1+m) (2+m)} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F(-2)]

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

[In]

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

[Out]

Exception raised: HeuristicGCDFailed >> no luck

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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