\(\int (a+b x)^m (c+d x)^{-2-m} (e+f x) (g+h x) \, dx\) [128]

Optimal result

Integrand size = 29, antiderivative size = 203 \[ \int (a+b x)^m (c+d x)^{-2-m} (e+f x) (g+h x) \, dx=\frac {(a+b x)^{1+m} (c+d x)^{-1-m} \left (b d^2 e g+b c^2 f h (2+m)-c d (b (f g+e h)+a f h (1+m))+d (b c-a d) f h (1+m) x\right )}{b d^2 (b c-a d) (1+m)}-\frac {(a d f h m+b (d (f g+e h)-c f h (2+m))) (a+b x)^m \left (-\frac {d (a+b x)}{b c-a d}\right )^{-m} (c+d x)^{-m} \operatorname {Hypergeometric2F1}\left (-m,-m,1-m,\frac {b (c+d x)}{b c-a d}\right )}{b d^3 m} \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.01, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {148, 72, 71} \[ \int (a+b x)^m (c+d x)^{-2-m} (e+f x) (g+h x) \, dx=-\frac {(a+b x)^{m+1} (c+d x)^{-m-1} \left (-d f h (m+1) x (b c-a d)+a c d f h (m+1)-b \left (c^2 f h (m+2)-c d (e h+f g)+d^2 e g\right )\right )}{b d^2 (m+1) (b c-a d)}-\frac {(a+b x)^m (c+d x)^{-m} \left (-\frac {d (a+b x)}{b c-a d}\right )^{-m} \operatorname {Hypergeometric2F1}\left (-m,-m,1-m,\frac {b (c+d x)}{b c-a d}\right ) (a d f h m-b c f h (m+2)+b d (e h+f g))}{b d^3 m} \]

[In]

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

[Out]

-(((a + b*x)^(1 + m)*(c + d*x)^(-1 - m)*(a*c*d*f*h*(1 + m) - b*(d^2*e*g - c*d*(f*g + e*h) + c^2*f*h*(2 + m)) -
d*(b*c - a*d)*f*h*(1 + m)*x))/(b*d^2*(b*c - a*d)*(1 + m))) - ((b*d*(f*g + e*h) + a*d*f*h*m - b*c*f*h*(2 + m))
*(a + b*x)^m*Hypergeometric2F1[-m, -m, 1 - m, (b*(c + d*x))/(b*c - a*d)])/(b*d^3*m*(-((d*(a + b*x))/(b*c - a*d
)))^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 148

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

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

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.98 \[ \int (a+b x)^m (c+d x)^{-2-m} (e+f x) (g+h x) \, dx=\frac {(a+b x)^m (c+d x)^{-m} \left (-\frac {d (a+b x) \left (a d f h (1+m) (c+d x)-b \left (d^2 e g+c^2 f h (2+m)+c d (-f g-e h+f h (1+m) x)\right )\right )}{c+d x}+\frac {(b c-a d) (1+m) (-b d (f g+e h)-a d f h m+b c f h (2+m)) \left (\frac {d (a+b x)}{-b c+a d}\right )^{-m} \operatorname {Hypergeometric2F1}\left (-m,-m,1-m,\frac {b (c+d x)}{b c-a d}\right )}{m}\right )}{b d^3 (b c-a d) (1+m)} \]

[In]

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

[Out]

((a + b*x)^m*(-((d*(a + b*x)*(a*d*f*h*(1 + m)*(c + d*x) - b*(d^2*e*g + c^2*f*h*(2 + m) + c*d*(-(f*g) - e*h + f
*h*(1 + m)*x))))/(c + d*x)) + ((b*c - a*d)*(1 + m)*(-(b*d*(f*g + e*h)) - a*d*f*h*m + b*c*f*h*(2 + m))*Hypergeo
metric2F1[-m, -m, 1 - m, (b*(c + d*x))/(b*c - a*d)])/(m*((d*(a + b*x))/(-(b*c) + a*d))^m)))/(b*d^3*(b*c - a*d)
*(1 + m)*(c + d*x)^m)

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

Sympy [F(-2)]

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

[In]

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

[Out]

Exception raised: HeuristicGCDFailed >> no luck

Maxima [F]

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

[In]

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

Giac [F]

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

[In]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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