\(\int \frac {(a+b x)^m}{(c+d x) (e+f x) (g+h x)} \, dx\) [122]

Optimal result

Integrand size = 29, antiderivative size = 224 \[ \int \frac {(a+b x)^m}{(c+d x) (e+f x) (g+h x)} \, dx=\frac {d^2 (a+b x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {d (a+b x)}{b c-a d}\right )}{(b c-a d) (d e-c f) (d g-c h) (1+m)}-\frac {f^2 (a+b x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {f (a+b x)}{b e-a f}\right )}{(b e-a f) (d e-c f) (f g-e h) (1+m)}+\frac {h^2 (a+b x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,-\frac {h (a+b x)}{b g-a h}\right )}{(b g-a h) (d g-c h) (f g-e h) (1+m)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {186, 70} \[ \int \frac {(a+b x)^m}{(c+d x) (e+f x) (g+h x)} \, dx=\frac {d^2 (a+b x)^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,-\frac {d (a+b x)}{b c-a d}\right )}{(m+1) (b c-a d) (d e-c f) (d g-c h)}-\frac {f^2 (a+b x)^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,-\frac {f (a+b x)}{b e-a f}\right )}{(m+1) (b e-a f) (d e-c f) (f g-e h)}+\frac {h^2 (a+b x)^{m+1} \operatorname {Hypergeometric2F1}\left (1,m+1,m+2,-\frac {h (a+b x)}{b g-a h}\right )}{(m+1) (b g-a h) (d g-c h) (f g-e h)} \]

[In]

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

[Out]

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

Rule 70

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

Rule 186

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_))^(q_), x
_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*(g + h*x)^q, x], x] /; FreeQ[{a, b, c, d,
e, f, g, h, m, n}, x] && IntegersQ[p, q]

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

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.86 \[ \int \frac {(a+b x)^m}{(c+d x) (e+f x) (g+h x)} \, dx=\frac {(a+b x)^{1+m} \left (\frac {d^2 \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {d (a+b x)}{-b c+a d}\right )}{(b c-a d) (-d e+c f) (-d g+c h)}+\frac {f^2 \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {f (a+b x)}{-b e+a f}\right )}{(b e-a f) (d e-c f) (-f g+e h)}+\frac {h^2 \operatorname {Hypergeometric2F1}\left (1,1+m,2+m,\frac {h (a+b x)}{-b g+a h}\right )}{(b g-a h) (d g-c h) (f g-e h)}\right )}{1+m} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F(-2)]

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

[In]

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

[Out]

Exception raised: HeuristicGCDFailed >> no luck

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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