\(\int \frac {(e+f x)^n}{x (a+b x) (c+d x)} \, dx\) [117]

Optimal result

Integrand size = 25, antiderivative size = 175 \[ \int \frac {(e+f x)^n}{x (a+b x) (c+d x)} \, dx=\frac {b^2 (e+f x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {b (e+f x)}{b e-a f}\right )}{a (b c-a d) (b e-a f) (1+n)}-\frac {d^2 (e+f x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {d (e+f x)}{d e-c f}\right )}{c (b c-a d) (d e-c f) (1+n)}-\frac {(e+f x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,1+\frac {f x}{e}\right )}{a c e (1+n)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {186, 67, 70} \[ \int \frac {(e+f x)^n}{x (a+b x) (c+d x)} \, dx=\frac {b^2 (e+f x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {b (e+f x)}{b e-a f}\right )}{a (n+1) (b c-a d) (b e-a f)}-\frac {d^2 (e+f x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {d (e+f x)}{d e-c f}\right )}{c (n+1) (b c-a d) (d e-c f)}-\frac {(e+f x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {f x}{e}+1\right )}{a c e (n+1)} \]

[In]

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

[Out]

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

Rule 67

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

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 {(e+f x)^n}{a c x}+\frac {b^2 (e+f x)^n}{a (-b c+a d) (a+b x)}+\frac {d^2 (e+f x)^n}{c (b c-a d) (c+d x)}\right ) \, dx \\ & = \frac {\int \frac {(e+f x)^n}{x} \, dx}{a c}-\frac {b^2 \int \frac {(e+f x)^n}{a+b x} \, dx}{a (b c-a d)}+\frac {d^2 \int \frac {(e+f x)^n}{c+d x} \, dx}{c (b c-a d)} \\ & = \frac {b^2 (e+f x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac {b (e+f x)}{b e-a f}\right )}{a (b c-a d) (b e-a f) (1+n)}-\frac {d^2 (e+f x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac {d (e+f x)}{d e-c f}\right )}{c (b c-a d) (d e-c f) (1+n)}-\frac {(e+f x)^{1+n} \, _2F_1\left (1,1+n;2+n;1+\frac {f x}{e}\right )}{a c e (1+n)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.97 \[ \int \frac {(e+f x)^n}{x (a+b x) (c+d x)} \, dx=-\frac {(e+f x)^{1+n} \left (b^2 c e (d e-c f) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {b (e+f x)}{b e-a f}\right )+(-b e+a f) \left (a d^2 e \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {d (e+f x)}{d e-c f}\right )-(b c-a d) (-d e+c f) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,1+\frac {f x}{e}\right )\right )\right )}{a c (-b c+a d) e (-b e+a f) (-d e+c f) (1+n)} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

int((f*x+e)^n/x/(b*x+a)/(d*x+c),x)

[Out]

int((f*x+e)^n/x/(b*x+a)/(d*x+c),x)

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

integrate((f*x+e)**n/x/(b*x+a)/(d*x+c),x)

[Out]

Integral((e + f*x)**n/(x*(a + b*x)*(c + d*x)), x)

Maxima [F]

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

[In]

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

[Out]

integrate((f*x + e)^n/((b*x + a)*(d*x + c)*x), x)

Giac [F]

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

[In]

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

[Out]

integrate((f*x + e)^n/((b*x + a)*(d*x + c)*x), x)

Mupad [F(-1)]

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

[In]

int((e + f*x)^n/(x*(a + b*x)*(c + d*x)),x)

[Out]

int((e + f*x)^n/(x*(a + b*x)*(c + d*x)), x)