\(\int \frac {(e+f x)^n}{x^2 (a+b x) (c+d x)} \, dx\) [118]

Optimal result

Integrand size = 25, antiderivative size = 222 \[ \int \frac {(e+f x)^n}{x^2 (a+b x) (c+d x)} \, dx=-\frac {b^3 (e+f x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {b (e+f x)}{b e-a f}\right )}{a^2 (b c-a d) (b e-a f) (1+n)}+\frac {d^3 (e+f x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {d (e+f x)}{d e-c f}\right )}{c^2 (b c-a d) (d e-c f) (1+n)}+\frac {(b c+a d) (e+f x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,1+\frac {f x}{e}\right )}{a^2 c^2 e (1+n)}+\frac {f (e+f x)^{1+n} \operatorname {Hypergeometric2F1}\left (2,1+n,2+n,1+\frac {f x}{e}\right )}{a c e^2 (1+n)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {186, 67, 70} \[ \int \frac {(e+f x)^n}{x^2 (a+b x) (c+d x)} \, dx=-\frac {b^3 (e+f x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {b (e+f x)}{b e-a f}\right )}{a^2 (n+1) (b c-a d) (b e-a f)}+\frac {(a d+b c) (e+f x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {f x}{e}+1\right )}{a^2 c^2 e (n+1)}+\frac {d^3 (e+f x)^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {d (e+f x)}{d e-c f}\right )}{c^2 (n+1) (b c-a d) (d e-c f)}+\frac {f (e+f x)^{n+1} \operatorname {Hypergeometric2F1}\left (2,n+1,n+2,\frac {f x}{e}+1\right )}{a c e^2 (n+1)} \]

[In]

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

[Out]

-((b^3*(e + f*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, (b*(e + f*x))/(b*e - a*f)])/(a^2*(b*c - a*d)*(b*e
- a*f)*(1 + n))) + (d^3*(e + f*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, (d*(e + f*x))/(d*e - c*f)])/(c^2*
(b*c - a*d)*(d*e - c*f)*(1 + n)) + ((b*c + a*d)*(e + f*x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, 1 + (f*x)
/e])/(a^2*c^2*e*(1 + n)) + (f*(e + f*x)^(1 + n)*Hypergeometric2F1[2, 1 + n, 2 + n, 1 + (f*x)/e])/(a*c*e^2*(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^2}+\frac {(-b c-a d) (e+f x)^n}{a^2 c^2 x}-\frac {b^3 (e+f x)^n}{a^2 (-b c+a d) (a+b x)}-\frac {d^3 (e+f x)^n}{c^2 (b c-a d) (c+d x)}\right ) \, dx \\ & = \frac {\int \frac {(e+f x)^n}{x^2} \, dx}{a c}+\frac {b^3 \int \frac {(e+f x)^n}{a+b x} \, dx}{a^2 (b c-a d)}-\frac {d^3 \int \frac {(e+f x)^n}{c+d x} \, dx}{c^2 (b c-a d)}-\frac {(b c+a d) \int \frac {(e+f x)^n}{x} \, dx}{a^2 c^2} \\ & = -\frac {b^3 (e+f x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac {b (e+f x)}{b e-a f}\right )}{a^2 (b c-a d) (b e-a f) (1+n)}+\frac {d^3 (e+f x)^{1+n} \, _2F_1\left (1,1+n;2+n;\frac {d (e+f x)}{d e-c f}\right )}{c^2 (b c-a d) (d e-c f) (1+n)}+\frac {(b c+a d) (e+f x)^{1+n} \, _2F_1\left (1,1+n;2+n;1+\frac {f x}{e}\right )}{a^2 c^2 e (1+n)}+\frac {f (e+f x)^{1+n} \, _2F_1\left (2,1+n;2+n;1+\frac {f x}{e}\right )}{a c e^2 (1+n)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.80 \[ \int \frac {(e+f x)^n}{x^2 (a+b x) (c+d x)} \, dx=\frac {(e+f x)^{1+n} \left (-\frac {b^3 \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {b (e+f x)}{b e-a f}\right )}{a^2 (b c-a d) (b e-a f)}+\frac {-\frac {d^3 \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {d (e+f x)}{d e-c f}\right )}{(b c-a d) (-d e+c f)}+\frac {(b c+a d) e \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,1+\frac {f x}{e}\right )+a c f \operatorname {Hypergeometric2F1}\left (2,1+n,2+n,1+\frac {f x}{e}\right )}{a^2 e^2}}{c^2}\right )}{1+n} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {(e+f x)^n}{x^2 (a+b x) (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{n}}{{\left (b x + a\right )} {\left (d x + c\right )} x^{2}} \,d x } \]

[In]

integrate((f*x+e)^n/x^2/(b*x+a)/(d*x+c),x, algorithm="fricas")

[Out]

integral((f*x + e)^n/(b*d*x^4 + a*c*x^2 + (b*c + a*d)*x^3), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {(e+f x)^n}{x^2 (a+b x) (c+d x)} \, dx=\text {Timed out} \]

[In]

integrate((f*x+e)**n/x**2/(b*x+a)/(d*x+c),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(e+f x)^n}{x^2 (a+b x) (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{n}}{{\left (b x + a\right )} {\left (d x + c\right )} x^{2}} \,d x } \]

[In]

integrate((f*x+e)^n/x^2/(b*x+a)/(d*x+c),x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {(e+f x)^n}{x^2 (a+b x) (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{n}}{{\left (b x + a\right )} {\left (d x + c\right )} x^{2}} \,d x } \]

[In]

integrate((f*x+e)^n/x^2/(b*x+a)/(d*x+c),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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