\(\int \frac {(d+e x)^m}{c d x+c e x^2} \, dx\) [438]

Optimal result

Integrand size = 21, antiderivative size = 32 \[ \int \frac {(d+e x)^m}{c d x+c e x^2} \, dx=-\frac {(d+e x)^m \operatorname {Hypergeometric2F1}\left (1,m,1+m,1+\frac {e x}{d}\right )}{c d m} \]

[Out]

-(e*x+d)^m*hypergeom([1, m],[1+m],1+e*x/d)/c/d/m

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {640, 12, 67} \[ \int \frac {(d+e x)^m}{c d x+c e x^2} \, dx=-\frac {(d+e x)^m \operatorname {Hypergeometric2F1}\left (1,m,m+1,\frac {e x}{d}+1\right )}{c d m} \]

[In]

Int[(d + e*x)^m/(c*d*x + c*e*x^2),x]

[Out]

-(((d + e*x)^m*Hypergeometric2F1[1, m, 1 + m, 1 + (e*x)/d])/(c*d*m))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 640

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c/e)*x)^p, x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&
IntegerQ[p]

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x)^m}{c d x+c e x^2} \, dx=-\frac {(d+e x)^m \operatorname {Hypergeometric2F1}\left (1,m,1+m,1+\frac {e x}{d}\right )}{c d m} \]

[In]

Integrate[(d + e*x)^m/(c*d*x + c*e*x^2),x]

[Out]

-(((d + e*x)^m*Hypergeometric2F1[1, m, 1 + m, 1 + (e*x)/d])/(c*d*m))

Maple [F]

\[\int \frac {\left (e x +d \right )^{m}}{c e \,x^{2}+c d x}d x\]

[In]

int((e*x+d)^m/(c*e*x^2+c*d*x),x)

[Out]

int((e*x+d)^m/(c*e*x^2+c*d*x),x)

Fricas [F]

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

[In]

integrate((e*x+d)^m/(c*e*x^2+c*d*x),x, algorithm="fricas")

[Out]

integral((e*x + d)^m/(c*e*x^2 + c*d*x), x)

Sympy [F]

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

[In]

integrate((e*x+d)**m/(c*e*x**2+c*d*x),x)

[Out]

Integral((d + e*x)**m/(d*x + e*x**2), x)/c

Maxima [F]

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

[In]

integrate((e*x+d)^m/(c*e*x^2+c*d*x),x, algorithm="maxima")

[Out]

integrate((e*x + d)^m/(c*e*x^2 + c*d*x), x)

Giac [F]

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

[In]

integrate((e*x+d)^m/(c*e*x^2+c*d*x),x, algorithm="giac")

[Out]

integrate((e*x + d)^m/(c*e*x^2 + c*d*x), x)

Mupad [F(-1)]

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

[In]

int((d + e*x)^m/(c*d*x + c*e*x^2),x)

[Out]

int((d + e*x)^m/(c*d*x + c*e*x^2), x)