\(\int \frac {(d+e x)^m}{a d e+(c d^2+a e^2) x+c d e x^2} \, dx\) [2087]

Optimal result

Integrand size = 35, antiderivative size = 54 \[ \int \frac {(d+e x)^m}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=-\frac {(d+e x)^m \operatorname {Hypergeometric2F1}\left (1,m,1+m,\frac {c d (d+e x)}{c d^2-a e^2}\right )}{\left (c d^2-a e^2\right ) m} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {640, 70} \[ \int \frac {(d+e x)^m}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=-\frac {(d+e x)^m \operatorname {Hypergeometric2F1}\left (1,m,m+1,\frac {c d (d+e x)}{c d^2-a e^2}\right )}{m \left (c d^2-a e^2\right )} \]

[In]

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

[Out]

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00 \[ \int \frac {(d+e x)^m}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=-\frac {(d+e x)^m \operatorname {Hypergeometric2F1}\left (1,m,1+m,\frac {c d (d+e x)}{c d^2-a e^2}\right )}{\left (c d^2-a e^2\right ) m} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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