### 3.2087 $$\int \frac{(d+e x)^m}{a d e+(c d^2+a e^2) x+c d e x^2} \, dx$$

Optimal. Leaf size=54 $-\frac{(d+e x)^m \, _2F_1\left (1,m;m+1;\frac{c d (d+e x)}{c d^2-a e^2}\right )}{m \left (c d^2-a e^2\right )}$

[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))

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Rubi [A]  time = 0.0243076, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 35, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.057, Rules used = {626, 68} $-\frac{(d+e x)^m \, _2F_1\left (1,m;m+1;\frac{c d (d+e x)}{c d^2-a e^2}\right )}{m \left (c d^2-a e^2\right )}$

Antiderivative was successfully veriﬁed.

[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 626

Int[((d_) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a
/d + (c*x)/e)^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]

Rule 68

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((b*c - a*d)^n*(a + b*x)^(m + 1)*Hype
rgeometric2F1[-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b^(n + 1)*(m + 1)), x] /; FreeQ[{a, b, c, d, m
}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] && IntegerQ[n]

Rubi steps

\begin{align*} \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{(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]  time = 0.012819, size = 54, normalized size = 1. $-\frac{(d+e x)^m \, _2F_1\left (1,m;m+1;\frac{c d (d+e x)}{c d^2-a e^2}\right )}{m \left (c d^2-a e^2\right )}$

Antiderivative was successfully veriﬁed.

[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))

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Maple [F]  time = 1.179, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex+d \right ) ^{m}}{ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (e x + d\right )}^{m}}{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d + e x\right )^{m}}{\left (d + e x\right ) \left (a e + c d x\right )}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)