3.2088 $$\int \frac{(d+e x)^m}{(a d e+(c d^2+a e^2) x+c d e x^2)^2} \, dx$$

Optimal. Leaf size=61 $-\frac{e (d+e x)^{m-1} \, _2F_1\left (2,m-1;m;\frac{c d (d+e x)}{c d^2-a e^2}\right )}{(1-m) \left (c d^2-a e^2\right )^2}$

[Out]

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

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Rubi [A]  time = 0.0223567, antiderivative size = 61, 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} $-\frac{e (d+e x)^{m-1} \, _2F_1\left (2,m-1;m;\frac{c d (d+e x)}{c d^2-a e^2}\right )}{(1-m) \left (c d^2-a e^2\right )^2}$

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)^2,x]

[Out]

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

Mathematica [A]  time = 0.0230798, size = 59, normalized size = 0.97 $\frac{e (d+e x)^{m-1} \, _2F_1\left (2,m-1;m;-\frac{c d (d+e x)}{a e^2-c d^2}\right )}{(m-1) \left (a e^2-c d^2\right )^2}$

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)^2,x]

[Out]

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

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Maple [F]  time = 1.269, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex+d \right ) ^{m}}{ \left ( ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2} \right ) ^{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)^2,x)

[Out]

int((e*x+d)^m/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^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}}{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{2}}\,{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)^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)^2, 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^{2} d^{2} e^{2} x^{4} + a^{2} d^{2} e^{2} + 2 \,{\left (c^{2} d^{3} e + a c d e^{3}\right )} x^{3} +{\left (c^{2} d^{4} + 4 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} x^{2} + 2 \,{\left (a c d^{3} e + a^{2} d e^{3}\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)^2,x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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)**2,x)

[Out]

Timed out

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