3.2089 \(\int \frac{(d+e x)^m}{(a d e+(c d^2+a e^2) x+c d e x^2)^3} \, dx\)

Optimal. Leaf size=64 \[ \frac{e^2 (d+e x)^{m-2} \, _2F_1\left (3,m-2;m-1;\frac{c d (d+e x)}{c d^2-a e^2}\right )}{(2-m) \left (c d^2-a e^2\right )^3} \]

[Out]

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

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Rubi [A]  time = 0.0272594, antiderivative size = 64, 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} \[ \frac{e^2 (d+e x)^{m-2} \, _2F_1\left (3,m-2;m-1;\frac{c d (d+e x)}{c d^2-a e^2}\right )}{(2-m) \left (c d^2-a e^2\right )^3} \]

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)^3,x]

[Out]

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

Mathematica [A]  time = 0.0228041, size = 63, normalized size = 0.98 \[ \frac{e^2 (d+e x)^{m-2} \, _2F_1\left (3,m-2;m-1;-\frac{c d (d+e x)}{a e^2-c d^2}\right )}{(m-2) \left (a e^2-c d^2\right )^3} \]

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)^3,x]

[Out]

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

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Maple [F]  time = 1.609, 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 ) ^{3}}}\, 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)^3,x)

[Out]

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

[Out]

integrate((e*x + d)^m/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^3, 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^{3} d^{3} e^{3} x^{6} + a^{3} d^{3} e^{3} + 3 \,{\left (c^{3} d^{4} e^{2} + a c^{2} d^{2} e^{4}\right )} x^{5} + 3 \,{\left (c^{3} d^{5} e + 3 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x^{4} +{\left (c^{3} d^{6} + 9 \, a c^{2} d^{4} e^{2} + 9 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} x^{3} + 3 \,{\left (a c^{2} d^{5} e + 3 \, a^{2} c d^{3} e^{3} + a^{3} d e^{5}\right )} x^{2} + 3 \,{\left (a^{2} c d^{4} e^{2} + a^{3} d^{2} e^{4}\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)^3,x, algorithm="fricas")

[Out]

integral((e*x + d)^m/(c^3*d^3*e^3*x^6 + a^3*d^3*e^3 + 3*(c^3*d^4*e^2 + a*c^2*d^2*e^4)*x^5 + 3*(c^3*d^5*e + 3*a
*c^2*d^3*e^3 + a^2*c*d*e^5)*x^4 + (c^3*d^6 + 9*a*c^2*d^4*e^2 + 9*a^2*c*d^2*e^4 + a^3*e^6)*x^3 + 3*(a*c^2*d^5*e
+ 3*a^2*c*d^3*e^3 + a^3*d*e^5)*x^2 + 3*(a^2*c*d^4*e^2 + a^3*d^2*e^4)*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)**3,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 )}^{3}}\,{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)^3,x, algorithm="giac")

[Out]

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