3.2090 \(\int \frac{(d+e x)^m}{(a d e+(c d^2+a e^2) x+c d e x^2)^4} \, dx\)

Optimal. Leaf size=65 \[ -\frac{e^3 (d+e x)^{m-3} \, _2F_1\left (4,m-3;m-2;\frac{c d (d+e x)}{c d^2-a e^2}\right )}{(3-m) \left (c d^2-a e^2\right )^4} \]

[Out]

-((e^3*(d + e*x)^(-3 + m)*Hypergeometric2F1[4, -3 + m, -2 + m, (c*d*(d + e*x))/(c*d^2 - a*e^2)])/((c*d^2 - a*e
^2)^4*(3 - m)))

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Rubi [A]  time = 0.0262358, antiderivative size = 65, 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} \[ -\frac{e^3 (d+e x)^{m-3} \, _2F_1\left (4,m-3;m-2;\frac{c d (d+e x)}{c d^2-a e^2}\right )}{(3-m) \left (c d^2-a e^2\right )^4} \]

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)^4,x]

[Out]

-((e^3*(d + e*x)^(-3 + m)*Hypergeometric2F1[4, -3 + m, -2 + m, (c*d*(d + e*x))/(c*d^2 - a*e^2)])/((c*d^2 - a*e
^2)^4*(3 - 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 )^4} \, dx &=\int \frac{(d+e x)^{-4+m}}{(a e+c d x)^4} \, dx\\ &=-\frac{e^3 (d+e x)^{-3+m} \, _2F_1\left (4,-3+m;-2+m;\frac{c d (d+e x)}{c d^2-a e^2}\right )}{\left (c d^2-a e^2\right )^4 (3-m)}\\ \end{align*}

Mathematica [A]  time = 0.0278587, size = 63, normalized size = 0.97 \[ \frac{e^3 (d+e x)^{m-3} \, _2F_1\left (4,m-3;m-2;-\frac{c d (d+e x)}{a e^2-c d^2}\right )}{(m-3) \left (a e^2-c d^2\right )^4} \]

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)^4,x]

[Out]

(e^3*(d + e*x)^(-3 + m)*Hypergeometric2F1[4, -3 + m, -2 + m, -((c*d*(d + e*x))/(-(c*d^2) + a*e^2))])/((-(c*d^2
) + a*e^2)^4*(-3 + m))

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Maple [F]  time = 1.294, 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 ) ^{4}}}\, 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)^4,x)

[Out]

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

[Out]

integrate((e*x + d)^m/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^4, 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^{4} d^{4} e^{4} x^{8} + a^{4} d^{4} e^{4} + 4 \,{\left (c^{4} d^{5} e^{3} + a c^{3} d^{3} e^{5}\right )} x^{7} + 2 \,{\left (3 \, c^{4} d^{6} e^{2} + 8 \, a c^{3} d^{4} e^{4} + 3 \, a^{2} c^{2} d^{2} e^{6}\right )} x^{6} + 4 \,{\left (c^{4} d^{7} e + 6 \, a c^{3} d^{5} e^{3} + 6 \, a^{2} c^{2} d^{3} e^{5} + a^{3} c d e^{7}\right )} x^{5} +{\left (c^{4} d^{8} + 16 \, a c^{3} d^{6} e^{2} + 36 \, a^{2} c^{2} d^{4} e^{4} + 16 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} x^{4} + 4 \,{\left (a c^{3} d^{7} e + 6 \, a^{2} c^{2} d^{5} e^{3} + 6 \, a^{3} c d^{3} e^{5} + a^{4} d e^{7}\right )} x^{3} + 2 \,{\left (3 \, a^{2} c^{2} d^{6} e^{2} + 8 \, a^{3} c d^{4} e^{4} + 3 \, a^{4} d^{2} e^{6}\right )} x^{2} + 4 \,{\left (a^{3} c d^{5} e^{3} + a^{4} d^{3} e^{5}\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)^4,x, algorithm="fricas")

[Out]

integral((e*x + d)^m/(c^4*d^4*e^4*x^8 + a^4*d^4*e^4 + 4*(c^4*d^5*e^3 + a*c^3*d^3*e^5)*x^7 + 2*(3*c^4*d^6*e^2 +
8*a*c^3*d^4*e^4 + 3*a^2*c^2*d^2*e^6)*x^6 + 4*(c^4*d^7*e + 6*a*c^3*d^5*e^3 + 6*a^2*c^2*d^3*e^5 + a^3*c*d*e^7)*
x^5 + (c^4*d^8 + 16*a*c^3*d^6*e^2 + 36*a^2*c^2*d^4*e^4 + 16*a^3*c*d^2*e^6 + a^4*e^8)*x^4 + 4*(a*c^3*d^7*e + 6*
a^2*c^2*d^5*e^3 + 6*a^3*c*d^3*e^5 + a^4*d*e^7)*x^3 + 2*(3*a^2*c^2*d^6*e^2 + 8*a^3*c*d^4*e^4 + 3*a^4*d^2*e^6)*x
^2 + 4*(a^3*c*d^5*e^3 + a^4*d^3*e^5)*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)**4,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 )}^{4}}\,{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)^4,x, algorithm="giac")

[Out]

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