3.2079 \(\int \frac{(d+e x)^m}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^4} \, dx\)

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)))

_______________________________________________________________________________________

Rubi [A]  time = 0.0912112, 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 \[ -\frac{e^3 (d+e x)^{m-3} \, _2F_1\left (4,m-3;m-2;\frac{c d (d+e x)}{c d^2-a e^2}\right )}{(3-m) \left (c d^2-a e^2\right )^4} \]

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)))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 26.3071, size = 53, normalized size = 0.82 \[ - \frac{e^{3} \left (d + e x\right )^{m - 3}{{}_{2}F_{1}\left (\begin{matrix} 4, m - 3 \\ m - 2 \end{matrix}\middle |{\frac{c d \left (- d - e x\right )}{a e^{2} - c d^{2}}} \right )}}{\left (- m + 3\right ) \left (a e^{2} - c d^{2}\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**m/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**4,x)

[Out]

-e**3*(d + e*x)**(m - 3)*hyper((4, m - 3), (m - 2,), c*d*(-d - e*x)/(a*e**2 - c*
d**2))/((-m + 3)*(a*e**2 - c*d**2)**4)

_______________________________________________________________________________________

Mathematica [B]  time = 2.55309, size = 504, normalized size = 7.75 \[ \frac{c^3 d^3 (d+e x)^m \left (\frac{\left (a e^3-c d^2 e\right )^3}{c^3 d^3 (m-3) (d+e x)^3}-\frac{4 e^3 \left (c d^2-a e^2\right )^2}{c^2 d^2 (m-2) (d+e x)^2}+\frac{10 e^2 \left (a e^2-c d^2\right ) \left (\frac{c d (d+e x)}{e (a e+c d x)}\right )^{-m} \, _2F_1\left (1-m,-m;2-m;\frac{a e^2-c d^2}{e (a e+c d x)}\right )}{(m-1) (a e+c d x)}+\frac{4 e \left (c d^2-a e^2\right )^2 \left (\frac{c d (d+e x)}{e (a e+c d x)}\right )^{-m} \, _2F_1\left (2-m,-m;3-m;\frac{a e^2-c d^2}{e (a e+c d x)}\right )}{(m-2) (a e+c d x)^2}-\frac{\left (c d^2-a e^2\right )^3 \left (\frac{c d (d+e x)}{e (a e+c d x)}\right )^{-m} \, _2F_1\left (3-m,-m;4-m;\frac{a e^2-c d^2}{e (a e+c d x)}\right )}{(m-3) (a e+c d x)^3}+\frac{20 e^3 \left (\frac{c d (d+e x)}{e (a e+c d x)}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{a e^2-c d^2}{e (a e+c d x)}\right )}{m}+\frac{10 e^3 \left (a e^2-c d^2\right )}{c d (m-1) (d+e x)}-\frac{20 e^3}{m}\right )}{\left (a e^2-c d^2\right )^7} \]

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]

(c^3*d^3*(d + e*x)^m*((-20*e^3)/m + (-(c*d^2*e) + a*e^3)^3/(c^3*d^3*(-3 + m)*(d
+ e*x)^3) - (4*e^3*(c*d^2 - a*e^2)^2)/(c^2*d^2*(-2 + m)*(d + e*x)^2) + (10*e^3*(
-(c*d^2) + a*e^2))/(c*d*(-1 + m)*(d + e*x)) + (10*e^2*(-(c*d^2) + a*e^2)*Hyperge
ometric2F1[1 - m, -m, 2 - m, (-(c*d^2) + a*e^2)/(e*(a*e + c*d*x))])/((-1 + m)*(a
*e + c*d*x)*((c*d*(d + e*x))/(e*(a*e + c*d*x)))^m) + (4*e*(c*d^2 - a*e^2)^2*Hype
rgeometric2F1[2 - m, -m, 3 - m, (-(c*d^2) + a*e^2)/(e*(a*e + c*d*x))])/((-2 + m)
*(a*e + c*d*x)^2*((c*d*(d + e*x))/(e*(a*e + c*d*x)))^m) - ((c*d^2 - a*e^2)^3*Hyp
ergeometric2F1[3 - m, -m, 4 - m, (-(c*d^2) + a*e^2)/(e*(a*e + c*d*x))])/((-3 + m
)*(a*e + c*d*x)^3*((c*d*(d + e*x))/(e*(a*e + c*d*x)))^m) + (20*e^3*Hypergeometri
c2F1[-m, -m, 1 - m, (-(c*d^2) + a*e^2)/(e*(a*e + c*d*x))])/(m*((c*d*(d + e*x))/(
e*(a*e + c*d*x)))^m)))/(-(c*d^2) + a*e^2)^7

_______________________________________________________________________________________

Maple [F]  time = 0.376, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex+d \right ) ^{m}}{ \left ( aed+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2} \right ) ^{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^m/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^4,x)

[Out]

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

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{m}}{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^m/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^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)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (e x + d\right )}^{m}}{c^{4} d^{4} e^{4} x^{8} + a^{4} d^{4} e^{4} + 4 \,{\left (c^{4} d^{5} e^{3} + a c^{3} d^{3} e^{5}\right )} x^{7} + 2 \,{\left (3 \, c^{4} d^{6} e^{2} + 8 \, a c^{3} d^{4} e^{4} + 3 \, a^{2} c^{2} d^{2} e^{6}\right )} x^{6} + 4 \,{\left (c^{4} d^{7} e + 6 \, a c^{3} d^{5} e^{3} + 6 \, a^{2} c^{2} d^{3} e^{5} + a^{3} c d e^{7}\right )} x^{5} +{\left (c^{4} d^{8} + 16 \, a c^{3} d^{6} e^{2} + 36 \, a^{2} c^{2} d^{4} e^{4} + 16 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} x^{4} + 4 \,{\left (a c^{3} d^{7} e + 6 \, a^{2} c^{2} d^{5} e^{3} + 6 \, a^{3} c d^{3} e^{5} + a^{4} d e^{7}\right )} x^{3} + 2 \,{\left (3 \, a^{2} c^{2} d^{6} e^{2} + 8 \, a^{3} c d^{4} e^{4} + 3 \, a^{4} d^{2} e^{6}\right )} x^{2} + 4 \,{\left (a^{3} c d^{5} e^{3} + a^{4} d^{3} e^{5}\right )} x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^m/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^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)

_______________________________________________________________________________________

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

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

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{m}}{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^m/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^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)