3.2078 \(\int \frac{(d+e x)^m}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx\)

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 = 0.0871909, 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 \[ \frac{e^2 (d+e x)^{m-2} \, _2F_1\left (3,m-2;m-1;\frac{c d (d+e x)}{c d^2-a e^2}\right )}{(2-m) \left (c d^2-a e^2\right )^3} \]

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))

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Rubi in Sympy [A]  time = 26.0049, size = 53, normalized size = 0.83 \[ - \frac{e^{2} \left (d + e x\right )^{m - 2}{{}_{2}F_{1}\left (\begin{matrix} 3, m - 2 \\ m - 1 \end{matrix}\middle |{\frac{c d \left (- d - e x\right )}{a e^{2} - c d^{2}}} \right )}}{\left (- m + 2\right ) \left (a e^{2} - c d^{2}\right )^{3}} \]

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)**3,x)

[Out]

-e**2*(d + e*x)**(m - 2)*hyper((3, m - 2), (m - 1,), c*d*(-d - e*x)/(a*e**2 - c*
d**2))/((-m + 2)*(a*e**2 - c*d**2)**3)

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Mathematica [B]  time = 2.18203, size = 368, normalized size = 5.75 \[ \frac{c^2 d^2 (d+e x)^m \left (\frac{\left (c d^2 e-a e^3\right )^2}{c^2 d^2 (m-2) (d+e x)^2}-\frac{3 e \left (a e^2-c d^2\right ) \left (\frac{c d (d+e x)}{e (a e+c d x)}\right )^{-m} \, _2F_1\left (1-m,-m;2-m;\frac{a e^2-c d^2}{e (a e+c d x)}\right )}{(m-1) (a e+c d x)}-\frac{\left (c d^2-a e^2\right )^2 \left (\frac{c d (d+e x)}{e (a e+c d x)}\right )^{-m} \, _2F_1\left (2-m,-m;3-m;\frac{a e^2-c d^2}{e (a e+c d x)}\right )}{(m-2) (a e+c d x)^2}-\frac{6 e^2 \left (\frac{c d (d+e x)}{e (a e+c d x)}\right )^{-m} \, _2F_1\left (-m,-m;1-m;\frac{a e^2-c d^2}{e (a e+c d x)}\right )}{m}+\frac{3 e^2 \left (c d^2-a e^2\right )}{c d (m-1) (d+e x)}+\frac{6 e^2}{m}\right )}{\left (a e^2-c d^2\right )^5} \]

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]

(c^2*d^2*(d + e*x)^m*((6*e^2)/m + (c*d^2*e - a*e^3)^2/(c^2*d^2*(-2 + m)*(d + e*x
)^2) + (3*e^2*(c*d^2 - a*e^2))/(c*d*(-1 + m)*(d + e*x)) - (3*e*(-(c*d^2) + a*e^2
)*Hypergeometric2F1[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) - ((c*d^2 - a*e^2)^2
*Hypergeometric2F1[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) - (6*e^2*Hypergeome
tric2F1[-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)^5

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Maple [F]  time = 0.389, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex+d \right ) ^{m}}{ \left ( aed+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2} \right ) ^{3}}}\, dx \]

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)^3,x)

[Out]

int((e*x+d)^m/(a*e*d+(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. \[ \int \frac{{\left (e x + d\right )}^{m}}{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{3}}\,{d x} \]

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)^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. \[{\rm integral}\left (\frac{{\left (e x + d\right )}^{m}}{c^{3} d^{3} e^{3} x^{6} + a^{3} d^{3} e^{3} + 3 \,{\left (c^{3} d^{4} e^{2} + a c^{2} d^{2} e^{4}\right )} x^{5} + 3 \,{\left (c^{3} d^{5} e + 3 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x^{4} +{\left (c^{3} d^{6} + 9 \, a c^{2} d^{4} e^{2} + 9 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} x^{3} + 3 \,{\left (a c^{2} d^{5} e + 3 \, a^{2} c d^{3} e^{3} + a^{3} d e^{5}\right )} x^{2} + 3 \,{\left (a^{2} c d^{4} e^{2} + a^{3} d^{2} e^{4}\right )} x}, x\right ) \]

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)^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. \[ \text{Timed out} \]

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)**3,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{m}}{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{3}}\,{d x} \]

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)^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)