3.1724 \(\int \frac{(d+e x)^m}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx\)

Optimal. Leaf size=53 \[ \frac{e^3 (d+e x)^{m+1} \, _2F_1\left (4,m+1;m+2;\frac{b (d+e x)}{b d-a e}\right )}{(m+1) (b d-a e)^4} \]

[Out]

(e^3*(d + e*x)^(1 + m)*Hypergeometric2F1[4, 1 + m, 2 + m, (b*(d + e*x))/(b*d - a
*e)])/((b*d - a*e)^4*(1 + m))

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Rubi [A]  time = 0.04912, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{e^3 (d+e x)^{m+1} \, _2F_1\left (4,m+1;m+2;\frac{b (d+e x)}{b d-a e}\right )}{(m+1) (b d-a e)^4} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^m/(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

(e^3*(d + e*x)^(1 + m)*Hypergeometric2F1[4, 1 + m, 2 + m, (b*(d + e*x))/(b*d - a
*e)])/((b*d - a*e)^4*(1 + m))

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Rubi in Sympy [A]  time = 20.6408, size = 42, normalized size = 0.79 \[ \frac{e^{3} \left (d + e x\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 4, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{b \left (- d - e x\right )}{a e - b d}} \right )}}{\left (m + 1\right ) \left (a e - b d\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e**3*(d + e*x)**(m + 1)*hyper((4, m + 1), (m + 2,), b*(-d - e*x)/(a*e - b*d))/((
m + 1)*(a*e - b*d)**4)

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Mathematica [A]  time = 0.0785091, size = 0, normalized size = 0. \[ \int \frac{(d+e x)^m}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[(d + e*x)^m/(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

Integrate[(d + e*x)^m/(a^2 + 2*a*b*x + b^2*x^2)^2, x]

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Maple [F]  time = 0.275, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex+d \right ) ^{m}}{ \left ({b}^{2}{x}^{2}+2\,abx+{a}^{2} \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^m/(b^2*x^2 + 2*a*b*x + a^2)^2,x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^m/(b^2*x^2 + 2*a*b*x + a^2)^2,x, algorithm="fricas")

[Out]

integral((e*x + d)^m/(b^4*x^4 + 4*a*b^3*x^3 + 6*a^2*b^2*x^2 + 4*a^3*b*x + a^4),
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/(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^m/(b^2*x^2 + 2*a*b*x + a^2)^2,x, algorithm="giac")

[Out]

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