3.1723 \(\int \frac{(d+e x)^m}{a^2+2 a b x+b^2 x^2} \, dx\)

Optimal. Leaf size=51 \[ \frac{e (d+e x)^{m+1} \, _2F_1\left (2,m+1;m+2;\frac{b (d+e x)}{b d-a e}\right )}{(m+1) (b d-a e)^2} \]

[Out]

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

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Rubi [A]  time = 0.0553475, antiderivative size = 51, 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 \[ \frac{e (d+e x)^{m+1} \, _2F_1\left (2,m+1;m+2;\frac{b (d+e x)}{b d-a e}\right )}{(m+1) (b d-a e)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 17.8685, size = 41, normalized size = 0.8 \[ \frac{e \left (d + e x\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 2, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{b \left (- d - e x\right )}{a e - b d}} \right )}}{\left (m + 1\right ) \left (a e - b d\right )^{2}} \]

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),x)

[Out]

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

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Mathematica [A]  time = 0.0540506, size = 0, normalized size = 0. \[ \int \frac{(d+e x)^m}{a^2+2 a b x+b^2 x^2} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.151, size = 0, normalized size = 0. \[ \int{\frac{ \left ( ex+d \right ) ^{m}}{{b}^{2}{x}^{2}+2\,abx+{a}^{2}}}\, dx \]

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),x)

[Out]

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

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

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),x, algorithm="maxima")

[Out]

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

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

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),x, algorithm="fricas")

[Out]

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

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

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),x)

[Out]

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

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

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),x, algorithm="giac")

[Out]

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