3.2149 \(\int \frac{(a+b x) (d+e x)^m}{a^2+2 a b x+b^2 x^2} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0369977, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ -\frac{(d+e x)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{b (d+e x)}{b d-a e}\right )}{(m+1) (b d-a e)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 21.1856, size = 37, normalized size = 0.73 \[ \frac{\left (d + e x\right )^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{b \left (- d - e x\right )}{a e - b d}} \right )}}{\left (m + 1\right ) \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0344961, size = 51, normalized size = 1. \[ -\frac{(d+e x)^{m+1} \, _2F_1\left (1,m+1;m+2;\frac{b (d+e x)}{b d-a e}\right )}{(m+1) (b d-a e)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.276, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) \left ( ex+d \right ) ^{m}}{{b}^{2}{x}^{2}+2\,abx+{a}^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x + a)*(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. \[{\rm integral}\left (\frac{{\left (e x + d\right )}^{m}}{b x + a}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((e*x + d)^m/(b*x + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((d + e*x)**m/(a + b*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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