3.840 \(\int \frac{x^m (A+B x)}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx\)

Optimal. Leaf size=81 \[ \frac{x^{m+1} (a B (m+1)+A b (2-m)) \, _2F_1\left (3,m+1;m+2;-\frac{b x}{a}\right )}{3 a^4 b (m+1)}+\frac{x^{m+1} (A b-a B)}{3 a b (a+b x)^3} \]

[Out]

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

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Rubi [A]  time = 0.0981282, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{x^{m+1} (a B (m+1)+A b (2-m)) \, _2F_1\left (3,m+1;m+2;-\frac{b x}{a}\right )}{3 a^4 b (m+1)}+\frac{x^{m+1} (A b-a B)}{3 a b (a+b x)^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 22.8603, size = 63, normalized size = 0.78 \[ \frac{x^{m + 1} \left (A b - B a\right )}{3 a b \left (a + b x\right )^{3}} + \frac{x^{m + 1} \left (A b \left (- m + 2\right ) + B a \left (m + 1\right )\right ){{}_{2}F_{1}\left (\begin{matrix} 3, m + 1 \\ m + 2 \end{matrix}\middle |{- \frac{b x}{a}} \right )}}{3 a^{4} b \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**m*(B*x+A)/(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

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

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Mathematica [A]  time = 0.0951725, size = 60, normalized size = 0.74 \[ \frac{x^{m+1} \left ((A b-a B) \, _2F_1\left (4,m+1;m+2;-\frac{b x}{a}\right )+a B \, _2F_1\left (3,m+1;m+2;-\frac{b x}{a}\right )\right )}{a^4 b (m+1)} \]

Antiderivative was successfully verified.

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

[Out]

(x^(1 + m)*(a*B*Hypergeometric2F1[3, 1 + m, 2 + m, -((b*x)/a)] + (A*b - a*B)*Hyp
ergeometric2F1[4, 1 + m, 2 + m, -((b*x)/a)]))/(a^4*b*(1 + m))

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Maple [F]  time = 0.189, size = 0, normalized size = 0. \[ \int{\frac{{x}^{m} \left ( Bx+A \right ) }{ \left ({b}^{2}{x}^{2}+2\,abx+{a}^{2} \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((B*x + A)*x^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. \[{\rm integral}\left (\frac{{\left (B x + A\right )} x^{m}}{b^{4} x^{4} + 4 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x + a^{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((B*x + A)*x^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]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m} \left (A + B x\right )}{\left (a + b x\right )^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**m*(B*x+A)/(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

Integral(x**m*(A + B*x)/(a + b*x)**4, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((B*x + A)*x^m/(b^2*x^2 + 2*a*b*x + a^2)^2, x)