3.938 \(\int \frac{(a+b x)^m}{\left (a^2-b^2 x^2\right )^2} \, dx\)

Optimal. Leaf size=44 \[ -\frac{(a+b x)^{m-1} \, _2F_1\left (2,m-1;m;\frac{a+b x}{2 a}\right )}{4 a^2 b (1-m)} \]

[Out]

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

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Rubi [A]  time = 0.0600023, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{(a+b x)^{m-1} \, _2F_1\left (2,m-1;m;\frac{a+b x}{2 a}\right )}{4 a^2 b (1-m)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 12.5397, size = 32, normalized size = 0.73 \[ - \frac{\left (a + b x\right )^{m - 1}{{}_{2}F_{1}\left (\begin{matrix} 2, m - 1 \\ m \end{matrix}\middle |{\frac{\frac{a}{2} + \frac{b x}{2}}{a}} \right )}}{4 a^{2} b \left (- m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [B]  time = 0.145731, size = 102, normalized size = 2.32 \[ \frac{(a+b x)^m \left (\frac{2 (a+b x) \, _2F_1\left (1,m+1;m+2;\frac{a+b x}{2 a}\right )}{m+1}+\frac{(a+b x) \, _2F_1\left (2,m+1;m+2;\frac{a+b x}{2 a}\right )}{m+1}+4 a \left (\frac{a}{(m-1) (a+b x)}+\frac{1}{m}\right )\right )}{16 a^4 b} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x)^m*(4*a*(m^(-1) + a/((-1 + m)*(a + b*x))) + (2*(a + b*x)*Hypergeometri
c2F1[1, 1 + m, 2 + m, (a + b*x)/(2*a)])/(1 + m) + ((a + b*x)*Hypergeometric2F1[2
, 1 + m, 2 + m, (a + b*x)/(2*a)])/(1 + m)))/(16*a^4*b)

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Maple [F]  time = 0.087, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{m}}{ \left ( -{b}^{2}{x}^{2}+{a}^{2} \right ) ^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x + a)^m/(b^4*x^4 - 2*a^2*b^2*x^2 + a^4), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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