3.939 \(\int \frac{(a+b x)^m}{\left (a^2-b^2 x^2\right )^3} \, dx\)

Optimal. Leaf size=46 \[ -\frac{(a+b x)^{m-2} \, _2F_1\left (3,m-2;m-1;\frac{a+b x}{2 a}\right )}{8 a^3 b (2-m)} \]

[Out]

-((a + b*x)^(-2 + m)*Hypergeometric2F1[3, -2 + m, -1 + m, (a + b*x)/(2*a)])/(8*a
^3*b*(2 - m))

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Rubi [A]  time = 0.061111, antiderivative size = 46, 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 \[ -\frac{(a+b x)^{m-2} \, _2F_1\left (3,m-2;m-1;\frac{a+b x}{2 a}\right )}{8 a^3 b (2-m)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^m/(a^2 - b^2*x^2)^3,x]

[Out]

-((a + b*x)^(-2 + m)*Hypergeometric2F1[3, -2 + m, -1 + m, (a + b*x)/(2*a)])/(8*a
^3*b*(2 - m))

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Rubi in Sympy [A]  time = 12.3577, size = 34, normalized size = 0.74 \[ - \frac{\left (a + b x\right )^{m - 2}{{}_{2}F_{1}\left (\begin{matrix} 3, m - 2 \\ m - 1 \end{matrix}\middle |{\frac{\frac{a}{2} + \frac{b x}{2}}{a}} \right )}}{8 a^{3} b \left (- m + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**m/(-b**2*x**2+a**2)**3,x)

[Out]

-(a + b*x)**(m - 2)*hyper((3, m - 2), (m - 1,), (a/2 + b*x/2)/a)/(8*a**3*b*(-m +
2))

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Mathematica [A]  time = 0.0650657, size = 44, normalized size = 0.96 \[ \frac{(a+b x)^{m-2} \, _2F_1\left (3,m-2;m-1;\frac{a+b x}{2 a}\right )}{8 a^3 b (m-2)} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^m/(a^2 - b^2*x^2)^3,x]

[Out]

((a + b*x)^(-2 + m)*Hypergeometric2F1[3, -2 + m, -1 + m, (a + b*x)/(2*a)])/(8*a^
3*b*(-2 + m))

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Maple [F]  time = 0.129, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{m}}{ \left ( -{b}^{2}{x}^{2}+{a}^{2} \right ) ^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^m/(-b^2*x^2+a^2)^3,x)

[Out]

int((b*x+a)^m/(-b^2*x^2+a^2)^3,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{{\left (b x + a\right )}^{m}}{{\left (b^{2} x^{2} - a^{2}\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*x + a)^m/(b^2*x^2 - a^2)^3,x, algorithm="maxima")

[Out]

-integrate((b*x + a)^m/(b^2*x^2 - a^2)^3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*x + a)^m/(b^2*x^2 - a^2)^3,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{\left (a + b x\right )^{m}}{- a^{6} + 3 a^{4} b^{2} x^{2} - 3 a^{2} b^{4} x^{4} + b^{6} x^{6}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**m/(-b**2*x**2+a**2)**3,x)

[Out]

-Integral((a + b*x)**m/(-a**6 + 3*a**4*b**2*x**2 - 3*a**2*b**4*x**4 + b**6*x**6)
, x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -\frac{{\left (b x + a\right )}^{m}}{{\left (b^{2} x^{2} - a^{2}\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*x + a)^m/(b^2*x^2 - a^2)^3,x, algorithm="giac")

[Out]

integrate(-(b*x + a)^m/(b^2*x^2 - a^2)^3, x)