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

Optimal. Leaf size=55 \[ \frac{(a-b x) \left (a^2-b^2 x^2\right )^p \, _2F_1\left (1,2 p+1;p+1;\frac{a+b x}{2 a}\right )}{2 a b p} \]

[Out]

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

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Rubi [A]  time = 0.0887822, antiderivative size = 73, normalized size of antiderivative = 1.33, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{2^{p-1} \left (\frac{b x}{a}+1\right )^{-p-1} \left (a^2-b^2 x^2\right )^{p+1} \, _2F_1\left (1-p,p+1;p+2;\frac{a-b x}{2 a}\right )}{a^2 b (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 24.7968, size = 41, normalized size = 0.75 \[ \frac{\left (\frac{\frac{a}{2} - \frac{b x}{2}}{a}\right )^{- p} \left (a^{2} - b^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, p \\ p + 1 \end{matrix}\middle |{\frac{\frac{a}{2} + \frac{b x}{2}}{a}} \right )}}{b p} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0659949, size = 75, normalized size = 1.36 \[ -\frac{2^{p-1} (a-b x) \left (\frac{b x}{a}+1\right )^{-p} \left (a^2-b^2 x^2\right )^p \, _2F_1\left (1-p,p+1;p+2;\frac{a-b x}{2 a}\right )}{a b (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.071, size = 0, normalized size = 0. \[ \int{\frac{ \left ( -{b}^{2}{x}^{2}+{a}^{2} \right ) ^{p}}{bx+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-b^2*x^2+a^2)^p/(b*x+a),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 13.2593, size = 321, normalized size = 5.84 \[ \begin{cases} \frac{0^{p} \log{\left (-1 + \frac{b^{2} x^{2}}{a^{2}} \right )}}{2 b} + \frac{0^{p} \operatorname{acoth}{\left (\frac{b x}{a} \right )}}{b} + \frac{a b^{2 p} p x^{2 p} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (- p + \frac{1}{2}\right ){{}_{2}F_{1}\left (\begin{matrix} - p + 1, - p + \frac{1}{2} \\ - p + \frac{3}{2} \end{matrix}\middle |{\frac{a^{2}}{b^{2} x^{2}}} \right )}}{2 b^{2} x \Gamma \left (- p + \frac{3}{2}\right ) \Gamma \left (p + 1\right )} + \frac{a^{2 p} b x^{2} \Gamma \left (p\right ) \Gamma \left (- p + 1\right ){{}_{3}F_{2}\left (\begin{matrix} 2, 1, - p + 1 \\ 2, 2 \end{matrix}\middle |{\frac{b^{2} x^{2} e^{2 i \pi }}{a^{2}}} \right )}}{2 a^{2} \Gamma \left (- p\right ) \Gamma \left (p + 1\right )} & \text{for}\: \left |{\frac{b^{2} x^{2}}{a^{2}}}\right | > 1 \\\frac{0^{p} \log{\left (1 - \frac{b^{2} x^{2}}{a^{2}} \right )}}{2 b} + \frac{0^{p} \operatorname{atanh}{\left (\frac{b x}{a} \right )}}{b} + \frac{a b^{2 p} p x^{2 p} e^{i \pi p} \Gamma \left (p\right ) \Gamma \left (- p + \frac{1}{2}\right ){{}_{2}F_{1}\left (\begin{matrix} - p + 1, - p + \frac{1}{2} \\ - p + \frac{3}{2} \end{matrix}\middle |{\frac{a^{2}}{b^{2} x^{2}}} \right )}}{2 b^{2} x \Gamma \left (- p + \frac{3}{2}\right ) \Gamma \left (p + 1\right )} + \frac{a^{2 p} b x^{2} \Gamma \left (p\right ) \Gamma \left (- p + 1\right ){{}_{3}F_{2}\left (\begin{matrix} 2, 1, - p + 1 \\ 2, 2 \end{matrix}\middle |{\frac{b^{2} x^{2} e^{2 i \pi }}{a^{2}}} \right )}}{2 a^{2} \Gamma \left (- p\right ) \Gamma \left (p + 1\right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((0**p*log(-1 + b**2*x**2/a**2)/(2*b) + 0**p*acoth(b*x/a)/b + a*b**(2*p
)*p*x**(2*p)*exp(I*pi*p)*gamma(p)*gamma(-p + 1/2)*hyper((-p + 1, -p + 1/2), (-p
+ 3/2,), a**2/(b**2*x**2))/(2*b**2*x*gamma(-p + 3/2)*gamma(p + 1)) + a**(2*p)*b*
x**2*gamma(p)*gamma(-p + 1)*hyper((2, 1, -p + 1), (2, 2), b**2*x**2*exp_polar(2*
I*pi)/a**2)/(2*a**2*gamma(-p)*gamma(p + 1)), Abs(b**2*x**2/a**2) > 1), (0**p*log
(1 - b**2*x**2/a**2)/(2*b) + 0**p*atanh(b*x/a)/b + a*b**(2*p)*p*x**(2*p)*exp(I*p
i*p)*gamma(p)*gamma(-p + 1/2)*hyper((-p + 1, -p + 1/2), (-p + 3/2,), a**2/(b**2*
x**2))/(2*b**2*x*gamma(-p + 3/2)*gamma(p + 1)) + a**(2*p)*b*x**2*gamma(p)*gamma(
-p + 1)*hyper((2, 1, -p + 1), (2, 2), b**2*x**2*exp_polar(2*I*pi)/a**2)/(2*a**2*
gamma(-p)*gamma(p + 1)), True))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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