3.948 \(\int (a+b x)^m \left (a^2-b^2 x^2\right )^p \, dx\)

Optimal. Leaf size=63 \[ \frac{(a+b x)^m \left (a^2-b^2 x^2\right )^{p+1} \, _2F_1\left (1,m+2 p+2;m+p+2;\frac{a+b x}{2 a}\right )}{2 a b (m+p+1)} \]

[Out]

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

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Rubi [A]  time = 0.147248, antiderivative size = 85, normalized size of antiderivative = 1.35, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ -\frac{2^{m+p} (a+b x)^m \left (a^2-b^2 x^2\right )^{p+1} \left (\frac{b x}{a}+1\right )^{-m-p-1} \, _2F_1\left (-m-p,p+1;p+2;\frac{a-b x}{2 a}\right )}{a b (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

-((2^(m + p)*(a + b*x)^m*(1 + (b*x)/a)^(-1 - m - p)*(a^2 - b^2*x^2)^(1 + p)*Hype
rgeometric2F1[-m - p, 1 + p, 2 + p, (a - b*x)/(2*a)])/(a*b*(1 + p)))

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Rubi in Sympy [A]  time = 26.8037, size = 82, normalized size = 1.3 \[ - \frac{\left (\frac{\frac{a}{2} + \frac{b x}{2}}{a}\right )^{- m - p} \left (a - b x\right )^{- p} \left (a - b x\right )^{p + 1} \left (a + b x\right )^{- p} \left (a + b x\right )^{m + p} \left (a^{2} - b^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - m - p, p + 1 \\ p + 2 \end{matrix}\middle |{\frac{\frac{a}{2} - \frac{b x}{2}}{a}} \right )}}{b \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.131301, size = 85, normalized size = 1.35 \[ \frac{2^{m+p} (b x-a) (a+b x)^m \left (a^2-b^2 x^2\right )^p \left (\frac{b x}{a}+1\right )^{-m-p} \, _2F_1\left (-m-p,p+1;p+2;\frac{a-b x}{2 a}\right )}{b (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

(2^(m + p)*(-a + b*x)*(a + b*x)^m*(1 + (b*x)/a)^(-m - p)*(a^2 - b^2*x^2)^p*Hyper
geometric2F1[-m - p, 1 + p, 2 + p, (a - b*x)/(2*a)])/(b*(1 + p))

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Maple [F]  time = 0.148, size = 0, normalized size = 0. \[ \int \left ( bx+a \right ) ^{m} \left ( -{b}^{2}{x}^{2}+{a}^{2} \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((-(-a + b*x)*(a + b*x))**p*(a + b*x)**m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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