3.962 \(\int (a+b x)^{3/2} \left (a^2-b^2 x^2\right )^p \, dx\)

Optimal. Leaf size=85 \[ -\frac{2^{p+\frac{3}{2}} \sqrt{a+b x} \left (\frac{b x}{a}+1\right )^{-p-\frac{3}{2}} \left (a^2-b^2 x^2\right )^{p+1} \, _2F_1\left (-p-\frac{3}{2},p+1;p+2;\frac{a-b x}{2 a}\right )}{b (p+1)} \]

[Out]

-((2^(3/2 + p)*Sqrt[a + b*x]*(1 + (b*x)/a)^(-3/2 - p)*(a^2 - b^2*x^2)^(1 + p)*Hy
pergeometric2F1[-3/2 - p, 1 + p, 2 + p, (a - b*x)/(2*a)])/(b*(1 + p)))

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Rubi [A]  time = 0.166048, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{2^{p+\frac{3}{2}} \sqrt{a+b x} \left (\frac{b x}{a}+1\right )^{-p-\frac{3}{2}} \left (a^2-b^2 x^2\right )^{p+1} \, _2F_1\left (-p-\frac{3}{2},p+1;p+2;\frac{a-b x}{2 a}\right )}{b (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

-((2^(3/2 + p)*Sqrt[a + b*x]*(1 + (b*x)/a)^(-3/2 - p)*(a^2 - b^2*x^2)^(1 + p)*Hy
pergeometric2F1[-3/2 - p, 1 + p, 2 + p, (a - b*x)/(2*a)])/(b*(1 + p)))

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Rubi in Sympy [A]  time = 28.1955, size = 90, normalized size = 1.06 \[ - \frac{2 a \left (\frac{\frac{a}{2} + \frac{b x}{2}}{a}\right )^{- p - \frac{1}{2}} \left (a - b x\right )^{- p} \left (a - b x\right )^{p + 1} \left (a + b x\right )^{- p} \left (a + b x\right )^{p + \frac{1}{2}} \left (a^{2} - b^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p - \frac{3}{2}, p + 1 \\ p + 2 \end{matrix}\middle |{\frac{\frac{a}{2} - \frac{b x}{2}}{a}} \right )}}{b \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.948257, size = 246, normalized size = 2.89 \[ \frac{a \sqrt{a+b x} (a-b x)^p \left (\frac{3 b^2 x^2 (a+b x)^p F_1\left (2;-p,-p-\frac{1}{2};3;\frac{b x}{a},-\frac{b x}{a}\right )}{6 a F_1\left (2;-p,-p-\frac{1}{2};3;\frac{b x}{a},-\frac{b x}{a}\right )+b x \left ((2 p+1) F_1\left (3;-p,\frac{1}{2}-p;4;\frac{b x}{a},-\frac{b x}{a}\right )-2 p F_1\left (3;1-p,-p-\frac{1}{2};4;\frac{b x}{a},-\frac{b x}{a}\right )\right )}-\frac{2^{p+\frac{1}{2}} (a-b x)^{1-p} \left (\frac{b x}{a}+1\right )^{-p-\frac{1}{2}} \left (a^2-b^2 x^2\right )^p \, _2F_1\left (-p-\frac{1}{2},p+1;p+2;\frac{a-b x}{2 a}\right )}{p+1}\right )}{b} \]

Warning: Unable to verify antiderivative.

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

[Out]

(a*(a - b*x)^p*Sqrt[a + b*x]*((3*b^2*x^2*(a + b*x)^p*AppellF1[2, -p, -1/2 - p, 3
, (b*x)/a, -((b*x)/a)])/(6*a*AppellF1[2, -p, -1/2 - p, 3, (b*x)/a, -((b*x)/a)] +
b*x*(-2*p*AppellF1[3, 1 - p, -1/2 - p, 4, (b*x)/a, -((b*x)/a)] + (1 + 2*p)*Appe
llF1[3, -p, 1/2 - p, 4, (b*x)/a, -((b*x)/a)])) - (2^(1/2 + p)*(a - b*x)^(1 - p)*
(1 + (b*x)/a)^(-1/2 - p)*(a^2 - b^2*x^2)^p*Hypergeometric2F1[-1/2 - p, 1 + p, 2
+ p, (a - b*x)/(2*a)])/(1 + p)))/b

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Maple [F]  time = 0.051, size = 0, normalized size = 0. \[ \int \left ( bx+a \right ) ^{{\frac{3}{2}}} \left ( -{b}^{2}{x}^{2}+{a}^{2} \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x + a)^(3/2)*(-b^2*x^2 + a^2)^p, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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