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

Optimal. Leaf size=79 \[ \frac{a^2 x \sqrt{a-b x^n} \sqrt{a+b x^n} \, _2F_1\left (-\frac{3}{2},\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );\frac{b^2 x^{2 n}}{a^2}\right )}{\sqrt{1-\frac{b^2 x^{2 n}}{a^2}}} \]

[Out]

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

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Rubi [A]  time = 0.0826049, antiderivative size = 79, 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 \[ \frac{a^2 x \sqrt{a-b x^n} \sqrt{a+b x^n} \, _2F_1\left (-\frac{3}{2},\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );\frac{b^2 x^{2 n}}{a^2}\right )}{\sqrt{1-\frac{b^2 x^{2 n}}{a^2}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 19.6892, size = 66, normalized size = 0.84 \[ \frac{a^{2} x \sqrt{a - b x^{n}} \sqrt{a + b x^{n}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, \frac{1}{2 n} \\ \frac{n + \frac{1}{2}}{n} \end{matrix}\middle |{\frac{b^{2} x^{2 n}}{a^{2}}} \right )}}{\sqrt{1 - \frac{b^{2} x^{2 n}}{a^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.262048, size = 151, normalized size = 1.91 \[ \frac{x \sqrt{a-b x^n} \sqrt{a+b x^n} \left (\left (a^2-b^2 x^{2 n}\right ) \left (a^2 (4 n+1)-b^2 (n+1) x^{2 n}\right )+3 a^4 n^2 \sqrt{1-\frac{b^2 x^{2 n}}{a^2}} \, _2F_1\left (\frac{1}{2},\frac{1}{2 n};1+\frac{1}{2 n};\frac{b^2 x^{2 n}}{a^2}\right )\right )}{(n+1) (3 n+1) \left (a^2-b^2 x^{2 n}\right )} \]

Antiderivative was successfully verified.

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

[Out]

(x*Sqrt[a - b*x^n]*Sqrt[a + b*x^n]*((a^2 - b^2*x^(2*n))*(a^2*(1 + 4*n) - b^2*(1
+ n)*x^(2*n)) + 3*a^4*n^2*Sqrt[1 - (b^2*x^(2*n))/a^2]*Hypergeometric2F1[1/2, 1/(
2*n), 1 + 1/(2*n), (b^2*x^(2*n))/a^2]))/((1 + n)*(1 + 3*n)*(a^2 - b^2*x^(2*n)))

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Maple [F]  time = 0.118, size = 0, normalized size = 0. \[ \int \left ( a-b{x}^{n} \right ) ^{{\frac{3}{2}}} \left ( a+b{x}^{n} \right ) ^{{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError

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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((a-b*x**n)**(3/2)*(a+b*x**n)**(3/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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