3.2490 \(\int x \left (a+b x^n\right )^{5/2} \, dx\)

Optimal. Leaf size=48 \[ \frac{x^2 \left (a+b x^n\right )^{7/2} \, _2F_1\left (1,\frac{7}{2}+\frac{2}{n};\frac{n+2}{n};-\frac{b x^n}{a}\right )}{2 a} \]

[Out]

(x^2*(a + b*x^n)^(7/2)*Hypergeometric2F1[1, 7/2 + 2/n, (2 + n)/n, -((b*x^n)/a)])
/(2*a)

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Rubi [A]  time = 0.0589421, antiderivative size = 60, normalized size of antiderivative = 1.25, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{a^2 x^2 \sqrt{a+b x^n} \, _2F_1\left (-\frac{5}{2},\frac{2}{n};\frac{n+2}{n};-\frac{b x^n}{a}\right )}{2 \sqrt{\frac{b x^n}{a}+1}} \]

Antiderivative was successfully verified.

[In]  Int[x*(a + b*x^n)^(5/2),x]

[Out]

(a^2*x^2*Sqrt[a + b*x^n]*Hypergeometric2F1[-5/2, 2/n, (2 + n)/n, -((b*x^n)/a)])/
(2*Sqrt[1 + (b*x^n)/a])

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Rubi in Sympy [A]  time = 6.4851, size = 48, normalized size = 1. \[ \frac{a^{2} x^{2} \sqrt{a + b x^{n}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{2}, \frac{2}{n} \\ \frac{n + 2}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{2 \sqrt{1 + \frac{b x^{n}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(a+b*x**n)**(5/2),x)

[Out]

a**2*x**2*sqrt(a + b*x**n)*hyper((-5/2, 2/n), ((n + 2)/n,), -b*x**n/a)/(2*sqrt(1
+ b*x**n/a))

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Mathematica [B]  time = 0.286405, size = 144, normalized size = 3. \[ \frac{x^2 \left (15 a^3 n^3 \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{2}{n};\frac{n+2}{n};-\frac{b x^n}{a}\right )+4 \left (a+b x^n\right ) \left (a^2 \left (23 n^2+36 n+16\right )+a b \left (11 n^2+52 n+32\right ) x^n+b^2 \left (3 n^2+16 n+16\right ) x^{2 n}\right )\right )}{2 (n+4) (3 n+4) (5 n+4) \sqrt{a+b x^n}} \]

Antiderivative was successfully verified.

[In]  Integrate[x*(a + b*x^n)^(5/2),x]

[Out]

(x^2*(4*(a + b*x^n)*(a^2*(16 + 36*n + 23*n^2) + a*b*(32 + 52*n + 11*n^2)*x^n + b
^2*(16 + 16*n + 3*n^2)*x^(2*n)) + 15*a^3*n^3*Sqrt[1 + (b*x^n)/a]*Hypergeometric2
F1[1/2, 2/n, (2 + n)/n, -((b*x^n)/a)]))/(2*(4 + n)*(4 + 3*n)*(4 + 5*n)*Sqrt[a +
b*x^n])

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Maple [F]  time = 0.054, size = 0, normalized size = 0. \[ \int x \left ( a+b{x}^{n} \right ) ^{{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(a+b*x^n)^(5/2),x)

[Out]

int(x*(a+b*x^n)^(5/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{n} + a\right )}^{\frac{5}{2}} x\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^(5/2)*x,x, algorithm="maxima")

[Out]

integrate((b*x^n + a)^(5/2)*x, 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)^(5/2)*x,x, algorithm="fricas")

[Out]

Exception raised: TypeError

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Sympy [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(x*(a+b*x**n)**(5/2),x)

[Out]

Exception raised: TypeError

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{n} + a\right )}^{\frac{5}{2}} x\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^(5/2)*x,x, algorithm="giac")

[Out]

integrate((b*x^n + a)^(5/2)*x, x)