3.2492 \(\int x (a+b x^n)^{3/2} \, dx\)

Optimal. Leaf size=48 \[ \frac{x^2 \left (a+b x^n\right )^{5/2} \, _2F_1\left (1,\frac{5}{2}+\frac{2}{n};\frac{n+2}{n};-\frac{b x^n}{a}\right )}{2 a} \]

[Out]

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

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Rubi [A]  time = 0.0152103, antiderivative size = 58, normalized size of antiderivative = 1.21, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {365, 364} \[ \frac{a x^2 \sqrt{a+b x^n} \, _2F_1\left (-\frac{3}{2},\frac{2}{n};\frac{n+2}{n};-\frac{b x^n}{a}\right )}{2 \sqrt{\frac{b x^n}{a}+1}} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x^n)^(3/2),x]

[Out]

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

Rule 365

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])
/(1 + (b*x^n)/a)^FracPart[p], Int[(c*x)^m*(1 + (b*x^n)/a)^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[
p, 0] &&  !(ILtQ[p, 0] || GtQ[a, 0])

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \int x \left (a+b x^n\right )^{3/2} \, dx &=\frac{\left (a \sqrt{a+b x^n}\right ) \int x \left (1+\frac{b x^n}{a}\right )^{3/2} \, dx}{\sqrt{1+\frac{b x^n}{a}}}\\ &=\frac{a x^2 \sqrt{a+b x^n} \, _2F_1\left (-\frac{3}{2},\frac{2}{n};\frac{2+n}{n};-\frac{b x^n}{a}\right )}{2 \sqrt{1+\frac{b x^n}{a}}}\\ \end{align*}

Mathematica [A]  time = 0.0151957, size = 58, normalized size = 1.21 \[ \frac{a x^2 \sqrt{a+b x^n} \, _2F_1\left (-\frac{3}{2},\frac{2}{n};1+\frac{2}{n};-\frac{b x^n}{a}\right )}{2 \sqrt{\frac{b x^n}{a}+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x^n)^(3/2),x]

[Out]

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

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Maple [F]  time = 0.047, size = 0, normalized size = 0. \begin{align*} \int x \left ( a+b{x}^{n} \right ) ^{{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x^n + a)^(3/2)*x, x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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Sympy [C]  time = 3.09395, size = 42, normalized size = 0.88 \begin{align*} \frac{a^{\frac{3}{2}} x^{2} \Gamma \left (\frac{2}{n}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, \frac{2}{n} \\ 1 + \frac{2}{n} \end{matrix}\middle |{\frac{b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (1 + \frac{2}{n}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**(3/2)*x**2*gamma(2/n)*hyper((-3/2, 2/n), (1 + 2/n,), b*x**n*exp_polar(I*pi)/a)/(n*gamma(1 + 2/n))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x^n + a)^(3/2)*x, x)