3.25.92 \(\int x (a+b x^n)^{3/2} \, dx\) [2492]

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

[Out]

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

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Rubi [A]
time = 0.01, 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 = {372, 371} \begin {gather*} \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}} \end {gather*}

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 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 372

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])

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*}

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Mathematica [A]
time = 0.02, size = 58, normalized size = 1.21 \begin {gather*} \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 {1+\frac {b x^n}{a}}} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \[\int x \left (a +b \,x^{n}\right )^{\frac {3}{2}}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

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Sympy [C] Result contains complex when optimal does not.
time = 1.13, size = 42, normalized size = 0.88 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int x\,{\left (a+b\,x^n\right )}^{3/2} \,d x \end {gather*}

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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