3.2497 \(\int x (a+b x^n)^{5/2} \, dx\)

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]

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

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Rubi [A]  time = 0.02, 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, Rules used = {365, 364} \[ \frac {a^2 x^2 \sqrt {a+b x^n} \, _2F_1\left (-\frac {5}{2},\frac {2}{n};\frac {n+2}{n};-\frac {b x^n}{a}\right )}{2 \sqrt {\frac {b x^n}{a}+1}} \]

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])

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])

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])

Rubi steps

\begin {align*} \int x \left (a+b x^n\right )^{5/2} \, dx &=\frac {\left (a^2 \sqrt {a+b x^n}\right ) \int x \left (1+\frac {b x^n}{a}\right )^{5/2} \, dx}{\sqrt {1+\frac {b x^n}{a}}}\\ &=\frac {a^2 x^2 \sqrt {a+b x^n} \, _2F_1\left (-\frac {5}{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 = 0.01, size = 60, normalized size = 1.25 \[ \frac {a^2 x^2 \sqrt {a+b x^n} \, _2F_1\left (-\frac {5}{2},\frac {2}{n};1+\frac {2}{n};-\frac {b x^n}{a}\right )}{2 \sqrt {\frac {b x^n}{a}+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x^n)^(5/2),x]

[Out]

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

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x^{n} + a\right )}^{\frac {5}{2}} x\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x^n + a)^(5/2)*x, x)

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maple [F]  time = 0.21, size = 0, normalized size = 0.00 \[ \int \left (b \,x^{n}+a \right )^{\frac {5}{2}} x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x^{n} + a\right )}^{\frac {5}{2}} x\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x^n + a)^(5/2)*x, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int x\,{\left (a+b\,x^n\right )}^{5/2} \,d x \]

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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sympy [C]  time = 6.90, size = 42, normalized size = 0.88 \[ \frac {a^{\frac {5}{2}} x^{2} \Gamma \left (\frac {2}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{2}, \frac {2}{n} \\ 1 + \frac {2}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (1 + \frac {2}{n}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**(5/2)*x**2*gamma(2/n)*hyper((-5/2, 2/n), (1 + 2/n,), b*x**n*exp_polar(I*pi)/a)/(n*gamma(1 + 2/n))

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