\(\int (a+b x^n)^{3/2} \, dx\) [2493]

Optimal result

Integrand size = 11, antiderivative size = 39 \[ \int \left (a+b x^n\right )^{3/2} \, dx=\frac {x \left (a+b x^n\right )^{5/2} \operatorname {Hypergeometric2F1}\left (1,\frac {5}{2}+\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )}{a} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.26, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {252, 251} \[ \int \left (a+b x^n\right )^{3/2} \, dx=\frac {a x \sqrt {a+b x^n} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )}{\sqrt {\frac {b x^n}{a}+1}} \]

[In]

Int[(a + b*x^n)^(3/2),x]

[Out]

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

Rule 251

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

Rule 252

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^IntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^Fra
cPart[p]), Int[(1 + b*(x^n/a))^p, x], x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILt
Q[Simplify[1/n + p], 0] &&  !(IntegerQ[p] || GtQ[a, 0])

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.26 \[ \int \left (a+b x^n\right )^{3/2} \, dx=\frac {a x \sqrt {a+b x^n} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )}{\sqrt {1+\frac {b x^n}{a}}} \]

[In]

Integrate[(a + b*x^n)^(3/2),x]

[Out]

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

Maple [F]

\[\int \left (a +b \,x^{n}\right )^{\frac {3}{2}}d x\]

[In]

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

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \left (a+b x^n\right )^{3/2} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((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)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.92 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.26 \[ \int \left (a+b x^n\right )^{3/2} \, dx=\frac {a^{\frac {1}{n}} a^{\frac {3}{2} - \frac {1}{n}} x \Gamma \left (\frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, \frac {1}{n} \\ 1 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (1 + \frac {1}{n}\right )} \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \left (a+b x^n\right )^{3/2} \, dx=\int { {\left (b x^{n} + a\right )}^{\frac {3}{2}} \,d x } \]

[In]

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

[Out]

integrate((b*x^n + a)^(3/2), x)

Giac [F]

\[ \int \left (a+b x^n\right )^{3/2} \, dx=\int { {\left (b x^{n} + a\right )}^{\frac {3}{2}} \,d x } \]

[In]

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

[Out]

integrate((b*x^n + a)^(3/2), x)

Mupad [B] (verification not implemented)

Time = 5.61 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.10 \[ \int \left (a+b x^n\right )^{3/2} \, dx=\frac {x\,{\left (a+b\,x^n\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},\frac {1}{n};\ \frac {1}{n}+1;\ -\frac {b\,x^n}{a}\right )}{{\left (\frac {b\,x^n}{a}+1\right )}^{3/2}} \]

[In]

int((a + b*x^n)^(3/2),x)

[Out]

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