3.27.60 \(\int \frac {x^{-1+2 n}}{\sqrt {a+b x^n}} \, dx\) [2660]

Optimal. Leaf size=42 \[ -\frac {2 a \sqrt {a+b x^n}}{b^2 n}+\frac {2 \left (a+b x^n\right )^{3/2}}{3 b^2 n} \]

[Out]

2/3*(a+b*x^n)^(3/2)/b^2/n-2*a*(a+b*x^n)^(1/2)/b^2/n

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Rubi [A]
time = 0.01, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {272, 45} \begin {gather*} \frac {2 \left (a+b x^n\right )^{3/2}}{3 b^2 n}-\frac {2 a \sqrt {a+b x^n}}{b^2 n} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(-1 + 2*n)/Sqrt[a + b*x^n],x]

[Out]

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

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

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

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Mathematica [A]
time = 0.03, size = 30, normalized size = 0.71 \begin {gather*} \frac {2 \left (-2 a+b x^n\right ) \sqrt {a+b x^n}}{3 b^2 n} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.25, size = 28, normalized size = 0.67

method	result	size
risch	\(-\frac {2 \left (-b \,x^{n}+2 a \right ) \sqrt {a +b \,x^{n}}}{3 b^{2} n}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1+2*n)/(a+b*x^n)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2/3*(-b*x^n+2*a)*(a+b*x^n)^(1/2)/b^2/n

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Maxima [A]
time = 0.30, size = 39, normalized size = 0.93 \begin {gather*} \frac {2 \, {\left (b^{2} x^{2 \, n} - a b x^{n} - 2 \, a^{2}\right )}}{3 \, \sqrt {b x^{n} + a} b^{2} n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+2*n)/(a+b*x^n)^(1/2),x, algorithm="maxima")

[Out]

2/3*(b^2*x^(2*n) - a*b*x^n - 2*a^2)/(sqrt(b*x^n + a)*b^2*n)

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Fricas [A]
time = 0.39, size = 26, normalized size = 0.62 \begin {gather*} \frac {2 \, \sqrt {b x^{n} + a} {\left (b x^{n} - 2 \, a\right )}}{3 \, b^{2} n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+2*n)/(a+b*x^n)^(1/2),x, algorithm="fricas")

[Out]

2/3*sqrt(b*x^n + a)*(b*x^n - 2*a)/(b^2*n)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 275 vs. \(2 (36) = 72\).
time = 4.38, size = 275, normalized size = 6.55 \begin {gather*} - \frac {4 a^{\frac {7}{2}} b^{\frac {3}{2}} x^{\frac {3 n}{2}} \sqrt {\frac {a x^{- n}}{b} + 1}}{3 a^{\frac {5}{2}} b^{3} n x^{n} + 3 a^{\frac {3}{2}} b^{4} n x^{2 n}} - \frac {2 a^{\frac {5}{2}} b^{\frac {5}{2}} x^{\frac {5 n}{2}} \sqrt {\frac {a x^{- n}}{b} + 1}}{3 a^{\frac {5}{2}} b^{3} n x^{n} + 3 a^{\frac {3}{2}} b^{4} n x^{2 n}} + \frac {2 a^{\frac {3}{2}} b^{\frac {7}{2}} x^{\frac {7 n}{2}} \sqrt {\frac {a x^{- n}}{b} + 1}}{3 a^{\frac {5}{2}} b^{3} n x^{n} + 3 a^{\frac {3}{2}} b^{4} n x^{2 n}} + \frac {4 a^{4} b x^{n}}{3 a^{\frac {5}{2}} b^{3} n x^{n} + 3 a^{\frac {3}{2}} b^{4} n x^{2 n}} + \frac {4 a^{3} b^{2} x^{2 n}}{3 a^{\frac {5}{2}} b^{3} n x^{n} + 3 a^{\frac {3}{2}} b^{4} n x^{2 n}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(-1+2*n)/(a+b*x**n)**(1/2),x)

[Out]

-4*a**(7/2)*b**(3/2)*x**(3*n/2)*sqrt(a/(b*x**n) + 1)/(3*a**(5/2)*b**3*n*x**n + 3*a**(3/2)*b**4*n*x**(2*n)) - 2
*a**(5/2)*b**(5/2)*x**(5*n/2)*sqrt(a/(b*x**n) + 1)/(3*a**(5/2)*b**3*n*x**n + 3*a**(3/2)*b**4*n*x**(2*n)) + 2*a
**(3/2)*b**(7/2)*x**(7*n/2)*sqrt(a/(b*x**n) + 1)/(3*a**(5/2)*b**3*n*x**n + 3*a**(3/2)*b**4*n*x**(2*n)) + 4*a**
4*b*x**n/(3*a**(5/2)*b**3*n*x**n + 3*a**(3/2)*b**4*n*x**(2*n)) + 4*a**3*b**2*x**(2*n)/(3*a**(5/2)*b**3*n*x**n
+ 3*a**(3/2)*b**4*n*x**(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^(-1+2*n)/(a+b*x^n)^(1/2),x, algorithm="giac")

[Out]

integrate(x^(2*n - 1)/sqrt(b*x^n + a), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x^{2\,n-1}}{\sqrt {a+b\,x^n}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(2*n - 1)/(a + b*x^n)^(1/2),x)

[Out]

int(x^(2*n - 1)/(a + b*x^n)^(1/2), x)

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