3.27.56 \(\int x^{-1-3 n} \sqrt {a+b x^n} \, dx\) [2656]

Optimal. Leaf size=113 \[ -\frac {x^{-3 n} \sqrt {a+b x^n}}{3 n}-\frac {b x^{-2 n} \sqrt {a+b x^n}}{12 a n}+\frac {b^2 x^{-n} \sqrt {a+b x^n}}{8 a^2 n}-\frac {b^3 \tanh ^{-1}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{8 a^{5/2} n} \]

[Out]

-1/8*b^3*arctanh((a+b*x^n)^(1/2)/a^(1/2))/a^(5/2)/n-1/3*(a+b*x^n)^(1/2)/n/(x^(3*n))-1/12*b*(a+b*x^n)^(1/2)/a/n
/(x^(2*n))+1/8*b^2*(a+b*x^n)^(1/2)/a^2/n/(x^n)

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Rubi [A]
time = 0.04, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {272, 43, 44, 65, 214} \begin {gather*} -\frac {b^3 \tanh ^{-1}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{8 a^{5/2} n}+\frac {b^2 x^{-n} \sqrt {a+b x^n}}{8 a^2 n}-\frac {x^{-3 n} \sqrt {a+b x^n}}{3 n}-\frac {b x^{-2 n} \sqrt {a+b x^n}}{12 a n} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(-1 - 3*n)*Sqrt[a + b*x^n],x]

[Out]

-1/3*Sqrt[a + b*x^n]/(n*x^(3*n)) - (b*Sqrt[a + b*x^n])/(12*a*n*x^(2*n)) + (b^2*Sqrt[a + b*x^n])/(8*a^2*n*x^n)
- (b^3*ArcTanh[Sqrt[a + b*x^n]/Sqrt[a]])/(8*a^(5/2)*n)

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + 1))), x] - Dist[d*(n/(b*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n
}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && GtQ[n, 0]

Rule 44

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

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 x^{-1-3 n} \sqrt {a+b x^n} \, dx &=\frac {\text {Subst}\left (\int \frac {\sqrt {a+b x}}{x^4} \, dx,x,x^n\right )}{n}\\ &=-\frac {x^{-3 n} \sqrt {a+b x^n}}{3 n}+\frac {b \text {Subst}\left (\int \frac {1}{x^3 \sqrt {a+b x}} \, dx,x,x^n\right )}{6 n}\\ &=-\frac {x^{-3 n} \sqrt {a+b x^n}}{3 n}-\frac {b x^{-2 n} \sqrt {a+b x^n}}{12 a n}-\frac {b^2 \text {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x}} \, dx,x,x^n\right )}{8 a n}\\ &=-\frac {x^{-3 n} \sqrt {a+b x^n}}{3 n}-\frac {b x^{-2 n} \sqrt {a+b x^n}}{12 a n}+\frac {b^2 x^{-n} \sqrt {a+b x^n}}{8 a^2 n}+\frac {b^3 \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^n\right )}{16 a^2 n}\\ &=-\frac {x^{-3 n} \sqrt {a+b x^n}}{3 n}-\frac {b x^{-2 n} \sqrt {a+b x^n}}{12 a n}+\frac {b^2 x^{-n} \sqrt {a+b x^n}}{8 a^2 n}+\frac {b^2 \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^n}\right )}{8 a^2 n}\\ &=-\frac {x^{-3 n} \sqrt {a+b x^n}}{3 n}-\frac {b x^{-2 n} \sqrt {a+b x^n}}{12 a n}+\frac {b^2 x^{-n} \sqrt {a+b x^n}}{8 a^2 n}-\frac {b^3 \tanh ^{-1}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{8 a^{5/2} n}\\ \end {align*}

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Mathematica [A]
time = 0.15, size = 81, normalized size = 0.72 \begin {gather*} \frac {\sqrt {a} x^{-3 n} \sqrt {a+b x^n} \left (-8 a^2-2 a b x^n+3 b^2 x^{2 n}\right )-3 b^3 \tanh ^{-1}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{24 a^{5/2} n} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((Sqrt[a]*Sqrt[a + b*x^n]*(-8*a^2 - 2*a*b*x^n + 3*b^2*x^(2*n)))/x^(3*n) - 3*b^3*ArcTanh[Sqrt[a + b*x^n]/Sqrt[a
]])/(24*a^(5/2)*n)

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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int x^{-1-3 n} \sqrt {a +b \,x^{n}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^(-1-3*n)*(a+b*x^n)^(1/2),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^(-1-3*n)*(a+b*x^n)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(b*x^n + a)*x^(-3*n - 1), x)

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Fricas [A]
time = 0.41, size = 183, normalized size = 1.62 \begin {gather*} \left [\frac {3 \, \sqrt {a} b^{3} x^{3 \, n} \log \left (\frac {b x^{n} - 2 \, \sqrt {b x^{n} + a} \sqrt {a} + 2 \, a}{x^{n}}\right ) + 2 \, {\left (3 \, a b^{2} x^{2 \, n} - 2 \, a^{2} b x^{n} - 8 \, a^{3}\right )} \sqrt {b x^{n} + a}}{48 \, a^{3} n x^{3 \, n}}, \frac {3 \, \sqrt {-a} b^{3} x^{3 \, n} \arctan \left (\frac {\sqrt {b x^{n} + a} \sqrt {-a}}{a}\right ) + {\left (3 \, a b^{2} x^{2 \, n} - 2 \, a^{2} b x^{n} - 8 \, a^{3}\right )} \sqrt {b x^{n} + a}}{24 \, a^{3} n x^{3 \, n}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-3*n)*(a+b*x^n)^(1/2),x, algorithm="fricas")

[Out]

[1/48*(3*sqrt(a)*b^3*x^(3*n)*log((b*x^n - 2*sqrt(b*x^n + a)*sqrt(a) + 2*a)/x^n) + 2*(3*a*b^2*x^(2*n) - 2*a^2*b
*x^n - 8*a^3)*sqrt(b*x^n + a))/(a^3*n*x^(3*n)), 1/24*(3*sqrt(-a)*b^3*x^(3*n)*arctan(sqrt(b*x^n + a)*sqrt(-a)/a
) + (3*a*b^2*x^(2*n) - 2*a^2*b*x^n - 8*a^3)*sqrt(b*x^n + a))/(a^3*n*x^(3*n))]

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3005 deep

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

[Out]

integrate(sqrt(b*x^n + a)*x^(-3*n - 1), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {a+b\,x^n}}{x^{3\,n+1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^n)^(1/2)/x^(3*n + 1),x)

[Out]

int((a + b*x^n)^(1/2)/x^(3*n + 1), x)

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