3.2649 \(\int x^{-1+4 n} \sqrt{a+b x^n} \, dx\)

Optimal. Leaf size=92 \[ \frac{6 a^2 \left (a+b x^n\right )^{5/2}}{5 b^4 n}-\frac{2 a^3 \left (a+b x^n\right )^{3/2}}{3 b^4 n}+\frac{2 \left (a+b x^n\right )^{9/2}}{9 b^4 n}-\frac{6 a \left (a+b x^n\right )^{7/2}}{7 b^4 n} \]

[Out]

(-2*a^3*(a + b*x^n)^(3/2))/(3*b^4*n) + (6*a^2*(a + b*x^n)^(5/2))/(5*b^4*n) - (6*a*(a + b*x^n)^(7/2))/(7*b^4*n)
+ (2*(a + b*x^n)^(9/2))/(9*b^4*n)

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Rubi [A]  time = 0.0430168, antiderivative size = 92, normalized size of antiderivative = 1., 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 = {266, 43} \[ \frac{6 a^2 \left (a+b x^n\right )^{5/2}}{5 b^4 n}-\frac{2 a^3 \left (a+b x^n\right )^{3/2}}{3 b^4 n}+\frac{2 \left (a+b x^n\right )^{9/2}}{9 b^4 n}-\frac{6 a \left (a+b x^n\right )^{7/2}}{7 b^4 n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 + 4*n)*Sqrt[a + b*x^n],x]

[Out]

(-2*a^3*(a + b*x^n)^(3/2))/(3*b^4*n) + (6*a^2*(a + b*x^n)^(5/2))/(5*b^4*n) - (6*a*(a + b*x^n)^(7/2))/(7*b^4*n)
+ (2*(a + b*x^n)^(9/2))/(9*b^4*n)

Rule 266

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]]

Rule 43

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])

Rubi steps

\begin{align*} \int x^{-1+4 n} \sqrt{a+b x^n} \, dx &=\frac{\operatorname{Subst}\left (\int x^3 \sqrt{a+b x} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{a^3 \sqrt{a+b x}}{b^3}+\frac{3 a^2 (a+b x)^{3/2}}{b^3}-\frac{3 a (a+b x)^{5/2}}{b^3}+\frac{(a+b x)^{7/2}}{b^3}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac{2 a^3 \left (a+b x^n\right )^{3/2}}{3 b^4 n}+\frac{6 a^2 \left (a+b x^n\right )^{5/2}}{5 b^4 n}-\frac{6 a \left (a+b x^n\right )^{7/2}}{7 b^4 n}+\frac{2 \left (a+b x^n\right )^{9/2}}{9 b^4 n}\\ \end{align*}

Mathematica [A]  time = 0.0335395, size = 57, normalized size = 0.62 \[ \frac{2 \left (a+b x^n\right )^{3/2} \left (24 a^2 b x^n-16 a^3-30 a b^2 x^{2 n}+35 b^3 x^{3 n}\right )}{315 b^4 n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 + 4*n)*Sqrt[a + b*x^n],x]

[Out]

(2*(a + b*x^n)^(3/2)*(-16*a^3 + 24*a^2*b*x^n - 30*a*b^2*x^(2*n) + 35*b^3*x^(3*n)))/(315*b^4*n)

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Maple [A]  time = 0.019, size = 67, normalized size = 0.7 \begin{align*} -{\frac{-70\, \left ({x}^{n} \right ) ^{4}{b}^{4}-10\,a \left ({x}^{n} \right ) ^{3}{b}^{3}+12\,{a}^{2} \left ({x}^{n} \right ) ^{2}{b}^{2}-16\,{a}^{3}{x}^{n}b+32\,{a}^{4}}{315\,{b}^{4}n}\sqrt{a+b{x}^{n}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1+4*n)*(a+b*x^n)^(1/2),x)

[Out]

-2/315*(-35*(x^n)^4*b^4-5*a*(x^n)^3*b^3+6*a^2*(x^n)^2*b^2-8*a^3*x^n*b+16*a^4)*(a+b*x^n)^(1/2)/b^4/n

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Maxima [A]  time = 1.04136, size = 89, normalized size = 0.97 \begin{align*} \frac{2 \,{\left (35 \, b^{4} x^{4 \, n} + 5 \, a b^{3} x^{3 \, n} - 6 \, a^{2} b^{2} x^{2 \, n} + 8 \, a^{3} b x^{n} - 16 \, a^{4}\right )} \sqrt{b x^{n} + a}}{315 \, b^{4} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+4*n)*(a+b*x^n)^(1/2),x, algorithm="maxima")

[Out]

2/315*(35*b^4*x^(4*n) + 5*a*b^3*x^(3*n) - 6*a^2*b^2*x^(2*n) + 8*a^3*b*x^n - 16*a^4)*sqrt(b*x^n + a)/(b^4*n)

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Fricas [A]  time = 0.983131, size = 147, normalized size = 1.6 \begin{align*} \frac{2 \,{\left (35 \, b^{4} x^{4 \, n} + 5 \, a b^{3} x^{3 \, n} - 6 \, a^{2} b^{2} x^{2 \, n} + 8 \, a^{3} b x^{n} - 16 \, a^{4}\right )} \sqrt{b x^{n} + a}}{315 \, b^{4} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+4*n)*(a+b*x^n)^(1/2),x, algorithm="fricas")

[Out]

2/315*(35*b^4*x^(4*n) + 5*a*b^3*x^(3*n) - 6*a^2*b^2*x^(2*n) + 8*a^3*b*x^n - 16*a^4)*sqrt(b*x^n + a)/(b^4*n)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(-1+4*n)*(a+b*x**n)**(1/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b x^{n} + a} x^{4 \, n - 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+4*n)*(a+b*x^n)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(b*x^n + a)*x^(4*n - 1), x)