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

Optimal. Leaf size=44 \[ \frac{4 x^{5 (1-n)} \left (a x^{-4 (1-n)}+b x^{5 n-4}\right )^{5/4}}{5 b n} \]

[Out]

(4*x^(5*(1 - n))*(a/x^(4*(1 - n)) + b*x^(-4 + 5*n))^(5/4))/(5*b*n)

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Rubi [A]  time = 0.0183645, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {1979, 2000} \[ \frac{4 x^{5 (1-n)} \left (a x^{-4 (1-n)}+b x^{5 n-4}\right )^{5/4}}{5 b n} \]

Antiderivative was successfully verified.

[In]

Int[(x^(4*(-1 + n))*(a + b*x^n))^(1/4),x]

[Out]

(4*x^(5*(1 - n))*(a/x^(4*(1 - n)) + b*x^(-4 + 5*n))^(5/4))/(5*b*n)

Rule 1979

Int[(u_)^(p_), x_Symbol] :> Int[ExpandToSum[u, x]^p, x] /; FreeQ[p, x] && GeneralizedBinomialQ[u, x] &&  !Gene
ralizedBinomialMatchQ[u, x]

Rule 2000

Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(a*x^j + b*x^n)^(p + 1)/(b*(n - j)*(p + 1)*x
^(n - 1)), x] /; FreeQ[{a, b, j, n, p}, x] &&  !IntegerQ[p] && NeQ[n, j] && EqQ[j*p - n + j + 1, 0]

Rubi steps

\begin{align*} \int \sqrt [4]{x^{4 (-1+n)} \left (a+b x^n\right )} \, dx &=\int \sqrt [4]{a x^{4 (-1+n)}+b x^{4 (-1+n)+n}} \, dx\\ &=\frac{4 x^{5 (1-n)} \left (a x^{-4 (1-n)}+b x^{-4+5 n}\right )^{5/4}}{5 b n}\\ \end{align*}

Mathematica [A]  time = 0.030217, size = 36, normalized size = 0.82 \[ \frac{4 x^{5-5 n} \left (x^{4 n-4} \left (a+b x^n\right )\right )^{5/4}}{5 b n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*x^(5 - 5*n)*(x^(-4 + 4*n)*(a + b*x^n))^(5/4))/(5*b*n)

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Maple [A]  time = 0.018, size = 40, normalized size = 0.9 \begin{align*}{\frac{4\,x \left ( a+b{x}^{n} \right ) }{5\,b{x}^{n}n}\sqrt [4]{{\frac{ \left ({x}^{n} \right ) ^{4} \left ( a+b{x}^{n} \right ) }{{x}^{4}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/5*(1/x^4*(x^n)^4*(a+b*x^n))^(1/4)*x/(x^n)*(a+b*x^n)/b/n

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Maxima [A]  time = 1.06904, size = 23, normalized size = 0.52 \begin{align*} \frac{4 \,{\left (b x^{n} + a\right )}^{\frac{5}{4}}}{5 \, b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/5*(b*x^n + a)^(5/4)/(b*n)

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Fricas [A]  time = 0.93438, size = 90, normalized size = 2.05 \begin{align*} \frac{4 \,{\left (b x x^{n} + a x\right )} \left (\frac{b x^{5 \, n} + a x^{4 \, n}}{x^{4}}\right )^{\frac{1}{4}}}{5 \, b n x^{n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/5*(b*x*x^n + a*x)*((b*x^(5*n) + a*x^(4*n))/x^4)^(1/4)/(b*n*x^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**(-4+4*n)*(a+b*x**n))**(1/4),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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