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

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

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Rubi [A]  time = 0.02, antiderivative size = 44, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {4 x^{5 (1-n)} \left (a x^{-4 (1-n)}+b x^{5 n-4}\right )^{5/4}}{5 b n} \end {gather*}

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*}

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Mathematica [A]  time = 0.03, size = 36, normalized size = 0.82 \begin {gather*} \frac {4 x^{5-5 n} \left (x^{4 n-4} \left (a+b x^n\right )\right )^{5/4}}{5 b n} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.08, size = 43, normalized size = 0.98 \begin {gather*} \frac {4 x^{1-n} \left (a+b x^n\right ) \sqrt [4]{x^{4 n-4} \left (a+b x^n\right )}}{5 b n} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.41, size = 44, normalized size = 1.00 \begin {gather*} \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 {gather*}

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left ({\left (b x^{n} + a\right )} x^{4 \, n - 4}\right )^{\frac {1}{4}}\,{d x} \end {gather*}

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)

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maple [A]  time = 0.07, size = 40, normalized size = 0.91 \begin {gather*} \frac {4 \left (\frac {\left (b \,x^{n}+a \right ) x^{4 n}}{x^{4}}\right )^{\frac {1}{4}} \left (b \,x^{n}+a \right ) x \,x^{-n}}{5 b n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.58, size = 17, normalized size = 0.39 \begin {gather*} \frac {4 \, {\left (b x^{n} + a\right )}^{\frac {5}{4}}}{5 \, b n} \end {gather*}

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int {\left (x^{4\,n-4}\,\left (a+b\,x^n\right )\right )}^{1/4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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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