3.7.36 \(\int \frac {(-1+x^2) \sqrt [4]{-x^3+x^4}}{x^8} \, dx\)

Optimal. Leaf size=50 \[ \frac {4 \sqrt [4]{x^4-x^3} \left (5248 x^6+1312 x^5+820 x^4+615 x^3-21255 x^2-663 x+13923\right )}{348075 x^7} \]

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Rubi [B]  time = 0.31, antiderivative size = 121, normalized size of antiderivative = 2.42, number of steps used = 12, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2052, 2016, 2014} \begin {gather*} -\frac {4 \left (x^4-x^3\right )^{5/4}}{25 x^{10}}-\frac {16 \left (x^4-x^3\right )^{5/4}}{105 x^9}+\frac {164 \left (x^4-x^3\right )^{5/4}}{1785 x^8}+\frac {656 \left (x^4-x^3\right )^{5/4}}{7735 x^7}+\frac {5248 \left (x^4-x^3\right )^{5/4}}{69615 x^6}+\frac {20992 \left (x^4-x^3\right )^{5/4}}{348075 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-1 + x^2)*(-x^3 + x^4)^(1/4))/x^8,x]

[Out]

(-4*(-x^3 + x^4)^(5/4))/(25*x^10) - (16*(-x^3 + x^4)^(5/4))/(105*x^9) + (164*(-x^3 + x^4)^(5/4))/(1785*x^8) +
(656*(-x^3 + x^4)^(5/4))/(7735*x^7) + (5248*(-x^3 + x^4)^(5/4))/(69615*x^6) + (20992*(-x^3 + x^4)^(5/4))/(3480
75*x^5)

Rule 2014

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

Rule 2016

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

Rule 2052

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(c*x)
^m*Pq*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] && (PolyQ[Pq, x] || PolyQ[Pq, x^n]) &&  !In
tegerQ[p] && NeQ[n, j]

Rubi steps

\begin {align*} \int \frac {\left (-1+x^2\right ) \sqrt [4]{-x^3+x^4}}{x^8} \, dx &=\int \left (-\frac {\sqrt [4]{-x^3+x^4}}{x^8}+\frac {\sqrt [4]{-x^3+x^4}}{x^6}\right ) \, dx\\ &=-\int \frac {\sqrt [4]{-x^3+x^4}}{x^8} \, dx+\int \frac {\sqrt [4]{-x^3+x^4}}{x^6} \, dx\\ &=-\frac {4 \left (-x^3+x^4\right )^{5/4}}{25 x^{10}}+\frac {4 \left (-x^3+x^4\right )^{5/4}}{17 x^8}+\frac {12}{17} \int \frac {\sqrt [4]{-x^3+x^4}}{x^5} \, dx-\frac {4}{5} \int \frac {\sqrt [4]{-x^3+x^4}}{x^7} \, dx\\ &=-\frac {4 \left (-x^3+x^4\right )^{5/4}}{25 x^{10}}-\frac {16 \left (-x^3+x^4\right )^{5/4}}{105 x^9}+\frac {4 \left (-x^3+x^4\right )^{5/4}}{17 x^8}+\frac {48 \left (-x^3+x^4\right )^{5/4}}{221 x^7}+\frac {96}{221} \int \frac {\sqrt [4]{-x^3+x^4}}{x^4} \, dx-\frac {64}{105} \int \frac {\sqrt [4]{-x^3+x^4}}{x^6} \, dx\\ &=-\frac {4 \left (-x^3+x^4\right )^{5/4}}{25 x^{10}}-\frac {16 \left (-x^3+x^4\right )^{5/4}}{105 x^9}+\frac {164 \left (-x^3+x^4\right )^{5/4}}{1785 x^8}+\frac {48 \left (-x^3+x^4\right )^{5/4}}{221 x^7}+\frac {128 \left (-x^3+x^4\right )^{5/4}}{663 x^6}+\frac {128}{663} \int \frac {\sqrt [4]{-x^3+x^4}}{x^3} \, dx-\frac {256}{595} \int \frac {\sqrt [4]{-x^3+x^4}}{x^5} \, dx\\ &=-\frac {4 \left (-x^3+x^4\right )^{5/4}}{25 x^{10}}-\frac {16 \left (-x^3+x^4\right )^{5/4}}{105 x^9}+\frac {164 \left (-x^3+x^4\right )^{5/4}}{1785 x^8}+\frac {656 \left (-x^3+x^4\right )^{5/4}}{7735 x^7}+\frac {128 \left (-x^3+x^4\right )^{5/4}}{663 x^6}+\frac {512 \left (-x^3+x^4\right )^{5/4}}{3315 x^5}-\frac {2048 \int \frac {\sqrt [4]{-x^3+x^4}}{x^4} \, dx}{7735}\\ &=-\frac {4 \left (-x^3+x^4\right )^{5/4}}{25 x^{10}}-\frac {16 \left (-x^3+x^4\right )^{5/4}}{105 x^9}+\frac {164 \left (-x^3+x^4\right )^{5/4}}{1785 x^8}+\frac {656 \left (-x^3+x^4\right )^{5/4}}{7735 x^7}+\frac {5248 \left (-x^3+x^4\right )^{5/4}}{69615 x^6}+\frac {512 \left (-x^3+x^4\right )^{5/4}}{3315 x^5}-\frac {8192 \int \frac {\sqrt [4]{-x^3+x^4}}{x^3} \, dx}{69615}\\ &=-\frac {4 \left (-x^3+x^4\right )^{5/4}}{25 x^{10}}-\frac {16 \left (-x^3+x^4\right )^{5/4}}{105 x^9}+\frac {164 \left (-x^3+x^4\right )^{5/4}}{1785 x^8}+\frac {656 \left (-x^3+x^4\right )^{5/4}}{7735 x^7}+\frac {5248 \left (-x^3+x^4\right )^{5/4}}{69615 x^6}+\frac {20992 \left (-x^3+x^4\right )^{5/4}}{348075 x^5}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 38, normalized size = 0.76 \begin {gather*} \frac {4 \left ((x-1) x^3\right )^{9/4} \left (5248 x^4+11808 x^3+19188 x^2+27183 x+13923\right )}{348075 x^{13}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-1 + x^2)*(-x^3 + x^4)^(1/4))/x^8,x]

[Out]

(4*((-1 + x)*x^3)^(9/4)*(13923 + 27183*x + 19188*x^2 + 11808*x^3 + 5248*x^4))/(348075*x^13)

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IntegrateAlgebraic [A]  time = 0.27, size = 50, normalized size = 1.00 \begin {gather*} \frac {4 \sqrt [4]{-x^3+x^4} \left (13923-663 x-21255 x^2+615 x^3+820 x^4+1312 x^5+5248 x^6\right )}{348075 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-1 + x^2)*(-x^3 + x^4)^(1/4))/x^8,x]

[Out]

(4*(-x^3 + x^4)^(1/4)*(13923 - 663*x - 21255*x^2 + 615*x^3 + 820*x^4 + 1312*x^5 + 5248*x^6))/(348075*x^7)

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fricas [A]  time = 0.53, size = 46, normalized size = 0.92 \begin {gather*} \frac {4 \, {\left (5248 \, x^{6} + 1312 \, x^{5} + 820 \, x^{4} + 615 \, x^{3} - 21255 \, x^{2} - 663 \, x + 13923\right )} {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{348075 \, x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-1)*(x^4-x^3)^(1/4)/x^8,x, algorithm="fricas")

[Out]

4/348075*(5248*x^6 + 1312*x^5 + 820*x^4 + 615*x^3 - 21255*x^2 - 663*x + 13923)*(x^4 - x^3)^(1/4)/x^7

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giac [A]  time = 0.21, size = 91, normalized size = 1.82 \begin {gather*} -\frac {4}{25} \, {\left (\frac {1}{x} - 1\right )}^{6} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} - \frac {20}{21} \, {\left (\frac {1}{x} - 1\right )}^{5} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} - \frac {36}{17} \, {\left (\frac {1}{x} - 1\right )}^{4} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} - \frac {28}{13} \, {\left (\frac {1}{x} - 1\right )}^{3} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} - \frac {8}{9} \, {\left (\frac {1}{x} - 1\right )}^{2} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-1)*(x^4-x^3)^(1/4)/x^8,x, algorithm="giac")

[Out]

-4/25*(1/x - 1)^6*(-1/x + 1)^(1/4) - 20/21*(1/x - 1)^5*(-1/x + 1)^(1/4) - 36/17*(1/x - 1)^4*(-1/x + 1)^(1/4) -
28/13*(1/x - 1)^3*(-1/x + 1)^(1/4) - 8/9*(1/x - 1)^2*(-1/x + 1)^(1/4)

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maple [A]  time = 0.12, size = 42, normalized size = 0.84

method	result	size
gosper	\(\frac {4 \left (x^{4}-x^{3}\right )^{\frac {1}{4}} \left (-1+x \right )^{2} \left (5248 x^{4}+11808 x^{3}+19188 x^{2}+27183 x +13923\right )}{348075 x^{7}}\)	\(42\)
trager	\(\frac {4 \left (x^{4}-x^{3}\right )^{\frac {1}{4}} \left (5248 x^{6}+1312 x^{5}+820 x^{4}+615 x^{3}-21255 x^{2}-663 x +13923\right )}{348075 x^{7}}\)	\(47\)
risch	\(\frac {4 \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}} \left (5248 x^{7}-3936 x^{6}-492 x^{5}-205 x^{4}-21870 x^{3}+20592 x^{2}+14586 x -13923\right )}{348075 x^{7} \left (-1+x \right )}\)	\(55\)
meijerg	\(\frac {4 \mathrm {signum}\left (-1+x \right )^{\frac {1}{4}} \hypergeom \left (\left [-\frac {25}{4}, -\frac {1}{4}\right ], \left [-\frac {21}{4}\right ], x\right )}{25 \left (-\mathrm {signum}\left (-1+x \right )\right )^{\frac {1}{4}} x^{\frac {25}{4}}}-\frac {4 \mathrm {signum}\left (-1+x \right )^{\frac {1}{4}} \left (-\frac {128}{195} x^{4}-\frac {32}{195} x^{3}-\frac {4}{39} x^{2}-\frac {1}{13} x +1\right ) \left (1-x \right )^{\frac {1}{4}}}{17 \left (-\mathrm {signum}\left (-1+x \right )\right )^{\frac {1}{4}} x^{\frac {17}{4}}}\)	\(74\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2-1)*(x^4-x^3)^(1/4)/x^8,x,method=_RETURNVERBOSE)

[Out]

4/348075*(x^4-x^3)^(1/4)*(-1+x)^2*(5248*x^4+11808*x^3+19188*x^2+27183*x+13923)/x^7

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (x^{2} - 1\right )}}{x^{8}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-1)*(x^4-x^3)^(1/4)/x^8,x, algorithm="maxima")

[Out]

integrate((x^4 - x^3)^(1/4)*(x^2 - 1)/x^8, x)

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mupad [B]  time = 1.23, size = 113, normalized size = 2.26 \begin {gather*} \frac {20992\,{\left (x^4-x^3\right )}^{1/4}}{348075\,x}+\frac {5248\,{\left (x^4-x^3\right )}^{1/4}}{348075\,x^2}+\frac {656\,{\left (x^4-x^3\right )}^{1/4}}{69615\,x^3}+\frac {164\,{\left (x^4-x^3\right )}^{1/4}}{23205\,x^4}-\frac {436\,{\left (x^4-x^3\right )}^{1/4}}{1785\,x^5}-\frac {4\,{\left (x^4-x^3\right )}^{1/4}}{525\,x^6}+\frac {4\,{\left (x^4-x^3\right )}^{1/4}}{25\,x^7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^2 - 1)*(x^4 - x^3)^(1/4))/x^8,x)

[Out]

(20992*(x^4 - x^3)^(1/4))/(348075*x) + (5248*(x^4 - x^3)^(1/4))/(348075*x^2) + (656*(x^4 - x^3)^(1/4))/(69615*
x^3) + (164*(x^4 - x^3)^(1/4))/(23205*x^4) - (436*(x^4 - x^3)^(1/4))/(1785*x^5) - (4*(x^4 - x^3)^(1/4))/(525*x
^6) + (4*(x^4 - x^3)^(1/4))/(25*x^7)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [4]{x^{3} \left (x - 1\right )} \left (x - 1\right ) \left (x + 1\right )}{x^{8}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2-1)*(x**4-x**3)**(1/4)/x**8,x)

[Out]

Integral((x**3*(x - 1))**(1/4)*(x - 1)*(x + 1)/x**8, x)

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