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

Optimal. Leaf size=48 \[ \frac {4 \sqrt [4]{x^4+x^3} \left (22748 x^6-5687 x^5-39955 x^4+960 x^3-780 x^2+663 x+13923\right )}{348075 x^7} \]

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Rubi [B]  time = 0.26, antiderivative size = 109, normalized size of antiderivative = 2.27, number of steps used = 10, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, 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 {256 \left (x^4+x^3\right )^{5/4}}{1785 x^8}-\frac {1024 \left (x^4+x^3\right )^{5/4}}{7735 x^7}-\frac {22748 \left (x^4+x^3\right )^{5/4}}{69615 x^6}+\frac {90992 \left (x^4+x^3\right )^{5/4}}{348075 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-1 + x^4)*(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) + (256*(x^3 + x^4)^(5/4))/(1785*x^8) - (102
4*(x^3 + x^4)^(5/4))/(7735*x^7) - (22748*(x^3 + x^4)^(5/4))/(69615*x^6) + (90992*(x^3 + x^4)^(5/4))/(348075*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^4\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^4}\right ) \, dx\\ &=-\int \frac {\sqrt [4]{x^3+x^4}}{x^8} \, dx+\int \frac {\sqrt [4]{x^3+x^4}}{x^4} \, dx\\ &=\frac {4 \left (x^3+x^4\right )^{5/4}}{25 x^{10}}-\frac {4 \left (x^3+x^4\right )^{5/4}}{9 x^6}-\frac {4}{9} \int \frac {\sqrt [4]{x^3+x^4}}{x^3} \, 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}}{9 x^6}+\frac {16 \left (x^3+x^4\right )^{5/4}}{45 x^5}-\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 {256 \left (x^3+x^4\right )^{5/4}}{1785 x^8}-\frac {4 \left (x^3+x^4\right )^{5/4}}{9 x^6}+\frac {16 \left (x^3+x^4\right )^{5/4}}{45 x^5}+\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 {256 \left (x^3+x^4\right )^{5/4}}{1785 x^8}-\frac {1024 \left (x^3+x^4\right )^{5/4}}{7735 x^7}-\frac {4 \left (x^3+x^4\right )^{5/4}}{9 x^6}+\frac {16 \left (x^3+x^4\right )^{5/4}}{45 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 {256 \left (x^3+x^4\right )^{5/4}}{1785 x^8}-\frac {1024 \left (x^3+x^4\right )^{5/4}}{7735 x^7}-\frac {22748 \left (x^3+x^4\right )^{5/4}}{69615 x^6}+\frac {16 \left (x^3+x^4\right )^{5/4}}{45 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 {256 \left (x^3+x^4\right )^{5/4}}{1785 x^8}-\frac {1024 \left (x^3+x^4\right )^{5/4}}{7735 x^7}-\frac {22748 \left (x^3+x^4\right )^{5/4}}{69615 x^6}+\frac {90992 \left (x^3+x^4\right )^{5/4}}{348075 x^5}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 38, normalized size = 0.79 \begin {gather*} \frac {4 \left (x^3 (x+1)\right )^{9/4} \left (22748 x^4-51183 x^3+39663 x^2-27183 x+13923\right )}{348075 x^{13}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*(x^3*(1 + x))^(9/4)*(13923 - 27183*x + 39663*x^2 - 51183*x^3 + 22748*x^4))/(348075*x^13)

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IntegrateAlgebraic [A]  time = 0.33, size = 48, normalized size = 1.00 \begin {gather*} \frac {4 \sqrt [4]{x^3+x^4} \left (13923+663 x-780 x^2+960 x^3-39955 x^4-5687 x^5+22748 x^6\right )}{348075 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*(x^3 + x^4)^(1/4)*(13923 + 663*x - 780*x^2 + 960*x^3 - 39955*x^4 - 5687*x^5 + 22748*x^6))/(348075*x^7)

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fricas [A]  time = 0.43, size = 44, normalized size = 0.92 \begin {gather*} \frac {4 \, {\left (22748 \, x^{6} - 5687 \, x^{5} - 39955 \, x^{4} + 960 \, x^{3} - 780 \, 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^4-1)*(x^4+x^3)^(1/4)/x^8,x, algorithm="fricas")

[Out]

4/348075*(22748*x^6 - 5687*x^5 - 39955*x^4 + 960*x^3 - 780*x^2 + 663*x + 13923)*(x^4 + x^3)^(1/4)/x^7

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giac [A]  time = 0.36, size = 46, normalized size = 0.96 \begin {gather*} \frac {4}{25} \, {\left (\frac {1}{x} + 1\right )}^{\frac {25}{4}} - \frac {20}{21} \, {\left (\frac {1}{x} + 1\right )}^{\frac {21}{4}} + \frac {40}{17} \, {\left (\frac {1}{x} + 1\right )}^{\frac {17}{4}} - \frac {40}{13} \, {\left (\frac {1}{x} + 1\right )}^{\frac {13}{4}} + \frac {16}{9} \, {\left (\frac {1}{x} + 1\right )}^{\frac {9}{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/25*(1/x + 1)^(25/4) - 20/21*(1/x + 1)^(21/4) + 40/17*(1/x + 1)^(17/4) - 40/13*(1/x + 1)^(13/4) + 16/9*(1/x +
1)^(9/4)

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maple [C]  time = 0.09, size = 36, normalized size = 0.75

method	result	size
meijerg	\(\frac {4 \hypergeom \left (\left [-\frac {25}{4}, -\frac {1}{4}\right ], \left [-\frac {21}{4}\right ], -x \right )}{25 x^{\frac {25}{4}}}-\frac {4 \left (-\frac {4}{5} x^{2}+\frac {1}{5} x +1\right ) \left (1+x \right )^{\frac {1}{4}}}{9 x^{\frac {9}{4}}}\)	\(36\)
gosper	\(\frac {4 \left (1+x \right )^{2} \left (22748 x^{4}-51183 x^{3}+39663 x^{2}-27183 x +13923\right ) \left (x^{4}+x^{3}\right )^{\frac {1}{4}}}{348075 x^{7}}\)	\(40\)
trager	\(\frac {4 \left (x^{4}+x^{3}\right )^{\frac {1}{4}} \left (22748 x^{6}-5687 x^{5}-39955 x^{4}+960 x^{3}-780 x^{2}+663 x +13923\right )}{348075 x^{7}}\)	\(45\)
risch	\(\frac {4 \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}} \left (22748 x^{7}+17061 x^{6}-45642 x^{5}-38995 x^{4}+180 x^{3}-117 x^{2}+14586 x +13923\right )}{348075 x^{7} \left (1+x \right )}\)	\(55\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/25*hypergeom([-25/4,-1/4],[-21/4],-x)/x^(25/4)-4/9*(-4/5*x^2+1/5*x+1)*(1+x)^(1/4)/x^(9/4)

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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^{4} - 1\right )}}{x^{8}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.10, size = 99, normalized size = 2.06 \begin {gather*} \frac {90992\,{\left (x^4+x^3\right )}^{1/4}}{348075\,x}-\frac {22748\,{\left (x^4+x^3\right )}^{1/4}}{348075\,x^2}-\frac {31964\,{\left (x^4+x^3\right )}^{1/4}}{69615\,x^3}+\frac {256\,{\left (x^4+x^3\right )}^{1/4}}{23205\,x^4}-\frac {16\,{\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^3 + x^4)^(1/4)*(x^4 - 1))/x^8,x)

[Out]

(90992*(x^3 + x^4)^(1/4))/(348075*x) - (22748*(x^3 + x^4)^(1/4))/(348075*x^2) - (31964*(x^3 + x^4)^(1/4))/(696
15*x^3) + (256*(x^3 + x^4)^(1/4))/(23205*x^4) - (16*(x^3 + x^4)^(1/4))/(1785*x^5) + (4*(x^3 + x^4)^(1/4))/(525
*x^6) + (4*(x^3 + x^4)^(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 ) \left (x^{2} + 1\right )}{x^{8}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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