3.4.67 \(\int \frac {(3+x^4) (-1+x^3+x^4)}{x^6 \sqrt [4]{-x+x^5}} \, dx\)

Optimal. Leaf size=30 \[ \frac {4 \left (3 x^4+7 x^3-3\right ) \left (x^5-x\right )^{3/4}}{21 x^6} \]

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Rubi [A]  time = 0.35, antiderivative size = 55, normalized size of antiderivative = 1.83, number of steps used = 21, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {2052, 2025, 2032, 365, 364} \begin {gather*} -\frac {4 \left (x^5-x\right )^{3/4}}{7 x^6}+\frac {4 \left (x^5-x\right )^{3/4}}{3 x^3}+\frac {4 \left (x^5-x\right )^{3/4}}{7 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((3 + x^4)*(-1 + x^3 + x^4))/(x^6*(-x + x^5)^(1/4)),x]

[Out]

(-4*(-x + x^5)^(3/4))/(7*x^6) + (4*(-x + x^5)^(3/4))/(3*x^3) + (4*(-x + x^5)^(3/4))/(7*x^2)

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 365

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])
/(1 + (b*x^n)/a)^FracPart[p], Int[(c*x)^m*(1 + (b*x^n)/a)^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[
p, 0] &&  !(ILtQ[p, 0] || GtQ[a, 0])

Rule 2025

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, m, p}, x] &&  !IntegerQ[p] && LtQ[0, j,
n] && (IntegersQ[j, n] || GtQ[c, 0]) && LtQ[m + j*p + 1, 0]

Rule 2032

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Dist[(c^IntPart[m]*(c*x)^FracP
art[m]*(a*x^j + b*x^n)^FracPart[p])/(x^(FracPart[m] + j*FracPart[p])*(a + b*x^(n - j))^FracPart[p]), Int[x^(m
+ j*p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[n, j] && PosQ[n
- j]

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 (3+x^4\right ) \left (-1+x^3+x^4\right )}{x^6 \sqrt [4]{-x+x^5}} \, dx &=\int \left (-\frac {3}{x^6 \sqrt [4]{-x+x^5}}+\frac {3}{x^3 \sqrt [4]{-x+x^5}}+\frac {2}{x^2 \sqrt [4]{-x+x^5}}+\frac {x}{\sqrt [4]{-x+x^5}}+\frac {x^2}{\sqrt [4]{-x+x^5}}\right ) \, dx\\ &=2 \int \frac {1}{x^2 \sqrt [4]{-x+x^5}} \, dx-3 \int \frac {1}{x^6 \sqrt [4]{-x+x^5}} \, dx+3 \int \frac {1}{x^3 \sqrt [4]{-x+x^5}} \, dx+\int \frac {x}{\sqrt [4]{-x+x^5}} \, dx+\int \frac {x^2}{\sqrt [4]{-x+x^5}} \, dx\\ &=-\frac {4 \left (-x+x^5\right )^{3/4}}{7 x^6}+\frac {4 \left (-x+x^5\right )^{3/4}}{3 x^3}+\frac {8 \left (-x+x^5\right )^{3/4}}{5 x^2}-\frac {9}{7} \int \frac {1}{x^2 \sqrt [4]{-x+x^5}} \, dx-\frac {14}{5} \int \frac {x^2}{\sqrt [4]{-x+x^5}} \, dx+\frac {\left (\sqrt [4]{x} \sqrt [4]{-1+x^4}\right ) \int \frac {x^{3/4}}{\sqrt [4]{-1+x^4}} \, dx}{\sqrt [4]{-x+x^5}}+\frac {\left (\sqrt [4]{x} \sqrt [4]{-1+x^4}\right ) \int \frac {x^{7/4}}{\sqrt [4]{-1+x^4}} \, dx}{\sqrt [4]{-x+x^5}}-\int \frac {x}{\sqrt [4]{-x+x^5}} \, dx\\ &=-\frac {4 \left (-x+x^5\right )^{3/4}}{7 x^6}+\frac {4 \left (-x+x^5\right )^{3/4}}{3 x^3}+\frac {4 \left (-x+x^5\right )^{3/4}}{7 x^2}+\frac {9}{5} \int \frac {x^2}{\sqrt [4]{-x+x^5}} \, dx+\frac {\left (\sqrt [4]{x} \sqrt [4]{1-x^4}\right ) \int \frac {x^{3/4}}{\sqrt [4]{1-x^4}} \, dx}{\sqrt [4]{-x+x^5}}+\frac {\left (\sqrt [4]{x} \sqrt [4]{1-x^4}\right ) \int \frac {x^{7/4}}{\sqrt [4]{1-x^4}} \, dx}{\sqrt [4]{-x+x^5}}-\frac {\left (\sqrt [4]{x} \sqrt [4]{-1+x^4}\right ) \int \frac {x^{3/4}}{\sqrt [4]{-1+x^4}} \, dx}{\sqrt [4]{-x+x^5}}-\frac {\left (14 \sqrt [4]{x} \sqrt [4]{-1+x^4}\right ) \int \frac {x^{7/4}}{\sqrt [4]{-1+x^4}} \, dx}{5 \sqrt [4]{-x+x^5}}\\ &=-\frac {4 \left (-x+x^5\right )^{3/4}}{7 x^6}+\frac {4 \left (-x+x^5\right )^{3/4}}{3 x^3}+\frac {4 \left (-x+x^5\right )^{3/4}}{7 x^2}+\frac {4 x^2 \sqrt [4]{1-x^4} \, _2F_1\left (\frac {1}{4},\frac {7}{16};\frac {23}{16};x^4\right )}{7 \sqrt [4]{-x+x^5}}+\frac {4 x^3 \sqrt [4]{1-x^4} \, _2F_1\left (\frac {1}{4},\frac {11}{16};\frac {27}{16};x^4\right )}{11 \sqrt [4]{-x+x^5}}-\frac {\left (\sqrt [4]{x} \sqrt [4]{1-x^4}\right ) \int \frac {x^{3/4}}{\sqrt [4]{1-x^4}} \, dx}{\sqrt [4]{-x+x^5}}-\frac {\left (14 \sqrt [4]{x} \sqrt [4]{1-x^4}\right ) \int \frac {x^{7/4}}{\sqrt [4]{1-x^4}} \, dx}{5 \sqrt [4]{-x+x^5}}+\frac {\left (9 \sqrt [4]{x} \sqrt [4]{-1+x^4}\right ) \int \frac {x^{7/4}}{\sqrt [4]{-1+x^4}} \, dx}{5 \sqrt [4]{-x+x^5}}\\ &=-\frac {4 \left (-x+x^5\right )^{3/4}}{7 x^6}+\frac {4 \left (-x+x^5\right )^{3/4}}{3 x^3}+\frac {4 \left (-x+x^5\right )^{3/4}}{7 x^2}-\frac {36 x^3 \sqrt [4]{1-x^4} \, _2F_1\left (\frac {1}{4},\frac {11}{16};\frac {27}{16};x^4\right )}{55 \sqrt [4]{-x+x^5}}+\frac {\left (9 \sqrt [4]{x} \sqrt [4]{1-x^4}\right ) \int \frac {x^{7/4}}{\sqrt [4]{1-x^4}} \, dx}{5 \sqrt [4]{-x+x^5}}\\ &=-\frac {4 \left (-x+x^5\right )^{3/4}}{7 x^6}+\frac {4 \left (-x+x^5\right )^{3/4}}{3 x^3}+\frac {4 \left (-x+x^5\right )^{3/4}}{7 x^2}\\ \end {align*}

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Mathematica [C]  time = 0.07, size = 117, normalized size = 3.90 \begin {gather*} \frac {4 \sqrt [4]{1-x^4} \left (165 \, _2F_1\left (-\frac {21}{16},\frac {1}{4};-\frac {5}{16};x^4\right )+x^3 \left (165 x^4 \, _2F_1\left (\frac {1}{4},\frac {7}{16};\frac {23}{16};x^4\right )-462 x \, _2F_1\left (-\frac {5}{16},\frac {1}{4};\frac {11}{16};x^4\right )-385 \, _2F_1\left (-\frac {9}{16},\frac {1}{4};\frac {7}{16};x^4\right )+105 x^5 \, _2F_1\left (\frac {1}{4},\frac {11}{16};\frac {27}{16};x^4\right )\right )\right )}{1155 x^5 \sqrt [4]{x \left (x^4-1\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((3 + x^4)*(-1 + x^3 + x^4))/(x^6*(-x + x^5)^(1/4)),x]

[Out]

(4*(1 - x^4)^(1/4)*(165*Hypergeometric2F1[-21/16, 1/4, -5/16, x^4] + x^3*(-385*Hypergeometric2F1[-9/16, 1/4, 7
/16, x^4] - 462*x*Hypergeometric2F1[-5/16, 1/4, 11/16, x^4] + 165*x^4*Hypergeometric2F1[1/4, 7/16, 23/16, x^4]
+ 105*x^5*Hypergeometric2F1[1/4, 11/16, 27/16, x^4])))/(1155*x^5*(x*(-1 + x^4))^(1/4))

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IntegrateAlgebraic [A]  time = 2.59, size = 30, normalized size = 1.00 \begin {gather*} \frac {4 \left (-3+7 x^3+3 x^4\right ) \left (-x+x^5\right )^{3/4}}{21 x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((3 + x^4)*(-1 + x^3 + x^4))/(x^6*(-x + x^5)^(1/4)),x]

[Out]

(4*(-3 + 7*x^3 + 3*x^4)*(-x + x^5)^(3/4))/(21*x^6)

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fricas [A]  time = 0.50, size = 26, normalized size = 0.87 \begin {gather*} \frac {4 \, {\left (x^{5} - x\right )}^{\frac {3}{4}} {\left (3 \, x^{4} + 7 \, x^{3} - 3\right )}}{21 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+3)*(x^4+x^3-1)/x^6/(x^5-x)^(1/4),x, algorithm="fricas")

[Out]

4/21*(x^5 - x)^(3/4)*(3*x^4 + 7*x^3 - 3)/x^6

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{4} + x^{3} - 1\right )} {\left (x^{4} + 3\right )}}{{\left (x^{5} - x\right )}^{\frac {1}{4}} x^{6}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+3)*(x^4+x^3-1)/x^6/(x^5-x)^(1/4),x, algorithm="giac")

[Out]

integrate((x^4 + x^3 - 1)*(x^4 + 3)/((x^5 - x)^(1/4)*x^6), x)

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maple [A]  time = 0.09, size = 27, normalized size = 0.90

method	result	size
trager	\(\frac {4 \left (3 x^{4}+7 x^{3}-3\right ) \left (x^{5}-x \right )^{\frac {3}{4}}}{21 x^{6}}\)	\(27\)
risch	\(\frac {\frac {4}{3} x^{7}-\frac {4}{3} x^{3}+\frac {4}{7} x^{8}-\frac {8}{7} x^{4}+\frac {4}{7}}{x^{5} \left (x \left (x^{4}-1\right )\right )^{\frac {1}{4}}}\)	\(37\)
gosper	\(\frac {4 \left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right ) \left (3 x^{4}+7 x^{3}-3\right )}{21 x^{5} \left (x^{5}-x \right )^{\frac {1}{4}}}\)	\(38\)
meijerg	\(\frac {4 \left (-\mathrm {signum}\left (x^{4}-1\right )\right )^{\frac {1}{4}} \hypergeom \left (\left [-\frac {21}{16}, \frac {1}{4}\right ], \left [-\frac {5}{16}\right ], x^{4}\right )}{7 \mathrm {signum}\left (x^{4}-1\right )^{\frac {1}{4}} x^{\frac {21}{4}}}-\frac {8 \left (-\mathrm {signum}\left (x^{4}-1\right )\right )^{\frac {1}{4}} \hypergeom \left (\left [-\frac {5}{16}, \frac {1}{4}\right ], \left [\frac {11}{16}\right ], x^{4}\right )}{5 \mathrm {signum}\left (x^{4}-1\right )^{\frac {1}{4}} x^{\frac {5}{4}}}-\frac {4 \left (-\mathrm {signum}\left (x^{4}-1\right )\right )^{\frac {1}{4}} \hypergeom \left (\left [-\frac {9}{16}, \frac {1}{4}\right ], \left [\frac {7}{16}\right ], x^{4}\right )}{3 \mathrm {signum}\left (x^{4}-1\right )^{\frac {1}{4}} x^{\frac {9}{4}}}+\frac {4 \left (-\mathrm {signum}\left (x^{4}-1\right )\right )^{\frac {1}{4}} \hypergeom \left (\left [\frac {1}{4}, \frac {11}{16}\right ], \left [\frac {27}{16}\right ], x^{4}\right ) x^{\frac {11}{4}}}{11 \mathrm {signum}\left (x^{4}-1\right )^{\frac {1}{4}}}+\frac {4 \left (-\mathrm {signum}\left (x^{4}-1\right )\right )^{\frac {1}{4}} \hypergeom \left (\left [\frac {1}{4}, \frac {7}{16}\right ], \left [\frac {23}{16}\right ], x^{4}\right ) x^{\frac {7}{4}}}{7 \mathrm {signum}\left (x^{4}-1\right )^{\frac {1}{4}}}\)	\(162\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4+3)*(x^4+x^3-1)/x^6/(x^5-x)^(1/4),x,method=_RETURNVERBOSE)

[Out]

4/21*(3*x^4+7*x^3-3)*(x^5-x)^(3/4)/x^6

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{4} + x^{3} - 1\right )} {\left (x^{4} + 3\right )}}{{\left (x^{5} - x\right )}^{\frac {1}{4}} x^{6}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4+3)*(x^4+x^3-1)/x^6/(x^5-x)^(1/4),x, algorithm="maxima")

[Out]

integrate((x^4 + x^3 - 1)*(x^4 + 3)/((x^5 - x)^(1/4)*x^6), x)

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mupad [B]  time = 0.25, size = 45, normalized size = 1.50 \begin {gather*} \frac {28\,x^3\,{\left (x^5-x\right )}^{3/4}-12\,{\left (x^5-x\right )}^{3/4}+12\,x^4\,{\left (x^5-x\right )}^{3/4}}{21\,x^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^4 + 3)*(x^3 + x^4 - 1))/(x^6*(x^5 - x)^(1/4)),x)

[Out]

(28*x^3*(x^5 - x)^(3/4) - 12*(x^5 - x)^(3/4) + 12*x^4*(x^5 - x)^(3/4))/(21*x^6)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x^{4} + 3\right ) \left (x^{4} + x^{3} - 1\right )}{x^{6} \sqrt [4]{x \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**4+3)*(x**4+x**3-1)/x**6/(x**5-x)**(1/4),x)

[Out]

Integral((x**4 + 3)*(x**4 + x**3 - 1)/(x**6*(x*(x - 1)*(x + 1)*(x**2 + 1))**(1/4)), x)

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