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

Optimal. Leaf size=18 \[ \frac {4 \left (x^5-x\right )^{7/4}}{7 x^7} \]

________________________________________________________________________________________

Rubi [B]  time = 0.26, antiderivative size = 37, normalized size of antiderivative = 2.06, number of steps used = 14, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2052, 2025, 2032, 365, 364} \begin {gather*} \frac {4 \left (x^5-x\right )^{3/4}}{7 x^2}-\frac {4 \left (x^5-x\right )^{3/4}}{7 x^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-1 + x^4)*(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))/(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 (-1+x^4\right ) \left (3+x^4\right )}{x^6 \sqrt [4]{-x+x^5}} \, dx &=\int \left (-\frac {3}{x^6 \sqrt [4]{-x+x^5}}+\frac {2}{x^2 \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+\int \frac {x^2}{\sqrt [4]{-x+x^5}} \, dx\\ &=-\frac {4 \left (-x+x^5\right )^{3/4}}{7 x^6}+\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^{7/4}}{\sqrt [4]{-1+x^4}} \, dx}{\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}}{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^{7/4}}{\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}}{7 x^2}+\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 (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}}{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}}{7 x^2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 18, normalized size = 1.00 \begin {gather*} \frac {4 \left (x \left (x^4-1\right )\right )^{7/4}}{7 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*(x*(-1 + x^4))^(7/4))/(7*x^7)

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.23, size = 18, normalized size = 1.00 \begin {gather*} \frac {4 \left (-x+x^5\right )^{7/4}}{7 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*(-x + x^5)^(7/4))/(7*x^7)

________________________________________________________________________________________

fricas [A]  time = 0.47, size = 19, normalized size = 1.06 \begin {gather*} \frac {4 \, {\left (x^{5} - x\right )}^{\frac {3}{4}} {\left (x^{4} - 1\right )}}{7 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/7*(x^5 - x)^(3/4)*(x^4 - 1)/x^6

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.10, size = 20, normalized size = 1.11

method	result	size
trager	\(\frac {4 \left (x^{4}-1\right ) \left (x^{5}-x \right )^{\frac {3}{4}}}{7 x^{6}}\)	\(20\)
risch	\(\frac {\frac {4}{7} x^{8}-\frac {8}{7} x^{4}+\frac {4}{7}}{x^{5} \left (x \left (x^{4}-1\right )\right )^{\frac {1}{4}}}\)	\(25\)
gosper	\(\frac {4 \left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right ) \left (x^{4}-1\right )}{7 x^{5} \left (x^{5}-x \right )^{\frac {1}{4}}}\)	\(31\)
meijerg	\(\frac {4 \left (-\mathrm {signum}\left (x^{4}-1\right )\right )^{\frac {1}{4}} \hypergeom \left (\left [-\frac {21}{16}, \frac {1}{4}\right ], \left [-\frac {5}{16}\right ], x^{4}\right )}{7 \mathrm {signum}\left (x^{4}-1\right )^{\frac {1}{4}} x^{\frac {21}{4}}}-\frac {8 \left (-\mathrm {signum}\left (x^{4}-1\right )\right )^{\frac {1}{4}} \hypergeom \left (\left [-\frac {5}{16}, \frac {1}{4}\right ], \left [\frac {11}{16}\right ], x^{4}\right )}{5 \mathrm {signum}\left (x^{4}-1\right )^{\frac {1}{4}} x^{\frac {5}{4}}}+\frac {4 \left (-\mathrm {signum}\left (x^{4}-1\right )\right )^{\frac {1}{4}} \hypergeom \left (\left [\frac {1}{4}, \frac {11}{16}\right ], \left [\frac {27}{16}\right ], x^{4}\right ) x^{\frac {11}{4}}}{11 \mathrm {signum}\left (x^{4}-1\right )^{\frac {1}{4}}}\)	\(98\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/7*(x^4-1)/x^6*(x^5-x)^(3/4)

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.28, size = 31, normalized size = 1.72 \begin {gather*} -\frac {4\,{\left (x^5-x\right )}^{3/4}-4\,x^4\,{\left (x^5-x\right )}^{3/4}}{7\,x^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \left (x^{4} + 3\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**4-1)*(x**4+3)/x**6/(x**5-x)**(1/4),x)

[Out]

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

________________________________________________________________________________________