3.3.85 \(\int \frac {(4+x^5) (-1+x^4+x^5)}{x^6 (-1+x^5)^{3/4}} \, dx\)

Optimal. Leaf size=26 \[ \frac {4 \sqrt [4]{x^5-1} \left (x^5+5 x^4-1\right )}{5 x^5} \]

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Rubi [A]  time = 0.10, antiderivative size = 44, normalized size of antiderivative = 1.69, number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1835, 1585, 12, 261} \begin {gather*} \frac {4 \sqrt [4]{x^5-1}}{x}-\frac {4 \sqrt [4]{x^5-1}}{5 x^5}+\frac {4}{5} \sqrt [4]{x^5-1} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*(-1 + x^5)^(1/4))/5 - (4*(-1 + x^5)^(1/4))/(5*x^5) + (4*(-1 + x^5)^(1/4))/x

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 1585

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(m +
n*p)*(a + b*x^(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, m, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] &
& PosQ[r - p]

Rule 1835

Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{Pq0 = Coeff[Pq, x, 0]}, Simp[(Pq
0*(c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*c*(m + 1)), x] + Dist[1/(2*a*c*(m + 1)), Int[(c*x)^(m + 1)*ExpandToSum
[(2*a*(m + 1)*(Pq - Pq0))/x - 2*b*Pq0*(m + n*(p + 1) + 1)*x^(n - 1), x]*(a + b*x^n)^p, x], x] /; NeQ[Pq0, 0]]
/; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[m, -1] && LeQ[n - 1, Expon[Pq, x]]

Rubi steps

\begin {align*} \int \frac {\left (4+x^5\right ) \left (-1+x^4+x^5\right )}{x^6 \left (-1+x^5\right )^{3/4}} \, dx &=-\frac {4 \sqrt [4]{-1+x^5}}{5 x^5}+\frac {1}{10} \int \frac {40 x^3+10 x^8+10 x^9}{x^5 \left (-1+x^5\right )^{3/4}} \, dx\\ &=-\frac {4 \sqrt [4]{-1+x^5}}{5 x^5}+\frac {1}{10} \int \frac {40+10 x^5+10 x^6}{x^2 \left (-1+x^5\right )^{3/4}} \, dx\\ &=-\frac {4 \sqrt [4]{-1+x^5}}{5 x^5}+\frac {4 \sqrt [4]{-1+x^5}}{x}+\frac {1}{20} \int \frac {20 x^4}{\left (-1+x^5\right )^{3/4}} \, dx\\ &=-\frac {4 \sqrt [4]{-1+x^5}}{5 x^5}+\frac {4 \sqrt [4]{-1+x^5}}{x}+\int \frac {x^4}{\left (-1+x^5\right )^{3/4}} \, dx\\ &=\frac {4}{5} \sqrt [4]{-1+x^5}-\frac {4 \sqrt [4]{-1+x^5}}{5 x^5}+\frac {4 \sqrt [4]{-1+x^5}}{x}\\ \end {align*}

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Mathematica [C]  time = 0.30, size = 238, normalized size = 9.15 \begin {gather*} \frac {1}{20} \left (-64 \sqrt [4]{x^5-1} \, _2F_1\left (\frac {1}{4},2;\frac {5}{4};1-x^5\right )-\frac {80 \left (1-x^5\right )^{3/4} \, _2F_1\left (-\frac {1}{5},\frac {3}{4};\frac {4}{5};x^5\right )}{\left (x^5-1\right )^{3/4} x}+\frac {5 \left (1-x^5\right )^{3/4} x^4 \, _2F_1\left (\frac {3}{4},\frac {4}{5};\frac {9}{5};x^5\right )}{\left (x^5-1\right )^{3/4}}+16 \sqrt [4]{x^5-1}-6 \sqrt {2} \log \left (\sqrt {x^5-1}-\sqrt {2} \sqrt [4]{x^5-1}+1\right )+6 \sqrt {2} \log \left (\sqrt {x^5-1}+\sqrt {2} \sqrt [4]{x^5-1}+1\right )-12 \sqrt {2} \tan ^{-1}\left (1-\sqrt {2} \sqrt [4]{x^5-1}\right )+12 \sqrt {2} \tan ^{-1}\left (\sqrt {2} \sqrt [4]{x^5-1}+1\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(16*(-1 + x^5)^(1/4) - 12*Sqrt[2]*ArcTan[1 - Sqrt[2]*(-1 + x^5)^(1/4)] + 12*Sqrt[2]*ArcTan[1 + Sqrt[2]*(-1 + x
^5)^(1/4)] - (80*(1 - x^5)^(3/4)*Hypergeometric2F1[-1/5, 3/4, 4/5, x^5])/(x*(-1 + x^5)^(3/4)) - 64*(-1 + x^5)^
(1/4)*Hypergeometric2F1[1/4, 2, 5/4, 1 - x^5] + (5*x^4*(1 - x^5)^(3/4)*Hypergeometric2F1[3/4, 4/5, 9/5, x^5])/
(-1 + x^5)^(3/4) - 6*Sqrt[2]*Log[1 - Sqrt[2]*(-1 + x^5)^(1/4) + Sqrt[-1 + x^5]] + 6*Sqrt[2]*Log[1 + Sqrt[2]*(-
1 + x^5)^(1/4) + Sqrt[-1 + x^5]])/20

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IntegrateAlgebraic [A]  time = 2.19, size = 26, normalized size = 1.00 \begin {gather*} \frac {4 \sqrt [4]{-1+x^5} \left (-1+5 x^4+x^5\right )}{5 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*(-1 + x^5)^(1/4)*(-1 + 5*x^4 + x^5))/(5*x^5)

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fricas [A]  time = 0.46, size = 22, normalized size = 0.85 \begin {gather*} \frac {4 \, {\left (x^{5} + 5 \, x^{4} - 1\right )} {\left (x^{5} - 1\right )}^{\frac {1}{4}}}{5 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/5*(x^5 + 5*x^4 - 1)*(x^5 - 1)^(1/4)/x^5

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.14, size = 23, normalized size = 0.88

method	result	size
trager	\(\frac {4 \left (x^{5}-1\right )^{\frac {1}{4}} \left (x^{5}+5 x^{4}-1\right )}{5 x^{5}}\)	\(23\)
risch	\(\frac {-\frac {8}{5} x^{5}+\frac {4}{5}+4 x^{9}-4 x^{4}+\frac {4}{5} x^{10}}{\left (x^{5}-1\right )^{\frac {3}{4}} x^{5}}\)	\(33\)
gosper	\(\frac {4 \left (x^{4}+x^{3}+x^{2}+x +1\right ) \left (-1+x \right ) \left (x^{5}+5 x^{4}-1\right )}{5 x^{5} \left (x^{5}-1\right )^{\frac {3}{4}}}\)	\(38\)
meijerg	\(\frac {\left (-\mathrm {signum}\left (x^{5}-1\right )\right )^{\frac {3}{4}} \hypergeom \left (\left [\frac {3}{4}, 1\right ], \relax [2], x^{5}\right ) x^{5}}{5 \mathrm {signum}\left (x^{5}-1\right )^{\frac {3}{4}}}+\frac {\left (-\mathrm {signum}\left (x^{5}-1\right )\right )^{\frac {3}{4}} \hypergeom \left (\left [\frac {3}{4}, \frac {4}{5}\right ], \left [\frac {9}{5}\right ], x^{5}\right ) x^{4}}{4 \mathrm {signum}\left (x^{5}-1\right )^{\frac {3}{4}}}+\frac {3 \left (-\mathrm {signum}\left (x^{5}-1\right )\right )^{\frac {3}{4}} \left (\left (-3 \ln \relax (2)+\frac {\pi }{2}+5 \ln \relax (x )+i \pi \right ) \Gamma \left (\frac {3}{4}\right )+\frac {3 \hypergeom \left (\left [1, 1, \frac {7}{4}\right ], \left [2, 2\right ], x^{5}\right ) \Gamma \left (\frac {3}{4}\right ) x^{5}}{4}\right )}{5 \Gamma \left (\frac {3}{4}\right ) \mathrm {signum}\left (x^{5}-1\right )^{\frac {3}{4}}}+\frac {4 \left (-\mathrm {signum}\left (x^{5}-1\right )\right )^{\frac {3}{4}} \left (\frac {\Gamma \left (\frac {3}{4}\right )}{x^{5}}-\frac {3 \left (\frac {1}{3}-3 \ln \relax (2)+\frac {\pi }{2}+5 \ln \relax (x )+i \pi \right ) \Gamma \left (\frac {3}{4}\right )}{4}-\frac {21 \hypergeom \left (\left [1, 1, \frac {11}{4}\right ], \left [2, 3\right ], x^{5}\right ) \Gamma \left (\frac {3}{4}\right ) x^{5}}{32}\right )}{5 \Gamma \left (\frac {3}{4}\right ) \mathrm {signum}\left (x^{5}-1\right )^{\frac {3}{4}}}-\frac {4 \left (-\mathrm {signum}\left (x^{5}-1\right )\right )^{\frac {3}{4}} \hypergeom \left (\left [-\frac {1}{5}, \frac {3}{4}\right ], \left [\frac {4}{5}\right ], x^{5}\right )}{\mathrm {signum}\left (x^{5}-1\right )^{\frac {3}{4}} x}\)	\(230\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/5*(x^5-1)^(1/4)*(x^5+5*x^4-1)/x^5

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\frac {3}{5} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, {\left (x^{5} - 1\right )}^{\frac {1}{4}}\right )}\right ) - \frac {3}{5} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, {\left (x^{5} - 1\right )}^{\frac {1}{4}}\right )}\right ) - \frac {3}{10} \, \sqrt {2} \log \left (\sqrt {2} {\left (x^{5} - 1\right )}^{\frac {1}{4}} + \sqrt {x^{5} - 1} + 1\right ) + \frac {3}{10} \, \sqrt {2} \log \left (-\sqrt {2} {\left (x^{5} - 1\right )}^{\frac {1}{4}} + \sqrt {x^{5} - 1} + 1\right ) - \frac {4 \, {\left (x^{5} - 1\right )}^{\frac {1}{4}}}{5 \, x^{5}} + \int \frac {{\left (x^{6} + x^{5} + 3 \, x + 4\right )} {\left (x^{4} + x^{3} + x^{2} + x + 1\right )}^{\frac {1}{4}} {\left (x - 1\right )}^{\frac {1}{4}}}{x^{7} - x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/5*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*(x^5 - 1)^(1/4))) - 3/5*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*
(x^5 - 1)^(1/4))) - 3/10*sqrt(2)*log(sqrt(2)*(x^5 - 1)^(1/4) + sqrt(x^5 - 1) + 1) + 3/10*sqrt(2)*log(-sqrt(2)*
(x^5 - 1)^(1/4) + sqrt(x^5 - 1) + 1) - 4/5*(x^5 - 1)^(1/4)/x^5 + integrate((x^6 + x^5 + 3*x + 4)*(x^4 + x^3 +
x^2 + x + 1)^(1/4)*(x - 1)^(1/4)/(x^7 - x^2), x)

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mupad [B]  time = 0.28, size = 27, normalized size = 1.04 \begin {gather*} \frac {4\,{\left (x^5-1\right )}^{5/4}+20\,x^4\,{\left (x^5-1\right )}^{1/4}}{5\,x^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(4*(x^5 - 1)^(5/4) + 20*x^4*(x^5 - 1)^(1/4))/(5*x^5)

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sympy [C]  time = 4.56, size = 151, normalized size = 5.81 \begin {gather*} \frac {x^{4} e^{- \frac {3 i \pi }{4}} \Gamma \left (\frac {4}{5}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {4}{5} \\ \frac {9}{5} \end {matrix}\middle | {x^{5}} \right )}}{5 \Gamma \left (\frac {9}{5}\right )} + \frac {4 \sqrt [4]{x^{5} - 1}}{5} - \frac {4 e^{\frac {i \pi }{4}} \Gamma \left (- \frac {1}{5}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{5}, \frac {3}{4} \\ \frac {4}{5} \end {matrix}\middle | {x^{5}} \right )}}{5 x \Gamma \left (\frac {4}{5}\right )} - \frac {3 \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{5}}} \right )}}{5 x^{\frac {15}{4}} \Gamma \left (\frac {7}{4}\right )} + \frac {4 \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{5}}} \right )}}{5 x^{\frac {35}{4}} \Gamma \left (\frac {11}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**4*exp(-3*I*pi/4)*gamma(4/5)*hyper((3/4, 4/5), (9/5,), x**5)/(5*gamma(9/5)) + 4*(x**5 - 1)**(1/4)/5 - 4*exp(
I*pi/4)*gamma(-1/5)*hyper((-1/5, 3/4), (4/5,), x**5)/(5*x*gamma(4/5)) - 3*gamma(3/4)*hyper((3/4, 3/4), (7/4,),
exp_polar(2*I*pi)/x**5)/(5*x**(15/4)*gamma(7/4)) + 4*gamma(7/4)*hyper((3/4, 7/4), (11/4,), exp_polar(2*I*pi)/
x**5)/(5*x**(35/4)*gamma(11/4))

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