3.9.20 \(\int \frac {(-3+x^4) (1-x^3+2 x^4-x^6-x^7+x^8)}{x^6 (1-x^3+x^4) \sqrt [4]{x+x^5}} \, dx\)

Optimal. Leaf size=62 \[ 2 \tan ^{-1}\left (\frac {\left (x^5+x\right )^{3/4}}{x^4+1}\right )+2 \tanh ^{-1}\left (\frac {\left (x^5+x\right )^{3/4}}{x^4+1}\right )+\frac {4 \left (x^5+x\right )^{3/4} \left (x^4+1\right )}{7 x^6} \]

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Rubi [F]  time = 2.71, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\left (-3+x^4\right ) \left (1-x^3+2 x^4-x^6-x^7+x^8\right )}{x^6 \left (1-x^3+x^4\right ) \sqrt [4]{x+x^5}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-3 + x^4)*(1 - x^3 + 2*x^4 - x^6 - x^7 + x^8))/(x^6*(1 - x^3 + x^4)*(x + x^5)^(1/4)),x]

[Out]

(4*(1 + x^4)^(1/4)*Hypergeometric2F1[-21/16, 1/4, -5/16, -x^4])/(7*x^5*(x + x^5)^(1/4)) + (8*(1 + x^4)^(1/4)*H
ypergeometric2F1[-5/16, 1/4, 11/16, -x^4])/(5*x*(x + x^5)^(1/4)) - (4*x*(1 + x^4)^(1/4)*Hypergeometric2F1[3/16
, 1/4, 19/16, -x^4])/(3*(x + x^5)^(1/4)) + (4*x^3*(1 + x^4)^(1/4)*Hypergeometric2F1[1/4, 11/16, 27/16, -x^4])/
(11*(x + x^5)^(1/4)) + (16*x^(1/4)*(1 + x^4)^(1/4)*Defer[Subst][Defer[Int][x^2/((1 + x^16)^(1/4)*(1 - x^12 + x
^16)), x], x, x^(1/4)])/(x + x^5)^(1/4) - (4*x^(1/4)*(1 + x^4)^(1/4)*Defer[Subst][Defer[Int][x^14/((1 + x^16)^
(1/4)*(1 - x^12 + x^16)), x], x, x^(1/4)])/(x + x^5)^(1/4)

Rubi steps

\begin {align*} \int \frac {\left (-3+x^4\right ) \left (1-x^3+2 x^4-x^6-x^7+x^8\right )}{x^6 \left (1-x^3+x^4\right ) \sqrt [4]{x+x^5}} \, dx &=\frac {\left (\sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \int \frac {\left (-3+x^4\right ) \left (1-x^3+2 x^4-x^6-x^7+x^8\right )}{x^{25/4} \sqrt [4]{1+x^4} \left (1-x^3+x^4\right )} \, dx}{\sqrt [4]{x+x^5}}\\ &=\frac {\left (4 \sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {\left (-3+x^{16}\right ) \left (1-x^{12}+2 x^{16}-x^{24}-x^{28}+x^{32}\right )}{x^{22} \sqrt [4]{1+x^{16}} \left (1-x^{12}+x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^5}}\\ &=\frac {\left (4 \sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \operatorname {Subst}\left (\int \left (-\frac {3}{x^{22} \sqrt [4]{1+x^{16}}}-\frac {2}{x^6 \sqrt [4]{1+x^{16}}}-\frac {x^2}{\sqrt [4]{1+x^{16}}}+\frac {x^{10}}{\sqrt [4]{1+x^{16}}}-\frac {x^2 \left (-4+x^{12}\right )}{\sqrt [4]{1+x^{16}} \left (1-x^{12}+x^{16}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^5}}\\ &=-\frac {\left (4 \sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt [4]{1+x^{16}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^5}}+\frac {\left (4 \sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^{10}}{\sqrt [4]{1+x^{16}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^5}}-\frac {\left (4 \sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (-4+x^{12}\right )}{\sqrt [4]{1+x^{16}} \left (1-x^{12}+x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^5}}-\frac {\left (8 \sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{x^6 \sqrt [4]{1+x^{16}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^5}}-\frac {\left (12 \sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{x^{22} \sqrt [4]{1+x^{16}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^5}}\\ &=\frac {4 \sqrt [4]{1+x^4} \, _2F_1\left (-\frac {21}{16},\frac {1}{4};-\frac {5}{16};-x^4\right )}{7 x^5 \sqrt [4]{x+x^5}}+\frac {8 \sqrt [4]{1+x^4} \, _2F_1\left (-\frac {5}{16},\frac {1}{4};\frac {11}{16};-x^4\right )}{5 x \sqrt [4]{x+x^5}}-\frac {4 x \sqrt [4]{1+x^4} \, _2F_1\left (\frac {3}{16},\frac {1}{4};\frac {19}{16};-x^4\right )}{3 \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 (4 \sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \operatorname {Subst}\left (\int \left (-\frac {4 x^2}{\sqrt [4]{1+x^{16}} \left (1-x^{12}+x^{16}\right )}+\frac {x^{14}}{\sqrt [4]{1+x^{16}} \left (1-x^{12}+x^{16}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^5}}\\ &=\frac {4 \sqrt [4]{1+x^4} \, _2F_1\left (-\frac {21}{16},\frac {1}{4};-\frac {5}{16};-x^4\right )}{7 x^5 \sqrt [4]{x+x^5}}+\frac {8 \sqrt [4]{1+x^4} \, _2F_1\left (-\frac {5}{16},\frac {1}{4};\frac {11}{16};-x^4\right )}{5 x \sqrt [4]{x+x^5}}-\frac {4 x \sqrt [4]{1+x^4} \, _2F_1\left (\frac {3}{16},\frac {1}{4};\frac {19}{16};-x^4\right )}{3 \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 (4 \sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^{14}}{\sqrt [4]{1+x^{16}} \left (1-x^{12}+x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^5}}+\frac {\left (16 \sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt [4]{1+x^{16}} \left (1-x^{12}+x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^5}}\\ \end {align*}

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Mathematica [F]  time = 0.63, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-3+x^4\right ) \left (1-x^3+2 x^4-x^6-x^7+x^8\right )}{x^6 \left (1-x^3+x^4\right ) \sqrt [4]{x+x^5}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((-3 + x^4)*(1 - x^3 + 2*x^4 - x^6 - x^7 + x^8))/(x^6*(1 - x^3 + x^4)*(x + x^5)^(1/4)),x]

[Out]

Integrate[((-3 + x^4)*(1 - x^3 + 2*x^4 - x^6 - x^7 + x^8))/(x^6*(1 - x^3 + x^4)*(x + x^5)^(1/4)), x]

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IntegrateAlgebraic [A]  time = 2.68, size = 62, normalized size = 1.00 \begin {gather*} \frac {4 \left (1+x^4\right ) \left (x+x^5\right )^{3/4}}{7 x^6}+2 \tan ^{-1}\left (\frac {\left (x+x^5\right )^{3/4}}{1+x^4}\right )+2 \tanh ^{-1}\left (\frac {\left (x+x^5\right )^{3/4}}{1+x^4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-3 + x^4)*(1 - x^3 + 2*x^4 - x^6 - x^7 + x^8))/(x^6*(1 - x^3 + x^4)*(x + x^5)^(1/4)),x]

[Out]

(4*(1 + x^4)*(x + x^5)^(3/4))/(7*x^6) + 2*ArcTan[(x + x^5)^(3/4)/(1 + x^4)] + 2*ArcTanh[(x + x^5)^(3/4)/(1 + x
^4)]

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fricas [B]  time = 40.66, size = 118, normalized size = 1.90 \begin {gather*} \frac {7 \, x^{6} \arctan \left (\frac {{\left (x^{5} + x\right )}^{\frac {3}{4}} x - {\left (x^{5} + x\right )}^{\frac {1}{4}} {\left (x^{4} + 1\right )}}{2 \, {\left (x^{5} + x\right )}}\right ) + 7 \, x^{6} \log \left (-\frac {x^{4} + x^{3} + 2 \, {\left (x^{5} + x\right )}^{\frac {1}{4}} x^{2} + 2 \, \sqrt {x^{5} + x} x + 2 \, {\left (x^{5} + x\right )}^{\frac {3}{4}} + 1}{x^{4} - x^{3} + 1}\right ) + 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^4-3)*(x^8-x^7-x^6+2*x^4-x^3+1)/x^6/(x^4-x^3+1)/(x^5+x)^(1/4),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{8} - x^{7} - x^{6} + 2 \, x^{4} - x^{3} + 1\right )} {\left (x^{4} - 3\right )}}{{\left (x^{5} + x\right )}^{\frac {1}{4}} {\left (x^{4} - x^{3} + 1\right )} x^{6}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-3)*(x^8-x^7-x^6+2*x^4-x^3+1)/x^6/(x^4-x^3+1)/(x^5+x)^(1/4),x, algorithm="giac")

[Out]

integrate((x^8 - x^7 - x^6 + 2*x^4 - x^3 + 1)*(x^4 - 3)/((x^5 + x)^(1/4)*(x^4 - x^3 + 1)*x^6), x)

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maple [C]  time = 5.40, size = 157, normalized size = 2.53

method	result	size
trager	\(\frac {4 \left (x^{4}+1\right ) \left (x^{5}+x \right )^{\frac {3}{4}}}{7 x^{6}}+\ln \left (\frac {x^{4}+2 \left (x^{5}+x \right )^{\frac {3}{4}}+2 \sqrt {x^{5}+x}\, x +2 x^{2} \left (x^{5}+x \right )^{\frac {1}{4}}+x^{3}+1}{x^{4}-x^{3}+1}\right )+\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{5}+x}\, x +\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \left (x^{5}+x \right )^{\frac {3}{4}}-2 x^{2} \left (x^{5}+x \right )^{\frac {1}{4}}+\RootOf \left (\textit {\_Z}^{2}+1\right )}{x^{4}-x^{3}+1}\right )\)	\(157\)
risch	\(\frac {\frac {4}{7} x^{8}+\frac {8}{7} x^{4}+\frac {4}{7}}{x^{5} \left (x \left (x^{4}+1\right )\right )^{\frac {1}{4}}}+\ln \left (-\frac {x^{4}+2 \left (x^{5}+x \right )^{\frac {3}{4}}+2 \sqrt {x^{5}+x}\, x +2 x^{2} \left (x^{5}+x \right )^{\frac {1}{4}}+x^{3}+1}{x^{4}-x^{3}+1}\right )+\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{5}+x}\, x +\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \left (x^{5}+x \right )^{\frac {3}{4}}-2 x^{2} \left (x^{5}+x \right )^{\frac {1}{4}}+\RootOf \left (\textit {\_Z}^{2}+1\right )}{x^{4}-x^{3}+1}\right )\)	\(165\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4-3)*(x^8-x^7-x^6+2*x^4-x^3+1)/x^6/(x^4-x^3+1)/(x^5+x)^(1/4),x,method=_RETURNVERBOSE)

[Out]

4/7*(x^4+1)*(x^5+x)^(3/4)/x^6+ln((x^4+2*(x^5+x)^(3/4)+2*(x^5+x)^(1/2)*x+2*x^2*(x^5+x)^(1/4)+x^3+1)/(x^4-x^3+1)
)+RootOf(_Z^2+1)*ln((RootOf(_Z^2+1)*x^4-2*RootOf(_Z^2+1)*(x^5+x)^(1/2)*x+RootOf(_Z^2+1)*x^3+2*(x^5+x)^(3/4)-2*
x^2*(x^5+x)^(1/4)+RootOf(_Z^2+1))/(x^4-x^3+1))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{8} - x^{7} - x^{6} + 2 \, x^{4} - x^{3} + 1\right )} {\left (x^{4} - 3\right )}}{{\left (x^{5} + x\right )}^{\frac {1}{4}} {\left (x^{4} - x^{3} + 1\right )} x^{6}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-3)*(x^8-x^7-x^6+2*x^4-x^3+1)/x^6/(x^4-x^3+1)/(x^5+x)^(1/4),x, algorithm="maxima")

[Out]

integrate((x^8 - x^7 - x^6 + 2*x^4 - x^3 + 1)*(x^4 - 3)/((x^5 + x)^(1/4)*(x^4 - x^3 + 1)*x^6), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int -\frac {\left (x^4-3\right )\,\left (-x^8+x^7+x^6-2\,x^4+x^3-1\right )}{x^6\,{\left (x^5+x\right )}^{1/4}\,\left (x^4-x^3+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((x^4 - 3)*(x^3 - 2*x^4 + x^6 + x^7 - x^8 - 1))/(x^6*(x + x^5)^(1/4)*(x^4 - x^3 + 1)),x)

[Out]

int(-((x^4 - 3)*(x^3 - 2*x^4 + x^6 + x^7 - x^8 - 1))/(x^6*(x + x^5)^(1/4)*(x^4 - x^3 + 1)), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**4-3)*(x**8-x**7-x**6+2*x**4-x**3+1)/x**6/(x**4-x**3+1)/(x**5+x)**(1/4),x)

[Out]

Timed out

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