3.19.67 \(\int \frac {(-1+x^3)^3 (1+x^3) \sqrt {2+3 x^6+2 x^{12}}}{x^7 (1+x^6)} \, dx\)

Optimal. Leaf size=128 \[ -\frac {\log \left (\sqrt {2} x^6+\sqrt {2 x^{12}+3 x^6+2}+\sqrt {2}\right )}{6 \sqrt {2}}+\frac {\sqrt {2 x^{12}+3 x^6+2} \left (x^6-4 x^3+1\right )}{6 x^6}-\frac {4}{3} \tan ^{-1}\left (\frac {x^3}{\sqrt {2} x^6+\sqrt {2 x^{12}+3 x^6+2}+\sqrt {2}}\right )+\frac {\log (x)}{2 \sqrt {2}} \]

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Rubi [F]  time = 2.32, 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 (-1+x^3\right )^3 \left (1+x^3\right ) \sqrt {2+3 x^6+2 x^{12}}}{x^7 \left (1+x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-1 + x^3)^3*(1 + x^3)*Sqrt[2 + 3*x^6 + 2*x^12])/(x^7*(1 + x^6)),x]

[Out]

Sqrt[2 + 3*x^6 + 2*x^12]/6 + Sqrt[2 + 3*x^6 + 2*x^12]/(6*x^6) - (2*Sqrt[2 + 3*x^6 + 2*x^12])/(3*x^3) + (4*x^3*
Sqrt[2 + 3*x^6 + 2*x^12])/(3*(1 + x^6)) - ArcSinh[(3 + 4*x^6)/Sqrt[7]]/(12*Sqrt[2]) - ArcTanh[(4 + 3*x^6)/(2*S
qrt[2]*Sqrt[2 + 3*x^6 + 2*x^12])]/(12*Sqrt[2]) - (4*Sqrt[2]*(1 + x^6)*Sqrt[(2 + 3*x^6 + 2*x^12)/(1 + x^6)^2]*E
llipticE[2*ArcTan[x^3], 1/8])/(3*Sqrt[2 + 3*x^6 + 2*x^12]) + (7*(1 + x^6)*Sqrt[(2 + 3*x^6 + 2*x^12)/(1 + x^6)^
2]*EllipticF[2*ArcTan[x^3], 1/8])/(3*Sqrt[2]*Sqrt[2 + 3*x^6 + 2*x^12]) + ((2*I)/3)*Defer[Int][Sqrt[2 + 3*x^6 +
2*x^12]/(I - x), x] + ((2*I)/3)*Defer[Int][Sqrt[2 + 3*x^6 + 2*x^12]/(I + x), x] + (2*(1 + I*Sqrt[3])*Defer[In
t][Sqrt[2 + 3*x^6 + 2*x^12]/(Sqrt[1 - I*Sqrt[3]] - Sqrt[2]*x), x])/(3*Sqrt[1 - I*Sqrt[3]]) + (2*(1 - I*Sqrt[3]
)*Defer[Int][Sqrt[2 + 3*x^6 + 2*x^12]/(Sqrt[1 + I*Sqrt[3]] - Sqrt[2]*x), x])/(3*Sqrt[1 + I*Sqrt[3]]) + (2*(1 +
I*Sqrt[3])*Defer[Int][Sqrt[2 + 3*x^6 + 2*x^12]/(Sqrt[1 - I*Sqrt[3]] + Sqrt[2]*x), x])/(3*Sqrt[1 - I*Sqrt[3]])
+ (2*(1 - I*Sqrt[3])*Defer[Int][Sqrt[2 + 3*x^6 + 2*x^12]/(Sqrt[1 + I*Sqrt[3]] + Sqrt[2]*x), x])/(3*Sqrt[1 + I
*Sqrt[3]])

Rubi steps

\begin {align*} \int \frac {\left (-1+x^3\right )^3 \left (1+x^3\right ) \sqrt {2+3 x^6+2 x^{12}}}{x^7 \left (1+x^6\right )} \, dx &=\int \left (-\frac {\sqrt {2+3 x^6+2 x^{12}}}{x^7}+\frac {2 \sqrt {2+3 x^6+2 x^{12}}}{x^4}+\frac {\sqrt {2+3 x^6+2 x^{12}}}{x}+\frac {4 \sqrt {2+3 x^6+2 x^{12}}}{3 \left (1+x^2\right )}-\frac {4 \left (1+x^2\right ) \sqrt {2+3 x^6+2 x^{12}}}{3 \left (1-x^2+x^4\right )}\right ) \, dx\\ &=\frac {4}{3} \int \frac {\sqrt {2+3 x^6+2 x^{12}}}{1+x^2} \, dx-\frac {4}{3} \int \frac {\left (1+x^2\right ) \sqrt {2+3 x^6+2 x^{12}}}{1-x^2+x^4} \, dx+2 \int \frac {\sqrt {2+3 x^6+2 x^{12}}}{x^4} \, dx-\int \frac {\sqrt {2+3 x^6+2 x^{12}}}{x^7} \, dx+\int \frac {\sqrt {2+3 x^6+2 x^{12}}}{x} \, dx\\ &=-\left (\frac {1}{6} \operatorname {Subst}\left (\int \frac {\sqrt {2+3 x+2 x^2}}{x^2} \, dx,x,x^6\right )\right )+\frac {1}{6} \operatorname {Subst}\left (\int \frac {\sqrt {2+3 x+2 x^2}}{x} \, dx,x,x^6\right )+\frac {2}{3} \operatorname {Subst}\left (\int \frac {\sqrt {2+3 x^2+2 x^4}}{x^2} \, dx,x,x^3\right )+\frac {4}{3} \int \left (\frac {i \sqrt {2+3 x^6+2 x^{12}}}{2 (i-x)}+\frac {i \sqrt {2+3 x^6+2 x^{12}}}{2 (i+x)}\right ) \, dx-\frac {4}{3} \int \left (\frac {\left (1-i \sqrt {3}\right ) \sqrt {2+3 x^6+2 x^{12}}}{-1-i \sqrt {3}+2 x^2}+\frac {\left (1+i \sqrt {3}\right ) \sqrt {2+3 x^6+2 x^{12}}}{-1+i \sqrt {3}+2 x^2}\right ) \, dx\\ &=\frac {1}{6} \sqrt {2+3 x^6+2 x^{12}}+\frac {\sqrt {2+3 x^6+2 x^{12}}}{6 x^6}-\frac {2 \sqrt {2+3 x^6+2 x^{12}}}{3 x^3}+\frac {2}{3} i \int \frac {\sqrt {2+3 x^6+2 x^{12}}}{i-x} \, dx+\frac {2}{3} i \int \frac {\sqrt {2+3 x^6+2 x^{12}}}{i+x} \, dx-\frac {1}{12} \operatorname {Subst}\left (\int \frac {-4-3 x}{x \sqrt {2+3 x+2 x^2}} \, dx,x,x^6\right )-\frac {1}{12} \operatorname {Subst}\left (\int \frac {3+4 x}{x \sqrt {2+3 x+2 x^2}} \, dx,x,x^6\right )+\frac {2}{3} \operatorname {Subst}\left (\int \frac {3+4 x^2}{\sqrt {2+3 x^2+2 x^4}} \, dx,x,x^3\right )-\frac {1}{3} \left (4 \left (1-i \sqrt {3}\right )\right ) \int \frac {\sqrt {2+3 x^6+2 x^{12}}}{-1-i \sqrt {3}+2 x^2} \, dx-\frac {1}{3} \left (4 \left (1+i \sqrt {3}\right )\right ) \int \frac {\sqrt {2+3 x^6+2 x^{12}}}{-1+i \sqrt {3}+2 x^2} \, dx\\ &=\frac {1}{6} \sqrt {2+3 x^6+2 x^{12}}+\frac {\sqrt {2+3 x^6+2 x^{12}}}{6 x^6}-\frac {2 \sqrt {2+3 x^6+2 x^{12}}}{3 x^3}+\frac {2}{3} i \int \frac {\sqrt {2+3 x^6+2 x^{12}}}{i-x} \, dx+\frac {2}{3} i \int \frac {\sqrt {2+3 x^6+2 x^{12}}}{i+x} \, dx+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{\sqrt {2+3 x+2 x^2}} \, dx,x,x^6\right )-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {2+3 x+2 x^2}} \, dx,x,x^6\right )-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{\sqrt {2+3 x+2 x^2}} \, dx,x,x^6\right )+\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {2+3 x+2 x^2}} \, dx,x,x^6\right )-\frac {8}{3} \operatorname {Subst}\left (\int \frac {1-x^2}{\sqrt {2+3 x^2+2 x^4}} \, dx,x,x^3\right )+\frac {14}{3} \operatorname {Subst}\left (\int \frac {1}{\sqrt {2+3 x^2+2 x^4}} \, dx,x,x^3\right )-\frac {1}{3} \left (4 \left (1-i \sqrt {3}\right )\right ) \int \left (\frac {\sqrt {1+i \sqrt {3}} \sqrt {2+3 x^6+2 x^{12}}}{2 \left (-1-i \sqrt {3}\right ) \left (\sqrt {1+i \sqrt {3}}-\sqrt {2} x\right )}+\frac {\sqrt {1+i \sqrt {3}} \sqrt {2+3 x^6+2 x^{12}}}{2 \left (-1-i \sqrt {3}\right ) \left (\sqrt {1+i \sqrt {3}}+\sqrt {2} x\right )}\right ) \, dx-\frac {1}{3} \left (4 \left (1+i \sqrt {3}\right )\right ) \int \left (\frac {\sqrt {1-i \sqrt {3}} \sqrt {2+3 x^6+2 x^{12}}}{2 \left (-1+i \sqrt {3}\right ) \left (\sqrt {1-i \sqrt {3}}-\sqrt {2} x\right )}+\frac {\sqrt {1-i \sqrt {3}} \sqrt {2+3 x^6+2 x^{12}}}{2 \left (-1+i \sqrt {3}\right ) \left (\sqrt {1-i \sqrt {3}}+\sqrt {2} x\right )}\right ) \, dx\\ &=\frac {1}{6} \sqrt {2+3 x^6+2 x^{12}}+\frac {\sqrt {2+3 x^6+2 x^{12}}}{6 x^6}-\frac {2 \sqrt {2+3 x^6+2 x^{12}}}{3 x^3}+\frac {4 x^3 \sqrt {2+3 x^6+2 x^{12}}}{3 \left (1+x^6\right )}-\frac {4 \sqrt {2} \left (1+x^6\right ) \sqrt {\frac {2+3 x^6+2 x^{12}}{\left (1+x^6\right )^2}} E\left (2 \tan ^{-1}\left (x^3\right )|\frac {1}{8}\right )}{3 \sqrt {2+3 x^6+2 x^{12}}}+\frac {7 \left (1+x^6\right ) \sqrt {\frac {2+3 x^6+2 x^{12}}{\left (1+x^6\right )^2}} F\left (2 \tan ^{-1}\left (x^3\right )|\frac {1}{8}\right )}{3 \sqrt {2} \sqrt {2+3 x^6+2 x^{12}}}+\frac {2}{3} i \int \frac {\sqrt {2+3 x^6+2 x^{12}}}{i-x} \, dx+\frac {2}{3} i \int \frac {\sqrt {2+3 x^6+2 x^{12}}}{i+x} \, dx+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{8-x^2} \, dx,x,\frac {4+3 x^6}{\sqrt {2+3 x^6+2 x^{12}}}\right )-\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{8-x^2} \, dx,x,\frac {4+3 x^6}{\sqrt {2+3 x^6+2 x^{12}}}\right )+\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{7}}} \, dx,x,3+4 x^6\right )}{4 \sqrt {14}}-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{7}}} \, dx,x,3+4 x^6\right )}{3 \sqrt {14}}+\frac {\left (2 \left (1-i \sqrt {3}\right )\right ) \int \frac {\sqrt {2+3 x^6+2 x^{12}}}{\sqrt {1+i \sqrt {3}}-\sqrt {2} x} \, dx}{3 \sqrt {1+i \sqrt {3}}}+\frac {\left (2 \left (1-i \sqrt {3}\right )\right ) \int \frac {\sqrt {2+3 x^6+2 x^{12}}}{\sqrt {1+i \sqrt {3}}+\sqrt {2} x} \, dx}{3 \sqrt {1+i \sqrt {3}}}+\frac {\left (2 \left (1+i \sqrt {3}\right )\right ) \int \frac {\sqrt {2+3 x^6+2 x^{12}}}{\sqrt {1-i \sqrt {3}}-\sqrt {2} x} \, dx}{3 \sqrt {1-i \sqrt {3}}}+\frac {\left (2 \left (1+i \sqrt {3}\right )\right ) \int \frac {\sqrt {2+3 x^6+2 x^{12}}}{\sqrt {1-i \sqrt {3}}+\sqrt {2} x} \, dx}{3 \sqrt {1-i \sqrt {3}}}\\ &=\frac {1}{6} \sqrt {2+3 x^6+2 x^{12}}+\frac {\sqrt {2+3 x^6+2 x^{12}}}{6 x^6}-\frac {2 \sqrt {2+3 x^6+2 x^{12}}}{3 x^3}+\frac {4 x^3 \sqrt {2+3 x^6+2 x^{12}}}{3 \left (1+x^6\right )}-\frac {\sinh ^{-1}\left (\frac {3+4 x^6}{\sqrt {7}}\right )}{12 \sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {4+3 x^6}{2 \sqrt {2} \sqrt {2+3 x^6+2 x^{12}}}\right )}{12 \sqrt {2}}-\frac {4 \sqrt {2} \left (1+x^6\right ) \sqrt {\frac {2+3 x^6+2 x^{12}}{\left (1+x^6\right )^2}} E\left (2 \tan ^{-1}\left (x^3\right )|\frac {1}{8}\right )}{3 \sqrt {2+3 x^6+2 x^{12}}}+\frac {7 \left (1+x^6\right ) \sqrt {\frac {2+3 x^6+2 x^{12}}{\left (1+x^6\right )^2}} F\left (2 \tan ^{-1}\left (x^3\right )|\frac {1}{8}\right )}{3 \sqrt {2} \sqrt {2+3 x^6+2 x^{12}}}+\frac {2}{3} i \int \frac {\sqrt {2+3 x^6+2 x^{12}}}{i-x} \, dx+\frac {2}{3} i \int \frac {\sqrt {2+3 x^6+2 x^{12}}}{i+x} \, dx+\frac {\left (2 \left (1-i \sqrt {3}\right )\right ) \int \frac {\sqrt {2+3 x^6+2 x^{12}}}{\sqrt {1+i \sqrt {3}}-\sqrt {2} x} \, dx}{3 \sqrt {1+i \sqrt {3}}}+\frac {\left (2 \left (1-i \sqrt {3}\right )\right ) \int \frac {\sqrt {2+3 x^6+2 x^{12}}}{\sqrt {1+i \sqrt {3}}+\sqrt {2} x} \, dx}{3 \sqrt {1+i \sqrt {3}}}+\frac {\left (2 \left (1+i \sqrt {3}\right )\right ) \int \frac {\sqrt {2+3 x^6+2 x^{12}}}{\sqrt {1-i \sqrt {3}}-\sqrt {2} x} \, dx}{3 \sqrt {1-i \sqrt {3}}}+\frac {\left (2 \left (1+i \sqrt {3}\right )\right ) \int \frac {\sqrt {2+3 x^6+2 x^{12}}}{\sqrt {1-i \sqrt {3}}+\sqrt {2} x} \, dx}{3 \sqrt {1-i \sqrt {3}}}\\ \end {align*}

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Mathematica [F]  time = 1.16, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-1+x^3\right )^3 \left (1+x^3\right ) \sqrt {2+3 x^6+2 x^{12}}}{x^7 \left (1+x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((-1 + x^3)^3*(1 + x^3)*Sqrt[2 + 3*x^6 + 2*x^12])/(x^7*(1 + x^6)),x]

[Out]

Integrate[((-1 + x^3)^3*(1 + x^3)*Sqrt[2 + 3*x^6 + 2*x^12])/(x^7*(1 + x^6)), x]

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IntegrateAlgebraic [A]  time = 3.75, size = 128, normalized size = 1.00 \begin {gather*} \frac {\left (1-4 x^3+x^6\right ) \sqrt {2+3 x^6+2 x^{12}}}{6 x^6}-\frac {4}{3} \tan ^{-1}\left (\frac {x^3}{\sqrt {2}+\sqrt {2} x^6+\sqrt {2+3 x^6+2 x^{12}}}\right )+\frac {\log (x)}{2 \sqrt {2}}-\frac {\log \left (\sqrt {2}+\sqrt {2} x^6+\sqrt {2+3 x^6+2 x^{12}}\right )}{6 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-1 + x^3)^3*(1 + x^3)*Sqrt[2 + 3*x^6 + 2*x^12])/(x^7*(1 + x^6)),x]

[Out]

((1 - 4*x^3 + x^6)*Sqrt[2 + 3*x^6 + 2*x^12])/(6*x^6) - (4*ArcTan[x^3/(Sqrt[2] + Sqrt[2]*x^6 + Sqrt[2 + 3*x^6 +
2*x^12])])/3 + Log[x]/(2*Sqrt[2]) - Log[Sqrt[2] + Sqrt[2]*x^6 + Sqrt[2 + 3*x^6 + 2*x^12]]/(6*Sqrt[2])

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fricas [A]  time = 0.57, size = 105, normalized size = 0.82 \begin {gather*} \frac {\sqrt {2} x^{6} \log \left (-\frac {4 \, x^{12} + 7 \, x^{6} - 2 \, \sqrt {2} \sqrt {2 \, x^{12} + 3 \, x^{6} + 2} {\left (x^{6} + 1\right )} + 4}{x^{6}}\right ) - 16 \, x^{6} \arctan \left (\frac {x^{3}}{\sqrt {2 \, x^{12} + 3 \, x^{6} + 2}}\right ) + 4 \, \sqrt {2 \, x^{12} + 3 \, x^{6} + 2} {\left (x^{6} - 4 \, x^{3} + 1\right )}}{24 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-1)^3*(x^3+1)*(2*x^12+3*x^6+2)^(1/2)/x^7/(x^6+1),x, algorithm="fricas")

[Out]

1/24*(sqrt(2)*x^6*log(-(4*x^12 + 7*x^6 - 2*sqrt(2)*sqrt(2*x^12 + 3*x^6 + 2)*(x^6 + 1) + 4)/x^6) - 16*x^6*arcta
n(x^3/sqrt(2*x^12 + 3*x^6 + 2)) + 4*sqrt(2*x^12 + 3*x^6 + 2)*(x^6 - 4*x^3 + 1))/x^6

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {2 \, x^{12} + 3 \, x^{6} + 2} {\left (x^{3} + 1\right )} {\left (x^{3} - 1\right )}^{3}}{{\left (x^{6} + 1\right )} x^{7}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-1)^3*(x^3+1)*(2*x^12+3*x^6+2)^(1/2)/x^7/(x^6+1),x, algorithm="giac")

[Out]

integrate(sqrt(2*x^12 + 3*x^6 + 2)*(x^3 + 1)*(x^3 - 1)^3/((x^6 + 1)*x^7), x)

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maple [C]  time = 2.31, size = 129, normalized size = 1.01

method	result	size
trager	\(\frac {\left (x^{6}-4 x^{3}+1\right ) \sqrt {2 x^{12}+3 x^{6}+2}}{6 x^{6}}-\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) x^{6}+\RootOf \left (\textit {\_Z}^{2}-2\right )+\sqrt {2 x^{12}+3 x^{6}+2}}{x^{3}}\right )}{12}+\frac {2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}+\sqrt {2 x^{12}+3 x^{6}+2}}{\left (x^{2}+1\right ) \left (x^{4}-x^{2}+1\right )}\right )}{3}\)	\(129\)
risch	\(-\frac {8 x^{15}-2 x^{12}+12 x^{9}-3 x^{6}+8 x^{3}-2}{6 x^{6} \sqrt {2 x^{12}+3 x^{6}+2}}+\frac {\sqrt {2 x^{12}+3 x^{6}+2}}{6}-\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) x^{6}+\RootOf \left (\textit {\_Z}^{2}-2\right )+\sqrt {2 x^{12}+3 x^{6}+2}}{x^{3}}\right )}{12}+\frac {2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}+\sqrt {2 x^{12}+3 x^{6}+2}}{\left (x^{2}+1\right ) \left (x^{4}-x^{2}+1\right )}\right )}{3}\)	\(162\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3-1)^3*(x^3+1)*(2*x^12+3*x^6+2)^(1/2)/x^7/(x^6+1),x,method=_RETURNVERBOSE)

[Out]

1/6*(x^6-4*x^3+1)*(2*x^12+3*x^6+2)^(1/2)/x^6-1/12*RootOf(_Z^2-2)*ln((RootOf(_Z^2-2)*x^6+RootOf(_Z^2-2)+(2*x^12
+3*x^6+2)^(1/2))/x^3)+2/3*RootOf(_Z^2+1)*ln((RootOf(_Z^2+1)*x^3+(2*x^12+3*x^6+2)^(1/2))/(x^2+1)/(x^4-x^2+1))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {2 \, x^{12} + 3 \, x^{6} + 2} {\left (x^{3} + 1\right )} {\left (x^{3} - 1\right )}^{3}}{{\left (x^{6} + 1\right )} x^{7}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-1)^3*(x^3+1)*(2*x^12+3*x^6+2)^(1/2)/x^7/(x^6+1),x, algorithm="maxima")

[Out]

integrate(sqrt(2*x^12 + 3*x^6 + 2)*(x^3 + 1)*(x^3 - 1)^3/((x^6 + 1)*x^7), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (x^3-1\right )}^3\,\left (x^3+1\right )\,\sqrt {2\,x^{12}+3\,x^6+2}}{x^7\,\left (x^6+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^3 - 1)^3*(x^3 + 1)*(3*x^6 + 2*x^12 + 2)^(1/2))/(x^7*(x^6 + 1)),x)

[Out]

int(((x^3 - 1)^3*(x^3 + 1)*(3*x^6 + 2*x^12 + 2)^(1/2))/(x^7*(x^6 + 1)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x - 1\right )^{3} \left (x + 1\right ) \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )^{3} \sqrt {2 x^{12} + 3 x^{6} + 2}}{x^{7} \left (x^{2} + 1\right ) \left (x^{4} - x^{2} + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**3-1)**3*(x**3+1)*(2*x**12+3*x**6+2)**(1/2)/x**7/(x**6+1),x)

[Out]

Integral((x - 1)**3*(x + 1)*(x**2 - x + 1)*(x**2 + x + 1)**3*sqrt(2*x**12 + 3*x**6 + 2)/(x**7*(x**2 + 1)*(x**4
- x**2 + 1)), x)

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