3.10.64 \(\int \frac {(-3+2 x^5) \sqrt {x+2 x^4+x^6}}{(1+x^5) (1+x^3+x^5)} \, dx\)

Optimal. Leaf size=73 \[ 2 \tanh ^{-1}\left (\frac {x \sqrt {x^6+2 x^4+x}}{x^5+2 x^3+1}\right )-2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt {x^6+2 x^4+x}}{x^5+2 x^3+1}\right ) \]

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Rubi [F]  time = 5.60, 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+2 x^5\right ) \sqrt {x+2 x^4+x^6}}{\left (1+x^5\right ) \left (1+x^3+x^5\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

((-I)*Sqrt[x + 2*x^4 + x^6]*Defer[Subst][Defer[Int][Sqrt[1 + 2*x^6 + x^10]/(I - x), x], x, Sqrt[x]])/(Sqrt[x]*
Sqrt[1 + 2*x^3 + x^5]) - (I*Sqrt[x + 2*x^4 + x^6]*Defer[Subst][Defer[Int][Sqrt[1 + 2*x^6 + x^10]/(I + x), x],
x, Sqrt[x]])/(Sqrt[x]*Sqrt[1 + 2*x^3 + x^5]) + (2*Sqrt[x + 2*x^4 + x^6]*Defer[Subst][Defer[Int][Sqrt[1 + 2*x^6
+ x^10]/(1 - x^2 + x^4 - x^6 + x^8), x], x, Sqrt[x]])/(Sqrt[x]*Sqrt[1 + 2*x^3 + x^5]) - (4*Sqrt[x + 2*x^4 + x
^6]*Defer[Subst][Defer[Int][(x^2*Sqrt[1 + 2*x^6 + x^10])/(1 - x^2 + x^4 - x^6 + x^8), x], x, Sqrt[x]])/(Sqrt[x
]*Sqrt[1 + 2*x^3 + x^5]) + (6*Sqrt[x + 2*x^4 + x^6]*Defer[Subst][Defer[Int][(x^4*Sqrt[1 + 2*x^6 + x^10])/(1 -
x^2 + x^4 - x^6 + x^8), x], x, Sqrt[x]])/(Sqrt[x]*Sqrt[1 + 2*x^3 + x^5]) + (2*Sqrt[x + 2*x^4 + x^6]*Defer[Subs
t][Defer[Int][(x^6*Sqrt[1 + 2*x^6 + x^10])/(1 - x^2 + x^4 - x^6 + x^8), x], x, Sqrt[x]])/(Sqrt[x]*Sqrt[1 + 2*x
^3 + x^5]) - (6*Sqrt[x + 2*x^4 + x^6]*Defer[Subst][Defer[Int][(x^2*Sqrt[1 + 2*x^6 + x^10])/(1 + x^6 + x^10), x
], x, Sqrt[x]])/(Sqrt[x]*Sqrt[1 + 2*x^3 + x^5]) - (10*Sqrt[x + 2*x^4 + x^6]*Defer[Subst][Defer[Int][(x^6*Sqrt[
1 + 2*x^6 + x^10])/(1 + x^6 + x^10), x], x, Sqrt[x]])/(Sqrt[x]*Sqrt[1 + 2*x^3 + x^5])

Rubi steps

\begin {align*} \int \frac {\left (-3+2 x^5\right ) \sqrt {x+2 x^4+x^6}}{\left (1+x^5\right ) \left (1+x^3+x^5\right )} \, dx &=\frac {\sqrt {x+2 x^4+x^6} \int \frac {\sqrt {x} \sqrt {1+2 x^3+x^5} \left (-3+2 x^5\right )}{\left (1+x^5\right ) \left (1+x^3+x^5\right )} \, dx}{\sqrt {x} \sqrt {1+2 x^3+x^5}}\\ &=\frac {\sqrt {x+2 x^4+x^6} \int \left (\frac {\sqrt {x} \sqrt {1+2 x^3+x^5}}{1+x}+\frac {\sqrt {x} \left (-1+2 x+2 x^2-x^3\right ) \sqrt {1+2 x^3+x^5}}{1-x+x^2-x^3+x^4}+\frac {\sqrt {x} \left (-3-5 x^2\right ) \sqrt {1+2 x^3+x^5}}{1+x^3+x^5}\right ) \, dx}{\sqrt {x} \sqrt {1+2 x^3+x^5}}\\ &=\frac {\sqrt {x+2 x^4+x^6} \int \frac {\sqrt {x} \sqrt {1+2 x^3+x^5}}{1+x} \, dx}{\sqrt {x} \sqrt {1+2 x^3+x^5}}+\frac {\sqrt {x+2 x^4+x^6} \int \frac {\sqrt {x} \left (-1+2 x+2 x^2-x^3\right ) \sqrt {1+2 x^3+x^5}}{1-x+x^2-x^3+x^4} \, dx}{\sqrt {x} \sqrt {1+2 x^3+x^5}}+\frac {\sqrt {x+2 x^4+x^6} \int \frac {\sqrt {x} \left (-3-5 x^2\right ) \sqrt {1+2 x^3+x^5}}{1+x^3+x^5} \, dx}{\sqrt {x} \sqrt {1+2 x^3+x^5}}\\ &=\frac {\left (2 \sqrt {x+2 x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {1+2 x^6+x^{10}}}{1+x^2} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+2 x^3+x^5}}+\frac {\left (2 \sqrt {x+2 x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (-1+2 x^2+2 x^4-x^6\right ) \sqrt {1+2 x^6+x^{10}}}{1-x^2+x^4-x^6+x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+2 x^3+x^5}}+\frac {\left (2 \sqrt {x+2 x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (-3-5 x^4\right ) \sqrt {1+2 x^6+x^{10}}}{1+x^6+x^{10}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+2 x^3+x^5}}\\ &=\frac {\left (2 \sqrt {x+2 x^4+x^6}\right ) \operatorname {Subst}\left (\int \left (\sqrt {1+2 x^6+x^{10}}-\frac {\sqrt {1+2 x^6+x^{10}}}{1+x^2}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+2 x^3+x^5}}+\frac {\left (2 \sqrt {x+2 x^4+x^6}\right ) \operatorname {Subst}\left (\int \left (-\sqrt {1+2 x^6+x^{10}}+\frac {\left (1-2 x^2+3 x^4+x^6\right ) \sqrt {1+2 x^6+x^{10}}}{1-x^2+x^4-x^6+x^8}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+2 x^3+x^5}}+\frac {\left (2 \sqrt {x+2 x^4+x^6}\right ) \operatorname {Subst}\left (\int \left (-\frac {3 x^2 \sqrt {1+2 x^6+x^{10}}}{1+x^6+x^{10}}-\frac {5 x^6 \sqrt {1+2 x^6+x^{10}}}{1+x^6+x^{10}}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+2 x^3+x^5}}\\ &=-\frac {\left (2 \sqrt {x+2 x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+2 x^6+x^{10}}}{1+x^2} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+2 x^3+x^5}}+\frac {\left (2 \sqrt {x+2 x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {\left (1-2 x^2+3 x^4+x^6\right ) \sqrt {1+2 x^6+x^{10}}}{1-x^2+x^4-x^6+x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+2 x^3+x^5}}-\frac {\left (6 \sqrt {x+2 x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {1+2 x^6+x^{10}}}{1+x^6+x^{10}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+2 x^3+x^5}}-\frac {\left (10 \sqrt {x+2 x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {x^6 \sqrt {1+2 x^6+x^{10}}}{1+x^6+x^{10}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+2 x^3+x^5}}\\ &=-\frac {\left (2 \sqrt {x+2 x^4+x^6}\right ) \operatorname {Subst}\left (\int \left (\frac {i \sqrt {1+2 x^6+x^{10}}}{2 (i-x)}+\frac {i \sqrt {1+2 x^6+x^{10}}}{2 (i+x)}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+2 x^3+x^5}}+\frac {\left (2 \sqrt {x+2 x^4+x^6}\right ) \operatorname {Subst}\left (\int \left (\frac {\sqrt {1+2 x^6+x^{10}}}{1-x^2+x^4-x^6+x^8}-\frac {2 x^2 \sqrt {1+2 x^6+x^{10}}}{1-x^2+x^4-x^6+x^8}+\frac {3 x^4 \sqrt {1+2 x^6+x^{10}}}{1-x^2+x^4-x^6+x^8}+\frac {x^6 \sqrt {1+2 x^6+x^{10}}}{1-x^2+x^4-x^6+x^8}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+2 x^3+x^5}}-\frac {\left (6 \sqrt {x+2 x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {1+2 x^6+x^{10}}}{1+x^6+x^{10}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+2 x^3+x^5}}-\frac {\left (10 \sqrt {x+2 x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {x^6 \sqrt {1+2 x^6+x^{10}}}{1+x^6+x^{10}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+2 x^3+x^5}}\\ &=-\frac {\left (i \sqrt {x+2 x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+2 x^6+x^{10}}}{i-x} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+2 x^3+x^5}}-\frac {\left (i \sqrt {x+2 x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+2 x^6+x^{10}}}{i+x} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+2 x^3+x^5}}+\frac {\left (2 \sqrt {x+2 x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+2 x^6+x^{10}}}{1-x^2+x^4-x^6+x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+2 x^3+x^5}}+\frac {\left (2 \sqrt {x+2 x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {x^6 \sqrt {1+2 x^6+x^{10}}}{1-x^2+x^4-x^6+x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+2 x^3+x^5}}-\frac {\left (4 \sqrt {x+2 x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {1+2 x^6+x^{10}}}{1-x^2+x^4-x^6+x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+2 x^3+x^5}}+\frac {\left (6 \sqrt {x+2 x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {x^4 \sqrt {1+2 x^6+x^{10}}}{1-x^2+x^4-x^6+x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+2 x^3+x^5}}-\frac {\left (6 \sqrt {x+2 x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {1+2 x^6+x^{10}}}{1+x^6+x^{10}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+2 x^3+x^5}}-\frac {\left (10 \sqrt {x+2 x^4+x^6}\right ) \operatorname {Subst}\left (\int \frac {x^6 \sqrt {1+2 x^6+x^{10}}}{1+x^6+x^{10}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {1+2 x^3+x^5}}\\ \end {align*}

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Mathematica [F]  time = 0.94, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-3+2 x^5\right ) \sqrt {x+2 x^4+x^6}}{\left (1+x^5\right ) \left (1+x^3+x^5\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.38, size = 73, normalized size = 1.00 \begin {gather*} 2 \tanh ^{-1}\left (\frac {x \sqrt {x+2 x^4+x^6}}{1+2 x^3+x^5}\right )-2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt {x+2 x^4+x^6}}{1+2 x^3+x^5}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.58, size = 111, normalized size = 1.52 \begin {gather*} \frac {1}{2} \, \sqrt {2} \log \left (-\frac {x^{10} + 16 \, x^{8} + 32 \, x^{6} + 2 \, x^{5} + 16 \, x^{3} - 4 \, \sqrt {2} {\left (x^{6} + 4 \, x^{4} + x\right )} \sqrt {x^{6} + 2 \, x^{4} + x} + 1}{x^{10} + 2 \, x^{5} + 1}\right ) + \log \left (-\frac {x^{5} + 3 \, x^{3} + 2 \, \sqrt {x^{6} + 2 \, x^{4} + x} x + 1}{x^{5} + x^{3} + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(2)*log(-(x^10 + 16*x^8 + 32*x^6 + 2*x^5 + 16*x^3 - 4*sqrt(2)*(x^6 + 4*x^4 + x)*sqrt(x^6 + 2*x^4 + x)
+ 1)/(x^10 + 2*x^5 + 1)) + log(-(x^5 + 3*x^3 + 2*sqrt(x^6 + 2*x^4 + x)*x + 1)/(x^5 + x^3 + 1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.68, size = 121, normalized size = 1.66

method	result	size
trager	\(\RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}-2\right ) x^{5}-4 \RootOf \left (\textit {\_Z}^{2}-2\right ) x^{3}+4 \sqrt {x^{6}+2 x^{4}+x}\, x -\RootOf \left (\textit {\_Z}^{2}-2\right )}{\left (1+x \right ) \left (x^{4}-x^{3}+x^{2}-x +1\right )}\right )-\ln \left (\frac {-x^{5}-3 x^{3}+2 \sqrt {x^{6}+2 x^{4}+x}\, x -1}{x^{5}+x^{3}+1}\right )\)	\(121\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (2\,x^5-3\right )\,\sqrt {x^6+2\,x^4+x}}{\left (x^5+1\right )\,\left (x^5+x^3+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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