3.30.46 \(\int \frac {x^3}{\sqrt {a+b x+c x^2+b x^3+a x^4} (1-x^6)} \, dx\)

Optimal. Leaf size=352 \[ \frac {\sqrt {-2 a-2 b-c} \tan ^{-1}\left (\frac {x \sqrt {-2 a-2 b-c}}{-\sqrt {a x^4+a+b x^3+b x+c x^2}+\sqrt {a} x^2-2 \sqrt {a} x+\sqrt {a}}\right )}{6 (2 a+2 b+c)}+\frac {\tan ^{-1}\left (\frac {x \sqrt {a-b-c}}{-\sqrt {a x^4+a+b x^3+b x+c x^2}+\sqrt {a} x^2-\sqrt {a} x+\sqrt {a}}\right )}{3 \sqrt {a-b-c}}-\frac {\tan ^{-1}\left (\frac {x \sqrt {a+b-c}}{-\sqrt {a x^4+a+b x^3+b x+c x^2}+\sqrt {a} x^2+\sqrt {a} x+\sqrt {a}}\right )}{3 \sqrt {a+b-c}}-\frac {\sqrt {-2 a+2 b-c} \tan ^{-1}\left (\frac {x \sqrt {-2 a+2 b-c}}{-\sqrt {a x^4+a+b x^3+b x+c x^2}+\sqrt {a} x^2+2 \sqrt {a} x+\sqrt {a}}\right )}{6 (2 a-2 b+c)} \]

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Rubi [F]  time = 2.75, 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 {x^3}{\sqrt {a+b x+c x^2+b x^3+a x^4} \left (1-x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

-1/6*Defer[Int][1/((-1 + x)*Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4]), x] - Defer[Int][1/((1 + x)*Sqrt[a + b*x +
c*x^2 + b*x^3 + a*x^4]), x]/6 + ((1 + I*Sqrt[3])*Defer[Int][1/((-1 - I*Sqrt[3] + 2*x)*Sqrt[a + b*x + c*x^2 + b
*x^3 + a*x^4]), x])/6 + ((1 - I*Sqrt[3])*Defer[Int][1/((1 - I*Sqrt[3] + 2*x)*Sqrt[a + b*x + c*x^2 + b*x^3 + a*
x^4]), x])/6 + ((1 - I*Sqrt[3])*Defer[Int][1/((-1 + I*Sqrt[3] + 2*x)*Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4]), x
])/6 + ((1 + I*Sqrt[3])*Defer[Int][1/((1 + I*Sqrt[3] + 2*x)*Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4]), x])/6

Rubi steps

\begin {align*} \int \frac {x^3}{\sqrt {a+b x+c x^2+b x^3+a x^4} \left (1-x^6\right )} \, dx &=\int \left (-\frac {x}{3 \left (-1+x^2\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}}+\frac {-2+x}{6 \left (1-x+x^2\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}}+\frac {2+x}{6 \left (1+x+x^2\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}}\right ) \, dx\\ &=\frac {1}{6} \int \frac {-2+x}{\left (1-x+x^2\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx+\frac {1}{6} \int \frac {2+x}{\left (1+x+x^2\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx-\frac {1}{3} \int \frac {x}{\left (-1+x^2\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx\\ &=\frac {1}{6} \int \left (\frac {1+i \sqrt {3}}{\left (-1-i \sqrt {3}+2 x\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}}+\frac {1-i \sqrt {3}}{\left (-1+i \sqrt {3}+2 x\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}}\right ) \, dx+\frac {1}{6} \int \left (\frac {1-i \sqrt {3}}{\left (1-i \sqrt {3}+2 x\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}}+\frac {1+i \sqrt {3}}{\left (1+i \sqrt {3}+2 x\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}}\right ) \, dx-\frac {1}{3} \int \left (\frac {1}{2 (-1+x) \sqrt {a+b x+c x^2+b x^3+a x^4}}+\frac {1}{2 (1+x) \sqrt {a+b x+c x^2+b x^3+a x^4}}\right ) \, dx\\ &=-\left (\frac {1}{6} \int \frac {1}{(-1+x) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx\right )-\frac {1}{6} \int \frac {1}{(1+x) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx+\frac {1}{6} \left (1-i \sqrt {3}\right ) \int \frac {1}{\left (1-i \sqrt {3}+2 x\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx+\frac {1}{6} \left (1-i \sqrt {3}\right ) \int \frac {1}{\left (-1+i \sqrt {3}+2 x\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx+\frac {1}{6} \left (1+i \sqrt {3}\right ) \int \frac {1}{\left (-1-i \sqrt {3}+2 x\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx+\frac {1}{6} \left (1+i \sqrt {3}\right ) \int \frac {1}{\left (1+i \sqrt {3}+2 x\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx\\ \end {align*}

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Mathematica [C]  time = 7.14, size = 12183, normalized size = 34.61 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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IntegrateAlgebraic [A]  time = 4.74, size = 352, normalized size = 1.00 \begin {gather*} \frac {\sqrt {-2 a-2 b-c} \tan ^{-1}\left (\frac {\sqrt {-2 a-2 b-c} x}{\sqrt {a}-2 \sqrt {a} x+\sqrt {a} x^2-\sqrt {a+b x+c x^2+b x^3+a x^4}}\right )}{6 (2 a+2 b+c)}+\frac {\tan ^{-1}\left (\frac {\sqrt {a-b-c} x}{\sqrt {a}-\sqrt {a} x+\sqrt {a} x^2-\sqrt {a+b x+c x^2+b x^3+a x^4}}\right )}{3 \sqrt {a-b-c}}-\frac {\tan ^{-1}\left (\frac {\sqrt {a+b-c} x}{\sqrt {a}+\sqrt {a} x+\sqrt {a} x^2-\sqrt {a+b x+c x^2+b x^3+a x^4}}\right )}{3 \sqrt {a+b-c}}-\frac {\sqrt {-2 a+2 b-c} \tan ^{-1}\left (\frac {\sqrt {-2 a+2 b-c} x}{\sqrt {a}+2 \sqrt {a} x+\sqrt {a} x^2-\sqrt {a+b x+c x^2+b x^3+a x^4}}\right )}{6 (2 a-2 b+c)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[-2*a - 2*b - c]*ArcTan[(Sqrt[-2*a - 2*b - c]*x)/(Sqrt[a] - 2*Sqrt[a]*x + Sqrt[a]*x^2 - Sqrt[a + b*x + c*
x^2 + b*x^3 + a*x^4])])/(6*(2*a + 2*b + c)) + ArcTan[(Sqrt[a - b - c]*x)/(Sqrt[a] - Sqrt[a]*x + Sqrt[a]*x^2 -
Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4])]/(3*Sqrt[a - b - c]) - ArcTan[(Sqrt[a + b - c]*x)/(Sqrt[a] + Sqrt[a]*x
+ Sqrt[a]*x^2 - Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4])]/(3*Sqrt[a + b - c]) - (Sqrt[-2*a + 2*b - c]*ArcTan[(Sq
rt[-2*a + 2*b - c]*x)/(Sqrt[a] + 2*Sqrt[a]*x + Sqrt[a]*x^2 - Sqrt[a + b*x + c*x^2 + b*x^3 + a*x^4])])/(6*(2*a
- 2*b + c))

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fricas [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^3/(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2)/(-x^6+1),x, algorithm="fricas")

[Out]

Timed out

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giac [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^3/(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2)/(-x^6+1),x, algorithm="giac")

[Out]

Timed out

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maple [F]  time = 0.09, size = 0, normalized size = 0.00 \[\int \frac {x^{3}}{\sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}\, \left (-x^{6}+1\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2)/(-x^6+1),x)

[Out]

int(x^3/(a*x^4+b*x^3+c*x^2+b*x+a)^(1/2)/(-x^6+1),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-x^3/((x^6 - 1)*(a + b*x + a*x^4 + b*x^3 + c*x^2)^(1/2)),x)

[Out]

-int(x^3/((x^6 - 1)*(a + b*x + a*x^4 + b*x^3 + c*x^2)^(1/2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {x^{3}}{x^{6} \sqrt {a x^{4} + a + b x^{3} + b x + c x^{2}} - \sqrt {a x^{4} + a + b x^{3} + b x + c x^{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(a*x**4+b*x**3+c*x**2+b*x+a)**(1/2)/(-x**6+1),x)

[Out]

-Integral(x**3/(x**6*sqrt(a*x**4 + a + b*x**3 + b*x + c*x**2) - sqrt(a*x**4 + a + b*x**3 + b*x + c*x**2)), x)

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