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

Optimal. Leaf size=174 \[ \frac {4 \tan ^{-1}\left (\frac {x \sqrt {a+b-c}}{-\sqrt {a x^4+a+b x^3+b x+c x^2}+\sqrt {a} x^2+\sqrt {a} x+\sqrt {a}}\right )}{3 \sqrt {a+b-c}}-\frac {2 \sqrt {-2 a-2 b-c} \tan ^{-1}\left (\frac {x \sqrt {-2 a-2 b-c}}{-\sqrt {a x^4+a+b x^3+b x+c x^2}+\sqrt {a} x^2-2 \sqrt {a} x+\sqrt {a}}\right )}{3 (2 a+2 b+c)} \]

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Rubi [F]  time = 1.54, 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 {1+x^3}{\left (-1+x^3\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [C]  time = 4.89, size = 5076, normalized size = 29.17 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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IntegrateAlgebraic [A]  time = 1.30, size = 174, normalized size = 1.00 \begin {gather*} -\frac {2 \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 )}{3 (2 a+2 b+c)}+\frac {4 \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}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*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])])/(3*(2*a + 2*b + c)) + (4*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])

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fricas [A]  time = 3.33, size = 1497, normalized size = 8.60

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/6*(2*(2*a + 2*b + c)*sqrt(-a - b + c)*log(-((8*a*b - b^2 - 4*a*c)*x^4 - 2*(8*a^2 - 4*a*b - 3*b^2 - 4*(a -
b)*c)*x^3 - (24*a^2 + 3*b^2 - 4*(5*a + 2*b)*c + 8*c^2)*x^2 - 4*sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a)*((2*a - b
)*x^2 + (4*a + b - 2*c)*x + 2*a - b)*sqrt(-a - b + c) + 8*a*b - b^2 - 4*a*c - 2*(8*a^2 - 4*a*b - 3*b^2 - 4*(a
- b)*c)*x)/(x^4 + 2*x^3 + 3*x^2 + 2*x + 1)) - sqrt(2*a + 2*b + c)*(a + b - c)*log(((24*a^2 + 16*a*b + b^2 + 4*
a*c)*x^4 - 4*(8*a^2 - 4*a*b - 3*b^2 - 2*(2*a + b)*c)*x^3 + 2*(24*a^2 + 3*b^2 - 4*(a - 2*b)*c + 4*c^2)*x^2 - 4*
sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a)*((4*a + b)*x^2 - 2*(2*a - b - c)*x + 4*a + b)*sqrt(2*a + 2*b + c) + 24*a
^2 + 16*a*b + b^2 + 4*a*c - 4*(8*a^2 - 4*a*b - 3*b^2 - 2*(2*a + b)*c)*x)/(x^4 - 4*x^3 + 6*x^2 - 4*x + 1)))/(2*
a^2 + 4*a*b + 2*b^2 - (a + b)*c - c^2), 1/3*((a + b - c)*sqrt(-2*a - 2*b - c)*arctan(1/2*sqrt(a*x^4 + b*x^3 +
c*x^2 + b*x + a)*((4*a + b)*x^2 - 2*(2*a - b - c)*x + 4*a + b)*sqrt(-2*a - 2*b - c)/((2*a^2 + 2*a*b + a*c)*x^4
+ (2*a*b + 2*b^2 + b*c)*x^3 + (2*(a + b)*c + c^2)*x^2 + 2*a^2 + 2*a*b + a*c + (2*a*b + 2*b^2 + b*c)*x)) - (2*
a + 2*b + c)*sqrt(-a - b + c)*log(-((8*a*b - b^2 - 4*a*c)*x^4 - 2*(8*a^2 - 4*a*b - 3*b^2 - 4*(a - b)*c)*x^3 -
(24*a^2 + 3*b^2 - 4*(5*a + 2*b)*c + 8*c^2)*x^2 - 4*sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a)*((2*a - b)*x^2 + (4*a
+ b - 2*c)*x + 2*a - b)*sqrt(-a - b + c) + 8*a*b - b^2 - 4*a*c - 2*(8*a^2 - 4*a*b - 3*b^2 - 4*(a - b)*c)*x)/(
x^4 + 2*x^3 + 3*x^2 + 2*x + 1)))/(2*a^2 + 4*a*b + 2*b^2 - (a + b)*c - c^2), -1/6*(4*(2*a + 2*b + c)*sqrt(a + b
- c)*arctan(-2*sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a)*sqrt(a + b - c)/((2*a - b)*x^2 + (4*a + b - 2*c)*x + 2*a
- b)) - sqrt(2*a + 2*b + c)*(a + b - c)*log(((24*a^2 + 16*a*b + b^2 + 4*a*c)*x^4 - 4*(8*a^2 - 4*a*b - 3*b^2 -
2*(2*a + b)*c)*x^3 + 2*(24*a^2 + 3*b^2 - 4*(a - 2*b)*c + 4*c^2)*x^2 - 4*sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a)
*((4*a + b)*x^2 - 2*(2*a - b - c)*x + 4*a + b)*sqrt(2*a + 2*b + c) + 24*a^2 + 16*a*b + b^2 + 4*a*c - 4*(8*a^2
- 4*a*b - 3*b^2 - 2*(2*a + b)*c)*x)/(x^4 - 4*x^3 + 6*x^2 - 4*x + 1)))/(2*a^2 + 4*a*b + 2*b^2 - (a + b)*c - c^2
), 1/3*((a + b - c)*sqrt(-2*a - 2*b - c)*arctan(1/2*sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a)*((4*a + b)*x^2 - 2*(
2*a - b - c)*x + 4*a + b)*sqrt(-2*a - 2*b - c)/((2*a^2 + 2*a*b + a*c)*x^4 + (2*a*b + 2*b^2 + b*c)*x^3 + (2*(a
+ b)*c + c^2)*x^2 + 2*a^2 + 2*a*b + a*c + (2*a*b + 2*b^2 + b*c)*x)) - 2*(2*a + 2*b + c)*sqrt(a + b - c)*arctan
(-2*sqrt(a*x^4 + b*x^3 + c*x^2 + b*x + a)*sqrt(a + b - c)/((2*a - b)*x^2 + (4*a + b - 2*c)*x + 2*a - b)))/(2*a
^2 + 4*a*b + 2*b^2 - (a + b)*c - c^2)]

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

[Out]

Timed out

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maple [F]  time = 0.12, size = 0, normalized size = 0.00 \[\int \frac {x^{3}+1}{\left (x^{3}-1\right ) \sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^3+1}{\left (x^3-1\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 + 1)/((x^3 - 1)*(a + b*x + a*x^4 + b*x^3 + c*x^2)^(1/2)),x)

[Out]

int((x^3 + 1)/((x^3 - 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 {\left (x + 1\right ) \left (x^{2} - x + 1\right )}{\left (x - 1\right ) \left (x^{2} + x + 1\right ) \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+1)/(x**3-1)/(a*x**4+b*x**3+c*x**2+b*x+a)**(1/2),x)

[Out]

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

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