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

Optimal. Leaf size=41 \[ 2 \tan ^{-1}\left (\frac {\sqrt [4]{a x^2+b x+c}}{x}\right )-2 \tanh ^{-1}\left (\frac {x}{\sqrt [4]{a x^2+b x+c}}\right ) \]

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Rubi [F]  time = 1.22, 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 {4 c+3 b x+2 a x^2}{\sqrt [4]{c+b x+a x^2} \left (-c-b x-a x^2+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 0.44, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {4 c+3 b x+2 a x^2}{\sqrt [4]{c+b x+a x^2} \left (-c-b x-a x^2+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.41, size = 41, normalized size = 1.00 \begin {gather*} 2 \tan ^{-1}\left (\frac {\sqrt [4]{c+b x+a x^2}}{x}\right )-2 \tanh ^{-1}\left (\frac {x}{\sqrt [4]{c+b x+a x^2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2*ArcTan[(c + b*x + a*x^2)^(1/4)/x] - 2*ArcTanh[x/(c + b*x + a*x^2)^(1/4)]

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

[Out]

Timed out

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maple [F]  time = 0.08, size = 0, normalized size = 0.00 \[\int \frac {2 a \,x^{2}+3 b x +4 c}{\left (a \,x^{2}+b x +c \right )^{\frac {1}{4}} \left (x^{4}-a \,x^{2}-b x -c \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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