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

Optimal. Leaf size=105 \[ \log \left (\sqrt [3]{a x^2+b x+c}+x\right )-\frac {1}{2} \log \left (-x \sqrt [3]{a x^2+b x+c}+\left (a x^2+b x+c\right )^{2/3}+x^2\right )+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{a x^2+b x+c}}{\sqrt [3]{a x^2+b x+c}-2 x}\right ) \]

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Rubi [F]  time = 1.13, 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 {3 c+2 b x+a x^2}{\sqrt [3]{c+b x+a x^2} \left (c+b x+a x^2+x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 0.38, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3 c+2 b x+a x^2}{\sqrt [3]{c+b x+a x^2} \left (c+b x+a x^2+x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.55, size = 105, normalized size = 1.00 \begin {gather*} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{c+b x+a x^2}}{-2 x+\sqrt [3]{c+b x+a x^2}}\right )+\log \left (x+\sqrt [3]{c+b x+a x^2}\right )-\frac {1}{2} \log \left (x^2-x \sqrt [3]{c+b x+a x^2}+\left (c+b x+a x^2\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[3]*ArcTan[(Sqrt[3]*(c + b*x + a*x^2)^(1/3))/(-2*x + (c + b*x + a*x^2)^(1/3))] + Log[x + (c + b*x + a*x^2)
^(1/3)] - Log[x^2 - x*(c + b*x + a*x^2)^(1/3) + (c + b*x + a*x^2)^(2/3)]/2

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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