3.95 \(\int \frac{1+x}{(1+x+x^2) \sqrt [3]{a+b x^3}} \, dx\)

Optimal. Leaf size=154 \[ \frac{\log \left (\sqrt [3]{a+b}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a+b}}-\frac{\log \left (x \sqrt [3]{a+b}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a+b}}+\frac{\tan ^{-1}\left (\frac{\frac{2 x \sqrt [3]{a+b}}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} \sqrt [3]{a+b}}+\frac{\tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a+b}}+1}{\sqrt{3}}\right )}{\sqrt{3} \sqrt [3]{a+b}} \]

[Out]

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

________________________________________________________________________________________

Rubi [F]  time = 0.299582, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{1+x}{\left (1+x+x^2\right ) \sqrt [3]{a+b x^3}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [F]  time = 0.28751, size = 0, normalized size = 0. \[ \int \frac{1+x}{\left (1+x+x^2\right ) \sqrt [3]{a+b x^3}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.088, size = 0, normalized size = 0. \begin{align*} \int{\frac{1+x}{{x}^{2}+x+1}{\frac{1}{\sqrt [3]{b{x}^{3}+a}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x + 1}{\sqrt [3]{a + b x^{3}} \left (x^{2} + x + 1\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((x + 1)/((a + b*x**3)**(1/3)*(x**2 + x + 1)), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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