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]

1/2*ln((a+b)^(1/3)-(b*x^3+a)^(1/3))/(a+b)^(1/3)-1/2*ln((a+b)^(1/3)*x-(b*x^3+a)^(1/3))/(a+b)^(1/3)+1/3*arctan(1
/3*(1+2*(a+b)^(1/3)*x/(b*x^3+a)^(1/3))*3^(1/2))/(a+b)^(1/3)*3^(1/2)+1/3*arctan(1/3*(1+2*(b*x^3+a)^(1/3)/(a+b)^
(1/3))*3^(1/2))/(a+b)^(1/3)*3^(1/2)

________________________________________________________________________________________

Rubi [F]  time = 0.30, 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 = {} \[ \int \frac {1+x}{\left (1+x+x^2\right ) \sqrt [3]{a+b x^3}} \, dx \]

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 = 0.20, size = 0, normalized size = 0.00 \[ \int \frac {1+x}{\left (1+x+x^2\right ) \sqrt [3]{a+b x^3}} \, dx \]

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]

________________________________________________________________________________________

fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

maple [F]  time = 0.21, size = 0, normalized size = 0.00 \[ \int \frac {x +1}{\left (x^{2}+x +1\right ) \left (b \,x^{3}+a \right )^{\frac {1}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

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)

________________________________________________________________________________________