\(\int \frac {3 c+2 b x+a x^2}{\sqrt [3]{c+b x+a x^2} (c+b x+a x^2+x^3)} \, dx\) [1504]

Optimal result

Integrand size = 43, antiderivative size = 105 \[ \int \frac {3 c+2 b x+a x^2}{\sqrt [3]{c+b x+a x^2} \left (c+b x+a x^2+x^3\right )} \, dx=\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{c+b x+a x^2}}{-2 x+\sqrt [3]{c+b x+a x^2}}\right )+\log \left (x+\sqrt [3]{c+b x+a x^2}\right )-\frac {1}{2} \log \left (x^2-x \sqrt [3]{c+b x+a x^2}+\left (c+b x+a x^2\right )^{2/3}\right ) \]

[Out]

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

Rubi [F]

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

[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*} \text {integral}& = \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*}

Mathematica [A] (verified)

Time = 0.52 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.95 \[ \int \frac {3 c+2 b x+a x^2}{\sqrt [3]{c+b x+a x^2} \left (c+b x+a x^2+x^3\right )} \, dx=\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{c+x (b+a x)}}{-2 x+\sqrt [3]{c+x (b+a x)}}\right )+\log \left (x+\sqrt [3]{c+x (b+a x)}\right )-\frac {1}{2} \log \left (x^2-x \sqrt [3]{c+x (b+a x)}+(c+x (b+a x))^{2/3}\right ) \]

[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]

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

Maple [F]

\[\int \frac {a \,x^{2}+2 b x +3 c}{\left (a \,x^{2}+b x +c \right )^{\frac {1}{3}} \left (a \,x^{2}+x^{3}+b x +c \right )}d x\]

[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)

Fricas [F(-1)]

Timed out. \[ \int \frac {3 c+2 b x+a x^2}{\sqrt [3]{c+b x+a x^2} \left (c+b x+a x^2+x^3\right )} \, dx=\text {Timed out} \]

[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

Sympy [F(-1)]

Timed out. \[ \int \frac {3 c+2 b x+a x^2}{\sqrt [3]{c+b x+a x^2} \left (c+b x+a x^2+x^3\right )} \, dx=\text {Timed out} \]

[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

Maxima [F]

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

[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)

Giac [F]

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

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {3 c+2 b x+a x^2}{\sqrt [3]{c+b x+a x^2} \left (c+b x+a x^2+x^3\right )} \, dx=\int \frac {a\,x^2+2\,b\,x+3\,c}{{\left (a\,x^2+b\,x+c\right )}^{1/3}\,\left (x^3+a\,x^2+b\,x+c\right )} \,d x \]

[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)