\(\int \frac {x^2 (10 b+9 a x)}{\sqrt [4]{b x^2+a x^3} (-b-a x+x^{10})} \, dx\) [585]

Optimal result

Integrand size = 40, antiderivative size = 45 \[ \int \frac {x^2 (10 b+9 a x)}{\sqrt [4]{b x^2+a x^3} \left (-b-a x+x^{10}\right )} \, dx=2 \arctan \left (\frac {\sqrt [4]{b x^2+a x^3}}{x^3}\right )-2 \text {arctanh}\left (\frac {x^3}{\sqrt [4]{b x^2+a x^3}}\right ) \]

[Out]

2*arctan((a*x^3+b*x^2)^(1/4)/x^3)-2*arctanh(x^3/(a*x^3+b*x^2)^(1/4))

Rubi [F]

\[ \int \frac {x^2 (10 b+9 a x)}{\sqrt [4]{b x^2+a x^3} \left (-b-a x+x^{10}\right )} \, dx=\int \frac {x^2 (10 b+9 a x)}{\sqrt [4]{b x^2+a x^3} \left (-b-a x+x^{10}\right )} \, dx \]

[In]

Int[(x^2*(10*b + 9*a*x))/((b*x^2 + a*x^3)^(1/4)*(-b - a*x + x^10)),x]

[Out]

(-20*b*Sqrt[x]*(b + a*x)^(1/4)*Defer[Subst][Defer[Int][x^4/((b + a*x^2)^(1/4)*(b + a*x^2 - x^20)), x], x, Sqrt
[x]])/(b*x^2 + a*x^3)^(1/4) - (18*a*Sqrt[x]*(b + a*x)^(1/4)*Defer[Subst][Defer[Int][x^6/((b + a*x^2)^(1/4)*(b
+ a*x^2 - x^20)), x], x, Sqrt[x]])/(b*x^2 + a*x^3)^(1/4)

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {x} \sqrt [4]{b+a x}\right ) \int \frac {x^{3/2} (10 b+9 a x)}{\sqrt [4]{b+a x} \left (-b-a x+x^{10}\right )} \, dx}{\sqrt [4]{b x^2+a x^3}} \\ & = \frac {\left (2 \sqrt {x} \sqrt [4]{b+a x}\right ) \text {Subst}\left (\int \frac {x^4 \left (10 b+9 a x^2\right )}{\sqrt [4]{b+a x^2} \left (-b-a x^2+x^{20}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^3}} \\ & = \frac {\left (2 \sqrt {x} \sqrt [4]{b+a x}\right ) \text {Subst}\left (\int \left (-\frac {10 b x^4}{\sqrt [4]{b+a x^2} \left (b+a x^2-x^{20}\right )}-\frac {9 a x^6}{\sqrt [4]{b+a x^2} \left (b+a x^2-x^{20}\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^3}} \\ & = -\frac {\left (18 a \sqrt {x} \sqrt [4]{b+a x}\right ) \text {Subst}\left (\int \frac {x^6}{\sqrt [4]{b+a x^2} \left (b+a x^2-x^{20}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^3}}-\frac {\left (20 b \sqrt {x} \sqrt [4]{b+a x}\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt [4]{b+a x^2} \left (b+a x^2-x^{20}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{b x^2+a x^3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 11.11 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00 \[ \int \frac {x^2 (10 b+9 a x)}{\sqrt [4]{b x^2+a x^3} \left (-b-a x+x^{10}\right )} \, dx=2 \arctan \left (\frac {\sqrt [4]{b x^2+a x^3}}{x^3}\right )-2 \text {arctanh}\left (\frac {x^3}{\sqrt [4]{b x^2+a x^3}}\right ) \]

[In]

Integrate[(x^2*(10*b + 9*a*x))/((b*x^2 + a*x^3)^(1/4)*(-b - a*x + x^10)),x]

[Out]

2*ArcTan[(b*x^2 + a*x^3)^(1/4)/x^3] - 2*ArcTanh[x^3/(b*x^2 + a*x^3)^(1/4)]

Maple [F]

\[\int \frac {x^{2} \left (9 a x +10 b \right )}{\left (a \,x^{3}+b \,x^{2}\right )^{\frac {1}{4}} \left (x^{10}-a x -b \right )}d x\]

[In]

int(x^2*(9*a*x+10*b)/(a*x^3+b*x^2)^(1/4)/(x^10-a*x-b),x)

[Out]

int(x^2*(9*a*x+10*b)/(a*x^3+b*x^2)^(1/4)/(x^10-a*x-b),x)

Fricas [F(-1)]

Timed out. \[ \int \frac {x^2 (10 b+9 a x)}{\sqrt [4]{b x^2+a x^3} \left (-b-a x+x^{10}\right )} \, dx=\text {Timed out} \]

[In]

integrate(x^2*(9*a*x+10*b)/(a*x^3+b*x^2)^(1/4)/(x^10-a*x-b),x, algorithm="fricas")

[Out]

Timed out

Sympy [F]

\[ \int \frac {x^2 (10 b+9 a x)}{\sqrt [4]{b x^2+a x^3} \left (-b-a x+x^{10}\right )} \, dx=\int \frac {x^{2} \cdot \left (9 a x + 10 b\right )}{\sqrt [4]{x^{2} \left (a x + b\right )} \left (- a x - b + x^{10}\right )}\, dx \]

[In]

integrate(x**2*(9*a*x+10*b)/(a*x**3+b*x**2)**(1/4)/(x**10-a*x-b),x)

[Out]

Integral(x**2*(9*a*x + 10*b)/((x**2*(a*x + b))**(1/4)*(-a*x - b + x**10)), x)

Maxima [F]

\[ \int \frac {x^2 (10 b+9 a x)}{\sqrt [4]{b x^2+a x^3} \left (-b-a x+x^{10}\right )} \, dx=\int { \frac {{\left (9 \, a x + 10 \, b\right )} x^{2}}{{\left (x^{10} - a x - b\right )} {\left (a x^{3} + b x^{2}\right )}^{\frac {1}{4}}} \,d x } \]

[In]

integrate(x^2*(9*a*x+10*b)/(a*x^3+b*x^2)^(1/4)/(x^10-a*x-b),x, algorithm="maxima")

[Out]

integrate((9*a*x + 10*b)*x^2/((x^10 - a*x - b)*(a*x^3 + b*x^2)^(1/4)), x)

Giac [F]

\[ \int \frac {x^2 (10 b+9 a x)}{\sqrt [4]{b x^2+a x^3} \left (-b-a x+x^{10}\right )} \, dx=\int { \frac {{\left (9 \, a x + 10 \, b\right )} x^{2}}{{\left (x^{10} - a x - b\right )} {\left (a x^{3} + b x^{2}\right )}^{\frac {1}{4}}} \,d x } \]

[In]

integrate(x^2*(9*a*x+10*b)/(a*x^3+b*x^2)^(1/4)/(x^10-a*x-b),x, algorithm="giac")

[Out]

integrate((9*a*x + 10*b)*x^2/((x^10 - a*x - b)*(a*x^3 + b*x^2)^(1/4)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 (10 b+9 a x)}{\sqrt [4]{b x^2+a x^3} \left (-b-a x+x^{10}\right )} \, dx=\int -\frac {x^2\,\left (10\,b+9\,a\,x\right )}{{\left (a\,x^3+b\,x^2\right )}^{1/4}\,\left (-x^{10}+a\,x+b\right )} \,d x \]

[In]

int(-(x^2*(10*b + 9*a*x))/((a*x^3 + b*x^2)^(1/4)*(b + a*x - x^10)),x)

[Out]

int(-(x^2*(10*b + 9*a*x))/((a*x^3 + b*x^2)^(1/4)*(b + a*x - x^10)), x)