\(\int \frac {3 b+a x^2}{(b+a x^2+x^3) \sqrt [4]{b x+a x^3}} \, dx\) [1441]

Optimal result

Integrand size = 35, antiderivative size = 102 \[ \int \frac {3 b+a x^2}{\left (b+a x^2+x^3\right ) \sqrt [4]{b x+a x^3}} \, dx=-\sqrt {2} \arctan \left (\frac {-\frac {x^2}{\sqrt {2}}+\frac {\sqrt {b x+a x^3}}{\sqrt {2}}}{x \sqrt [4]{b x+a x^3}}\right )+\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{b x+a x^3}}{x^2+\sqrt {b x+a x^3}}\right ) \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

(12*b*x^(1/4)*(b + a*x^2)^(1/4)*Defer[Subst][Defer[Int][x^2/((b + a*x^8)^(1/4)*(b + a*x^8 + x^12)), x], x, x^(
1/4)])/(b*x + a*x^3)^(1/4) + (4*a*x^(1/4)*(b + a*x^2)^(1/4)*Defer[Subst][Defer[Int][x^10/((b + a*x^8)^(1/4)*(b
+ a*x^8 + x^12)), x], x, x^(1/4)])/(b*x + a*x^3)^(1/4)

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt [4]{x} \sqrt [4]{b+a x^2}\right ) \int \frac {3 b+a x^2}{\sqrt [4]{x} \sqrt [4]{b+a x^2} \left (b+a x^2+x^3\right )} \, dx}{\sqrt [4]{b x+a x^3}} \\ & = \frac {\left (4 \sqrt [4]{x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {x^2 \left (3 b+a x^8\right )}{\sqrt [4]{b+a x^8} \left (b+a x^8+x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x+a x^3}} \\ & = \frac {\left (4 \sqrt [4]{x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \left (\frac {3 b x^2}{\sqrt [4]{b+a x^8} \left (b+a x^8+x^{12}\right )}+\frac {a x^{10}}{\sqrt [4]{b+a x^8} \left (b+a x^8+x^{12}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x+a x^3}} \\ & = \frac {\left (4 a \sqrt [4]{x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {x^{10}}{\sqrt [4]{b+a x^8} \left (b+a x^8+x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x+a x^3}}+\frac {\left (12 b \sqrt [4]{x} \sqrt [4]{b+a x^2}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt [4]{b+a x^8} \left (b+a x^8+x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b x+a x^3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 10.66 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.89 \[ \int \frac {3 b+a x^2}{\left (b+a x^2+x^3\right ) \sqrt [4]{b x+a x^3}} \, dx=\sqrt {2} \left (-\arctan \left (\frac {-x^2+\sqrt {x \left (b+a x^2\right )}}{\sqrt {2} x \sqrt [4]{x \left (b+a x^2\right )}}\right )+\text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{x \left (b+a x^2\right )}}{x^2+\sqrt {x \left (b+a x^2\right )}}\right )\right ) \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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