\(\int \frac {2 b+a x^3}{\sqrt {-b+a x^3} (-b+x^2+a x^3)} \, dx\) [318]

Optimal result

Integrand size = 37, antiderivative size = 28 \[ \int \frac {2 b+a x^3}{\sqrt {-b+a x^3} \left (-b+x^2+a x^3\right )} \, dx=2 \arctan \left (\frac {x \sqrt {-b+a x^3}}{b-a x^3}\right ) \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

(-2*Sqrt[2 - Sqrt[3]]*(b^(1/3) - a^(1/3)*x)*Sqrt[(b^(2/3) + a^(1/3)*b^(1/3)*x + a^(2/3)*x^2)/((1 - Sqrt[3])*b^
(1/3) - a^(1/3)*x)^2]*EllipticF[ArcSin[((1 + Sqrt[3])*b^(1/3) - a^(1/3)*x)/((1 - Sqrt[3])*b^(1/3) - a^(1/3)*x)
], -7 + 4*Sqrt[3]])/(3^(1/4)*a^(1/3)*Sqrt[-((b^(1/3)*(b^(1/3) - a^(1/3)*x))/((1 - Sqrt[3])*b^(1/3) - a^(1/3)*x
)^2)]*Sqrt[-b + a*x^3]) - 3*b*Defer[Int][1/((b - x^2 - a*x^3)*Sqrt[-b + a*x^3]), x] - Defer[Int][x^2/(Sqrt[-b
+ a*x^3]*(-b + x^2 + a*x^3)), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{\sqrt {-b+a x^3}}+\frac {3 b-x^2}{\sqrt {-b+a x^3} \left (-b+x^2+a x^3\right )}\right ) \, dx \\ & = \int \frac {1}{\sqrt {-b+a x^3}} \, dx+\int \frac {3 b-x^2}{\sqrt {-b+a x^3} \left (-b+x^2+a x^3\right )} \, dx \\ & = -\frac {2 \sqrt {2-\sqrt {3}} \left (\sqrt [3]{b}-\sqrt [3]{a} x\right ) \sqrt {\frac {b^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{b}-\sqrt [3]{a} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b}-\sqrt [3]{a} x}{\left (1-\sqrt {3}\right ) \sqrt [3]{b}-\sqrt [3]{a} x}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt [3]{a} \sqrt {-\frac {\sqrt [3]{b} \left (\sqrt [3]{b}-\sqrt [3]{a} x\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{b}-\sqrt [3]{a} x\right )^2}} \sqrt {-b+a x^3}}+\int \left (-\frac {3 b}{\left (b-x^2-a x^3\right ) \sqrt {-b+a x^3}}-\frac {x^2}{\sqrt {-b+a x^3} \left (-b+x^2+a x^3\right )}\right ) \, dx \\ & = -\frac {2 \sqrt {2-\sqrt {3}} \left (\sqrt [3]{b}-\sqrt [3]{a} x\right ) \sqrt {\frac {b^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{b}-\sqrt [3]{a} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b}-\sqrt [3]{a} x}{\left (1-\sqrt {3}\right ) \sqrt [3]{b}-\sqrt [3]{a} x}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt [3]{a} \sqrt {-\frac {\sqrt [3]{b} \left (\sqrt [3]{b}-\sqrt [3]{a} x\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{b}-\sqrt [3]{a} x\right )^2}} \sqrt {-b+a x^3}}-(3 b) \int \frac {1}{\left (b-x^2-a x^3\right ) \sqrt {-b+a x^3}} \, dx-\int \frac {x^2}{\sqrt {-b+a x^3} \left (-b+x^2+a x^3\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.78 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {2 b+a x^3}{\sqrt {-b+a x^3} \left (-b+x^2+a x^3\right )} \, dx=2 \arctan \left (\frac {x \sqrt {-b+a x^3}}{b-a x^3}\right ) \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.90 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.68

method	result	size
default	\(2 \arctan \left (\frac {\sqrt {a \,x^{3}-b}}{x}\right )\)	\(19\)
pseudoelliptic	\(2 \arctan \left (\frac {\sqrt {a \,x^{3}-b}}{x}\right )\)	\(19\)
elliptic	\(\text {Expression too large to display}\)	\(786\)

[In]

int((a*x^3+2*b)/(a*x^3-b)^(1/2)/(a*x^3+x^2-b),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 40 vs. \(2 (17) = 34\).

Time = 0.25 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.43 \[ \int \frac {2 b+a x^3}{\sqrt {-b+a x^3} \left (-b+x^2+a x^3\right )} \, dx=\arctan \left (\frac {{\left (a x^{3} - x^{2} - b\right )} \sqrt {a x^{3} - b}}{2 \, {\left (a x^{4} - b x\right )}}\right ) \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 7.16 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.61 \[ \int \frac {2 b+a x^3}{\sqrt {-b+a x^3} \left (-b+x^2+a x^3\right )} \, dx=\ln \left (\frac {b-a\,x^3+x^2-x\,\sqrt {a\,x^3-b}\,2{}\mathrm {i}}{a\,x^3+x^2-b}\right )\,1{}\mathrm {i} \]

[In]

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

[Out]

log((b - a*x^3 + x^2 - x*(a*x^3 - b)^(1/2)*2i)/(a*x^3 - b + x^2))*1i