\(\int \frac {-2 b+a x^3}{\sqrt {b+a x^3} (b+c x^2+a x^3)} \, dx\) [277]

Optimal result

Integrand size = 35, antiderivative size = 26 \[ \int \frac {-2 b+a x^3}{\sqrt {b+a x^3} \left (b+c x^2+a x^3\right )} \, dx=-\frac {2 \arctan \left (\frac {\sqrt {c} x}{\sqrt {b+a x^3}}\right )}{\sqrt {c}} \]

[Out]

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

Rubi [F]

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

[In]

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

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{\sqrt {b+a x^3}}-\frac {3 b+c x^2}{\sqrt {b+a x^3} \left (b+c x^2+a x^3\right )}\right ) \, dx \\ & = \int \frac {1}{\sqrt {b+a x^3}} \, dx-\int \frac {3 b+c x^2}{\sqrt {b+a x^3} \left (b+c 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}{\sqrt {b+a x^3} \left (b+c x^2+a x^3\right )}+\frac {c x^2}{\sqrt {b+a x^3} \left (b+c 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}{\sqrt {b+a x^3} \left (b+c x^2+a x^3\right )} \, dx-c \int \frac {x^2}{\sqrt {b+a x^3} \left (b+c x^2+a x^3\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.80 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {-2 b+a x^3}{\sqrt {b+a x^3} \left (b+c x^2+a x^3\right )} \, dx=-\frac {2 \arctan \left (\frac {\sqrt {c} x}{\sqrt {b+a x^3}}\right )}{\sqrt {c}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 2.63 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88

method	result	size
default	\(\frac {2 \arctan \left (\frac {\sqrt {a \,x^{3}+b}}{\sqrt {c}\, x}\right )}{\sqrt {c}}\)	\(23\)
pseudoelliptic	\(\frac {2 \arctan \left (\frac {\sqrt {a \,x^{3}+b}}{\sqrt {c}\, x}\right )}{\sqrt {c}}\)	\(23\)
elliptic	\(\text {Expression too large to display}\)	\(841\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 45 vs. \(2 (20) = 40\).

Time = 0.28 (sec) , antiderivative size = 169, normalized size of antiderivative = 6.50 \[ \int \frac {-2 b+a x^3}{\sqrt {b+a x^3} \left (b+c x^2+a x^3\right )} \, dx=\left [-\frac {\sqrt {-c} \log \left (\frac {a^{2} x^{6} - 6 \, a c x^{5} + c^{2} x^{4} + 2 \, a b x^{3} - 6 \, b c x^{2} + b^{2} - 4 \, {\left (a x^{4} - c x^{3} + b x\right )} \sqrt {a x^{3} + b} \sqrt {-c}}{a^{2} x^{6} + 2 \, a c x^{5} + c^{2} x^{4} + 2 \, a b x^{3} + 2 \, b c x^{2} + b^{2}}\right )}{2 \, c}, \frac {\arctan \left (\frac {{\left (a x^{3} - c x^{2} + b\right )} \sqrt {a x^{3} + b} \sqrt {c}}{2 \, {\left (a c x^{4} + b c x\right )}}\right )}{\sqrt {c}}\right ] \]

[In]

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

[Out]

[-1/2*sqrt(-c)*log((a^2*x^6 - 6*a*c*x^5 + c^2*x^4 + 2*a*b*x^3 - 6*b*c*x^2 + b^2 - 4*(a*x^4 - c*x^3 + b*x)*sqrt
(a*x^3 + b)*sqrt(-c))/(a^2*x^6 + 2*a*c*x^5 + c^2*x^4 + 2*a*b*x^3 + 2*b*c*x^2 + b^2))/c, arctan(1/2*(a*x^3 - c*
x^2 + b)*sqrt(a*x^3 + b)*sqrt(c)/(a*c*x^4 + b*c*x))/sqrt(c)]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 8.62 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.96 \[ \int \frac {-2 b+a x^3}{\sqrt {b+a x^3} \left (b+c x^2+a x^3\right )} \, dx=\frac {\ln \left (\frac {b+a\,x^3-c\,x^2+\sqrt {c}\,x\,\sqrt {a\,x^3+b}\,2{}\mathrm {i}}{a\,x^3+c\,x^2+b}\right )\,1{}\mathrm {i}}{\sqrt {c}} \]

[In]

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

[Out]

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