\(\int \frac {-2 b+3 a x^5}{\sqrt {b+a x^5} (b+x^2+a x^5)} \, dx\) [99]

Optimal result

Integrand size = 34, antiderivative size = 16 \[ \int \frac {-2 b+3 a x^5}{\sqrt {b+a x^5} \left (b+x^2+a x^5\right )} \, dx=-2 \arctan \left (\frac {x}{\sqrt {b+a x^5}}\right ) \]

[Out]

-2*arctan(x/(a*x^5+b)^(1/2))

Rubi [F]

\[ \int \frac {-2 b+3 a x^5}{\sqrt {b+a x^5} \left (b+x^2+a x^5\right )} \, dx=\int \frac {-2 b+3 a x^5}{\sqrt {b+a x^5} \left (b+x^2+a x^5\right )} \, dx \]

[In]

Int[(-2*b + 3*a*x^5)/(Sqrt[b + a*x^5]*(b + x^2 + a*x^5)),x]

[Out]

(3*x*Sqrt[1 + (a*x^5)/b]*Hypergeometric2F1[1/5, 1/2, 6/5, -((a*x^5)/b)])/Sqrt[b + a*x^5] - 5*b*Defer[Int][1/(S
qrt[b + a*x^5]*(b + x^2 + a*x^5)), x] - 3*Defer[Int][x^2/(Sqrt[b + a*x^5]*(b + x^2 + a*x^5)), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {3}{\sqrt {b+a x^5}}-\frac {5 b+3 x^2}{\sqrt {b+a x^5} \left (b+x^2+a x^5\right )}\right ) \, dx \\ & = 3 \int \frac {1}{\sqrt {b+a x^5}} \, dx-\int \frac {5 b+3 x^2}{\sqrt {b+a x^5} \left (b+x^2+a x^5\right )} \, dx \\ & = \frac {\left (3 \sqrt {1+\frac {a x^5}{b}}\right ) \int \frac {1}{\sqrt {1+\frac {a x^5}{b}}} \, dx}{\sqrt {b+a x^5}}-\int \left (\frac {5 b}{\sqrt {b+a x^5} \left (b+x^2+a x^5\right )}+\frac {3 x^2}{\sqrt {b+a x^5} \left (b+x^2+a x^5\right )}\right ) \, dx \\ & = \frac {3 x \sqrt {1+\frac {a x^5}{b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{5},\frac {1}{2},\frac {6}{5},-\frac {a x^5}{b}\right )}{\sqrt {b+a x^5}}-3 \int \frac {x^2}{\sqrt {b+a x^5} \left (b+x^2+a x^5\right )} \, dx-(5 b) \int \frac {1}{\sqrt {b+a x^5} \left (b+x^2+a x^5\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 2.75 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {-2 b+3 a x^5}{\sqrt {b+a x^5} \left (b+x^2+a x^5\right )} \, dx=-2 \arctan \left (\frac {x}{\sqrt {b+a x^5}}\right ) \]

[In]

Integrate[(-2*b + 3*a*x^5)/(Sqrt[b + a*x^5]*(b + x^2 + a*x^5)),x]

[Out]

-2*ArcTan[x/Sqrt[b + a*x^5]]

Maple [A] (verified)

Time = 1.72 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06

method	result	size
pseudoelliptic	\(2 \arctan \left (\frac {\sqrt {a \,x^{5}+b}}{x}\right )\)	\(17\)

[In]

int((3*a*x^5-2*b)/(a*x^5+b)^(1/2)/(a*x^5+x^2+b),x,method=_RETURNVERBOSE)

[Out]

2*arctan(1/x*(a*x^5+b)^(1/2))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (14) = 28\).

Time = 0.29 (sec) , antiderivative size = 35, normalized size of antiderivative = 2.19 \[ \int \frac {-2 b+3 a x^5}{\sqrt {b+a x^5} \left (b+x^2+a x^5\right )} \, dx=\arctan \left (\frac {{\left (a x^{5} - x^{2} + b\right )} \sqrt {a x^{5} + b}}{2 \, {\left (a x^{6} + b x\right )}}\right ) \]

[In]

integrate((3*a*x^5-2*b)/(a*x^5+b)^(1/2)/(a*x^5+x^2+b),x, algorithm="fricas")

[Out]

arctan(1/2*(a*x^5 - x^2 + b)*sqrt(a*x^5 + b)/(a*x^6 + b*x))

Sympy [F]

\[ \int \frac {-2 b+3 a x^5}{\sqrt {b+a x^5} \left (b+x^2+a x^5\right )} \, dx=\int \frac {3 a x^{5} - 2 b}{\sqrt {a x^{5} + b} \left (a x^{5} + b + x^{2}\right )}\, dx \]

[In]

integrate((3*a*x**5-2*b)/(a*x**5+b)**(1/2)/(a*x**5+x**2+b),x)

[Out]

Integral((3*a*x**5 - 2*b)/(sqrt(a*x**5 + b)*(a*x**5 + b + x**2)), x)

Maxima [F]

\[ \int \frac {-2 b+3 a x^5}{\sqrt {b+a x^5} \left (b+x^2+a x^5\right )} \, dx=\int { \frac {3 \, a x^{5} - 2 \, b}{{\left (a x^{5} + x^{2} + b\right )} \sqrt {a x^{5} + b}} \,d x } \]

[In]

integrate((3*a*x^5-2*b)/(a*x^5+b)^(1/2)/(a*x^5+x^2+b),x, algorithm="maxima")

[Out]

integrate((3*a*x^5 - 2*b)/((a*x^5 + x^2 + b)*sqrt(a*x^5 + b)), x)

Giac [F]

\[ \int \frac {-2 b+3 a x^5}{\sqrt {b+a x^5} \left (b+x^2+a x^5\right )} \, dx=\int { \frac {3 \, a x^{5} - 2 \, b}{{\left (a x^{5} + x^{2} + b\right )} \sqrt {a x^{5} + b}} \,d x } \]

[In]

integrate((3*a*x^5-2*b)/(a*x^5+b)^(1/2)/(a*x^5+x^2+b),x, algorithm="giac")

[Out]

integrate((3*a*x^5 - 2*b)/((a*x^5 + x^2 + b)*sqrt(a*x^5 + b)), x)

Mupad [B] (verification not implemented)

Time = 7.13 (sec) , antiderivative size = 42, normalized size of antiderivative = 2.62 \[ \int \frac {-2 b+3 a x^5}{\sqrt {b+a x^5} \left (b+x^2+a x^5\right )} \, dx=\ln \left (\frac {b+a\,x^5-x^2+x\,\sqrt {a\,x^5+b}\,2{}\mathrm {i}}{a\,x^5+x^2+b}\right )\,1{}\mathrm {i} \]

[In]

int(-(2*b - 3*a*x^5)/((b + a*x^5)^(1/2)*(b + a*x^5 + x^2)),x)

[Out]

log((b + x*(b + a*x^5)^(1/2)*2i + a*x^5 - x^2)/(b + a*x^5 + x^2))*1i