\(\int \frac {x^4 (-9+5 x^4)}{\sqrt {-x+x^5} (1-x^4+a x^9)} \, dx\) [340]

Optimal result

Integrand size = 36, antiderivative size = 28 \[ \int \frac {x^4 \left (-9+5 x^4\right )}{\sqrt {-x+x^5} \left (1-x^4+a x^9\right )} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {-x+x^5}}{\sqrt {a} x^5}\right )}{\sqrt {a}} \]

[Out]

-2*arctanh((x^5-x)^(1/2)/a^(1/2)/x^5)/a^(1/2)

Rubi [F]

\[ \int \frac {x^4 \left (-9+5 x^4\right )}{\sqrt {-x+x^5} \left (1-x^4+a x^9\right )} \, dx=\int \frac {x^4 \left (-9+5 x^4\right )}{\sqrt {-x+x^5} \left (1-x^4+a x^9\right )} \, dx \]

[In]

Int[(x^4*(-9 + 5*x^4))/(Sqrt[-x + x^5]*(1 - x^4 + a*x^9)),x]

[Out]

(-18*Sqrt[x]*Sqrt[-1 + x^4]*Defer[Subst][Defer[Int][x^8/(Sqrt[-1 + x^8]*(1 - x^8 + a*x^18)), x], x, Sqrt[x]])/
Sqrt[-x + x^5] + (10*Sqrt[x]*Sqrt[-1 + x^4]*Defer[Subst][Defer[Int][x^16/(Sqrt[-1 + x^8]*(1 - x^8 + a*x^18)),
x], x, Sqrt[x]])/Sqrt[-x + x^5]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {x} \sqrt {-1+x^4}\right ) \int \frac {x^{7/2} \left (-9+5 x^4\right )}{\sqrt {-1+x^4} \left (1-x^4+a x^9\right )} \, dx}{\sqrt {-x+x^5}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {-1+x^4}\right ) \text {Subst}\left (\int \frac {x^8 \left (-9+5 x^8\right )}{\sqrt {-1+x^8} \left (1-x^8+a x^{18}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^5}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {-1+x^4}\right ) \text {Subst}\left (\int \left (-\frac {9 x^8}{\sqrt {-1+x^8} \left (1-x^8+a x^{18}\right )}+\frac {5 x^{16}}{\sqrt {-1+x^8} \left (1-x^8+a x^{18}\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^5}} \\ & = \frac {\left (10 \sqrt {x} \sqrt {-1+x^4}\right ) \text {Subst}\left (\int \frac {x^{16}}{\sqrt {-1+x^8} \left (1-x^8+a x^{18}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^5}}-\frac {\left (18 \sqrt {x} \sqrt {-1+x^4}\right ) \text {Subst}\left (\int \frac {x^8}{\sqrt {-1+x^8} \left (1-x^8+a x^{18}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^5}} \\ \end{align*}

Mathematica [F]

\[ \int \frac {x^4 \left (-9+5 x^4\right )}{\sqrt {-x+x^5} \left (1-x^4+a x^9\right )} \, dx=\int \frac {x^4 \left (-9+5 x^4\right )}{\sqrt {-x+x^5} \left (1-x^4+a x^9\right )} \, dx \]

[In]

Integrate[(x^4*(-9 + 5*x^4))/(Sqrt[-x + x^5]*(1 - x^4 + a*x^9)),x]

[Out]

Integrate[(x^4*(-9 + 5*x^4))/(Sqrt[-x + x^5]*(1 - x^4 + a*x^9)), x]

Maple [F]

\[\int \frac {x^{4} \left (5 x^{4}-9\right )}{\sqrt {x^{5}-x}\, \left (a \,x^{9}-x^{4}+1\right )}d x\]

[In]

int(x^4*(5*x^4-9)/(x^5-x)^(1/2)/(a*x^9-x^4+1),x)

[Out]

int(x^4*(5*x^4-9)/(x^5-x)^(1/2)/(a*x^9-x^4+1),x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (22) = 44\).

Time = 0.36 (sec) , antiderivative size = 146, normalized size of antiderivative = 5.21 \[ \int \frac {x^4 \left (-9+5 x^4\right )}{\sqrt {-x+x^5} \left (1-x^4+a x^9\right )} \, dx=\left [\frac {\log \left (\frac {a^{2} x^{18} + 6 \, a x^{13} - 6 \, a x^{9} + x^{8} - 2 \, x^{4} - 4 \, {\left (a x^{13} + x^{8} - x^{4}\right )} \sqrt {x^{5} - x} \sqrt {a} + 1}{a^{2} x^{18} - 2 \, a x^{13} + 2 \, a x^{9} + x^{8} - 2 \, x^{4} + 1}\right )}{2 \, \sqrt {a}}, \frac {\sqrt {-a} \arctan \left (\frac {{\left (a x^{9} + x^{4} - 1\right )} \sqrt {x^{5} - x} \sqrt {-a}}{2 \, {\left (a x^{9} - a x^{5}\right )}}\right )}{a}\right ] \]

[In]

integrate(x^4*(5*x^4-9)/(x^5-x)^(1/2)/(a*x^9-x^4+1),x, algorithm="fricas")

[Out]

[1/2*log((a^2*x^18 + 6*a*x^13 - 6*a*x^9 + x^8 - 2*x^4 - 4*(a*x^13 + x^8 - x^4)*sqrt(x^5 - x)*sqrt(a) + 1)/(a^2
*x^18 - 2*a*x^13 + 2*a*x^9 + x^8 - 2*x^4 + 1))/sqrt(a), sqrt(-a)*arctan(1/2*(a*x^9 + x^4 - 1)*sqrt(x^5 - x)*sq
rt(-a)/(a*x^9 - a*x^5))/a]

Sympy [F]

\[ \int \frac {x^4 \left (-9+5 x^4\right )}{\sqrt {-x+x^5} \left (1-x^4+a x^9\right )} \, dx=\int \frac {x^{4} \cdot \left (5 x^{4} - 9\right )}{\sqrt {x \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )} \left (a x^{9} - x^{4} + 1\right )}\, dx \]

[In]

integrate(x**4*(5*x**4-9)/(x**5-x)**(1/2)/(a*x**9-x**4+1),x)

[Out]

Integral(x**4*(5*x**4 - 9)/(sqrt(x*(x - 1)*(x + 1)*(x**2 + 1))*(a*x**9 - x**4 + 1)), x)

Maxima [F]

\[ \int \frac {x^4 \left (-9+5 x^4\right )}{\sqrt {-x+x^5} \left (1-x^4+a x^9\right )} \, dx=\int { \frac {{\left (5 \, x^{4} - 9\right )} x^{4}}{{\left (a x^{9} - x^{4} + 1\right )} \sqrt {x^{5} - x}} \,d x } \]

[In]

integrate(x^4*(5*x^4-9)/(x^5-x)^(1/2)/(a*x^9-x^4+1),x, algorithm="maxima")

[Out]

integrate((5*x^4 - 9)*x^4/((a*x^9 - x^4 + 1)*sqrt(x^5 - x)), x)

Giac [F]

\[ \int \frac {x^4 \left (-9+5 x^4\right )}{\sqrt {-x+x^5} \left (1-x^4+a x^9\right )} \, dx=\int { \frac {{\left (5 \, x^{4} - 9\right )} x^{4}}{{\left (a x^{9} - x^{4} + 1\right )} \sqrt {x^{5} - x}} \,d x } \]

[In]

integrate(x^4*(5*x^4-9)/(x^5-x)^(1/2)/(a*x^9-x^4+1),x, algorithm="giac")

[Out]

integrate((5*x^4 - 9)*x^4/((a*x^9 - x^4 + 1)*sqrt(x^5 - x)), x)

Mupad [B] (verification not implemented)

Time = 6.07 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.71 \[ \int \frac {x^4 \left (-9+5 x^4\right )}{\sqrt {-x+x^5} \left (1-x^4+a x^9\right )} \, dx=\frac {\ln \left (\frac {a\,x^9+x^4-2\,\sqrt {a}\,x^4\,\sqrt {x\,\left (x^4-1\right )}-1}{4\,a\,x^9-4\,x^4+4}\right )}{\sqrt {a}} \]

[In]

int((x^4*(5*x^4 - 9))/((x^5 - x)^(1/2)*(a*x^9 - x^4 + 1)),x)

[Out]

log((a*x^9 + x^4 - 2*a^(1/2)*x^4*(x*(x^4 - 1))^(1/2) - 1)/(4*a*x^9 - 4*x^4 + 4))/a^(1/2)