\(\int \frac {x^4 (9+4 x^5)}{\sqrt {x+x^6} (-1-x^5+a x^9)} \, dx\) [297]

Optimal result

Integrand size = 34, antiderivative size = 26 \[ \int \frac {x^4 \left (9+4 x^5\right )}{\sqrt {x+x^6} \left (-1-x^5+a x^9\right )} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {x+x^6}}{\sqrt {a} x^5}\right )}{\sqrt {a}} \]

[Out]

-2*arctanh((x^6+x)^(1/2)/a^(1/2)/x^5)/a^(1/2)

Rubi [F]

\[ \int \frac {x^4 \left (9+4 x^5\right )}{\sqrt {x+x^6} \left (-1-x^5+a x^9\right )} \, dx=\int \frac {x^4 \left (9+4 x^5\right )}{\sqrt {x+x^6} \left (-1-x^5+a x^9\right )} \, dx \]

[In]

Int[(x^4*(9 + 4*x^5))/(Sqrt[x + x^6]*(-1 - x^5 + a*x^9)),x]

[Out]

(8*x*Sqrt[1 + x^5]*Hypergeometric2F1[1/10, 1/2, 11/10, -x^5])/(a*Sqrt[x + x^6]) + (8*Sqrt[x]*Sqrt[1 + x^5]*Def
er[Subst][Defer[Int][1/(Sqrt[1 + x^10]*(-1 - x^10 + a*x^18)), x], x, Sqrt[x]])/(a*Sqrt[x + x^6]) + (18*Sqrt[x]
*Sqrt[1 + x^5]*Defer[Subst][Defer[Int][x^8/(Sqrt[1 + x^10]*(-1 - x^10 + a*x^18)), x], x, Sqrt[x]])/Sqrt[x + x^
6] + (8*Sqrt[x]*Sqrt[1 + x^5]*Defer[Subst][Defer[Int][x^10/(Sqrt[1 + x^10]*(-1 - x^10 + a*x^18)), x], x, Sqrt[
x]])/(a*Sqrt[x + x^6])

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {x} \sqrt {1+x^5}\right ) \int \frac {x^{7/2} \left (9+4 x^5\right )}{\sqrt {1+x^5} \left (-1-x^5+a x^9\right )} \, dx}{\sqrt {x+x^6}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {1+x^5}\right ) \text {Subst}\left (\int \frac {x^8 \left (9+4 x^{10}\right )}{\sqrt {1+x^{10}} \left (-1-x^{10}+a x^{18}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^6}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {1+x^5}\right ) \text {Subst}\left (\int \left (\frac {4}{a \sqrt {1+x^{10}}}+\frac {4+9 a x^8+4 x^{10}}{a \sqrt {1+x^{10}} \left (-1-x^{10}+a x^{18}\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^6}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {1+x^5}\right ) \text {Subst}\left (\int \frac {4+9 a x^8+4 x^{10}}{\sqrt {1+x^{10}} \left (-1-x^{10}+a x^{18}\right )} \, dx,x,\sqrt {x}\right )}{a \sqrt {x+x^6}}+\frac {\left (8 \sqrt {x} \sqrt {1+x^5}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+x^{10}}} \, dx,x,\sqrt {x}\right )}{a \sqrt {x+x^6}} \\ & = \frac {8 x \sqrt {1+x^5} \operatorname {Hypergeometric2F1}\left (\frac {1}{10},\frac {1}{2},\frac {11}{10},-x^5\right )}{a \sqrt {x+x^6}}+\frac {\left (2 \sqrt {x} \sqrt {1+x^5}\right ) \text {Subst}\left (\int \left (\frac {4}{\sqrt {1+x^{10}} \left (-1-x^{10}+a x^{18}\right )}+\frac {9 a x^8}{\sqrt {1+x^{10}} \left (-1-x^{10}+a x^{18}\right )}+\frac {4 x^{10}}{\sqrt {1+x^{10}} \left (-1-x^{10}+a x^{18}\right )}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {x+x^6}} \\ & = \frac {8 x \sqrt {1+x^5} \operatorname {Hypergeometric2F1}\left (\frac {1}{10},\frac {1}{2},\frac {11}{10},-x^5\right )}{a \sqrt {x+x^6}}+\frac {\left (18 \sqrt {x} \sqrt {1+x^5}\right ) \text {Subst}\left (\int \frac {x^8}{\sqrt {1+x^{10}} \left (-1-x^{10}+a x^{18}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^6}}+\frac {\left (8 \sqrt {x} \sqrt {1+x^5}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+x^{10}} \left (-1-x^{10}+a x^{18}\right )} \, dx,x,\sqrt {x}\right )}{a \sqrt {x+x^6}}+\frac {\left (8 \sqrt {x} \sqrt {1+x^5}\right ) \text {Subst}\left (\int \frac {x^{10}}{\sqrt {1+x^{10}} \left (-1-x^{10}+a x^{18}\right )} \, dx,x,\sqrt {x}\right )}{a \sqrt {x+x^6}} \\ \end{align*}

Mathematica [F]

\[ \int \frac {x^4 \left (9+4 x^5\right )}{\sqrt {x+x^6} \left (-1-x^5+a x^9\right )} \, dx=\int \frac {x^4 \left (9+4 x^5\right )}{\sqrt {x+x^6} \left (-1-x^5+a x^9\right )} \, dx \]

[In]

Integrate[(x^4*(9 + 4*x^5))/(Sqrt[x + x^6]*(-1 - x^5 + a*x^9)),x]

[Out]

Integrate[(x^4*(9 + 4*x^5))/(Sqrt[x + x^6]*(-1 - x^5 + a*x^9)), x]

Maple [F]

\[\int \frac {x^{4} \left (4 x^{5}+9\right )}{\sqrt {x^{6}+x}\, \left (a \,x^{9}-x^{5}-1\right )}d x\]

[In]

int(x^4*(4*x^5+9)/(x^6+x)^(1/2)/(a*x^9-x^5-1),x)

[Out]

int(x^4*(4*x^5+9)/(x^6+x)^(1/2)/(a*x^9-x^5-1),x)

Fricas [A] (verification not implemented)

none

Time = 0.65 (sec) , antiderivative size = 132, normalized size of antiderivative = 5.08 \[ \int \frac {x^4 \left (9+4 x^5\right )}{\sqrt {x+x^6} \left (-1-x^5+a x^9\right )} \, dx=\left [\frac {\log \left (-\frac {a^{2} x^{18} + 6 \, a x^{14} + 6 \, a x^{9} + x^{10} + 2 \, x^{5} - 4 \, {\left (a x^{13} + x^{9} + x^{4}\right )} \sqrt {x^{6} + x} \sqrt {a} + 1}{a^{2} x^{18} - 2 \, a x^{14} - 2 \, a x^{9} + x^{10} + 2 \, x^{5} + 1}\right )}{2 \, \sqrt {a}}, \frac {\sqrt {-a} \arctan \left (\frac {2 \, \sqrt {x^{6} + x} \sqrt {-a} x^{4}}{a x^{9} + x^{5} + 1}\right )}{a}\right ] \]

[In]

integrate(x^4*(4*x^5+9)/(x^6+x)^(1/2)/(a*x^9-x^5-1),x, algorithm="fricas")

[Out]

[1/2*log(-(a^2*x^18 + 6*a*x^14 + 6*a*x^9 + x^10 + 2*x^5 - 4*(a*x^13 + x^9 + x^4)*sqrt(x^6 + x)*sqrt(a) + 1)/(a
^2*x^18 - 2*a*x^14 - 2*a*x^9 + x^10 + 2*x^5 + 1))/sqrt(a), sqrt(-a)*arctan(2*sqrt(x^6 + x)*sqrt(-a)*x^4/(a*x^9
+ x^5 + 1))/a]

Sympy [F]

\[ \int \frac {x^4 \left (9+4 x^5\right )}{\sqrt {x+x^6} \left (-1-x^5+a x^9\right )} \, dx=\int \frac {x^{4} \cdot \left (4 x^{5} + 9\right )}{\sqrt {x \left (x + 1\right ) \left (x^{4} - x^{3} + x^{2} - x + 1\right )} \left (a x^{9} - x^{5} - 1\right )}\, dx \]

[In]

integrate(x**4*(4*x**5+9)/(x**6+x)**(1/2)/(a*x**9-x**5-1),x)

[Out]

Integral(x**4*(4*x**5 + 9)/(sqrt(x*(x + 1)*(x**4 - x**3 + x**2 - x + 1))*(a*x**9 - x**5 - 1)), x)

Maxima [F]

\[ \int \frac {x^4 \left (9+4 x^5\right )}{\sqrt {x+x^6} \left (-1-x^5+a x^9\right )} \, dx=\int { \frac {{\left (4 \, x^{5} + 9\right )} x^{4}}{{\left (a x^{9} - x^{5} - 1\right )} \sqrt {x^{6} + x}} \,d x } \]

[In]

integrate(x^4*(4*x^5+9)/(x^6+x)^(1/2)/(a*x^9-x^5-1),x, algorithm="maxima")

[Out]

integrate((4*x^5 + 9)*x^4/((a*x^9 - x^5 - 1)*sqrt(x^6 + x)), x)

Giac [F]

\[ \int \frac {x^4 \left (9+4 x^5\right )}{\sqrt {x+x^6} \left (-1-x^5+a x^9\right )} \, dx=\int { \frac {{\left (4 \, x^{5} + 9\right )} x^{4}}{{\left (a x^{9} - x^{5} - 1\right )} \sqrt {x^{6} + x}} \,d x } \]

[In]

integrate(x^4*(4*x^5+9)/(x^6+x)^(1/2)/(a*x^9-x^5-1),x, algorithm="giac")

[Out]

integrate((4*x^5 + 9)*x^4/((a*x^9 - x^5 - 1)*sqrt(x^6 + x)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 \left (9+4 x^5\right )}{\sqrt {x+x^6} \left (-1-x^5+a x^9\right )} \, dx=\int -\frac {x^4\,\left (4\,x^5+9\right )}{\sqrt {x^6+x}\,\left (-a\,x^9+x^5+1\right )} \,d x \]

[In]

int(-(x^4*(4*x^5 + 9))/((x + x^6)^(1/2)*(x^5 - a*x^9 + 1)),x)

[Out]

int(-(x^4*(4*x^5 + 9))/((x + x^6)^(1/2)*(x^5 - a*x^9 + 1)), x)