3.31.13 \(\int \frac {x^5 (-7 b+9 a x^2)}{\sqrt [4]{-b x^3+a x^5} (1-b x^7+a x^9)} \, dx\) [3013]

3.31.13.1 Optimal result

Integrand size = 45, antiderivative size = 413 \[ \int \frac {x^5 \left (-7 b+9 a x^2\right )}{\sqrt [4]{-b x^3+a x^5} \left (1-b x^7+a x^9\right )} \, dx=-\sqrt {2} \arctan \left (\frac {-2^{2/3} x \sqrt [4]{-b x^3+a x^5}+x^2 \sqrt [4]{-b x^3+a x^5}}{2 \sqrt [6]{2}-\sqrt {2} x+2^{2/3} x \sqrt [4]{-b x^3+a x^5}-x^2 \sqrt [4]{-b x^3+a x^5}}\right )+\sqrt {2} \arctan \left (\frac {-2^{2/3} x \sqrt [4]{-b x^3+a x^5}+x^2 \sqrt [4]{-b x^3+a x^5}}{-2 \sqrt [6]{2}+\sqrt {2} x+2^{2/3} x \sqrt [4]{-b x^3+a x^5}-x^2 \sqrt [4]{-b x^3+a x^5}}\right )-\sqrt {2} \text {arctanh}\left (\frac {-2 2^{5/6} x \sqrt [4]{-b x^3+a x^5}+4 \sqrt [6]{2} x^2 \sqrt [4]{-b x^3+a x^5}-\sqrt {2} x^3 \sqrt [4]{-b x^3+a x^5}}{-2 \sqrt [3]{2}+2\ 2^{2/3} x-x^2-2 \sqrt [3]{2} x^2 \sqrt {-b x^3+a x^5}+2\ 2^{2/3} x^3 \sqrt {-b x^3+a x^5}-x^4 \sqrt {-b x^3+a x^5}}\right ) \]

output

-2^(1/2)*arctan((-2^(2/3)*x*(a*x^5-b*x^3)^(1/4)+x^2*(a*x^5-b*x^3)^(1/4))/( 
2*2^(1/6)-x*2^(1/2)+2^(2/3)*x*(a*x^5-b*x^3)^(1/4)-x^2*(a*x^5-b*x^3)^(1/4)) 
)+2^(1/2)*arctan((-2^(2/3)*x*(a*x^5-b*x^3)^(1/4)+x^2*(a*x^5-b*x^3)^(1/4))/ 
(-2*2^(1/6)+x*2^(1/2)+2^(2/3)*x*(a*x^5-b*x^3)^(1/4)-x^2*(a*x^5-b*x^3)^(1/4 
)))-2^(1/2)*arctanh((-2*2^(5/6)*x*(a*x^5-b*x^3)^(1/4)+4*2^(1/6)*x^2*(a*x^5 
-b*x^3)^(1/4)-2^(1/2)*(a*x^5-b*x^3)^(1/4)*x^3)/(-2*2^(1/3)+2*2^(2/3)*x-x^2 
-2*2^(1/3)*x^2*(a*x^5-b*x^3)^(1/2)+2*2^(2/3)*x^3*(a*x^5-b*x^3)^(1/2)-x^4*( 
a*x^5-b*x^3)^(1/2)))
 

3.31.13.2 Mathematica [F]

\[ \int \frac {x^5 \left (-7 b+9 a x^2\right )}{\sqrt [4]{-b x^3+a x^5} \left (1-b x^7+a x^9\right )} \, dx=\int \frac {x^5 \left (-7 b+9 a x^2\right )}{\sqrt [4]{-b x^3+a x^5} \left (1-b x^7+a x^9\right )} \, dx \]

input

Integrate[(x^5*(-7*b + 9*a*x^2))/((-(b*x^3) + a*x^5)^(1/4)*(1 - b*x^7 + a* 
x^9)),x]
 

output

Integrate[(x^5*(-7*b + 9*a*x^2))/((-(b*x^3) + a*x^5)^(1/4)*(1 - b*x^7 + a* 
x^9)), x]
 

3.31.13.3 Rubi [F]

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {x^5 \left (9 a x^2-7 b\right )}{\sqrt [4]{a x^5-b x^3} \left (a x^9-b x^7+1\right )} \, dx\)

\(\Big \downarrow \) 2467

\(\displaystyle \frac {x^{3/4} \sqrt [4]{a x^2-b} \int -\frac {x^{17/4} \left (7 b-9 a x^2\right )}{\sqrt [4]{a x^2-b} \left (a x^9-b x^7+1\right )}dx}{\sqrt [4]{a x^5-b x^3}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {x^{3/4} \sqrt [4]{a x^2-b} \int \frac {x^{17/4} \left (7 b-9 a x^2\right )}{\sqrt [4]{a x^2-b} \left (a x^9-b x^7+1\right )}dx}{\sqrt [4]{a x^5-b x^3}}\)

\(\Big \downarrow \) 2035

\(\displaystyle -\frac {4 x^{3/4} \sqrt [4]{a x^2-b} \int \frac {x^5 \left (7 b-9 a x^2\right )}{\sqrt [4]{a x^2-b} \left (a x^9-b x^7+1\right )}d\sqrt [4]{x}}{\sqrt [4]{a x^5-b x^3}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {4 x^{3/4} \sqrt [4]{a x^2-b} \int \left (-\frac {9 a x^7}{\sqrt [4]{a x^2-b} \left (a x^9-b x^7+1\right )}-\frac {7 b x^5}{\sqrt [4]{a x^2-b} \left (-a x^9+b x^7-1\right )}\right )d\sqrt [4]{x}}{\sqrt [4]{a x^5-b x^3}}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {4 x^{3/4} \sqrt [4]{a x^2-b} \left (-9 a \int \frac {x^7}{\sqrt [4]{a x^2-b} \left (a x^9-b x^7+1\right )}d\sqrt [4]{x}-7 b \int \frac {x^5}{\sqrt [4]{a x^2-b} \left (-a x^9+b x^7-1\right )}d\sqrt [4]{x}\right )}{\sqrt [4]{a x^5-b x^3}}\)

input

Int[(x^5*(-7*b + 9*a*x^2))/((-(b*x^3) + a*x^5)^(1/4)*(1 - b*x^7 + a*x^9)), 
x]
 

output

$Aborted
 

3.31.13.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2035

Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k   Subst 
[Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti 
onQ[m] && AlgebraicFunctionQ[Fx, x]
 

rule 2467

Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F 
racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p])   Int[x^( 
p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P 
olyQ[Px, x] &&  !IntegerQ[p] &&  !MonomialQ[Px, x] &&  !PolyQ[Fx, x]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

3.31.13.4 Maple [F]

\[\int \frac {x^{5} \left (9 a \,x^{2}-7 b \right )}{\left (a \,x^{5}-b \,x^{3}\right )^{\frac {1}{4}} \left (a \,x^{9}-b \,x^{7}+1\right )}d x\]

input

int(x^5*(9*a*x^2-7*b)/(a*x^5-b*x^3)^(1/4)/(a*x^9-b*x^7+1),x)
 

output

int(x^5*(9*a*x^2-7*b)/(a*x^5-b*x^3)^(1/4)/(a*x^9-b*x^7+1),x)
 

3.31.13.5 Fricas [F(-1)]

Timed out. \[ \int \frac {x^5 \left (-7 b+9 a x^2\right )}{\sqrt [4]{-b x^3+a x^5} \left (1-b x^7+a x^9\right )} \, dx=\text {Timed out} \]

input

integrate(x^5*(9*a*x^2-7*b)/(a*x^5-b*x^3)^(1/4)/(a*x^9-b*x^7+1),x, algorit 
hm="fricas")
 

output

Timed out
 

3.31.13.6 Sympy [F]

\[ \int \frac {x^5 \left (-7 b+9 a x^2\right )}{\sqrt [4]{-b x^3+a x^5} \left (1-b x^7+a x^9\right )} \, dx=\int \frac {x^{5} \cdot \left (9 a x^{2} - 7 b\right )}{\sqrt [4]{x^{3} \left (a x^{2} - b\right )} \left (a x^{9} - b x^{7} + 1\right )}\, dx \]

input

integrate(x**5*(9*a*x**2-7*b)/(a*x**5-b*x**3)**(1/4)/(a*x**9-b*x**7+1),x)
 

output

Integral(x**5*(9*a*x**2 - 7*b)/((x**3*(a*x**2 - b))**(1/4)*(a*x**9 - b*x** 
7 + 1)), x)
 

3.31.13.7 Maxima [F]

\[ \int \frac {x^5 \left (-7 b+9 a x^2\right )}{\sqrt [4]{-b x^3+a x^5} \left (1-b x^7+a x^9\right )} \, dx=\int { \frac {{\left (9 \, a x^{2} - 7 \, b\right )} x^{5}}{{\left (a x^{9} - b x^{7} + 1\right )} {\left (a x^{5} - b x^{3}\right )}^{\frac {1}{4}}} \,d x } \]

input

integrate(x^5*(9*a*x^2-7*b)/(a*x^5-b*x^3)^(1/4)/(a*x^9-b*x^7+1),x, algorit 
hm="maxima")
 

output

integrate((9*a*x^2 - 7*b)*x^5/((a*x^9 - b*x^7 + 1)*(a*x^5 - b*x^3)^(1/4)), 
 x)
 

3.31.13.8 Giac [F]

\[ \int \frac {x^5 \left (-7 b+9 a x^2\right )}{\sqrt [4]{-b x^3+a x^5} \left (1-b x^7+a x^9\right )} \, dx=\int { \frac {{\left (9 \, a x^{2} - 7 \, b\right )} x^{5}}{{\left (a x^{9} - b x^{7} + 1\right )} {\left (a x^{5} - b x^{3}\right )}^{\frac {1}{4}}} \,d x } \]

input

integrate(x^5*(9*a*x^2-7*b)/(a*x^5-b*x^3)^(1/4)/(a*x^9-b*x^7+1),x, algorit 
hm="giac")
 

output

integrate((9*a*x^2 - 7*b)*x^5/((a*x^9 - b*x^7 + 1)*(a*x^5 - b*x^3)^(1/4)), 
 x)
 

3.31.13.9 Mupad [F(-1)]

Timed out. \[ \int \frac {x^5 \left (-7 b+9 a x^2\right )}{\sqrt [4]{-b x^3+a x^5} \left (1-b x^7+a x^9\right )} \, dx=-\int \frac {x^5\,\left (7\,b-9\,a\,x^2\right )}{{\left (a\,x^5-b\,x^3\right )}^{1/4}\,\left (a\,x^9-b\,x^7+1\right )} \,d x \]

input

int(-(x^5*(7*b - 9*a*x^2))/((a*x^5 - b*x^3)^(1/4)*(a*x^9 - b*x^7 + 1)),x)
 

output

-int((x^5*(7*b - 9*a*x^2))/((a*x^5 - b*x^3)^(1/4)*(a*x^9 - b*x^7 + 1)), x)