3.31.66 \(\int \frac {x^3 (-5 b+6 a x)}{\sqrt [4]{-b x+a x^2} (c-b x^5+a x^6)} \, dx\) [3066]

3.31.66.1 Optimal result

Integrand size = 41, antiderivative size = 477 \[ \int \frac {x^3 (-5 b+6 a x)}{\sqrt [4]{-b x+a x^2} \left (c-b x^5+a x^6\right )} \, dx=-\frac {\sqrt {2} \arctan \left (\frac {-2^{2/3} x \sqrt [4]{-b x+a x^2}+\sqrt [3]{c} x^2 \sqrt [4]{-b x+a x^2}}{2 \sqrt [6]{2} \sqrt [4]{c}-\sqrt {2} c^{7/12} x+2^{2/3} x \sqrt [4]{-b x+a x^2}-\sqrt [3]{c} x^2 \sqrt [4]{-b x+a x^2}}\right )}{\sqrt [4]{c}}+\frac {\sqrt {2} \arctan \left (\frac {-2^{2/3} x \sqrt [4]{-b x+a x^2}+\sqrt [3]{c} x^2 \sqrt [4]{-b x+a x^2}}{-2 \sqrt [6]{2} \sqrt [4]{c}+\sqrt {2} c^{7/12} x+2^{2/3} x \sqrt [4]{-b x+a x^2}-\sqrt [3]{c} x^2 \sqrt [4]{-b x+a x^2}}\right )}{\sqrt [4]{c}}-\frac {\sqrt {2} \text {arctanh}\left (\frac {2\ 2^{5/6} \sqrt [4]{c} x \sqrt [4]{-b x+a x^2}-4 \sqrt [6]{2} c^{7/12} x^2 \sqrt [4]{-b x+a x^2}+\sqrt {2} c^{11/12} x^3 \sqrt [4]{-b x+a x^2}}{2 \sqrt [3]{2} \sqrt {c}-2\ 2^{2/3} c^{5/6} x+c^{7/6} x^2+2 \sqrt [3]{2} x^2 \sqrt {-b x+a x^2}-2\ 2^{2/3} \sqrt [3]{c} x^3 \sqrt {-b x+a x^2}+c^{2/3} x^4 \sqrt {-b x+a x^2}}\right )}{\sqrt [4]{c}} \]

output

-2^(1/2)*arctan((-2^(2/3)*x*(a*x^2-b*x)^(1/4)+c^(1/3)*x^2*(a*x^2-b*x)^(1/4 
))/(2*2^(1/6)*c^(1/4)-2^(1/2)*c^(7/12)*x+2^(2/3)*x*(a*x^2-b*x)^(1/4)-c^(1/ 
3)*x^2*(a*x^2-b*x)^(1/4)))/c^(1/4)+2^(1/2)*arctan((-2^(2/3)*x*(a*x^2-b*x)^ 
(1/4)+c^(1/3)*x^2*(a*x^2-b*x)^(1/4))/(-2*2^(1/6)*c^(1/4)+2^(1/2)*c^(7/12)* 
x+2^(2/3)*x*(a*x^2-b*x)^(1/4)-c^(1/3)*x^2*(a*x^2-b*x)^(1/4)))/c^(1/4)-2^(1 
/2)*arctanh((2*2^(5/6)*c^(1/4)*x*(a*x^2-b*x)^(1/4)-4*2^(1/6)*c^(7/12)*x^2* 
(a*x^2-b*x)^(1/4)+2^(1/2)*c^(11/12)*x^3*(a*x^2-b*x)^(1/4))/(2*2^(1/3)*c^(1 
/2)-2*2^(2/3)*c^(5/6)*x+c^(7/6)*x^2+2*2^(1/3)*x^2*(a*x^2-b*x)^(1/2)-2*2^(2 
/3)*c^(1/3)*x^3*(a*x^2-b*x)^(1/2)+c^(2/3)*x^4*(a*x^2-b*x)^(1/2)))/c^(1/4)
 

3.31.66.2 Mathematica [F]

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

input

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

output

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

3.31.66.3 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.

\(\displaystyle \int \frac {x^3 (6 a x-5 b)}{\sqrt [4]{a x^2-b x} \left (a x^6-b x^5+c\right )} \, dx\)

\(\Big \downarrow \) 2467

\(\displaystyle \frac {\sqrt [4]{x} \sqrt [4]{a x-b} \int -\frac {x^{11/4} (5 b-6 a x)}{\sqrt [4]{a x-b} \left (a x^6-b x^5+c\right )}dx}{\sqrt [4]{a x^2-b x}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\sqrt [4]{x} \sqrt [4]{a x-b} \int \frac {x^{11/4} (5 b-6 a x)}{\sqrt [4]{a x-b} \left (a x^6-b x^5+c\right )}dx}{\sqrt [4]{a x^2-b x}}\)

\(\Big \downarrow \) 2035

\(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{a x-b} \int \frac {x^{7/2} (5 b-6 a x)}{\sqrt [4]{a x-b} \left (a x^6-b x^5+c\right )}d\sqrt [4]{x}}{\sqrt [4]{a x^2-b x}}\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{a x-b} \int \left (-\frac {6 a x^{9/2}}{\sqrt [4]{a x-b} \left (a x^6-b x^5+c\right )}-\frac {5 b x^{7/2}}{\sqrt [4]{a x-b} \left (-a x^6+b x^5-c\right )}\right )d\sqrt [4]{x}}{\sqrt [4]{a x^2-b x}}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{a x-b} \left (-5 b \int \frac {x^{7/2}}{\sqrt [4]{a x-b} \left (-a x^6+b x^5-c\right )}d\sqrt [4]{x}-6 a \int \frac {x^{9/2}}{\sqrt [4]{a x-b} \left (a x^6-b x^5+c\right )}d\sqrt [4]{x}\right )}{\sqrt [4]{a x^2-b x}}\)

input

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

output

$Aborted
 

3.31.66.3.1 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] 
]
 

3.31.66.4 Maple [F]

\[\int \frac {x^{3} \left (6 a x -5 b \right )}{\left (a \,x^{2}-b x \right )^{\frac {1}{4}} \left (a \,x^{6}-b \,x^{5}+c \right )}d x\]

input

int(x^3*(6*a*x-5*b)/(a*x^2-b*x)^(1/4)/(a*x^6-b*x^5+c),x)
 

output

int(x^3*(6*a*x-5*b)/(a*x^2-b*x)^(1/4)/(a*x^6-b*x^5+c),x)
 

3.31.66.5 Fricas [F(-1)]

Timed out. \[ \int \frac {x^3 (-5 b+6 a x)}{\sqrt [4]{-b x+a x^2} \left (c-b x^5+a x^6\right )} \, dx=\text {Timed out} \]

input

integrate(x^3*(6*a*x-5*b)/(a*x^2-b*x)^(1/4)/(a*x^6-b*x^5+c),x, algorithm=" 
fricas")
 

output

Timed out
 

3.31.66.6 Sympy [F]

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

input

integrate(x**3*(6*a*x-5*b)/(a*x**2-b*x)**(1/4)/(a*x**6-b*x**5+c),x)
 

output

Integral(x**3*(6*a*x - 5*b)/((x*(a*x - b))**(1/4)*(a*x**6 - b*x**5 + c)), 
x)
 

3.31.66.7 Maxima [F]

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

input

integrate(x^3*(6*a*x-5*b)/(a*x^2-b*x)^(1/4)/(a*x^6-b*x^5+c),x, algorithm=" 
maxima")
 

output

integrate((6*a*x - 5*b)*x^3/((a*x^6 - b*x^5 + c)*(a*x^2 - b*x)^(1/4)), x)
 

3.31.66.8 Giac [F]

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

input

integrate(x^3*(6*a*x-5*b)/(a*x^2-b*x)^(1/4)/(a*x^6-b*x^5+c),x, algorithm=" 
giac")
 

output

integrate((6*a*x - 5*b)*x^3/((a*x^6 - b*x^5 + c)*(a*x^2 - b*x)^(1/4)), x)
 

3.31.66.9 Mupad [F(-1)]

Timed out. \[ \int \frac {x^3 (-5 b+6 a x)}{\sqrt [4]{-b x+a x^2} \left (c-b x^5+a x^6\right )} \, dx=\int -\frac {x^3\,\left (5\,b-6\,a\,x\right )}{{\left (a\,x^2-b\,x\right )}^{1/4}\,\left (a\,x^6-b\,x^5+c\right )} \,d x \]

input

int(-(x^3*(5*b - 6*a*x))/((a*x^2 - b*x)^(1/4)*(c + a*x^6 - b*x^5)),x)
 

output

int(-(x^3*(5*b - 6*a*x))/((a*x^2 - b*x)^(1/4)*(c + a*x^6 - b*x^5)), x)