\(\int \frac {x^3 (b+2 a x^5)}{\sqrt [4]{b x+a x^6} (-1+b x^5+a x^{10})} \, dx\) [536]

Optimal result

Integrand size = 39, antiderivative size = 41 \[ \int \frac {x^3 \left (b+2 a x^5\right )}{\sqrt [4]{b x+a x^6} \left (-1+b x^5+a x^{10}\right )} \, dx=\frac {2}{5} \arctan \left (x \sqrt [4]{b x+a x^6}\right )-\frac {2}{5} \text {arctanh}\left (x \sqrt [4]{b x+a x^6}\right ) \] Output:

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

Mathematica [F]

\[ \int \frac {x^3 \left (b+2 a x^5\right )}{\sqrt [4]{b x+a x^6} \left (-1+b x^5+a x^{10}\right )} \, dx=\int \frac {x^3 \left (b+2 a x^5\right )}{\sqrt [4]{b x+a x^6} \left (-1+b x^5+a x^{10}\right )} \, dx \] Input:

Integrate[(x^3*(b + 2*a*x^5))/((b*x + a*x^6)^(1/4)*(-1 + b*x^5 + a*x^10)), 
x]
 

Output:

Integrate[(x^3*(b + 2*a*x^5))/((b*x + a*x^6)^(1/4)*(-1 + b*x^5 + a*x^10)), 
 x]
 

Rubi [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.

Time = 1.81 (sec) , antiderivative size = 408, normalized size of antiderivative = 9.95, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.128, Rules used = {2467, 25, 2035, 7279, 2009}

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.

\(\displaystyle \int \frac {x^3 \left (2 a x^5+b\right )}{\sqrt [4]{a x^6+b x} \left (a x^{10}+b x^5-1\right )} \, dx\)

\(\Big \downarrow \) 2467

\(\displaystyle \frac {\sqrt [4]{x} \sqrt [4]{a x^5+b} \int -\frac {x^{11/4} \left (2 a x^5+b\right )}{\sqrt [4]{a x^5+b} \left (-a x^{10}-b x^5+1\right )}dx}{\sqrt [4]{a x^6+b x}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\sqrt [4]{x} \sqrt [4]{a x^5+b} \int \frac {x^{11/4} \left (2 a x^5+b\right )}{\sqrt [4]{a x^5+b} \left (-a x^{10}-b x^5+1\right )}dx}{\sqrt [4]{a x^6+b x}}\)

\(\Big \downarrow \) 2035

\(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{a x^5+b} \int \frac {x^{7/2} \left (2 a x^5+b\right )}{\sqrt [4]{a x^5+b} \left (-a x^{10}-b x^5+1\right )}d\sqrt [4]{x}}{\sqrt [4]{a x^6+b x}}\)

\(\Big \downarrow \) 7279

\(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{a x^5+b} \int \left (-\frac {2 a x^{17/2}}{\sqrt [4]{a x^5+b} \left (a x^{10}+b x^5-1\right )}-\frac {b x^{7/2}}{\sqrt [4]{a x^5+b} \left (a x^{10}+b x^5-1\right )}\right )d\sqrt [4]{x}}{\sqrt [4]{a x^6+b x}}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {4 \sqrt [4]{x} \sqrt [4]{a x^5+b} \left (\frac {2 a x^{15/4} \sqrt [4]{\frac {a x^5}{b}+1} \operatorname {AppellF1}\left (\frac {3}{4},1,\frac {1}{4},\frac {7}{4},-\frac {2 a x^5}{b-\sqrt {b^2+4 a}},-\frac {a x^5}{b}\right )}{15 \sqrt {4 a+b^2} \sqrt [4]{a x^5+b}}+\frac {2 a b x^{15/4} \sqrt [4]{\frac {a x^5}{b}+1} \operatorname {AppellF1}\left (\frac {3}{4},1,\frac {1}{4},\frac {7}{4},-\frac {2 a x^5}{b-\sqrt {b^2+4 a}},-\frac {a x^5}{b}\right )}{15 \left (-b \sqrt {4 a+b^2}+4 a+b^2\right ) \sqrt [4]{a x^5+b}}-\frac {2 a x^{15/4} \sqrt [4]{\frac {a x^5}{b}+1} \operatorname {AppellF1}\left (\frac {3}{4},1,\frac {1}{4},\frac {7}{4},-\frac {2 a x^5}{b+\sqrt {b^2+4 a}},-\frac {a x^5}{b}\right )}{15 \sqrt {4 a+b^2} \sqrt [4]{a x^5+b}}+\frac {2 a b x^{15/4} \sqrt [4]{\frac {a x^5}{b}+1} \operatorname {AppellF1}\left (\frac {3}{4},1,\frac {1}{4},\frac {7}{4},-\frac {2 a x^5}{b+\sqrt {b^2+4 a}},-\frac {a x^5}{b}\right )}{15 \left (b \left (\sqrt {4 a+b^2}+b\right )+4 a\right ) \sqrt [4]{a x^5+b}}\right )}{\sqrt [4]{a x^6+b x}}\)

Input:

Int[(x^3*(b + 2*a*x^5))/((b*x + a*x^6)^(1/4)*(-1 + b*x^5 + a*x^10)),x]
 

Output:

(-4*x^(1/4)*(b + a*x^5)^(1/4)*((2*a*x^(15/4)*(1 + (a*x^5)/b)^(1/4)*AppellF 
1[3/4, 1, 1/4, 7/4, (-2*a*x^5)/(b - Sqrt[4*a + b^2]), -((a*x^5)/b)])/(15*S 
qrt[4*a + b^2]*(b + a*x^5)^(1/4)) + (2*a*b*x^(15/4)*(1 + (a*x^5)/b)^(1/4)* 
AppellF1[3/4, 1, 1/4, 7/4, (-2*a*x^5)/(b - Sqrt[4*a + b^2]), -((a*x^5)/b)] 
)/(15*(4*a + b^2 - b*Sqrt[4*a + b^2])*(b + a*x^5)^(1/4)) - (2*a*x^(15/4)*( 
1 + (a*x^5)/b)^(1/4)*AppellF1[3/4, 1, 1/4, 7/4, (-2*a*x^5)/(b + Sqrt[4*a + 
 b^2]), -((a*x^5)/b)])/(15*Sqrt[4*a + b^2]*(b + a*x^5)^(1/4)) + (2*a*b*x^( 
15/4)*(1 + (a*x^5)/b)^(1/4)*AppellF1[3/4, 1, 1/4, 7/4, (-2*a*x^5)/(b + Sqr 
t[4*a + b^2]), -((a*x^5)/b)])/(15*(4*a + b*(b + Sqrt[4*a + b^2]))*(b + a*x 
^5)^(1/4))))/(b*x + a*x^6)^(1/4)
 

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 7279

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ 
{v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su 
mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
 

Maple [F]

\[\int \frac {x^{3} \left (2 a \,x^{5}+b \right )}{\left (a \,x^{6}+b x \right )^{\frac {1}{4}} \left (a \,x^{10}+b \,x^{5}-1\right )}d x\]

Input:

int(x^3*(2*a*x^5+b)/(a*x^6+b*x)^(1/4)/(a*x^10+b*x^5-1),x)
 

Output:

int(x^3*(2*a*x^5+b)/(a*x^6+b*x)^(1/4)/(a*x^10+b*x^5-1),x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {x^3 \left (b+2 a x^5\right )}{\sqrt [4]{b x+a x^6} \left (-1+b x^5+a x^{10}\right )} \, dx=\text {Timed out} \] Input:

integrate(x^3*(2*a*x^5+b)/(a*x^6+b*x)^(1/4)/(a*x^10+b*x^5-1),x, algorithm= 
"fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^3 \left (b+2 a x^5\right )}{\sqrt [4]{b x+a x^6} \left (-1+b x^5+a x^{10}\right )} \, dx=\text {Timed out} \] Input:

integrate(x**3*(2*a*x**5+b)/(a*x**6+b*x)**(1/4)/(a*x**10+b*x**5-1),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {x^3 \left (b+2 a x^5\right )}{\sqrt [4]{b x+a x^6} \left (-1+b x^5+a x^{10}\right )} \, dx=\int { \frac {{\left (2 \, a x^{5} + b\right )} x^{3}}{{\left (a x^{10} + b x^{5} - 1\right )} {\left (a x^{6} + b x\right )}^{\frac {1}{4}}} \,d x } \] Input:

integrate(x^3*(2*a*x^5+b)/(a*x^6+b*x)^(1/4)/(a*x^10+b*x^5-1),x, algorithm= 
"maxima")
 

Output:

integrate((2*a*x^5 + b)*x^3/((a*x^10 + b*x^5 - 1)*(a*x^6 + b*x)^(1/4)), x)
 

Giac [F]

\[ \int \frac {x^3 \left (b+2 a x^5\right )}{\sqrt [4]{b x+a x^6} \left (-1+b x^5+a x^{10}\right )} \, dx=\int { \frac {{\left (2 \, a x^{5} + b\right )} x^{3}}{{\left (a x^{10} + b x^{5} - 1\right )} {\left (a x^{6} + b x\right )}^{\frac {1}{4}}} \,d x } \] Input:

integrate(x^3*(2*a*x^5+b)/(a*x^6+b*x)^(1/4)/(a*x^10+b*x^5-1),x, algorithm= 
"giac")
 

Output:

integrate((2*a*x^5 + b)*x^3/((a*x^10 + b*x^5 - 1)*(a*x^6 + b*x)^(1/4)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3 \left (b+2 a x^5\right )}{\sqrt [4]{b x+a x^6} \left (-1+b x^5+a x^{10}\right )} \, dx=\int \frac {x^3\,\left (2\,a\,x^5+b\right )}{{\left (a\,x^6+b\,x\right )}^{1/4}\,\left (a\,x^{10}+b\,x^5-1\right )} \,d x \] Input:

int((x^3*(b + 2*a*x^5))/((b*x + a*x^6)^(1/4)*(a*x^10 + b*x^5 - 1)),x)
 

Output:

int((x^3*(b + 2*a*x^5))/((b*x + a*x^6)^(1/4)*(a*x^10 + b*x^5 - 1)), x)
 

Reduce [F]

\[ \int \frac {x^3 \left (b+2 a x^5\right )}{\sqrt [4]{b x+a x^6} \left (-1+b x^5+a x^{10}\right )} \, dx=2 \left (\int \frac {x^{8}}{x^{\frac {41}{4}} \left (a \,x^{5}+b \right )^{\frac {1}{4}} a +x^{\frac {21}{4}} \left (a \,x^{5}+b \right )^{\frac {1}{4}} b -x^{\frac {1}{4}} \left (a \,x^{5}+b \right )^{\frac {1}{4}}}d x \right ) a +\left (\int \frac {x^{3}}{x^{\frac {41}{4}} \left (a \,x^{5}+b \right )^{\frac {1}{4}} a +x^{\frac {21}{4}} \left (a \,x^{5}+b \right )^{\frac {1}{4}} b -x^{\frac {1}{4}} \left (a \,x^{5}+b \right )^{\frac {1}{4}}}d x \right ) b \] Input:

int(x^3*(2*a*x^5+b)/(a*x^6+b*x)^(1/4)/(a*x^10+b*x^5-1),x)
 

Output:

2*int(x**8/(x**(1/4)*(a*x**5 + b)**(1/4)*a*x**10 + x**(1/4)*(a*x**5 + b)** 
(1/4)*b*x**5 - x**(1/4)*(a*x**5 + b)**(1/4)),x)*a + int(x**3/(x**(1/4)*(a* 
x**5 + b)**(1/4)*a*x**10 + x**(1/4)*(a*x**5 + b)**(1/4)*b*x**5 - x**(1/4)* 
(a*x**5 + b)**(1/4)),x)*b