\(\int \frac {(-b+a x^5)^{3/4}}{x^6} \, dx\) [2076]

Optimal result

Integrand size = 17, antiderivative size = 150 \[ \int \frac {\left (-b+a x^5\right )^{3/4}}{x^6} \, dx=-\frac {\left (-b+a x^5\right )^{3/4}}{5 x^5}-\frac {3 a \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^5}}{-\sqrt {b}+\sqrt {-b+a x^5}}\right )}{10 \sqrt {2} \sqrt [4]{b}}-\frac {3 a \text {arctanh}\left (\frac {\frac {\sqrt [4]{b}}{\sqrt {2}}+\frac {\sqrt {-b+a x^5}}{\sqrt {2} \sqrt [4]{b}}}{\sqrt [4]{-b+a x^5}}\right )}{10 \sqrt {2} \sqrt [4]{b}} \] Output:

-1/5*(a*x^5-b)^(3/4)/x^5-3/20*a*arctan(2^(1/2)*b^(1/4)*(a*x^5-b)^(1/4)/(-b 
^(1/2)+(a*x^5-b)^(1/2)))*2^(1/2)/b^(1/4)-3/20*a*arctanh((1/2*b^(1/4)*2^(1/ 
2)+1/2*(a*x^5-b)^(1/2)*2^(1/2)/b^(1/4))/(a*x^5-b)^(1/4))*2^(1/2)/b^(1/4)
 

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.99 \[ \int \frac {\left (-b+a x^5\right )^{3/4}}{x^6} \, dx=-\frac {\left (-b+a x^5\right )^{3/4}}{5 x^5}+\frac {3 a \arctan \left (\frac {-\frac {\sqrt [4]{b}}{\sqrt {2}}+\frac {\sqrt {-b+a x^5}}{\sqrt {2} \sqrt [4]{b}}}{\sqrt [4]{-b+a x^5}}\right )}{10 \sqrt {2} \sqrt [4]{b}}-\frac {3 a \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt [4]{-b+a x^5}}{\sqrt {b}+\sqrt {-b+a x^5}}\right )}{10 \sqrt {2} \sqrt [4]{b}} \] Input:

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

Output:

-1/5*(-b + a*x^5)^(3/4)/x^5 + (3*a*ArcTan[(-(b^(1/4)/Sqrt[2]) + Sqrt[-b + 
a*x^5]/(Sqrt[2]*b^(1/4)))/(-b + a*x^5)^(1/4)])/(10*Sqrt[2]*b^(1/4)) - (3*a 
*ArcTanh[(Sqrt[2]*b^(1/4)*(-b + a*x^5)^(1/4))/(Sqrt[b] + Sqrt[-b + a*x^5]) 
])/(10*Sqrt[2]*b^(1/4))
 

Rubi [A] (warning: unable to verify)

Time = 0.37 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.41, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.706, Rules used = {798, 51, 73, 27, 826, 1476, 1082, 217, 1479, 25, 27, 1103}

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.

\(\displaystyle \int \frac {\left (a x^5-b\right )^{3/4}}{x^6} \, dx\)

\(\Big \downarrow \) 798

\(\displaystyle \frac {1}{5} \int \frac {\left (a x^5-b\right )^{3/4}}{x^{10}}dx^5\)

\(\Big \downarrow \) 51

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

\(\Big \downarrow \) 73

\(\displaystyle \frac {1}{5} \left (3 \int \frac {a x^{10}}{x^{20}+b}d\sqrt [4]{a x^5-b}-\frac {\left (a x^5-b\right )^{3/4}}{x^5}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{5} \left (3 a \int \frac {x^{10}}{x^{20}+b}d\sqrt [4]{a x^5-b}-\frac {\left (a x^5-b\right )^{3/4}}{x^5}\right )\)

\(\Big \downarrow \) 826

\(\displaystyle \frac {1}{5} \left (3 a \left (\frac {1}{2} \int \frac {x^{10}+\sqrt {b}}{x^{20}+b}d\sqrt [4]{a x^5-b}-\frac {1}{2} \int \frac {\sqrt {b}-x^{10}}{x^{20}+b}d\sqrt [4]{a x^5-b}\right )-\frac {\left (a x^5-b\right )^{3/4}}{x^5}\right )\)

\(\Big \downarrow \) 1476

\(\displaystyle \frac {1}{5} \left (3 a \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{x^{10}+\sqrt {b}-\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^5-b}}d\sqrt [4]{a x^5-b}+\frac {1}{2} \int \frac {1}{x^{10}+\sqrt {b}+\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^5-b}}d\sqrt [4]{a x^5-b}\right )-\frac {1}{2} \int \frac {\sqrt {b}-x^{10}}{x^{20}+b}d\sqrt [4]{a x^5-b}\right )-\frac {\left (a x^5-b\right )^{3/4}}{x^5}\right )\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {1}{5} \left (3 a \left (\frac {1}{2} \left (\frac {\int \frac {1}{-x^{10}-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{a x^5-b}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\int \frac {1}{-x^{10}-1}d\left (\frac {\sqrt {2} \sqrt [4]{a x^5-b}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b}}\right )-\frac {1}{2} \int \frac {\sqrt {b}-x^{10}}{x^{20}+b}d\sqrt [4]{a x^5-b}\right )-\frac {\left (a x^5-b\right )^{3/4}}{x^5}\right )\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {1}{5} \left (3 a \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a x^5-b}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x^5-b}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b}}\right )-\frac {1}{2} \int \frac {\sqrt {b}-x^{10}}{x^{20}+b}d\sqrt [4]{a x^5-b}\right )-\frac {\left (a x^5-b\right )^{3/4}}{x^5}\right )\)

\(\Big \downarrow \) 1479

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1103

\(\displaystyle \frac {1}{5} \left (3 a \left (\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a x^5-b}}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a x^5-b}}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{b}}\right )+\frac {1}{2} \left (\frac {\log \left (-\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^5-b}+\sqrt {b}+x^{10}\right )}{2 \sqrt {2} \sqrt [4]{b}}-\frac {\log \left (\sqrt {2} \sqrt [4]{b} \sqrt [4]{a x^5-b}+\sqrt {b}+x^{10}\right )}{2 \sqrt {2} \sqrt [4]{b}}\right )\right )-\frac {\left (a x^5-b\right )^{3/4}}{x^5}\right )\)

Input:

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

Output:

(-((-b + a*x^5)^(3/4)/x^5) + 3*a*((-(ArcTan[1 - (Sqrt[2]*(-b + a*x^5)^(1/4 
))/b^(1/4)]/(Sqrt[2]*b^(1/4))) + ArcTan[1 + (Sqrt[2]*(-b + a*x^5)^(1/4))/b 
^(1/4)]/(Sqrt[2]*b^(1/4)))/2 + (Log[Sqrt[b] + x^10 - Sqrt[2]*b^(1/4)*(-b + 
 a*x^5)^(1/4)]/(2*Sqrt[2]*b^(1/4)) - Log[Sqrt[b] + x^10 + Sqrt[2]*b^(1/4)* 
(-b + a*x^5)^(1/4)]/(2*Sqrt[2]*b^(1/4)))/2))/5
 

Defintions of rubi rules used

rule 25

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

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) 
Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x 
] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 798

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n   Subst 
[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, 
b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
 

rule 826

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 
2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s)   Int[(r + s*x^2)/(a + b*x^ 
4), x], x] - Simp[1/(2*s)   Int[(r - s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{ 
a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] 
 && AtomQ[SplitProduct[SumBaseQ, b]]))
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1476

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
2*(d/e), 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 
e/(2*c)   Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] 
 && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
 

rule 1479

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
-2*(d/e), 2]}, Simp[e/(2*c*q)   Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], 
 x] + Simp[e/(2*c*q)   Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F 
reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
 

Maple [A] (verified)

Time = 0.46 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.16

method	result	size
pseudoelliptic	\(\frac {3 \ln \left (\frac {\sqrt {a \,x^{5}-b}-b^{\frac {1}{4}} \left (a \,x^{5}-b \right )^{\frac {1}{4}} \sqrt {2}+\sqrt {b}}{\sqrt {a \,x^{5}-b}+b^{\frac {1}{4}} \left (a \,x^{5}-b \right )^{\frac {1}{4}} \sqrt {2}+\sqrt {b}}\right ) \sqrt {2}\, a \,x^{5}+6 \arctan \left (\frac {\sqrt {2}\, \left (a \,x^{5}-b \right )^{\frac {1}{4}}+b^{\frac {1}{4}}}{b^{\frac {1}{4}}}\right ) \sqrt {2}\, a \,x^{5}+6 \arctan \left (\frac {\sqrt {2}\, \left (a \,x^{5}-b \right )^{\frac {1}{4}}-b^{\frac {1}{4}}}{b^{\frac {1}{4}}}\right ) \sqrt {2}\, a \,x^{5}-8 \left (a \,x^{5}-b \right )^{\frac {3}{4}} b^{\frac {1}{4}}}{40 x^{5} b^{\frac {1}{4}}}\)	\(174\)

Input:

int((a*x^5-b)^(3/4)/x^6,x,method=_RETURNVERBOSE)
 

Output:

1/40*(3*ln(((a*x^5-b)^(1/2)-b^(1/4)*(a*x^5-b)^(1/4)*2^(1/2)+b^(1/2))/((a*x 
^5-b)^(1/2)+b^(1/4)*(a*x^5-b)^(1/4)*2^(1/2)+b^(1/2)))*2^(1/2)*a*x^5+6*arct 
an((2^(1/2)*(a*x^5-b)^(1/4)+b^(1/4))/b^(1/4))*2^(1/2)*a*x^5+6*arctan((2^(1 
/2)*(a*x^5-b)^(1/4)-b^(1/4))/b^(1/4))*2^(1/2)*a*x^5-8*(a*x^5-b)^(3/4)*b^(1 
/4))/x^5/b^(1/4)
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.09 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.35 \[ \int \frac {\left (-b+a x^5\right )^{3/4}}{x^6} \, dx=\frac {3 \, \left (-\frac {a^{4}}{b}\right )^{\frac {1}{4}} x^{5} \log \left (27 \, {\left (a x^{5} - b\right )}^{\frac {1}{4}} a^{3} + 27 \, \left (-\frac {a^{4}}{b}\right )^{\frac {3}{4}} b\right ) - 3 i \, \left (-\frac {a^{4}}{b}\right )^{\frac {1}{4}} x^{5} \log \left (27 \, {\left (a x^{5} - b\right )}^{\frac {1}{4}} a^{3} + 27 i \, \left (-\frac {a^{4}}{b}\right )^{\frac {3}{4}} b\right ) + 3 i \, \left (-\frac {a^{4}}{b}\right )^{\frac {1}{4}} x^{5} \log \left (27 \, {\left (a x^{5} - b\right )}^{\frac {1}{4}} a^{3} - 27 i \, \left (-\frac {a^{4}}{b}\right )^{\frac {3}{4}} b\right ) - 3 \, \left (-\frac {a^{4}}{b}\right )^{\frac {1}{4}} x^{5} \log \left (27 \, {\left (a x^{5} - b\right )}^{\frac {1}{4}} a^{3} - 27 \, \left (-\frac {a^{4}}{b}\right )^{\frac {3}{4}} b\right ) - 4 \, {\left (a x^{5} - b\right )}^{\frac {3}{4}}}{20 \, x^{5}} \] Input:

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

Output:

1/20*(3*(-a^4/b)^(1/4)*x^5*log(27*(a*x^5 - b)^(1/4)*a^3 + 27*(-a^4/b)^(3/4 
)*b) - 3*I*(-a^4/b)^(1/4)*x^5*log(27*(a*x^5 - b)^(1/4)*a^3 + 27*I*(-a^4/b) 
^(3/4)*b) + 3*I*(-a^4/b)^(1/4)*x^5*log(27*(a*x^5 - b)^(1/4)*a^3 - 27*I*(-a 
^4/b)^(3/4)*b) - 3*(-a^4/b)^(1/4)*x^5*log(27*(a*x^5 - b)^(1/4)*a^3 - 27*(- 
a^4/b)^(3/4)*b) - 4*(a*x^5 - b)^(3/4))/x^5
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.78 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.29 \[ \int \frac {\left (-b+a x^5\right )^{3/4}}{x^6} \, dx=- \frac {a^{\frac {3}{4}} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b e^{2 i \pi }}{a x^{5}}} \right )}}{5 x^{\frac {5}{4}} \Gamma \left (\frac {5}{4}\right )} \] Input:

integrate((a*x**5-b)**(3/4)/x**6,x)
 

Output:

-a**(3/4)*gamma(1/4)*hyper((-3/4, 1/4), (5/4,), b*exp_polar(2*I*pi)/(a*x** 
5))/(5*x**(5/4)*gamma(5/4))
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.21 \[ \int \frac {\left (-b+a x^5\right )^{3/4}}{x^6} \, dx=\frac {3}{40} \, {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} + 2 \, {\left (a x^{5} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{b^{\frac {1}{4}}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} - 2 \, {\left (a x^{5} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{b^{\frac {1}{4}}} - \frac {\sqrt {2} \log \left (\sqrt {2} {\left (a x^{5} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{5} - b} + \sqrt {b}\right )}{b^{\frac {1}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} {\left (a x^{5} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{5} - b} + \sqrt {b}\right )}{b^{\frac {1}{4}}}\right )} a - \frac {{\left (a x^{5} - b\right )}^{\frac {3}{4}}}{5 \, x^{5}} \] Input:

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

Output:

3/40*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*b^(1/4) + 2*(a*x^5 - b)^(1/4)) 
/b^(1/4))/b^(1/4) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*b^(1/4) - 2*(a* 
x^5 - b)^(1/4))/b^(1/4))/b^(1/4) - sqrt(2)*log(sqrt(2)*(a*x^5 - b)^(1/4)*b 
^(1/4) + sqrt(a*x^5 - b) + sqrt(b))/b^(1/4) + sqrt(2)*log(-sqrt(2)*(a*x^5 
- b)^(1/4)*b^(1/4) + sqrt(a*x^5 - b) + sqrt(b))/b^(1/4))*a - 1/5*(a*x^5 - 
b)^(3/4)/x^5
 

Giac [A] (verification not implemented)

Time = 0.44 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.23 \[ \int \frac {\left (-b+a x^5\right )^{3/4}}{x^6} \, dx=\frac {1}{40} \, {\left (\frac {6 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} + 2 \, {\left (a x^{5} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{b^{\frac {1}{4}}} + \frac {6 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} b^{\frac {1}{4}} - 2 \, {\left (a x^{5} - b\right )}^{\frac {1}{4}}\right )}}{2 \, b^{\frac {1}{4}}}\right )}{b^{\frac {1}{4}}} - \frac {3 \, \sqrt {2} \log \left (\sqrt {2} {\left (a x^{5} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{5} - b} + \sqrt {b}\right )}{b^{\frac {1}{4}}} + \frac {3 \, \sqrt {2} \log \left (-\sqrt {2} {\left (a x^{5} - b\right )}^{\frac {1}{4}} b^{\frac {1}{4}} + \sqrt {a x^{5} - b} + \sqrt {b}\right )}{b^{\frac {1}{4}}} - \frac {8 \, {\left (a x^{5} - b\right )}^{\frac {3}{4}}}{a x^{5}}\right )} a \] Input:

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

Output:

1/40*(6*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*b^(1/4) + 2*(a*x^5 - b)^(1/4)) 
/b^(1/4))/b^(1/4) + 6*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*b^(1/4) - 2*(a* 
x^5 - b)^(1/4))/b^(1/4))/b^(1/4) - 3*sqrt(2)*log(sqrt(2)*(a*x^5 - b)^(1/4) 
*b^(1/4) + sqrt(a*x^5 - b) + sqrt(b))/b^(1/4) + 3*sqrt(2)*log(-sqrt(2)*(a* 
x^5 - b)^(1/4)*b^(1/4) + sqrt(a*x^5 - b) + sqrt(b))/b^(1/4) - 8*(a*x^5 - b 
)^(3/4)/(a*x^5))*a
 

Mupad [B] (verification not implemented)

Time = 8.35 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.46 \[ \int \frac {\left (-b+a x^5\right )^{3/4}}{x^6} \, dx=\frac {3\,a\,\mathrm {atan}\left (\frac {{\left (a\,x^5-b\right )}^{1/4}}{{\left (-b\right )}^{1/4}}\right )}{10\,{\left (-b\right )}^{1/4}}-\frac {{\left (a\,x^5-b\right )}^{3/4}}{5\,x^5}-\frac {3\,a\,\mathrm {atanh}\left (\frac {{\left (a\,x^5-b\right )}^{1/4}}{{\left (-b\right )}^{1/4}}\right )}{10\,{\left (-b\right )}^{1/4}} \] Input:

int((a*x^5 - b)^(3/4)/x^6,x)
 

Output:

(3*a*atan((a*x^5 - b)^(1/4)/(-b)^(1/4)))/(10*(-b)^(1/4)) - (a*x^5 - b)^(3/ 
4)/(5*x^5) - (3*a*atanh((a*x^5 - b)^(1/4)/(-b)^(1/4)))/(10*(-b)^(1/4))
 

Reduce [F]

\[ \int \frac {\left (-b+a x^5\right )^{3/4}}{x^6} \, dx=\frac {-4 \left (a \,x^{5}-b \right )^{\frac {3}{4}}+15 \left (\int \frac {\left (a \,x^{5}-b \right )^{\frac {3}{4}}}{a \,x^{6}-b x}d x \right ) a \,x^{5}}{20 x^{5}} \] Input:

int((a*x^5-b)^(3/4)/x^6,x)
 

Output:

( - 4*(a*x**5 - b)**(3/4) + 15*int((a*x**5 - b)**(3/4)/(a*x**6 - b*x),x)*a 
*x**5)/(20*x**5)