\(\int \frac {g+h x}{(a+b x) (c+d x) (e+f x)^{2/3}} \, dx\) [221]

Optimal result

Integrand size = 29, antiderivative size = 386 \[ \int \frac {g+h x}{(a+b x) (c+d x) (e+f x)^{2/3}} \, dx=-\frac {\sqrt {3} (b g-a h) \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} \sqrt [3]{e+f x}}{\sqrt [3]{b e-a f}}}{\sqrt {3}}\right )}{\sqrt [3]{b} (b c-a d) (b e-a f)^{2/3}}+\frac {\sqrt {3} (d g-c h) \arctan \left (\frac {1+\frac {2 \sqrt [3]{d} \sqrt [3]{e+f x}}{\sqrt [3]{d e-c f}}}{\sqrt {3}}\right )}{\sqrt [3]{d} (b c-a d) (d e-c f)^{2/3}}-\frac {(b g-a h) \log (a+b x)}{2 \sqrt [3]{b} (b c-a d) (b e-a f)^{2/3}}+\frac {(d g-c h) \log (c+d x)}{2 \sqrt [3]{d} (b c-a d) (d e-c f)^{2/3}}+\frac {3 (b g-a h) \log \left (\sqrt [3]{b e-a f}-\sqrt [3]{b} \sqrt [3]{e+f x}\right )}{2 \sqrt [3]{b} (b c-a d) (b e-a f)^{2/3}}-\frac {3 (d g-c h) \log \left (\sqrt [3]{d e-c f}-\sqrt [3]{d} \sqrt [3]{e+f x}\right )}{2 \sqrt [3]{d} (b c-a d) (d e-c f)^{2/3}} \] Output:

-3^(1/2)*(-a*h+b*g)*arctan(1/3*(1+2*b^(1/3)*(f*x+e)^(1/3)/(-a*f+b*e)^(1/3) 
)*3^(1/2))/b^(1/3)/(-a*d+b*c)/(-a*f+b*e)^(2/3)+3^(1/2)*(-c*h+d*g)*arctan(1 
/3*(1+2*d^(1/3)*(f*x+e)^(1/3)/(-c*f+d*e)^(1/3))*3^(1/2))/d^(1/3)/(-a*d+b*c 
)/(-c*f+d*e)^(2/3)-1/2*(-a*h+b*g)*ln(b*x+a)/b^(1/3)/(-a*d+b*c)/(-a*f+b*e)^ 
(2/3)+1/2*(-c*h+d*g)*ln(d*x+c)/d^(1/3)/(-a*d+b*c)/(-c*f+d*e)^(2/3)+3/2*(-a 
*h+b*g)*ln((-a*f+b*e)^(1/3)-b^(1/3)*(f*x+e)^(1/3))/b^(1/3)/(-a*d+b*c)/(-a* 
f+b*e)^(2/3)-3/2*(-c*h+d*g)*ln((-c*f+d*e)^(1/3)-d^(1/3)*(f*x+e)^(1/3))/d^( 
1/3)/(-a*d+b*c)/(-c*f+d*e)^(2/3)
 

Mathematica [A] (verified)

Time = 1.88 (sec) , antiderivative size = 466, normalized size of antiderivative = 1.21 \[ \int \frac {g+h x}{(a+b x) (c+d x) (e+f x)^{2/3}} \, dx=\frac {-2 \sqrt {3} \sqrt [3]{d} (-d e+c f)^{2/3} (b g-a h) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{e+f x}}{\sqrt [3]{-b e+a f}}}{\sqrt {3}}\right )+2 \sqrt {3} \sqrt [3]{b} (-b e+a f)^{2/3} (d g-c h) \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{e+f x}}{\sqrt [3]{-d e+c f}}}{\sqrt {3}}\right )+2 \sqrt [3]{d} (-d e+c f)^{2/3} (b g-a h) \log \left (\sqrt [3]{-b e+a f}+\sqrt [3]{b} \sqrt [3]{e+f x}\right )-2 \sqrt [3]{b} (-b e+a f)^{2/3} (d g-c h) \log \left (\sqrt [3]{-d e+c f}+\sqrt [3]{d} \sqrt [3]{e+f x}\right )-\sqrt [3]{d} (-d e+c f)^{2/3} (b g-a h) \log \left ((-b e+a f)^{2/3}-\sqrt [3]{b} \sqrt [3]{-b e+a f} \sqrt [3]{e+f x}+b^{2/3} (e+f x)^{2/3}\right )+\sqrt [3]{b} (-b e+a f)^{2/3} (d g-c h) \log \left ((-d e+c f)^{2/3}-\sqrt [3]{d} \sqrt [3]{-d e+c f} \sqrt [3]{e+f x}+d^{2/3} (e+f x)^{2/3}\right )}{2 \sqrt [3]{b} \sqrt [3]{d} (b c-a d) (-b e+a f)^{2/3} (-d e+c f)^{2/3}} \] Input:

Integrate[(g + h*x)/((a + b*x)*(c + d*x)*(e + f*x)^(2/3)),x]
 

Output:

(-2*Sqrt[3]*d^(1/3)*(-(d*e) + c*f)^(2/3)*(b*g - a*h)*ArcTan[(1 - (2*b^(1/3 
)*(e + f*x)^(1/3))/(-(b*e) + a*f)^(1/3))/Sqrt[3]] + 2*Sqrt[3]*b^(1/3)*(-(b 
*e) + a*f)^(2/3)*(d*g - c*h)*ArcTan[(1 - (2*d^(1/3)*(e + f*x)^(1/3))/(-(d* 
e) + c*f)^(1/3))/Sqrt[3]] + 2*d^(1/3)*(-(d*e) + c*f)^(2/3)*(b*g - a*h)*Log 
[(-(b*e) + a*f)^(1/3) + b^(1/3)*(e + f*x)^(1/3)] - 2*b^(1/3)*(-(b*e) + a*f 
)^(2/3)*(d*g - c*h)*Log[(-(d*e) + c*f)^(1/3) + d^(1/3)*(e + f*x)^(1/3)] - 
d^(1/3)*(-(d*e) + c*f)^(2/3)*(b*g - a*h)*Log[(-(b*e) + a*f)^(2/3) - b^(1/3 
)*(-(b*e) + a*f)^(1/3)*(e + f*x)^(1/3) + b^(2/3)*(e + f*x)^(2/3)] + b^(1/3 
)*(-(b*e) + a*f)^(2/3)*(d*g - c*h)*Log[(-(d*e) + c*f)^(2/3) - d^(1/3)*(-(d 
*e) + c*f)^(1/3)*(e + f*x)^(1/3) + d^(2/3)*(e + f*x)^(2/3)])/(2*b^(1/3)*d^ 
(1/3)*(b*c - a*d)*(-(b*e) + a*f)^(2/3)*(-(d*e) + c*f)^(2/3))
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 320, normalized size of antiderivative = 0.83, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {174, 69, 16, 1082, 217}

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.

\(\displaystyle \int \frac {g+h x}{(a+b x) (c+d x) (e+f x)^{2/3}} \, dx\)

\(\Big \downarrow \) 174

\(\displaystyle \frac {(b g-a h) \int \frac {1}{(a+b x) (e+f x)^{2/3}}dx}{b c-a d}-\frac {(d g-c h) \int \frac {1}{(c+d x) (e+f x)^{2/3}}dx}{b c-a d}\)

\(\Big \downarrow \) 69

\(\displaystyle \frac {(b g-a h) \left (-\frac {3 \int \frac {1}{\frac {(b e-a f)^{2/3}}{b^{2/3}}+\frac {\sqrt [3]{e+f x} \sqrt [3]{b e-a f}}{\sqrt [3]{b}}+(e+f x)^{2/3}}d\sqrt [3]{e+f x}}{2 b^{2/3} \sqrt [3]{b e-a f}}-\frac {3 \int \frac {1}{\frac {\sqrt [3]{b e-a f}}{\sqrt [3]{b}}-\sqrt [3]{e+f x}}d\sqrt [3]{e+f x}}{2 \sqrt [3]{b} (b e-a f)^{2/3}}-\frac {\log (a+b x)}{2 \sqrt [3]{b} (b e-a f)^{2/3}}\right )}{b c-a d}-\frac {(d g-c h) \left (-\frac {3 \int \frac {1}{\frac {(d e-c f)^{2/3}}{d^{2/3}}+\frac {\sqrt [3]{e+f x} \sqrt [3]{d e-c f}}{\sqrt [3]{d}}+(e+f x)^{2/3}}d\sqrt [3]{e+f x}}{2 d^{2/3} \sqrt [3]{d e-c f}}-\frac {3 \int \frac {1}{\frac {\sqrt [3]{d e-c f}}{\sqrt [3]{d}}-\sqrt [3]{e+f x}}d\sqrt [3]{e+f x}}{2 \sqrt [3]{d} (d e-c f)^{2/3}}-\frac {\log (c+d x)}{2 \sqrt [3]{d} (d e-c f)^{2/3}}\right )}{b c-a d}\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {(b g-a h) \left (-\frac {3 \int \frac {1}{\frac {(b e-a f)^{2/3}}{b^{2/3}}+\frac {\sqrt [3]{e+f x} \sqrt [3]{b e-a f}}{\sqrt [3]{b}}+(e+f x)^{2/3}}d\sqrt [3]{e+f x}}{2 b^{2/3} \sqrt [3]{b e-a f}}-\frac {\log (a+b x)}{2 \sqrt [3]{b} (b e-a f)^{2/3}}+\frac {3 \log \left (\sqrt [3]{b e-a f}-\sqrt [3]{b} \sqrt [3]{e+f x}\right )}{2 \sqrt [3]{b} (b e-a f)^{2/3}}\right )}{b c-a d}-\frac {(d g-c h) \left (-\frac {3 \int \frac {1}{\frac {(d e-c f)^{2/3}}{d^{2/3}}+\frac {\sqrt [3]{e+f x} \sqrt [3]{d e-c f}}{\sqrt [3]{d}}+(e+f x)^{2/3}}d\sqrt [3]{e+f x}}{2 d^{2/3} \sqrt [3]{d e-c f}}-\frac {\log (c+d x)}{2 \sqrt [3]{d} (d e-c f)^{2/3}}+\frac {3 \log \left (\sqrt [3]{d e-c f}-\sqrt [3]{d} \sqrt [3]{e+f x}\right )}{2 \sqrt [3]{d} (d e-c f)^{2/3}}\right )}{b c-a d}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {(b g-a h) \left (\frac {3 \int \frac {1}{-(e+f x)^{2/3}-3}d\left (\frac {2 \sqrt [3]{b} \sqrt [3]{e+f x}}{\sqrt [3]{b e-a f}}+1\right )}{\sqrt [3]{b} (b e-a f)^{2/3}}-\frac {\log (a+b x)}{2 \sqrt [3]{b} (b e-a f)^{2/3}}+\frac {3 \log \left (\sqrt [3]{b e-a f}-\sqrt [3]{b} \sqrt [3]{e+f x}\right )}{2 \sqrt [3]{b} (b e-a f)^{2/3}}\right )}{b c-a d}-\frac {(d g-c h) \left (\frac {3 \int \frac {1}{-(e+f x)^{2/3}-3}d\left (\frac {2 \sqrt [3]{d} \sqrt [3]{e+f x}}{\sqrt [3]{d e-c f}}+1\right )}{\sqrt [3]{d} (d e-c f)^{2/3}}-\frac {\log (c+d x)}{2 \sqrt [3]{d} (d e-c f)^{2/3}}+\frac {3 \log \left (\sqrt [3]{d e-c f}-\sqrt [3]{d} \sqrt [3]{e+f x}\right )}{2 \sqrt [3]{d} (d e-c f)^{2/3}}\right )}{b c-a d}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {(b g-a h) \left (-\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{b} \sqrt [3]{e+f x}}{\sqrt [3]{b e-a f}}+1}{\sqrt {3}}\right )}{\sqrt [3]{b} (b e-a f)^{2/3}}-\frac {\log (a+b x)}{2 \sqrt [3]{b} (b e-a f)^{2/3}}+\frac {3 \log \left (\sqrt [3]{b e-a f}-\sqrt [3]{b} \sqrt [3]{e+f x}\right )}{2 \sqrt [3]{b} (b e-a f)^{2/3}}\right )}{b c-a d}-\frac {(d g-c h) \left (-\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{d} \sqrt [3]{e+f x}}{\sqrt [3]{d e-c f}}+1}{\sqrt {3}}\right )}{\sqrt [3]{d} (d e-c f)^{2/3}}-\frac {\log (c+d x)}{2 \sqrt [3]{d} (d e-c f)^{2/3}}+\frac {3 \log \left (\sqrt [3]{d e-c f}-\sqrt [3]{d} \sqrt [3]{e+f x}\right )}{2 \sqrt [3]{d} (d e-c f)^{2/3}}\right )}{b c-a d}\)

Input:

Int[(g + h*x)/((a + b*x)*(c + d*x)*(e + f*x)^(2/3)),x]
 

Output:

((b*g - a*h)*(-((Sqrt[3]*ArcTan[(1 + (2*b^(1/3)*(e + f*x)^(1/3))/(b*e - a* 
f)^(1/3))/Sqrt[3]])/(b^(1/3)*(b*e - a*f)^(2/3))) - Log[a + b*x]/(2*b^(1/3) 
*(b*e - a*f)^(2/3)) + (3*Log[(b*e - a*f)^(1/3) - b^(1/3)*(e + f*x)^(1/3)]) 
/(2*b^(1/3)*(b*e - a*f)^(2/3))))/(b*c - a*d) - ((d*g - c*h)*(-((Sqrt[3]*Ar 
cTan[(1 + (2*d^(1/3)*(e + f*x)^(1/3))/(d*e - c*f)^(1/3))/Sqrt[3]])/(d^(1/3 
)*(d*e - c*f)^(2/3))) - Log[c + d*x]/(2*d^(1/3)*(d*e - c*f)^(2/3)) + (3*Lo 
g[(d*e - c*f)^(1/3) - d^(1/3)*(e + f*x)^(1/3)])/(2*d^(1/3)*(d*e - c*f)^(2/ 
3))))/(b*c - a*d)
 

Defintions of rubi rules used

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

rule 69

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[ 
{q = Rt[(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q^2), 
 x] + (-Simp[3/(2*b*q)   Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1 
/3)], x] - Simp[3/(2*b*q^2)   Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], 
 x])] /; FreeQ[{a, b, c, d}, x] && PosQ[(b*c - a*d)/b]
 

rule 174

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* 
((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d)   Int[(e + f*x)^ 
p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d)   Int[(e + f*x)^p/(c + d 
*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, 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 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]
 

Maple [A] (verified)

Time = 0.65 (sec) , antiderivative size = 364, normalized size of antiderivative = 0.94

method	result	size
derivativedivides	\(\frac {3 \left (\frac {\ln \left (\left (f x +e \right )^{\frac {1}{3}}+\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}}\right )}{3 d \left (\frac {c f -d e}{d}\right )^{\frac {2}{3}}}-\frac {\ln \left (\left (f x +e \right )^{\frac {2}{3}}-\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}} \left (f x +e \right )^{\frac {1}{3}}+\left (\frac {c f -d e}{d}\right )^{\frac {2}{3}}\right )}{6 d \left (\frac {c f -d e}{d}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (f x +e \right )^{\frac {1}{3}}}{\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 d \left (\frac {c f -d e}{d}\right )^{\frac {2}{3}}}\right ) \left (-c h +d g \right )}{a d -b c}+\frac {3 \left (\frac {\ln \left (\left (f x +e \right )^{\frac {1}{3}}+\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a f -b e}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\left (f x +e \right )^{\frac {2}{3}}-\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}} \left (f x +e \right )^{\frac {1}{3}}+\left (\frac {a f -b e}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a f -b e}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (f x +e \right )^{\frac {1}{3}}}{\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a f -b e}{b}\right )^{\frac {2}{3}}}\right ) \left (a h -b g \right )}{a d -b c}\)	\(364\)
default	\(\frac {3 \left (\frac {\ln \left (\left (f x +e \right )^{\frac {1}{3}}+\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}}\right )}{3 d \left (\frac {c f -d e}{d}\right )^{\frac {2}{3}}}-\frac {\ln \left (\left (f x +e \right )^{\frac {2}{3}}-\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}} \left (f x +e \right )^{\frac {1}{3}}+\left (\frac {c f -d e}{d}\right )^{\frac {2}{3}}\right )}{6 d \left (\frac {c f -d e}{d}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (f x +e \right )^{\frac {1}{3}}}{\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 d \left (\frac {c f -d e}{d}\right )^{\frac {2}{3}}}\right ) \left (-c h +d g \right )}{a d -b c}+\frac {3 \left (\frac {\ln \left (\left (f x +e \right )^{\frac {1}{3}}+\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a f -b e}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\left (f x +e \right )^{\frac {2}{3}}-\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}} \left (f x +e \right )^{\frac {1}{3}}+\left (\frac {a f -b e}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a f -b e}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (f x +e \right )^{\frac {1}{3}}}{\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a f -b e}{b}\right )^{\frac {2}{3}}}\right ) \left (a h -b g \right )}{a d -b c}\)	\(364\)
pseudoelliptic	\(\frac {-\frac {\ln \left (\left (f x +e \right )^{\frac {2}{3}}-\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}} \left (f x +e \right )^{\frac {1}{3}}+\left (\frac {a f -b e}{b}\right )^{\frac {2}{3}}\right ) \left (a h -b g \right ) d \left (\frac {c f -d e}{d}\right )^{\frac {2}{3}}}{2}+\frac {\ln \left (\left (f x +e \right )^{\frac {2}{3}}-\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}} \left (f x +e \right )^{\frac {1}{3}}+\left (\frac {c f -d e}{d}\right )^{\frac {2}{3}}\right ) \left (c h -d g \right ) b \left (\frac {a f -b e}{b}\right )^{\frac {2}{3}}}{2}+\sqrt {3}\, \left (\frac {c f -d e}{d}\right )^{\frac {2}{3}} d \left (a h -b g \right ) \arctan \left (\frac {2 \sqrt {3}\, \left (f x +e \right )^{\frac {1}{3}}}{3 \left (\frac {a f -b e}{b}\right )^{\frac {1}{3}}}-\frac {\sqrt {3}}{3}\right )-\sqrt {3}\, \left (\frac {a f -b e}{b}\right )^{\frac {2}{3}} b \left (c h -d g \right ) \arctan \left (\frac {2 \sqrt {3}\, \left (f x +e \right )^{\frac {1}{3}}}{3 \left (\frac {c f -d e}{d}\right )^{\frac {1}{3}}}-\frac {\sqrt {3}}{3}\right )+\ln \left (\left (f x +e \right )^{\frac {1}{3}}+\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}}\right ) \left (a h -b g \right ) d \left (\frac {c f -d e}{d}\right )^{\frac {2}{3}}-\left (\frac {a f -b e}{b}\right )^{\frac {2}{3}} b \ln \left (\left (f x +e \right )^{\frac {1}{3}}+\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}}\right ) \left (c h -d g \right )}{\left (\frac {c f -d e}{d}\right )^{\frac {2}{3}} \left (\frac {a f -b e}{b}\right )^{\frac {2}{3}} \left (a d -b c \right ) b d}\)	\(405\)

Input:

int((h*x+g)/(b*x+a)/(d*x+c)/(f*x+e)^(2/3),x,method=_RETURNVERBOSE)
 

Output:

3*(1/3/d/((c*f-d*e)/d)^(2/3)*ln((f*x+e)^(1/3)+((c*f-d*e)/d)^(1/3))-1/6/d/( 
(c*f-d*e)/d)^(2/3)*ln((f*x+e)^(2/3)-((c*f-d*e)/d)^(1/3)*(f*x+e)^(1/3)+((c* 
f-d*e)/d)^(2/3))+1/3/d/((c*f-d*e)/d)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/( 
(c*f-d*e)/d)^(1/3)*(f*x+e)^(1/3)-1)))*(-c*h+d*g)/(a*d-b*c)+3*(1/3/b/((a*f- 
b*e)/b)^(2/3)*ln((f*x+e)^(1/3)+((a*f-b*e)/b)^(1/3))-1/6/b/((a*f-b*e)/b)^(2 
/3)*ln((f*x+e)^(2/3)-((a*f-b*e)/b)^(1/3)*(f*x+e)^(1/3)+((a*f-b*e)/b)^(2/3) 
)+1/3/b/((a*f-b*e)/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/((a*f-b*e)/b)^(1 
/3)*(f*x+e)^(1/3)-1)))*(a*h-b*g)/(a*d-b*c)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1455 vs. \(2 (324) = 648\).

Time = 49.49 (sec) , antiderivative size = 6174, normalized size of antiderivative = 15.99 \[ \int \frac {g+h x}{(a+b x) (c+d x) (e+f x)^{2/3}} \, dx=\text {Too large to display} \] Input:

integrate((h*x+g)/(b*x+a)/(d*x+c)/(f*x+e)^(2/3),x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F]

\[ \int \frac {g+h x}{(a+b x) (c+d x) (e+f x)^{2/3}} \, dx=\int \frac {g + h x}{\left (a + b x\right ) \left (c + d x\right ) \left (e + f x\right )^{\frac {2}{3}}}\, dx \] Input:

integrate((h*x+g)/(b*x+a)/(d*x+c)/(f*x+e)**(2/3),x)
 

Output:

Integral((g + h*x)/((a + b*x)*(c + d*x)*(e + f*x)**(2/3)), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {g+h x}{(a+b x) (c+d x) (e+f x)^{2/3}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((h*x+g)/(b*x+a)/(d*x+c)/(f*x+e)^(2/3),x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(c*f-d*e>0)', see `assume?` for m 
ore detail
 

Giac [A] (verification not implemented)

Time = 0.40 (sec) , antiderivative size = 644, normalized size of antiderivative = 1.67 \[ \int \frac {g+h x}{(a+b x) (c+d x) (e+f x)^{2/3}} \, dx =\text {Too large to display} \] Input:

integrate((h*x+g)/(b*x+a)/(d*x+c)/(f*x+e)^(2/3),x, algorithm="giac")
 

Output:

(b*g - a*h)*((b*e - a*f)/b)^(1/3)*log(abs((f*x + e)^(1/3) - ((b*e - a*f)/b 
)^(1/3)))/(b^2*c*e - a*b*d*e - a*b*c*f + a^2*d*f) - (d*g - c*h)*((d*e - c* 
f)/d)^(1/3)*log(abs((f*x + e)^(1/3) - ((d*e - c*f)/d)^(1/3)))/(b*c*d*e - a 
*d^2*e - b*c^2*f + a*c*d*f) - (sqrt(3)*(b^3*e - a*b^2*f)^(1/3)*b*g - sqrt( 
3)*(b^3*e - a*b^2*f)^(1/3)*a*h)*arctan(1/3*sqrt(3)*(2*(f*x + e)^(1/3) + (( 
b*e - a*f)/b)^(1/3))/((b*e - a*f)/b)^(1/3))/((b^3*e - a*b^2*f)*c - (a*b^2* 
e - a^2*b*f)*d) - (sqrt(3)*(d^3*e - c*d^2*f)^(1/3)*d*g - sqrt(3)*(d^3*e - 
c*d^2*f)^(1/3)*c*h)*arctan(1/3*sqrt(3)*(2*(f*x + e)^(1/3) + ((d*e - c*f)/d 
)^(1/3))/((d*e - c*f)/d)^(1/3))/((d^3*e - c*d^2*f)*a - (c*d^2*e - c^2*d*f) 
*b) - 1/2*((b^3*e - a*b^2*f)^(1/3)*b*g - (b^3*e - a*b^2*f)^(1/3)*a*h)*log( 
(f*x + e)^(2/3) + (f*x + e)^(1/3)*((b*e - a*f)/b)^(1/3) + ((b*e - a*f)/b)^ 
(2/3))/((b^3*e - a*b^2*f)*c - (a*b^2*e - a^2*b*f)*d) - 1/2*((d^3*e - c*d^2 
*f)^(1/3)*d*g - (d^3*e - c*d^2*f)^(1/3)*c*h)*log((f*x + e)^(2/3) + (f*x + 
e)^(1/3)*((d*e - c*f)/d)^(1/3) + ((d*e - c*f)/d)^(2/3))/((d^3*e - c*d^2*f) 
*a - (c*d^2*e - c^2*d*f)*b)
 

Mupad [B] (verification not implemented)

Time = 4.68 (sec) , antiderivative size = 13287, normalized size of antiderivative = 34.42 \[ \int \frac {g+h x}{(a+b x) (c+d x) (e+f x)^{2/3}} \, dx=\text {Too large to display} \] Input:

int((g + h*x)/((e + f*x)^(2/3)*(a + b*x)*(c + d*x)),x)
 

Output:

log((-(c^3*h^3 - d^3*g^3 + 3*c*d^2*g^2*h - 3*c^2*d*g*h^2)/(a^3*d^6*e^2 - b 
^3*c^5*d*f^2 + a^3*c^2*d^4*f^2 - b^3*c^3*d^3*e^2 - 2*a^3*c*d^5*e*f - 3*a^2 
*b*c*d^5*e^2 + 2*b^3*c^4*d^2*e*f + 3*a*b^2*c^2*d^4*e^2 + 3*a*b^2*c^4*d^2*f 
^2 - 3*a^2*b*c^3*d^3*f^2 - 6*a*b^2*c^3*d^3*e*f + 6*a^2*b*c^2*d^4*e*f))^(1/ 
3)*((-(c^3*h^3 - d^3*g^3 + 3*c*d^2*g^2*h - 3*c^2*d*g*h^2)/(a^3*d^6*e^2 - b 
^3*c^5*d*f^2 + a^3*c^2*d^4*f^2 - b^3*c^3*d^3*e^2 - 2*a^3*c*d^5*e*f - 3*a^2 
*b*c*d^5*e^2 + 2*b^3*c^4*d^2*e*f + 3*a*b^2*c^2*d^4*e^2 + 3*a*b^2*c^4*d^2*f 
^2 - 3*a^2*b*c^3*d^3*f^2 - 6*a*b^2*c^3*d^3*e*f + 6*a^2*b*c^2*d^4*e*f))^(2/ 
3)*((e + f*x)^(1/3)*(243*a^5*b^3*d^8*f^6*g + 243*b^8*c^5*d^3*f^6*g - 729*a 
*b^7*c^4*d^4*f^6*g - 729*a^4*b^4*c*d^7*f^6*g - 243*a*b^7*c^5*d^3*f^6*h - 2 
43*a^5*b^3*c*d^7*f^6*h - 486*a^4*b^4*d^8*e*f^5*g - 486*b^8*c^4*d^4*e*f^5*g 
 + 486*a^2*b^6*c^3*d^5*f^6*g + 486*a^3*b^5*c^2*d^6*f^6*g + 972*a^2*b^6*c^4 
*d^4*f^6*h - 1458*a^3*b^5*c^3*d^5*f^6*h + 972*a^4*b^4*c^2*d^6*f^6*h + 243* 
a^4*b^4*d^8*e^2*f^4*h + 243*b^8*c^4*d^4*e^2*f^4*h + 1458*a^2*b^6*c^2*d^6*e 
^2*f^4*h + 1944*a*b^7*c^3*d^5*e*f^5*g + 1944*a^3*b^5*c*d^7*e*f^5*g - 2916* 
a^2*b^6*c^2*d^6*e*f^5*g - 972*a*b^7*c^3*d^5*e^2*f^4*h - 972*a^3*b^5*c*d^7* 
e^2*f^4*h) + (-(c^3*h^3 - d^3*g^3 + 3*c*d^2*g^2*h - 3*c^2*d*g*h^2)/(a^3*d^ 
6*e^2 - b^3*c^5*d*f^2 + a^3*c^2*d^4*f^2 - b^3*c^3*d^3*e^2 - 2*a^3*c*d^5*e* 
f - 3*a^2*b*c*d^5*e^2 + 2*b^3*c^4*d^2*e*f + 3*a*b^2*c^2*d^4*e^2 + 3*a*b^2* 
c^4*d^2*f^2 - 3*a^2*b*c^3*d^3*f^2 - 6*a*b^2*c^3*d^3*e*f + 6*a^2*b*c^2*d...
 

Reduce [B] (verification not implemented)

Time = 0.55 (sec) , antiderivative size = 1207, normalized size of antiderivative = 3.13 \[ \int \frac {g+h x}{(a+b x) (c+d x) (e+f x)^{2/3}} \, dx =\text {Too large to display} \] Input:

int((h*x+g)/(b*x+a)/(d*x+c)/(f*x+e)^(2/3),x)
 

Output:

( - 2*d**(1/3)*(c*f - d*e)**(2/3)*sqrt(3)*atan((b**(1/6)*(a*f - b*e)**(1/6 
)*sqrt(3) - 2*b**(1/3)*(e + f*x)**(1/6))/(b**(1/6)*(a*f - b*e)**(1/6)))*a* 
h + 2*d**(1/3)*(c*f - d*e)**(2/3)*sqrt(3)*atan((b**(1/6)*(a*f - b*e)**(1/6 
)*sqrt(3) - 2*b**(1/3)*(e + f*x)**(1/6))/(b**(1/6)*(a*f - b*e)**(1/6)))*b* 
g - 2*d**(1/3)*(c*f - d*e)**(2/3)*sqrt(3)*atan((b**(1/6)*(a*f - b*e)**(1/6 
)*sqrt(3) + 2*b**(1/3)*(e + f*x)**(1/6))/(b**(1/6)*(a*f - b*e)**(1/6)))*a* 
h + 2*d**(1/3)*(c*f - d*e)**(2/3)*sqrt(3)*atan((b**(1/6)*(a*f - b*e)**(1/6 
)*sqrt(3) + 2*b**(1/3)*(e + f*x)**(1/6))/(b**(1/6)*(a*f - b*e)**(1/6)))*b* 
g + 2*b**(1/3)*(a*f - b*e)**(2/3)*sqrt(3)*atan((d**(1/6)*(c*f - d*e)**(1/6 
)*sqrt(3) - 2*d**(1/3)*(e + f*x)**(1/6))/(d**(1/6)*(c*f - d*e)**(1/6)))*c* 
h - 2*b**(1/3)*(a*f - b*e)**(2/3)*sqrt(3)*atan((d**(1/6)*(c*f - d*e)**(1/6 
)*sqrt(3) - 2*d**(1/3)*(e + f*x)**(1/6))/(d**(1/6)*(c*f - d*e)**(1/6)))*d* 
g + 2*b**(1/3)*(a*f - b*e)**(2/3)*sqrt(3)*atan((d**(1/6)*(c*f - d*e)**(1/6 
)*sqrt(3) + 2*d**(1/3)*(e + f*x)**(1/6))/(d**(1/6)*(c*f - d*e)**(1/6)))*c* 
h - 2*b**(1/3)*(a*f - b*e)**(2/3)*sqrt(3)*atan((d**(1/6)*(c*f - d*e)**(1/6 
)*sqrt(3) + 2*d**(1/3)*(e + f*x)**(1/6))/(d**(1/6)*(c*f - d*e)**(1/6)))*d* 
g - 2*b**(1/3)*(a*f - b*e)**(2/3)*log((c*f - d*e)**(1/3) + d**(1/3)*(e + f 
*x)**(1/3))*c*h + 2*b**(1/3)*(a*f - b*e)**(2/3)*log((c*f - d*e)**(1/3) + d 
**(1/3)*(e + f*x)**(1/3))*d*g + b**(1/3)*(a*f - b*e)**(2/3)*log( - d**(1/6 
)*(e + f*x)**(1/6)*(c*f - d*e)**(1/6)*sqrt(3) + (c*f - d*e)**(1/3) + d*...