\(\int \frac {g+h x}{(a+b x) (c+d x) (e+f x)^{4/3}} \, dx\) [217]

Optimal result

Integrand size = 29, antiderivative size = 425 \[ \int \frac {g+h x}{(a+b x) (c+d x) (e+f x)^{4/3}} \, dx=-\frac {3 (f g-e h)}{(b e-a f) (d e-c f) \sqrt [3]{e+f x}}+\frac {\sqrt {3} \sqrt [3]{b} (b g-a h) \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} \sqrt [3]{e+f x}}{\sqrt [3]{b e-a f}}}{\sqrt {3}}\right )}{(b c-a d) (b e-a f)^{4/3}}-\frac {\sqrt {3} \sqrt [3]{d} (d g-c h) \arctan \left (\frac {1+\frac {2 \sqrt [3]{d} \sqrt [3]{e+f x}}{\sqrt [3]{d e-c f}}}{\sqrt {3}}\right )}{(b c-a d) (d e-c f)^{4/3}}-\frac {\sqrt [3]{b} (b g-a h) \log (a+b x)}{2 (b c-a d) (b e-a f)^{4/3}}+\frac {\sqrt [3]{d} (d g-c h) \log (c+d x)}{2 (b c-a d) (d e-c f)^{4/3}}+\frac {3 \sqrt [3]{b} (b g-a h) \log \left (\sqrt [3]{b e-a f}-\sqrt [3]{b} \sqrt [3]{e+f x}\right )}{2 (b c-a d) (b e-a f)^{4/3}}-\frac {3 \sqrt [3]{d} (d g-c h) \log \left (\sqrt [3]{d e-c f}-\sqrt [3]{d} \sqrt [3]{e+f x}\right )}{2 (b c-a d) (d e-c f)^{4/3}} \] Output:

(3*e*h-3*f*g)/(-a*f+b*e)/(-c*f+d*e)/(f*x+e)^(1/3)+3^(1/2)*b^(1/3)*(-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))/(-a*d+ 
b*c)/(-a*f+b*e)^(4/3)-3^(1/2)*d^(1/3)*(-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))/(-a*d+b*c)/(-c*f+d*e)^(4/3)-1/2*b^ 
(1/3)*(-a*h+b*g)*ln(b*x+a)/(-a*d+b*c)/(-a*f+b*e)^(4/3)+1/2*d^(1/3)*(-c*h+d 
*g)*ln(d*x+c)/(-a*d+b*c)/(-c*f+d*e)^(4/3)+3/2*b^(1/3)*(-a*h+b*g)*ln((-a*f+ 
b*e)^(1/3)-b^(1/3)*(f*x+e)^(1/3))/(-a*d+b*c)/(-a*f+b*e)^(4/3)-3/2*d^(1/3)* 
(-c*h+d*g)*ln((-c*f+d*e)^(1/3)-d^(1/3)*(f*x+e)^(1/3))/(-a*d+b*c)/(-c*f+d*e 
)^(4/3)
 

Mathematica [A] (verified)

Time = 3.57 (sec) , antiderivative size = 519, normalized size of antiderivative = 1.22 \[ \int \frac {g+h x}{(a+b x) (c+d x) (e+f x)^{4/3}} \, dx=\frac {3 (-f g+e h)}{(b e-a f) (d e-c f) \sqrt [3]{e+f x}}+\frac {\sqrt {3} \sqrt [3]{b} (b g-a h) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{e+f x}}{\sqrt [3]{-b e+a f}}}{\sqrt {3}}\right )}{(b c-a d) (-b e+a f)^{4/3}}-\frac {\sqrt {3} \sqrt [3]{d} (d g-c h) \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{e+f x}}{\sqrt [3]{-d e+c f}}}{\sqrt {3}}\right )}{(b c-a d) (-d e+c f)^{4/3}}+\frac {\sqrt [3]{b} (b g-a h) \log \left (\sqrt [3]{-b e+a f}+\sqrt [3]{b} \sqrt [3]{e+f x}\right )}{(b c-a d) (-b e+a f)^{4/3}}+\frac {\sqrt [3]{d} (d g-c h) \log \left (\sqrt [3]{-d e+c f}+\sqrt [3]{d} \sqrt [3]{e+f x}\right )}{(-b c+a d) (-d e+c f)^{4/3}}-\frac {\sqrt [3]{b} (b g-a h) \log \left ((-b e+a f)^{2/3}-\sqrt [3]{b} \sqrt [3]{-b e+a f} \sqrt [3]{e+f x}+b^{2/3} (e+f x)^{2/3}\right )}{2 (b c-a d) (-b e+a f)^{4/3}}-\frac {\sqrt [3]{d} (d g-c h) \log \left ((-d e+c f)^{2/3}-\sqrt [3]{d} \sqrt [3]{-d e+c f} \sqrt [3]{e+f x}+d^{2/3} (e+f x)^{2/3}\right )}{2 (-b c+a d) (-d e+c f)^{4/3}} \] Input:

Integrate[(g + h*x)/((a + b*x)*(c + d*x)*(e + f*x)^(4/3)),x]
 

Output:

(3*(-(f*g) + e*h))/((b*e - a*f)*(d*e - c*f)*(e + f*x)^(1/3)) + (Sqrt[3]*b^ 
(1/3)*(b*g - a*h)*ArcTan[(1 - (2*b^(1/3)*(e + f*x)^(1/3))/(-(b*e) + a*f)^( 
1/3))/Sqrt[3]])/((b*c - a*d)*(-(b*e) + a*f)^(4/3)) - (Sqrt[3]*d^(1/3)*(d*g 
 - c*h)*ArcTan[(1 - (2*d^(1/3)*(e + f*x)^(1/3))/(-(d*e) + c*f)^(1/3))/Sqrt 
[3]])/((b*c - a*d)*(-(d*e) + c*f)^(4/3)) + (b^(1/3)*(b*g - a*h)*Log[(-(b*e 
) + a*f)^(1/3) + b^(1/3)*(e + f*x)^(1/3)])/((b*c - a*d)*(-(b*e) + a*f)^(4/ 
3)) + (d^(1/3)*(d*g - c*h)*Log[(-(d*e) + c*f)^(1/3) + d^(1/3)*(e + f*x)^(1 
/3)])/((-(b*c) + a*d)*(-(d*e) + c*f)^(4/3)) - (b^(1/3)*(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)])/(2*(b*c - a*d)*(-(b*e) + a*f)^(4/3)) - (d^(1/3)*(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*c) + a*d)*(-(d*e) + c*f)^(4/3))
 

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 386, normalized size of antiderivative = 0.91, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {174, 61, 67, 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.

\(\displaystyle \int \frac {g+h x}{(a+b x) (c+d x) (e+f x)^{4/3}} \, dx\)

\(\Big \downarrow \) 174

\(\displaystyle \frac {(b g-a h) \int \frac {1}{(a+b x) (e+f x)^{4/3}}dx}{b c-a d}-\frac {(d g-c h) \int \frac {1}{(c+d x) (e+f x)^{4/3}}dx}{b c-a d}\)

\(\Big \downarrow \) 61

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

\(\Big \downarrow \) 67

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

Input:

Int[(g + h*x)/((a + b*x)*(c + d*x)*(e + f*x)^(4/3)),x]
 

Output:

((b*g - a*h)*(3/((b*e - a*f)*(e + f*x)^(1/3)) + (b*((Sqrt[3]*ArcTan[(1 + ( 
2*b^(1/3)*(e + f*x)^(1/3))/(b*e - a*f)^(1/3))/Sqrt[3]])/(b^(2/3)*(b*e - a* 
f)^(1/3)) - Log[a + b*x]/(2*b^(2/3)*(b*e - a*f)^(1/3)) + (3*Log[(b*e - a*f 
)^(1/3) - b^(1/3)*(e + f*x)^(1/3)])/(2*b^(2/3)*(b*e - a*f)^(1/3))))/(b*e - 
 a*f)))/(b*c - a*d) - ((d*g - c*h)*(3/((d*e - c*f)*(e + f*x)^(1/3)) + (d*( 
(Sqrt[3]*ArcTan[(1 + (2*d^(1/3)*(e + f*x)^(1/3))/(d*e - c*f)^(1/3))/Sqrt[3 
]])/(d^(2/3)*(d*e - c*f)^(1/3)) - Log[c + d*x]/(2*d^(2/3)*(d*e - c*f)^(1/3 
)) + (3*Log[(d*e - c*f)^(1/3) - d^(1/3)*(e + f*x)^(1/3)])/(2*d^(2/3)*(d*e 
- c*f)^(1/3))))/(d*e - c*f)))/(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 61

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( 
m + n + 2)/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], 
x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0 
] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d 
, m, n, x]
 

rule 67

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[ 
{q = Rt[(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q), x 
] + (Simp[3/(2*b)   Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/3)], 
 x] - Simp[3/(2*b*q)   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 = 0.72 (sec) , antiderivative size = 423, normalized size of antiderivative = 1.00

method	result	size
derivativedivides	\(\frac {3 \left (-\frac {\ln \left (\left (f x +e \right )^{\frac {1}{3}}+\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}}\right )}{3 d \left (\frac {c f -d e}{d}\right )^{\frac {1}{3}}}+\frac {\ln \left (\left (f x +e \right )^{\frac {2}{3}}-\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}} \left (f x +e \right )^{\frac {1}{3}}+\left (\frac {c f -d e}{d}\right )^{\frac {2}{3}}\right )}{6 d \left (\frac {c f -d e}{d}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (f x +e \right )^{\frac {1}{3}}}{\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 d \left (\frac {c f -d e}{d}\right )^{\frac {1}{3}}}\right ) d \left (c h -d g \right )}{\left (a d -b c \right ) \left (c f -d e \right )}-\frac {3 \left (-e h +f g \right )}{\left (c f -d e \right ) \left (a f -b e \right ) \left (f x +e \right )^{\frac {1}{3}}}-\frac {3 \left (-\frac {\ln \left (\left (f x +e \right )^{\frac {1}{3}}+\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a f -b e}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (\left (f x +e \right )^{\frac {2}{3}}-\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}} \left (f x +e \right )^{\frac {1}{3}}+\left (\frac {a f -b e}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a f -b e}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (f x +e \right )^{\frac {1}{3}}}{\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a f -b e}{b}\right )^{\frac {1}{3}}}\right ) b \left (a h -b g \right )}{\left (a d -b c \right ) \left (a f -b e \right )}\)	\(423\)
default	\(\frac {3 \left (-\frac {\ln \left (\left (f x +e \right )^{\frac {1}{3}}+\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}}\right )}{3 d \left (\frac {c f -d e}{d}\right )^{\frac {1}{3}}}+\frac {\ln \left (\left (f x +e \right )^{\frac {2}{3}}-\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}} \left (f x +e \right )^{\frac {1}{3}}+\left (\frac {c f -d e}{d}\right )^{\frac {2}{3}}\right )}{6 d \left (\frac {c f -d e}{d}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (f x +e \right )^{\frac {1}{3}}}{\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 d \left (\frac {c f -d e}{d}\right )^{\frac {1}{3}}}\right ) d \left (c h -d g \right )}{\left (a d -b c \right ) \left (c f -d e \right )}-\frac {3 \left (-e h +f g \right )}{\left (c f -d e \right ) \left (a f -b e \right ) \left (f x +e \right )^{\frac {1}{3}}}-\frac {3 \left (-\frac {\ln \left (\left (f x +e \right )^{\frac {1}{3}}+\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a f -b e}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (\left (f x +e \right )^{\frac {2}{3}}-\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}} \left (f x +e \right )^{\frac {1}{3}}+\left (\frac {a f -b e}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a f -b e}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (f x +e \right )^{\frac {1}{3}}}{\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a f -b e}{b}\right )^{\frac {1}{3}}}\right ) b \left (a h -b g \right )}{\left (a d -b c \right ) \left (a f -b e \right )}\)	\(423\)
pseudoelliptic	\(-\frac {\ln \left (\left (f x +e \right )^{\frac {1}{3}}+\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}}\right ) \left (c h -d g \right )}{\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}} \left (a d -b c \right ) \left (c f -d e \right )}+\frac {\ln \left (\left (f x +e \right )^{\frac {2}{3}}-\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}} \left (f x +e \right )^{\frac {1}{3}}+\left (\frac {c f -d e}{d}\right )^{\frac {2}{3}}\right ) \left (c h -d g \right )}{2 \left (\frac {c f -d e}{d}\right )^{\frac {1}{3}} \left (a d -b c \right ) \left (c f -d e \right )}+\frac {\sqrt {3}\, \arctan \left (\frac {2 \sqrt {3}\, \left (f x +e \right )^{\frac {1}{3}}}{3 \left (\frac {c f -d e}{d}\right )^{\frac {1}{3}}}-\frac {\sqrt {3}}{3}\right ) \left (c h -d g \right )}{\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}} \left (a d -b c \right ) \left (c f -d e \right )}+\frac {3 e h -3 f g}{\left (c f -d e \right ) \left (a f -b e \right ) \left (f x +e \right )^{\frac {1}{3}}}+\frac {\ln \left (\left (f x +e \right )^{\frac {1}{3}}+\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}}\right ) \left (a h -b g \right )}{\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}} \left (a f -b e \right ) \left (a d -b c \right )}-\frac {\ln \left (\left (f x +e \right )^{\frac {2}{3}}-\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}} \left (f x +e \right )^{\frac {1}{3}}+\left (\frac {a f -b e}{b}\right )^{\frac {2}{3}}\right ) \left (a h -b g \right )}{2 \left (\frac {a f -b e}{b}\right )^{\frac {1}{3}} \left (a f -b e \right ) \left (a d -b c \right )}-\frac {\sqrt {3}\, \arctan \left (\frac {2 \sqrt {3}\, \left (f x +e \right )^{\frac {1}{3}}}{3 \left (\frac {a f -b e}{b}\right )^{\frac {1}{3}}}-\frac {\sqrt {3}}{3}\right ) \left (a h -b g \right )}{\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}} \left (a f -b e \right ) \left (a d -b c \right )}\)	\(511\)

Input:

int((h*x+g)/(b*x+a)/(d*x+c)/(f*x+e)^(4/3),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [F(-1)]

Timed out. \[ \int \frac {g+h x}{(a+b x) (c+d x) (e+f x)^{4/3}} \, dx=\text {Timed out} \] Input:

integrate((h*x+g)/(b*x+a)/(d*x+c)/(f*x+e)^(4/3),x, algorithm="fricas")
                                                                                    
                                                                                    
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {g+h x}{(a+b x) (c+d x) (e+f x)^{4/3}} \, dx=\int \frac {g + h x}{\left (a + b x\right ) \left (c + d x\right ) \left (e + f x\right )^{\frac {4}{3}}}\, dx \] Input:

integrate((h*x+g)/(b*x+a)/(d*x+c)/(f*x+e)**(4/3),x)
 

Output:

Integral((g + h*x)/((a + b*x)*(c + d*x)*(e + f*x)**(4/3)), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {g+h x}{(a+b x) (c+d x) (e+f x)^{4/3}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((h*x+g)/(b*x+a)/(d*x+c)/(f*x+e)^(4/3),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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 912 vs. \(2 (361) = 722\).

Time = 0.80 (sec) , antiderivative size = 912, normalized size of antiderivative = 2.15 \[ \int \frac {g+h x}{(a+b x) (c+d x) (e+f x)^{4/3}} \, dx =\text {Too large to display} \] Input:

integrate((h*x+g)/(b*x+a)/(d*x+c)/(f*x+e)^(4/3),x, algorithm="giac")
 

Output:

(b^2*g*((b*e - a*f)/b)^(1/3) - a*b*h*((b*e - a*f)/b)^(1/3))*((b*e - a*f)/b 
)^(1/3)*log(abs((f*x + e)^(1/3) - ((b*e - a*f)/b)^(1/3)))/(b^3*c*e^2 - a*b 
^2*d*e^2 - 2*a*b^2*c*e*f + 2*a^2*b*d*e*f + a^2*b*c*f^2 - a^3*d*f^2) - (d^2 
*g*((d*e - c*f)/d)^(1/3) - c*d*h*((d*e - c*f)/d)^(1/3))*((d*e - c*f)/d)^(1 
/3)*log(abs((f*x + e)^(1/3) - ((d*e - c*f)/d)^(1/3)))/(b*c*d^2*e^2 - a*d^3 
*e^2 - 2*b*c^2*d*e*f + 2*a*c*d^2*e*f + b*c^3*f^2 - a*c^2*d*f^2) + (sqrt(3) 
*(b^3*e - a*b^2*f)^(2/3)*b*g - sqrt(3)*(b^3*e - a*b^2*f)^(2/3)*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^4*e^2 - 2*a*b^3*e*f + a^2*b^2*f^2)*c - (a*b^3*e^2 - 2*a^2*b^2*e* 
f + a^3*b*f^2)*d) + (sqrt(3)*(d^3*e - c*d^2*f)^(2/3)*d*g - sqrt(3)*(d^3*e 
- c*d^2*f)^(2/3)*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^4*e^2 - 2*c*d^3*e*f + c^2*d^2*f^2)*a 
 - (c*d^3*e^2 - 2*c^2*d^2*e*f + c^3*d*f^2)*b) - 1/2*((b^3*e - a*b^2*f)^(2/ 
3)*b*g - (b^3*e - a*b^2*f)^(2/3)*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^4*e^2 - 2*a*b^3*e*f + 
 a^2*b^2*f^2)*c - (a*b^3*e^2 - 2*a^2*b^2*e*f + a^3*b*f^2)*d) - 1/2*((d^3*e 
 - c*d^2*f)^(2/3)*d*g - (d^3*e - c*d^2*f)^(2/3)*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^4*e^2 
- 2*c*d^3*e*f + c^2*d^2*f^2)*a - (c*d^3*e^2 - 2*c^2*d^2*e*f + c^3*d*f^2)*b 
) - 3*(f*g - e*h)/((b*d*e^2 - b*c*e*f - a*d*e*f + a*c*f^2)*(f*x + e)^(1...
 

Mupad [B] (verification not implemented)

Time = 17.24 (sec) , antiderivative size = 373297, normalized size of antiderivative = 878.35 \[ \int \frac {g+h x}{(a+b x) (c+d x) (e+f x)^{4/3}} \, dx=\text {Too large to display} \] Input:

int((g + h*x)/((e + f*x)^(4/3)*(a + b*x)*(c + d*x)),x)
 

Output:

log((((4*(b^4*d^4*g^6 + 3*a^2*b^2*d^4*g^4*h^2 + 3*b^4*c^2*d^2*g^4*h^2 + a^ 
3*b*c^3*d*h^6 - 3*a*b^3*d^4*g^5*h - 3*b^4*c*d^3*g^5*h - a^3*b*d^4*g^3*h^3 
- b^4*c^3*d*g^3*h^3 + 9*a*b^3*c*d^3*g^4*h^2 + 3*a*b^3*c^3*d*g^2*h^4 - 3*a^ 
2*b^2*c^3*d*g*h^5 + 3*a^3*b*c*d^3*g^2*h^4 - 3*a^3*b*c^2*d^2*g*h^5 - 9*a*b^ 
3*c^2*d^2*g^3*h^3 - 9*a^2*b^2*c*d^3*g^3*h^3 + 9*a^2*b^2*c^2*d^2*g^2*h^4)*( 
a^4*b^6*c^10*f^8 + a^6*b^4*d^10*e^8 + a^10*c^4*d^6*f^8 + b^10*c^6*d^4*e^8 
+ a^10*d^10*e^4*f^4 + b^10*c^10*e^4*f^4 + 15*a^2*b^8*c^4*d^6*e^8 - 20*a^3* 
b^7*c^3*d^7*e^8 + 15*a^4*b^6*c^2*d^8*e^8 + 15*a^6*b^4*c^8*d^2*f^8 - 20*a^7 
*b^3*c^7*d^3*f^8 + 15*a^8*b^2*c^6*d^4*f^8 + 6*a^2*b^8*c^10*e^2*f^6 + 6*a^8 
*b^2*d^10*e^6*f^2 + 6*a^10*c^2*d^8*e^2*f^6 + 6*b^10*c^8*d^2*e^6*f^2 - 6*a* 
b^9*c^5*d^5*e^8 - 6*a^5*b^5*c*d^9*e^8 - 6*a^5*b^5*c^9*d*f^8 - 6*a^9*b*c^5* 
d^5*f^8 - 4*a*b^9*c^10*e^3*f^5 - 4*a^3*b^7*c^10*e*f^7 - 4*a^7*b^3*d^10*e^7 
*f - 4*a^9*b*d^10*e^5*f^3 - 4*a^10*c*d^9*e^3*f^5 - 4*a^10*c^3*d^7*e*f^7 - 
4*b^10*c^7*d^3*e^7*f - 4*b^10*c^9*d*e^5*f^3 + 20*a*b^9*c^6*d^4*e^7*f + 10* 
a*b^9*c^9*d*e^4*f^4 + 20*a^4*b^6*c^9*d*e*f^7 + 20*a^6*b^4*c*d^9*e^7*f + 10 
*a^9*b*c*d^9*e^4*f^4 + 20*a^9*b*c^4*d^6*e*f^7 - 20*a*b^9*c^7*d^3*e^6*f^2 - 
 36*a^2*b^8*c^5*d^5*e^7*f + 20*a^3*b^7*c^4*d^6*e^7*f - 20*a^3*b^7*c^9*d*e^ 
2*f^6 + 20*a^4*b^6*c^3*d^7*e^7*f - 36*a^5*b^5*c^2*d^8*e^7*f - 36*a^5*b^5*c 
^8*d^2*e*f^7 + 20*a^6*b^4*c^7*d^3*e*f^7 - 20*a^7*b^3*c*d^9*e^6*f^2 + 20*a^ 
7*b^3*c^6*d^4*e*f^7 - 36*a^8*b^2*c^5*d^5*e*f^7 - 20*a^9*b*c^3*d^7*e^2*f...
 

Reduce [B] (verification not implemented)

Time = 0.84 (sec) , antiderivative size = 2870, normalized size of antiderivative = 6.75 \[ \int \frac {g+h x}{(a+b x) (c+d x) (e+f x)^{4/3}} \, dx =\text {Too large to display} \] Input:

int((h*x+g)/(b*x+a)/(d*x+c)/(f*x+e)^(4/3),x)
 

Output:

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