\(\int \frac {\sqrt [3]{e+f x} (g+h x)}{(a+b x)^2 (c+d x)} \, dx\) [224]

Optimal result

Integrand size = 29, antiderivative size = 528 \[ \int \frac {\sqrt [3]{e+f x} (g+h x)}{(a+b x)^2 (c+d x)} \, dx=-\frac {(b g-a h) \sqrt [3]{e+f x}}{b (b c-a d) (a+b x)}-\frac {\left (a^2 d f h+2 a b f (d g-2 c h)-b^2 (3 d e g-c f g-3 c e h)\right ) \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} \sqrt [3]{e+f x}}{\sqrt [3]{b e-a f}}}{\sqrt {3}}\right )}{\sqrt {3} b^{4/3} (b c-a d)^2 (b e-a f)^{2/3}}-\frac {\sqrt {3} \sqrt [3]{d e-c f} (d g-c h) \arctan \left (\frac {1+\frac {2 \sqrt [3]{d} \sqrt [3]{e+f x}}{\sqrt [3]{d e-c f}}}{\sqrt {3}}\right )}{\sqrt [3]{d} (b c-a d)^2}-\frac {\left (a^2 d f h+2 a b f (d g-2 c h)-b^2 (3 d e g-c f g-3 c e h)\right ) \log (a+b x)}{6 b^{4/3} (b c-a d)^2 (b e-a f)^{2/3}}-\frac {\sqrt [3]{d e-c f} (d g-c h) \log (c+d x)}{2 \sqrt [3]{d} (b c-a d)^2}+\frac {\left (a^2 d f h+2 a b f (d g-2 c h)-b^2 (3 d e g-c f g-3 c e h)\right ) \log \left (\sqrt [3]{b e-a f}-\sqrt [3]{b} \sqrt [3]{e+f x}\right )}{2 b^{4/3} (b c-a d)^2 (b e-a f)^{2/3}}+\frac {3 \sqrt [3]{d e-c f} (d g-c h) \log \left (\sqrt [3]{d e-c f}-\sqrt [3]{d} \sqrt [3]{e+f x}\right )}{2 \sqrt [3]{d} (b c-a d)^2} \] Output:

-(-a*h+b*g)*(f*x+e)^(1/3)/b/(-a*d+b*c)/(b*x+a)-1/3*(a^2*d*f*h+2*a*b*f*(-2* 
c*h+d*g)-b^2*(-3*c*e*h-c*f*g+3*d*e*g))*arctan(1/3*(1+2*b^(1/3)*(f*x+e)^(1/ 
3)/(-a*f+b*e)^(1/3))*3^(1/2))*3^(1/2)/b^(4/3)/(-a*d+b*c)^2/(-a*f+b*e)^(2/3 
)-3^(1/2)*(-c*f+d*e)^(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))/d^(1/3)/(-a*d+b*c)^2-1/6*(a^2*d*f*h+2*a*b*f*( 
-2*c*h+d*g)-b^2*(-3*c*e*h-c*f*g+3*d*e*g))*ln(b*x+a)/b^(4/3)/(-a*d+b*c)^2/( 
-a*f+b*e)^(2/3)-1/2*(-c*f+d*e)^(1/3)*(-c*h+d*g)*ln(d*x+c)/d^(1/3)/(-a*d+b* 
c)^2+1/2*(a^2*d*f*h+2*a*b*f*(-2*c*h+d*g)-b^2*(-3*c*e*h-c*f*g+3*d*e*g))*ln( 
(-a*f+b*e)^(1/3)-b^(1/3)*(f*x+e)^(1/3))/b^(4/3)/(-a*d+b*c)^2/(-a*f+b*e)^(2 
/3)+3/2*(-c*f+d*e)^(1/3)*(-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)^2
 

Mathematica [A] (verified)

Time = 3.33 (sec) , antiderivative size = 565, normalized size of antiderivative = 1.07 \[ \int \frac {\sqrt [3]{e+f x} (g+h x)}{(a+b x)^2 (c+d x)} \, dx=-\frac {\frac {6 (b c-a d) (b g-a h) \sqrt [3]{e+f x}}{b (a+b x)}+\frac {2 \sqrt {3} \left (a^2 d f h+2 a b f (d g-2 c h)+b^2 (-3 d e g+c f g+3 c e h)\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{e+f x}}{\sqrt [3]{-b e+a f}}}{\sqrt {3}}\right )}{b^{4/3} (-b e+a f)^{2/3}}-\frac {6 \sqrt {3} \sqrt [3]{-d e+c f} (d g-c h) \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{e+f x}}{\sqrt [3]{-d e+c f}}}{\sqrt {3}}\right )}{\sqrt [3]{d}}-\frac {2 \left (a^2 d f h+2 a b f (d g-2 c h)+b^2 (-3 d e g+c f g+3 c e h)\right ) \log \left (\sqrt [3]{-b e+a f}+\sqrt [3]{b} \sqrt [3]{e+f x}\right )}{b^{4/3} (-b e+a f)^{2/3}}+\frac {6 \sqrt [3]{-d e+c f} (d g-c h) \log \left (\sqrt [3]{-d e+c f}+\sqrt [3]{d} \sqrt [3]{e+f x}\right )}{\sqrt [3]{d}}+\frac {\left (a^2 d f h+2 a b f (d g-2 c h)+b^2 (-3 d e g+c f g+3 c e h)\right ) \log \left ((-b e+a f)^{2/3}-\sqrt [3]{b} \sqrt [3]{-b e+a f} \sqrt [3]{e+f x}+b^{2/3} (e+f x)^{2/3}\right )}{b^{4/3} (-b e+a f)^{2/3}}-\frac {3 \sqrt [3]{-d e+c f} (d g-c h) \log \left ((-d e+c f)^{2/3}-\sqrt [3]{d} \sqrt [3]{-d e+c f} \sqrt [3]{e+f x}+d^{2/3} (e+f x)^{2/3}\right )}{\sqrt [3]{d}}}{6 (b c-a d)^2} \] Input:

Integrate[((e + f*x)^(1/3)*(g + h*x))/((a + b*x)^2*(c + d*x)),x]
 

Output:

-1/6*((6*(b*c - a*d)*(b*g - a*h)*(e + f*x)^(1/3))/(b*(a + b*x)) + (2*Sqrt[ 
3]*(a^2*d*f*h + 2*a*b*f*(d*g - 2*c*h) + b^2*(-3*d*e*g + c*f*g + 3*c*e*h))* 
ArcTan[(1 - (2*b^(1/3)*(e + f*x)^(1/3))/(-(b*e) + a*f)^(1/3))/Sqrt[3]])/(b 
^(4/3)*(-(b*e) + a*f)^(2/3)) - (6*Sqrt[3]*(-(d*e) + c*f)^(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]])/d 
^(1/3) - (2*(a^2*d*f*h + 2*a*b*f*(d*g - 2*c*h) + b^2*(-3*d*e*g + c*f*g + 3 
*c*e*h))*Log[(-(b*e) + a*f)^(1/3) + b^(1/3)*(e + f*x)^(1/3)])/(b^(4/3)*(-( 
b*e) + a*f)^(2/3)) + (6*(-(d*e) + c*f)^(1/3)*(d*g - c*h)*Log[(-(d*e) + c*f 
)^(1/3) + d^(1/3)*(e + f*x)^(1/3)])/d^(1/3) + ((a^2*d*f*h + 2*a*b*f*(d*g - 
 2*c*h) + b^2*(-3*d*e*g + c*f*g + 3*c*e*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^( 
4/3)*(-(b*e) + a*f)^(2/3)) - (3*(-(d*e) + c*f)^(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)])/d^(1/3))/(b*c - a*d)^2
 

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 421, normalized size of antiderivative = 0.80, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {166, 27, 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.

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

\(\Big \downarrow \) 166

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 174

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

\(\Big \downarrow \) 69

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

Input:

Int[((e + f*x)^(1/3)*(g + h*x))/((a + b*x)^2*(c + d*x)),x]
 

Output:

-(((b*g - a*h)*(e + f*x)^(1/3))/(b*(b*c - a*d)*(a + b*x))) - (-(((a^2*d*f* 
h + 2*a*b*f*(d*g - 2*c*h) - b^2*(3*d*e*g - c*f*g - 3*c*e*h))*(-((Sqrt[3]*A 
rcTan[(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*L 
og[(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)) - (3*b*(d*e - c*f)*(d*g - c*h)*(-((Sqrt[3]*ArcTan[(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*Log[(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))/(3*b*(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 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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 166

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + 
 d*x)^n*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Simp[1/(b*(b*e - 
a*f)*(m + 1))   Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b* 
c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h 
)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, 
e, f, g, h, p}, x] && ILtQ[m, -1] && GtQ[n, 0]
 

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 = 0.90 (sec) , antiderivative size = 415, normalized size of antiderivative = 0.79

method	result	size
pseudoelliptic	\(\frac {-3 \left (\left (f x +e \right )^{\frac {1}{3}} \left (\frac {c f -d e}{d}\right )^{\frac {2}{3}} d \left (a h -b g \right ) \left (a d -b c \right )-\left (\arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (f x +e \right )^{\frac {1}{3}}}{\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}}}-1\right )}{3}\right ) \sqrt {3}+\ln \left (\left (f x +e \right )^{\frac {1}{3}}+\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}}\right )-\frac {\ln \left (\left (f x +e \right )^{\frac {2}{3}}-\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}} \left (f x +e \right )^{\frac {1}{3}}+\left (\frac {c f -d e}{d}\right )^{\frac {2}{3}}\right )}{2}\right ) \left (c f -d e \right ) \left (b x +a \right ) \left (c h -d g \right ) b \right ) b \left (\frac {a f -b e}{b}\right )^{\frac {2}{3}}+\left (\arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (f x +e \right )^{\frac {1}{3}}}{\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right ) \sqrt {3}+\ln \left (\left (f x +e \right )^{\frac {1}{3}}+\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}}\right )-\frac {\ln \left (\left (f x +e \right )^{\frac {2}{3}}-\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}} \left (f x +e \right )^{\frac {1}{3}}+\left (\frac {a f -b e}{b}\right )^{\frac {2}{3}}\right )}{2}\right ) \left (\left (a^{2} f h +2 b f g a -3 b^{2} e g \right ) d -4 \left (\frac {\left (-3 e h -f g \right ) b}{4}+a f h \right ) c b \right ) d \left (b x +a \right ) \left (\frac {c f -d e}{d}\right )^{\frac {2}{3}}}{3 \left (\frac {c f -d e}{d}\right )^{\frac {2}{3}} \left (\frac {a f -b e}{b}\right )^{\frac {2}{3}} d \left (a d -b c \right )^{2} b^{2} \left (b x +a \right )}\)	\(415\)
derivativedivides	\(3 f \left (\frac {\left (\frac {\ln \left (\left (f x +e \right )^{\frac {1}{3}}+\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}}\right )}{3 d \left (\frac {c f -d e}{d}\right )^{\frac {2}{3}}}-\frac {\ln \left (\left (f x +e \right )^{\frac {2}{3}}-\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}} \left (f x +e \right )^{\frac {1}{3}}+\left (\frac {c f -d e}{d}\right )^{\frac {2}{3}}\right )}{6 d \left (\frac {c f -d e}{d}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (f x +e \right )^{\frac {1}{3}}}{\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 d \left (\frac {c f -d e}{d}\right )^{\frac {2}{3}}}\right ) \left (c f -d e \right ) \left (c h -d g \right )}{f \left (a d -b c \right )^{2}}-\frac {\frac {f \left (a^{2} d h -a b c h -a b d g +b^{2} c g \right ) \left (f x +e \right )^{\frac {1}{3}}}{3 b \left (\left (f x +e \right ) b +a f -b e \right )}+\frac {\left (a^{2} d f h -4 a b c f h +2 a b d f g +3 b^{2} c e h +b^{2} c f g -3 b^{2} d e g \right ) \left (-\frac {\ln \left (\left (f x +e \right )^{\frac {1}{3}}+\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a f -b e}{b}\right )^{\frac {2}{3}}}+\frac {\ln \left (\left (f x +e \right )^{\frac {2}{3}}-\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}} \left (f x +e \right )^{\frac {1}{3}}+\left (\frac {a f -b e}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a f -b e}{b}\right )^{\frac {2}{3}}}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (f x +e \right )^{\frac {1}{3}}}{\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a f -b e}{b}\right )^{\frac {2}{3}}}\right )}{3 b}}{\left (a d -b c \right )^{2} f}\right )\)	\(478\)
default	\(3 f \left (\frac {\left (\frac {\ln \left (\left (f x +e \right )^{\frac {1}{3}}+\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}}\right )}{3 d \left (\frac {c f -d e}{d}\right )^{\frac {2}{3}}}-\frac {\ln \left (\left (f x +e \right )^{\frac {2}{3}}-\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}} \left (f x +e \right )^{\frac {1}{3}}+\left (\frac {c f -d e}{d}\right )^{\frac {2}{3}}\right )}{6 d \left (\frac {c f -d e}{d}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (f x +e \right )^{\frac {1}{3}}}{\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 d \left (\frac {c f -d e}{d}\right )^{\frac {2}{3}}}\right ) \left (c f -d e \right ) \left (c h -d g \right )}{f \left (a d -b c \right )^{2}}-\frac {\frac {f \left (a^{2} d h -a b c h -a b d g +b^{2} c g \right ) \left (f x +e \right )^{\frac {1}{3}}}{3 b \left (\left (f x +e \right ) b +a f -b e \right )}+\frac {\left (a^{2} d f h -4 a b c f h +2 a b d f g +3 b^{2} c e h +b^{2} c f g -3 b^{2} d e g \right ) \left (-\frac {\ln \left (\left (f x +e \right )^{\frac {1}{3}}+\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a f -b e}{b}\right )^{\frac {2}{3}}}+\frac {\ln \left (\left (f x +e \right )^{\frac {2}{3}}-\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}} \left (f x +e \right )^{\frac {1}{3}}+\left (\frac {a f -b e}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a f -b e}{b}\right )^{\frac {2}{3}}}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (f x +e \right )^{\frac {1}{3}}}{\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a f -b e}{b}\right )^{\frac {2}{3}}}\right )}{3 b}}{\left (a d -b c \right )^{2} f}\right )\)	\(478\)

Input:

int((f*x+e)^(1/3)*(h*x+g)/(b*x+a)^2/(d*x+c),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [F(-1)]

Timed out. \[ \int \frac {\sqrt [3]{e+f x} (g+h x)}{(a+b x)^2 (c+d x)} \, dx=\text {Timed out} \] Input:

integrate((f*x+e)^(1/3)*(h*x+g)/(b*x+a)^2/(d*x+c),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

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

integrate((f*x+e)**(1/3)*(h*x+g)/(b*x+a)**2/(d*x+c),x)
 

Output:

Integral((e + f*x)**(1/3)*(g + h*x)/((a + b*x)**2*(c + d*x)), x)
 

Maxima [F(-2)]

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

integrate((f*x+e)^(1/3)*(h*x+g)/(b*x+a)^2/(d*x+c),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 = 0.66 (sec) , antiderivative size = 888, normalized size of antiderivative = 1.68 \[ \int \frac {\sqrt [3]{e+f x} (g+h x)}{(a+b x)^2 (c+d x)} \, dx =\text {Too large to display} \] Input:

integrate((f*x+e)^(1/3)*(h*x+g)/(b*x+a)^2/(d*x+c),x, algorithm="giac")
 

Output:

-1/3*(3*b^2*d*e*g - b^2*c*f*g - 2*a*b*d*f*g - 3*b^2*c*e*h + 4*a*b*c*f*h - 
a^2*d*f*h)*((b*e - a*f)/b)^(1/3)*log(abs((f*x + e)^(1/3) - ((b*e - a*f)/b) 
^(1/3)))/(b^4*c^2*e - 2*a*b^3*c*d*e + a^2*b^2*d^2*e - a*b^3*c^2*f + 2*a^2* 
b^2*c*d*f - a^3*b*d^2*f) + (d^2*e*g - c*d*f*g - c*d*e*h + c^2*f*h)*((d*e - 
 c*f)/d)^(1/3)*log(abs((f*x + e)^(1/3) - ((d*e - c*f)/d)^(1/3)))/(b^2*c^2* 
d*e - 2*a*b*c*d^2*e + a^2*d^3*e - b^2*c^3*f + 2*a*b*c^2*d*f - a^2*c*d^2*f) 
 - (b^2*c*f*g + a^2*d*f*h - (3*b^2*e - 2*a*b*f)*d*g + (3*b^2*e - 4*a*b*f)* 
c*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))/(sqrt(3)*(b^3*e - a*b^2*f)^(2/3)*b^2*c^2 - 2*sqrt(3)*(b^3 
*e - a*b^2*f)^(2/3)*a*b*c*d + sqrt(3)*(b^3*e - a*b^2*f)^(2/3)*a^2*d^2) - ( 
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))/(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3) - 1/6*(b^2*c*f*g + a^2*d* 
f*h - (3*b^2*e - 2*a*b*f)*d*g + (3*b^2*e - 4*a*b*f)*c*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)^(2/3)*b^2*c^2 - 2*(b^3*e - a*b^2*f)^(2/3)*a*b*c*d + (b^3*e - 
a*b^2*f)^(2/3)*a^2*d^2) - 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))/(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3) - ((f*x + e 
)^(1/3)*b*f*g - (f*x + e)^(1/3)*a*f*h)/((b^2*c - a*b*d)*((f*x + e)*b - ...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt [3]{e+f x} (g+h x)}{(a+b x)^2 (c+d x)} \, dx=\text {Hanged} \] Input:

int(((e + f*x)^(1/3)*(g + h*x))/((a + b*x)^2*(c + d*x)),x)
 

Output:

\text{Hanged}
 

Reduce [B] (verification not implemented)

Time = 1.42 (sec) , antiderivative size = 6541, normalized size of antiderivative = 12.39 \[ \int \frac {\sqrt [3]{e+f x} (g+h x)}{(a+b x)^2 (c+d x)} \, dx =\text {Too large to display} \] Input:

int((f*x+e)^(1/3)*(h*x+g)/(b*x+a)^2/(d*x+c),x)
 

Output:

( - 2*d**(1/3)*(a*f - b*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* 
*3*d*f*h + 8*d**(1/3)*(a*f - b*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**2*b*c*f*h - 4*d**(1/3)*(a*f - b*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**2*b*d*f*g - 2*d**(1/3)*(a*f - b*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**2*b*d*f*h*x - 6*d**(1/3)*(a*f - b*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*b**2*c*e*h - 2*d**(1/3)*(a*f - b*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*b**2*c*f*g + 8*d**(1/3)*(a*f 
- b*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*b**2*c*f*h*x + 6*d** 
(1/3)*(a*f - b*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*b**2*d*e* 
g - 4*d**(1/3)*(a*f - b*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* 
b**2*d*f*g*x - 6*d**(1/3)*(a*f - b*e)**(1/3)*sqrt(3)*atan((b**(1/6)*(a*...