Integrand size = 29, antiderivative size = 448 \[ \int \frac {(e+f x)^{5/3} (g+h x)}{(a+b x) (c+d x)} \, dx=-\frac {3 (a d f h-b (d f g+d e h-c f h)) (e+f x)^{2/3}}{2 b^2 d^2}+\frac {3 h (e+f x)^{5/3}}{5 b d}+\frac {\sqrt {3} (b e-a f)^{5/3} (b g-a h) \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} \sqrt [3]{e+f x}}{\sqrt [3]{b e-a f}}}{\sqrt {3}}\right )}{b^{8/3} (b c-a d)}-\frac {\sqrt {3} (d e-c f)^{5/3} (d g-c h) \arctan \left (\frac {1+\frac {2 \sqrt [3]{d} \sqrt [3]{e+f x}}{\sqrt [3]{d e-c f}}}{\sqrt {3}}\right )}{d^{8/3} (b c-a d)}-\frac {(b e-a f)^{5/3} (b g-a h) \log (a+b x)}{2 b^{8/3} (b c-a d)}+\frac {(d e-c f)^{5/3} (d g-c h) \log (c+d x)}{2 d^{8/3} (b c-a d)}+\frac {3 (b e-a f)^{5/3} (b g-a h) \log \left (\sqrt [3]{b e-a f}-\sqrt [3]{b} \sqrt [3]{e+f x}\right )}{2 b^{8/3} (b c-a d)}-\frac {3 (d e-c f)^{5/3} (d g-c h) \log \left (\sqrt [3]{d e-c f}-\sqrt [3]{d} \sqrt [3]{e+f x}\right )}{2 d^{8/3} (b c-a d)} \] Output:
-3/2*(a*d*f*h-b*(-c*f*h+d*e*h+d*f*g))*(f*x+e)^(2/3)/b^2/d^2+3/5*h*(f*x+e)^ (5/3)/b/d+3^(1/2)*(-a*f+b*e)^(5/3)*(-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))/b^(8/3)/(-a*d+b*c)-3^(1/2)*(-c*f+d*e) ^(5/3)*(-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^(8/3)/(-a*d+b*c)-1/2*(-a*f+b*e)^(5/3)*(-a*h+b*g)*ln(b*x+a)/b^(8 /3)/(-a*d+b*c)+1/2*(-c*f+d*e)^(5/3)*(-c*h+d*g)*ln(d*x+c)/d^(8/3)/(-a*d+b*c )+3/2*(-a*f+b*e)^(5/3)*(-a*h+b*g)*ln((-a*f+b*e)^(1/3)-b^(1/3)*(f*x+e)^(1/3 ))/b^(8/3)/(-a*d+b*c)-3/2*(-c*f+d*e)^(5/3)*(-c*h+d*g)*ln((-c*f+d*e)^(1/3)- d^(1/3)*(f*x+e)^(1/3))/d^(8/3)/(-a*d+b*c)
Time = 2.24 (sec) , antiderivative size = 502, normalized size of antiderivative = 1.12 \[ \int \frac {(e+f x)^{5/3} (g+h x)}{(a+b x) (c+d x)} \, dx=\frac {3 b^{2/3} d^{2/3} (b c-a d) (e+f x)^{2/3} (-5 b c f h-5 a d f h+b d (5 f g+7 e h+2 f h x))-10 \sqrt {3} d^{8/3} (-b e+a f)^{5/3} (b g-a h) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{e+f x}}{\sqrt [3]{-b e+a f}}}{\sqrt {3}}\right )+10 \sqrt {3} b^{8/3} (-d e+c f)^{5/3} (d g-c h) \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{e+f x}}{\sqrt [3]{-d e+c f}}}{\sqrt {3}}\right )-10 d^{8/3} (-b e+a f)^{5/3} (b g-a h) \log \left (\sqrt [3]{-b e+a f}+\sqrt [3]{b} \sqrt [3]{e+f x}\right )+10 b^{8/3} (-d e+c f)^{5/3} (d g-c h) \log \left (\sqrt [3]{-d e+c f}+\sqrt [3]{d} \sqrt [3]{e+f x}\right )+5 d^{8/3} (-b e+a f)^{5/3} (b g-a h) \log \left ((-b e+a f)^{2/3}-\sqrt [3]{b} \sqrt [3]{-b e+a f} \sqrt [3]{e+f x}+b^{2/3} (e+f x)^{2/3}\right )-5 b^{8/3} (-d e+c f)^{5/3} (d g-c h) \log \left ((-d e+c f)^{2/3}-\sqrt [3]{d} \sqrt [3]{-d e+c f} \sqrt [3]{e+f x}+d^{2/3} (e+f x)^{2/3}\right )}{10 b^{8/3} d^{8/3} (b c-a d)} \] Input:
Integrate[((e + f*x)^(5/3)*(g + h*x))/((a + b*x)*(c + d*x)),x]
Output:
(3*b^(2/3)*d^(2/3)*(b*c - a*d)*(e + f*x)^(2/3)*(-5*b*c*f*h - 5*a*d*f*h + b *d*(5*f*g + 7*e*h + 2*f*h*x)) - 10*Sqrt[3]*d^(8/3)*(-(b*e) + a*f)^(5/3)*(b *g - a*h)*ArcTan[(1 - (2*b^(1/3)*(e + f*x)^(1/3))/(-(b*e) + a*f)^(1/3))/Sq rt[3]] + 10*Sqrt[3]*b^(8/3)*(-(d*e) + c*f)^(5/3)*(d*g - c*h)*ArcTan[(1 - ( 2*d^(1/3)*(e + f*x)^(1/3))/(-(d*e) + c*f)^(1/3))/Sqrt[3]] - 10*d^(8/3)*(-( b*e) + a*f)^(5/3)*(b*g - a*h)*Log[(-(b*e) + a*f)^(1/3) + b^(1/3)*(e + f*x) ^(1/3)] + 10*b^(8/3)*(-(d*e) + c*f)^(5/3)*(d*g - c*h)*Log[(-(d*e) + c*f)^( 1/3) + d^(1/3)*(e + f*x)^(1/3)] + 5*d^(8/3)*(-(b*e) + a*f)^(5/3)*(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)] - 5*b^(8/3)*(-(d*e) + c*f)^(5/3)*(d*g - c*h)*L og[(-(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)])/(10*b^(8/3)*d^(8/3)*(b*c - a*d))
Time = 0.52 (sec) , antiderivative size = 434, normalized size of antiderivative = 0.97, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {174, 60, 60, 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.
| \(\displaystyle \int \frac {(e+f x)^{5/3} (g+h x)}{(a+b x) (c+d x)} \, dx\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {(b g-a h) \int \frac {(e+f x)^{5/3}}{a+b x}dx}{b c-a d}-\frac {(d g-c h) \int \frac {(e+f x)^{5/3}}{c+d x}dx}{b c-a d}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(b g-a h) \left (\frac {(b e-a f) \int \frac {(e+f x)^{2/3}}{a+b x}dx}{b}+\frac {3 (e+f x)^{5/3}}{5 b}\right )}{b c-a d}-\frac {(d g-c h) \left (\frac {(d e-c f) \int \frac {(e+f x)^{2/3}}{c+d x}dx}{d}+\frac {3 (e+f x)^{5/3}}{5 d}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(b g-a h) \left (\frac {(b e-a f) \left (\frac {(b e-a f) \int \frac {1}{(a+b x) \sqrt [3]{e+f x}}dx}{b}+\frac {3 (e+f x)^{2/3}}{2 b}\right )}{b}+\frac {3 (e+f x)^{5/3}}{5 b}\right )}{b c-a d}-\frac {(d g-c h) \left (\frac {(d e-c f) \left (\frac {(d e-c f) \int \frac {1}{(c+d x) \sqrt [3]{e+f x}}dx}{d}+\frac {3 (e+f x)^{2/3}}{2 d}\right )}{d}+\frac {3 (e+f x)^{5/3}}{5 d}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 67 |
| \(\displaystyle \frac {(b g-a h) \left (\frac {(b e-a f) \left (\frac {(b e-a f) \left (-\frac {3 \int \frac {1}{\frac {\sqrt [3]{b e-a f}}{\sqrt [3]{b}}-\sqrt [3]{e+f x}}d\sqrt [3]{e+f x}}{2 b^{2/3} \sqrt [3]{b e-a f}}+\frac {3 \int \frac {1}{\frac {(b e-a f)^{2/3}}{b^{2/3}}+\frac {\sqrt [3]{e+f x} \sqrt [3]{b e-a f}}{\sqrt [3]{b}}+(e+f x)^{2/3}}d\sqrt [3]{e+f x}}{2 b}-\frac {\log (a+b x)}{2 b^{2/3} \sqrt [3]{b e-a f}}\right )}{b}+\frac {3 (e+f x)^{2/3}}{2 b}\right )}{b}+\frac {3 (e+f x)^{5/3}}{5 b}\right )}{b c-a d}-\frac {(d g-c h) \left (\frac {(d e-c f) \left (\frac {(d e-c f) \left (-\frac {3 \int \frac {1}{\frac {\sqrt [3]{d e-c f}}{\sqrt [3]{d}}-\sqrt [3]{e+f x}}d\sqrt [3]{e+f x}}{2 d^{2/3} \sqrt [3]{d e-c f}}+\frac {3 \int \frac {1}{\frac {(d e-c f)^{2/3}}{d^{2/3}}+\frac {\sqrt [3]{e+f x} \sqrt [3]{d e-c f}}{\sqrt [3]{d}}+(e+f x)^{2/3}}d\sqrt [3]{e+f x}}{2 d}-\frac {\log (c+d x)}{2 d^{2/3} \sqrt [3]{d e-c f}}\right )}{d}+\frac {3 (e+f x)^{2/3}}{2 d}\right )}{d}+\frac {3 (e+f x)^{5/3}}{5 d}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {(b g-a h) \left (\frac {(b e-a f) \left (\frac {(b e-a f) \left (\frac {3 \int \frac {1}{\frac {(b e-a f)^{2/3}}{b^{2/3}}+\frac {\sqrt [3]{e+f x} \sqrt [3]{b e-a f}}{\sqrt [3]{b}}+(e+f x)^{2/3}}d\sqrt [3]{e+f x}}{2 b}-\frac {\log (a+b x)}{2 b^{2/3} \sqrt [3]{b e-a f}}+\frac {3 \log \left (\sqrt [3]{b e-a f}-\sqrt [3]{b} \sqrt [3]{e+f x}\right )}{2 b^{2/3} \sqrt [3]{b e-a f}}\right )}{b}+\frac {3 (e+f x)^{2/3}}{2 b}\right )}{b}+\frac {3 (e+f x)^{5/3}}{5 b}\right )}{b c-a d}-\frac {(d g-c h) \left (\frac {(d e-c f) \left (\frac {(d e-c f) \left (\frac {3 \int \frac {1}{\frac {(d e-c f)^{2/3}}{d^{2/3}}+\frac {\sqrt [3]{e+f x} \sqrt [3]{d e-c f}}{\sqrt [3]{d}}+(e+f x)^{2/3}}d\sqrt [3]{e+f x}}{2 d}-\frac {\log (c+d x)}{2 d^{2/3} \sqrt [3]{d e-c f}}+\frac {3 \log \left (\sqrt [3]{d e-c f}-\sqrt [3]{d} \sqrt [3]{e+f x}\right )}{2 d^{2/3} \sqrt [3]{d e-c f}}\right )}{d}+\frac {3 (e+f x)^{2/3}}{2 d}\right )}{d}+\frac {3 (e+f x)^{5/3}}{5 d}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {(b g-a h) \left (\frac {(b e-a f) \left (\frac {(b e-a f) \left (-\frac {3 \int \frac {1}{-(e+f x)^{2/3}-3}d\left (\frac {2 \sqrt [3]{b} \sqrt [3]{e+f x}}{\sqrt [3]{b e-a f}}+1\right )}{b^{2/3} \sqrt [3]{b e-a f}}-\frac {\log (a+b x)}{2 b^{2/3} \sqrt [3]{b e-a f}}+\frac {3 \log \left (\sqrt [3]{b e-a f}-\sqrt [3]{b} \sqrt [3]{e+f x}\right )}{2 b^{2/3} \sqrt [3]{b e-a f}}\right )}{b}+\frac {3 (e+f x)^{2/3}}{2 b}\right )}{b}+\frac {3 (e+f x)^{5/3}}{5 b}\right )}{b c-a d}-\frac {(d g-c h) \left (\frac {(d e-c f) \left (\frac {(d e-c f) \left (-\frac {3 \int \frac {1}{-(e+f x)^{2/3}-3}d\left (\frac {2 \sqrt [3]{d} \sqrt [3]{e+f x}}{\sqrt [3]{d e-c f}}+1\right )}{d^{2/3} \sqrt [3]{d e-c f}}-\frac {\log (c+d x)}{2 d^{2/3} \sqrt [3]{d e-c f}}+\frac {3 \log \left (\sqrt [3]{d e-c f}-\sqrt [3]{d} \sqrt [3]{e+f x}\right )}{2 d^{2/3} \sqrt [3]{d e-c f}}\right )}{d}+\frac {3 (e+f x)^{2/3}}{2 d}\right )}{d}+\frac {3 (e+f x)^{5/3}}{5 d}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {(b g-a h) \left (\frac {(b e-a f) \left (\frac {(b e-a f) \left (\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{b} \sqrt [3]{e+f x}}{\sqrt [3]{b e-a f}}+1}{\sqrt {3}}\right )}{b^{2/3} \sqrt [3]{b e-a f}}-\frac {\log (a+b x)}{2 b^{2/3} \sqrt [3]{b e-a f}}+\frac {3 \log \left (\sqrt [3]{b e-a f}-\sqrt [3]{b} \sqrt [3]{e+f x}\right )}{2 b^{2/3} \sqrt [3]{b e-a f}}\right )}{b}+\frac {3 (e+f x)^{2/3}}{2 b}\right )}{b}+\frac {3 (e+f x)^{5/3}}{5 b}\right )}{b c-a d}-\frac {(d g-c h) \left (\frac {(d e-c f) \left (\frac {(d e-c f) \left (\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{d} \sqrt [3]{e+f x}}{\sqrt [3]{d e-c f}}+1}{\sqrt {3}}\right )}{d^{2/3} \sqrt [3]{d e-c f}}-\frac {\log (c+d x)}{2 d^{2/3} \sqrt [3]{d e-c f}}+\frac {3 \log \left (\sqrt [3]{d e-c f}-\sqrt [3]{d} \sqrt [3]{e+f x}\right )}{2 d^{2/3} \sqrt [3]{d e-c f}}\right )}{d}+\frac {3 (e+f x)^{2/3}}{2 d}\right )}{d}+\frac {3 (e+f x)^{5/3}}{5 d}\right )}{b c-a d}\) |
Input:
Int[((e + f*x)^(5/3)*(g + h*x))/((a + b*x)*(c + d*x)),x]
Output:
((b*g - a*h)*((3*(e + f*x)^(5/3))/(5*b) + ((b*e - a*f)*((3*(e + f*x)^(2/3) )/(2*b) + ((b*e - a*f)*((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))/b))/(b*c - a*d) - ((d*g - c*h) *((3*(e + f*x)^(5/3))/(5*d) + ((d*e - c*f)*((3*(e + f*x)^(2/3))/(2*d) + (( d*e - c*f)*((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))/d))/(b*c - a*d)
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
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]
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]
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])
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]
Time = 1.78 (sec) , antiderivative size = 432, normalized size of antiderivative = 0.96
| method | result | size |
| pseudoelliptic | \(\frac {\left (-3 \left (f x +e \right )^{\frac {2}{3}} \left (\left (\left (\left (-\frac {2 h x}{5}-g \right ) f -\frac {7 e h}{5}\right ) b +a f h \right ) d +b c f h \right ) b \left (a d -b c \right ) \left (\frac {a f -b e}{b}\right )^{\frac {1}{3}}+\left (a f -b e \right )^{2} d^{2} \left (2 \arctan \left (\frac {\sqrt {3}\, \left (2 \left (f x +e \right )^{\frac {1}{3}}-\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a f -b e}{b}\right )^{\frac {1}{3}}}\right ) \sqrt {3}+\ln \left (\left (f x +e \right )^{\frac {2}{3}}-\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}} \left (f x +e \right )^{\frac {1}{3}}+\left (\frac {a f -b e}{b}\right )^{\frac {2}{3}}\right )-2 \ln \left (\left (f x +e \right )^{\frac {1}{3}}+\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}}\right )\right ) \left (a h -b g \right )\right ) d \left (\frac {c f -d e}{d}\right )^{\frac {1}{3}}-\left (2 \arctan \left (\frac {\sqrt {3}\, \left (2 \left (f x +e \right )^{\frac {1}{3}}-\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {c f -d e}{d}\right )^{\frac {1}{3}}}\right ) \sqrt {3}+\ln \left (\left (f x +e \right )^{\frac {2}{3}}-\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}} \left (f x +e \right )^{\frac {1}{3}}+\left (\frac {c f -d e}{d}\right )^{\frac {2}{3}}\right )-2 \ln \left (\left (f x +e \right )^{\frac {1}{3}}+\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}}\right )\right ) \left (c f -d e \right )^{2} \left (\frac {a f -b e}{b}\right )^{\frac {1}{3}} \left (c h -d g \right ) b^{3}}{2 \left (\frac {c f -d e}{d}\right )^{\frac {1}{3}} \left (\frac {a f -b e}{b}\right )^{\frac {1}{3}} b^{3} d^{3} \left (a d -b c \right )}\) | \(432\) |
| derivativedivides | \(-\frac {3 \left (-\frac {h \left (f x +e \right )^{\frac {5}{3}} b d}{5}+\frac {\left (a d f h +b c f h -b d e h -b g d f \right ) \left (f x +e \right )^{\frac {2}{3}}}{2}\right )}{b^{2} d^{2}}+\frac {3 \left (-\frac {\ln \left (\left (f x +e \right )^{\frac {1}{3}}+\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a f -b e}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (\left (f x +e \right )^{\frac {2}{3}}-\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}} \left (f x +e \right )^{\frac {1}{3}}+\left (\frac {a f -b e}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a f -b e}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (f x +e \right )^{\frac {1}{3}}}{\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a f -b e}{b}\right )^{\frac {1}{3}}}\right ) \left (f^{2} a^{3} h -2 a^{2} b e f h -a^{2} b \,f^{2} g +a \,b^{2} e^{2} h +2 a \,b^{2} e f g -b^{3} e^{2} g \right )}{b^{2} \left (a d -b c \right )}-\frac {3 \left (-\frac {\ln \left (\left (f x +e \right )^{\frac {1}{3}}+\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}}\right )}{3 d \left (\frac {c f -d e}{d}\right )^{\frac {1}{3}}}+\frac {\ln \left (\left (f x +e \right )^{\frac {2}{3}}-\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}} \left (f x +e \right )^{\frac {1}{3}}+\left (\frac {c f -d e}{d}\right )^{\frac {2}{3}}\right )}{6 d \left (\frac {c f -d e}{d}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (f x +e \right )^{\frac {1}{3}}}{\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 d \left (\frac {c f -d e}{d}\right )^{\frac {1}{3}}}\right ) \left (c^{3} f^{2} h -2 c^{2} d e f h -c^{2} d \,f^{2} g +c \,d^{2} e^{2} h +2 c \,d^{2} e f g -d^{3} e^{2} g \right )}{d^{2} \left (a d -b c \right )}\) | \(517\) |
| default | \(-\frac {3 \left (-\frac {h \left (f x +e \right )^{\frac {5}{3}} b d}{5}+\frac {\left (a d f h +b c f h -b d e h -b g d f \right ) \left (f x +e \right )^{\frac {2}{3}}}{2}\right )}{b^{2} d^{2}}+\frac {3 \left (-\frac {\ln \left (\left (f x +e \right )^{\frac {1}{3}}+\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a f -b e}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (\left (f x +e \right )^{\frac {2}{3}}-\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}} \left (f x +e \right )^{\frac {1}{3}}+\left (\frac {a f -b e}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a f -b e}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (f x +e \right )^{\frac {1}{3}}}{\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a f -b e}{b}\right )^{\frac {1}{3}}}\right ) \left (f^{2} a^{3} h -2 a^{2} b e f h -a^{2} b \,f^{2} g +a \,b^{2} e^{2} h +2 a \,b^{2} e f g -b^{3} e^{2} g \right )}{b^{2} \left (a d -b c \right )}-\frac {3 \left (-\frac {\ln \left (\left (f x +e \right )^{\frac {1}{3}}+\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}}\right )}{3 d \left (\frac {c f -d e}{d}\right )^{\frac {1}{3}}}+\frac {\ln \left (\left (f x +e \right )^{\frac {2}{3}}-\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}} \left (f x +e \right )^{\frac {1}{3}}+\left (\frac {c f -d e}{d}\right )^{\frac {2}{3}}\right )}{6 d \left (\frac {c f -d e}{d}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (f x +e \right )^{\frac {1}{3}}}{\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 d \left (\frac {c f -d e}{d}\right )^{\frac {1}{3}}}\right ) \left (c^{3} f^{2} h -2 c^{2} d e f h -c^{2} d \,f^{2} g +c \,d^{2} e^{2} h +2 c \,d^{2} e f g -d^{3} e^{2} g \right )}{d^{2} \left (a d -b c \right )}\) | \(517\) |
| risch | \(-\frac {3 \left (-2 h b d f x +5 a d f h +5 b c f h -7 b d e h -5 b g d f \right ) \left (f x +e \right )^{\frac {2}{3}}}{10 b^{2} d^{2}}+\frac {\frac {3 \left (-\frac {\ln \left (\left (f x +e \right )^{\frac {1}{3}}+\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a f -b e}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (\left (f x +e \right )^{\frac {2}{3}}-\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}} \left (f x +e \right )^{\frac {1}{3}}+\left (\frac {a f -b e}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a f -b e}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (f x +e \right )^{\frac {1}{3}}}{\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a f -b e}{b}\right )^{\frac {1}{3}}}\right ) d^{2} \left (f^{2} a^{3} h -2 a^{2} b e f h -a^{2} b \,f^{2} g +a \,b^{2} e^{2} h +2 a \,b^{2} e f g -b^{3} e^{2} g \right )}{a d -b c}-\frac {3 \left (-\frac {\ln \left (\left (f x +e \right )^{\frac {1}{3}}+\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}}\right )}{3 d \left (\frac {c f -d e}{d}\right )^{\frac {1}{3}}}+\frac {\ln \left (\left (f x +e \right )^{\frac {2}{3}}-\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}} \left (f x +e \right )^{\frac {1}{3}}+\left (\frac {c f -d e}{d}\right )^{\frac {2}{3}}\right )}{6 d \left (\frac {c f -d e}{d}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (f x +e \right )^{\frac {1}{3}}}{\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 d \left (\frac {c f -d e}{d}\right )^{\frac {1}{3}}}\right ) b^{2} \left (c^{3} f^{2} h -2 c^{2} d e f h -c^{2} d \,f^{2} g +c \,d^{2} e^{2} h +2 c \,d^{2} e f g -d^{3} e^{2} g \right )}{a d -b c}}{b^{2} d^{2}}\) | \(519\) |
Input:
int((f*x+e)^(5/3)*(h*x+g)/(b*x+a)/(d*x+c),x,method=_RETURNVERBOSE)
Output:
1/2/((c*f-d*e)/d)^(1/3)*((-3*(f*x+e)^(2/3)*((((-2/5*h*x-g)*f-7/5*e*h)*b+a* f*h)*d+b*c*f*h)*b*(a*d-b*c)*((a*f-b*e)/b)^(1/3)+(a*f-b*e)^2*d^2*(2*arctan( 1/3*3^(1/2)*(2*(f*x+e)^(1/3)-((a*f-b*e)/b)^(1/3))/((a*f-b*e)/b)^(1/3))*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 ))-2*ln((f*x+e)^(1/3)+((a*f-b*e)/b)^(1/3)))*(a*h-b*g))*d*((c*f-d*e)/d)^(1/ 3)-(2*arctan(1/3*3^(1/2)*(2*(f*x+e)^(1/3)-((c*f-d*e)/d)^(1/3))/((c*f-d*e)/ d)^(1/3))*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))-2*ln((f*x+e)^(1/3)+((c*f-d*e)/d)^(1/3)))*(c*f-d*e)^2*((a*f -b*e)/b)^(1/3)*(c*h-d*g)*b^3)/((a*f-b*e)/b)^(1/3)/b^3/d^3/(a*d-b*c)
Timed out. \[ \int \frac {(e+f x)^{5/3} (g+h x)}{(a+b x) (c+d x)} \, dx=\text {Timed out} \] Input:
integrate((f*x+e)^(5/3)*(h*x+g)/(b*x+a)/(d*x+c),x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {(e+f x)^{5/3} (g+h x)}{(a+b x) (c+d x)} \, dx=\text {Timed out} \] Input:
integrate((f*x+e)**(5/3)*(h*x+g)/(b*x+a)/(d*x+c),x)
Output:
Timed out
Exception generated. \[ \int \frac {(e+f x)^{5/3} (g+h x)}{(a+b x) (c+d x)} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((f*x+e)^(5/3)*(h*x+g)/(b*x+a)/(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
Leaf count of result is larger than twice the leaf count of optimal. 1026 vs. \(2 (378) = 756\).
Time = 0.42 (sec) , antiderivative size = 1026, normalized size of antiderivative = 2.29 \[ \int \frac {(e+f x)^{5/3} (g+h x)}{(a+b x) (c+d x)} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)^(5/3)*(h*x+g)/(b*x+a)/(d*x+c),x, algorithm="giac")
Output:
(b^3*e^2*g*((b*e - a*f)/b)^(1/3) - 2*a*b^2*e*f*g*((b*e - a*f)/b)^(1/3) + a ^2*b*f^2*g*((b*e - a*f)/b)^(1/3) - a*b^2*e^2*h*((b*e - a*f)/b)^(1/3) + 2*a ^2*b*e*f*h*((b*e - a*f)/b)^(1/3) - a^3*f^2*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^4*c*e - a*b^3*d*e - a*b^3*c*f + a^2*b^2*d*f) - (d^3*e^2*g*((d*e - c*f)/d)^(1/3) - 2*c*d^2*e*f*g*((d*e - c*f)/d)^(1/3) + c^2*d*f^2*g*((d*e - c*f)/d)^(1/3) - c*d^2*e^2*h*((d*e - c*f)/d)^(1/3) + 2*c^2*d*e*f*h*((d*e - c*f)/d)^(1/3) - c^3*f^2*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^3*e - a*d^4*e - b*c^2*d^2*f + a*c *d^3*f) + ((b^3*e - a*b^2*f)^(2/3)*(sqrt(3)*b^2*e - sqrt(3)*a*b*f)*g - (b^ 3*e - a*b^2*f)^(2/3)*(sqrt(3)*a*b*e - sqrt(3)*a^2*f)*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^5*c - a*b^4*d) - ((d^3*e - c*d^2*f)^(2/3)*(sqrt(3)*d^2*e - sqrt(3)*c*d*f)*g - (d^3*e - c*d^2*f)^(2/3)*(sqrt(3)*c*d*e - sqrt(3)*c^2*f)*h)*arctan(1/3*sqr t(3)*(2*(f*x + e)^(1/3) + ((d*e - c*f)/d)^(1/3))/((d*e - c*f)/d)^(1/3))/(b *c*d^4 - a*d^5) - 1/2*((b^3*e - a*b^2*f)^(2/3)*(b^2*e - a*b*f)*g - (b^3*e - a*b^2*f)^(2/3)*(a*b*e - a^2*f)*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^5*c - a*b^4*d) + 1/2*((d ^3*e - c*d^2*f)^(2/3)*(d^2*e - c*d*f)*g - (d^3*e - c*d^2*f)^(2/3)*(c*d*e - c^2*f)*h)*log((f*x + e)^(2/3) + (f*x + e)^(1/3)*((d*e - c*f)/d)^(1/3) ...
Timed out. \[ \int \frac {(e+f x)^{5/3} (g+h x)}{(a+b x) (c+d x)} \, dx=\text {Hanged} \] Input:
int(((e + f*x)^(5/3)*(g + h*x))/((a + b*x)*(c + d*x)),x)
Output:
\text{Hanged}
Time = 0.84 (sec) , antiderivative size = 4185, normalized size of antiderivative = 9.34 \[ \int \frac {(e+f x)^{5/3} (g+h x)}{(a+b x) (c+d x)} \, dx =\text {Too large to display} \] Input:
int((f*x+e)^(5/3)*(h*x+g)/(b*x+a)/(d*x+c),x)
Output:
( - 10*d**(2/3)*(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 **3*d**2*f**2*h + 20*d**(2/3)*(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**2*b*d**2*e*f*h + 10*d**(2/3)*(c*f - d*e)**(1/3)*sqrt(3)*at an((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**2*f**2*g - 10*d**(2/3)*(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*b**2*d**2*e**2*h - 20*d**(2/ 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*b**2*d**2*e* f*g + 10*d**(2/3)*(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**3*d**2*e**2*g - 10*d**(2/3)*(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**3*d**2*f**2*h + 20*d**(2/3)*(c*f - d*e)**(1/3)*sqrt(3)*a tan((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**2*e*f*h + 10*d**(2/3)*(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**2*b*d**2*f**2*g - 10*d**...