Integrand size = 29, antiderivative size = 495 \[ \int \frac {g+h x}{(a+b x) (c+d x) (e+f x)^{8/3}} \, dx=-\frac {3 (f g-e h)}{5 (b e-a f) (d e-c f) (e+f x)^{5/3}}+\frac {3 \left (b c f^2 g+a f^2 (d g-c h)-b d e (2 f g-e h)\right )}{2 (b e-a f)^2 (d e-c f)^2 (e+f x)^{2/3}}-\frac {\sqrt {3} b^{5/3} (b g-a h) \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} \sqrt [3]{e+f x}}{\sqrt [3]{b e-a f}}}{\sqrt {3}}\right )}{(b c-a d) (b e-a f)^{8/3}}+\frac {\sqrt {3} d^{5/3} (d g-c h) \arctan \left (\frac {1+\frac {2 \sqrt [3]{d} \sqrt [3]{e+f x}}{\sqrt [3]{d e-c f}}}{\sqrt {3}}\right )}{(b c-a d) (d e-c f)^{8/3}}-\frac {b^{5/3} (b g-a h) \log (a+b x)}{2 (b c-a d) (b e-a f)^{8/3}}+\frac {d^{5/3} (d g-c h) \log (c+d x)}{2 (b c-a d) (d e-c f)^{8/3}}+\frac {3 b^{5/3} (b g-a h) \log \left (\sqrt [3]{b e-a f}-\sqrt [3]{b} \sqrt [3]{e+f x}\right )}{2 (b c-a d) (b e-a f)^{8/3}}-\frac {3 d^{5/3} (d g-c h) \log \left (\sqrt [3]{d e-c f}-\sqrt [3]{d} \sqrt [3]{e+f x}\right )}{2 (b c-a d) (d e-c f)^{8/3}} \] Output:
1/5*(3*e*h-3*f*g)/(-a*f+b*e)/(-c*f+d*e)/(f*x+e)^(5/3)+3/2*(b*c*f^2*g+a*f^2 *(-c*h+d*g)-b*d*e*(-e*h+2*f*g))/(-a*f+b*e)^2/(-c*f+d*e)^2/(f*x+e)^(2/3)-3^ (1/2)*b^(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))/(-a*d+b*c)/(-a*f+b*e)^(8/3)+3^(1/2)*d^(5/3)*(-c*h+d*g)*arc tan(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)^(8/3)-1/2*b^(5/3)*(-a*h+b*g)*ln(b*x+a)/(-a*d+b*c)/(-a*f+b*e)^(8/ 3)+1/2*d^(5/3)*(-c*h+d*g)*ln(d*x+c)/(-a*d+b*c)/(-c*f+d*e)^(8/3)+3/2*b^(5/3 )*(-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)^(8/3)-3/2*d^(5/3)*(-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)^(8/3)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 10.06 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.23 \[ \int \frac {g+h x}{(a+b x) (c+d x) (e+f x)^{8/3}} \, dx=\frac {3 \left (\frac {(b g-a h) \operatorname {Hypergeometric2F1}\left (-\frac {5}{3},1,-\frac {2}{3},\frac {b (e+f x)}{b e-a f}\right )}{b e-a f}+\frac {(-d g+c h) \operatorname {Hypergeometric2F1}\left (-\frac {5}{3},1,-\frac {2}{3},\frac {d (e+f x)}{d e-c f}\right )}{d e-c f}\right )}{5 (b c-a d) (e+f x)^{5/3}} \] Input:
Integrate[(g + h*x)/((a + b*x)*(c + d*x)*(e + f*x)^(8/3)),x]
Output:
(3*(((b*g - a*h)*Hypergeometric2F1[-5/3, 1, -2/3, (b*(e + f*x))/(b*e - a*f )])/(b*e - a*f) + ((-(d*g) + c*h)*Hypergeometric2F1[-5/3, 1, -2/3, (d*(e + f*x))/(d*e - c*f)])/(d*e - c*f)))/(5*(b*c - a*d)*(e + f*x)^(5/3))
Time = 0.50 (sec) , antiderivative size = 464, normalized size of antiderivative = 0.94, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {174, 61, 61, 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.
| \(\displaystyle \int \frac {g+h x}{(a+b x) (c+d x) (e+f x)^{8/3}} \, dx\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {(b g-a h) \int \frac {1}{(a+b x) (e+f x)^{8/3}}dx}{b c-a d}-\frac {(d g-c h) \int \frac {1}{(c+d x) (e+f x)^{8/3}}dx}{b c-a d}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {(b g-a h) \left (\frac {b \int \frac {1}{(a+b x) (e+f x)^{5/3}}dx}{b e-a f}+\frac {3}{5 (e+f x)^{5/3} (b e-a f)}\right )}{b c-a d}-\frac {(d g-c h) \left (\frac {d \int \frac {1}{(c+d x) (e+f x)^{5/3}}dx}{d e-c f}+\frac {3}{5 (e+f x)^{5/3} (d e-c f)}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {(b g-a h) \left (\frac {b \left (\frac {b \int \frac {1}{(a+b x) (e+f x)^{2/3}}dx}{b e-a f}+\frac {3}{2 (e+f x)^{2/3} (b e-a f)}\right )}{b e-a f}+\frac {3}{5 (e+f x)^{5/3} (b e-a f)}\right )}{b c-a d}-\frac {(d g-c h) \left (\frac {d \left (\frac {d \int \frac {1}{(c+d x) (e+f x)^{2/3}}dx}{d e-c f}+\frac {3}{2 (e+f x)^{2/3} (d e-c f)}\right )}{d e-c f}+\frac {3}{5 (e+f x)^{5/3} (d e-c f)}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 69 |
| \(\displaystyle \frac {(b g-a h) \left (\frac {b \left (\frac {b \left (-\frac {3 \int \frac {1}{\frac {(b e-a f)^{2/3}}{b^{2/3}}+\frac {\sqrt [3]{e+f x} \sqrt [3]{b e-a f}}{\sqrt [3]{b}}+(e+f x)^{2/3}}d\sqrt [3]{e+f x}}{2 b^{2/3} \sqrt [3]{b e-a f}}-\frac {3 \int \frac {1}{\frac {\sqrt [3]{b e-a f}}{\sqrt [3]{b}}-\sqrt [3]{e+f x}}d\sqrt [3]{e+f x}}{2 \sqrt [3]{b} (b e-a f)^{2/3}}-\frac {\log (a+b x)}{2 \sqrt [3]{b} (b e-a f)^{2/3}}\right )}{b e-a f}+\frac {3}{2 (e+f x)^{2/3} (b e-a f)}\right )}{b e-a f}+\frac {3}{5 (e+f x)^{5/3} (b e-a f)}\right )}{b c-a d}-\frac {(d g-c h) \left (\frac {d \left (\frac {d \left (-\frac {3 \int \frac {1}{\frac {(d e-c f)^{2/3}}{d^{2/3}}+\frac {\sqrt [3]{e+f x} \sqrt [3]{d e-c f}}{\sqrt [3]{d}}+(e+f x)^{2/3}}d\sqrt [3]{e+f x}}{2 d^{2/3} \sqrt [3]{d e-c f}}-\frac {3 \int \frac {1}{\frac {\sqrt [3]{d e-c f}}{\sqrt [3]{d}}-\sqrt [3]{e+f x}}d\sqrt [3]{e+f x}}{2 \sqrt [3]{d} (d e-c f)^{2/3}}-\frac {\log (c+d x)}{2 \sqrt [3]{d} (d e-c f)^{2/3}}\right )}{d e-c f}+\frac {3}{2 (e+f x)^{2/3} (d e-c f)}\right )}{d e-c f}+\frac {3}{5 (e+f x)^{5/3} (d e-c f)}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {(b g-a h) \left (\frac {b \left (\frac {b \left (-\frac {3 \int \frac {1}{\frac {(b e-a f)^{2/3}}{b^{2/3}}+\frac {\sqrt [3]{e+f x} \sqrt [3]{b e-a f}}{\sqrt [3]{b}}+(e+f x)^{2/3}}d\sqrt [3]{e+f x}}{2 b^{2/3} \sqrt [3]{b e-a f}}-\frac {\log (a+b x)}{2 \sqrt [3]{b} (b e-a f)^{2/3}}+\frac {3 \log \left (\sqrt [3]{b e-a f}-\sqrt [3]{b} \sqrt [3]{e+f x}\right )}{2 \sqrt [3]{b} (b e-a f)^{2/3}}\right )}{b e-a f}+\frac {3}{2 (e+f x)^{2/3} (b e-a f)}\right )}{b e-a f}+\frac {3}{5 (e+f x)^{5/3} (b e-a f)}\right )}{b c-a d}-\frac {(d g-c h) \left (\frac {d \left (\frac {d \left (-\frac {3 \int \frac {1}{\frac {(d e-c f)^{2/3}}{d^{2/3}}+\frac {\sqrt [3]{e+f x} \sqrt [3]{d e-c f}}{\sqrt [3]{d}}+(e+f x)^{2/3}}d\sqrt [3]{e+f x}}{2 d^{2/3} \sqrt [3]{d e-c f}}-\frac {\log (c+d x)}{2 \sqrt [3]{d} (d e-c f)^{2/3}}+\frac {3 \log \left (\sqrt [3]{d e-c f}-\sqrt [3]{d} \sqrt [3]{e+f x}\right )}{2 \sqrt [3]{d} (d e-c f)^{2/3}}\right )}{d e-c f}+\frac {3}{2 (e+f x)^{2/3} (d e-c f)}\right )}{d e-c f}+\frac {3}{5 (e+f x)^{5/3} (d e-c f)}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {(b g-a h) \left (\frac {b \left (\frac {b \left (\frac {3 \int \frac {1}{-(e+f x)^{2/3}-3}d\left (\frac {2 \sqrt [3]{b} \sqrt [3]{e+f x}}{\sqrt [3]{b e-a f}}+1\right )}{\sqrt [3]{b} (b e-a f)^{2/3}}-\frac {\log (a+b x)}{2 \sqrt [3]{b} (b e-a f)^{2/3}}+\frac {3 \log \left (\sqrt [3]{b e-a f}-\sqrt [3]{b} \sqrt [3]{e+f x}\right )}{2 \sqrt [3]{b} (b e-a f)^{2/3}}\right )}{b e-a f}+\frac {3}{2 (e+f x)^{2/3} (b e-a f)}\right )}{b e-a f}+\frac {3}{5 (e+f x)^{5/3} (b e-a f)}\right )}{b c-a d}-\frac {(d g-c h) \left (\frac {d \left (\frac {d \left (\frac {3 \int \frac {1}{-(e+f x)^{2/3}-3}d\left (\frac {2 \sqrt [3]{d} \sqrt [3]{e+f x}}{\sqrt [3]{d e-c f}}+1\right )}{\sqrt [3]{d} (d e-c f)^{2/3}}-\frac {\log (c+d x)}{2 \sqrt [3]{d} (d e-c f)^{2/3}}+\frac {3 \log \left (\sqrt [3]{d e-c f}-\sqrt [3]{d} \sqrt [3]{e+f x}\right )}{2 \sqrt [3]{d} (d e-c f)^{2/3}}\right )}{d e-c f}+\frac {3}{2 (e+f x)^{2/3} (d e-c f)}\right )}{d e-c f}+\frac {3}{5 (e+f x)^{5/3} (d e-c f)}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {(b g-a h) \left (\frac {b \left (\frac {b \left (-\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{b} \sqrt [3]{e+f x}}{\sqrt [3]{b e-a f}}+1}{\sqrt {3}}\right )}{\sqrt [3]{b} (b e-a f)^{2/3}}-\frac {\log (a+b x)}{2 \sqrt [3]{b} (b e-a f)^{2/3}}+\frac {3 \log \left (\sqrt [3]{b e-a f}-\sqrt [3]{b} \sqrt [3]{e+f x}\right )}{2 \sqrt [3]{b} (b e-a f)^{2/3}}\right )}{b e-a f}+\frac {3}{2 (e+f x)^{2/3} (b e-a f)}\right )}{b e-a f}+\frac {3}{5 (e+f x)^{5/3} (b e-a f)}\right )}{b c-a d}-\frac {(d g-c h) \left (\frac {d \left (\frac {d \left (-\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{d} \sqrt [3]{e+f x}}{\sqrt [3]{d e-c f}}+1}{\sqrt {3}}\right )}{\sqrt [3]{d} (d e-c f)^{2/3}}-\frac {\log (c+d x)}{2 \sqrt [3]{d} (d e-c f)^{2/3}}+\frac {3 \log \left (\sqrt [3]{d e-c f}-\sqrt [3]{d} \sqrt [3]{e+f x}\right )}{2 \sqrt [3]{d} (d e-c f)^{2/3}}\right )}{d e-c f}+\frac {3}{2 (e+f x)^{2/3} (d e-c f)}\right )}{d e-c f}+\frac {3}{5 (e+f x)^{5/3} (d e-c f)}\right )}{b c-a d}\) |
Input:
Int[(g + h*x)/((a + b*x)*(c + d*x)*(e + f*x)^(8/3)),x]
Output:
((b*g - a*h)*(3/(5*(b*e - a*f)*(e + f*x)^(5/3)) + (b*(3/(2*(b*e - a*f)*(e + f*x)^(2/3)) + (b*(-((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*e - a*f)))/(b*e - a*f)))/(b*c - a*d) - ((d*g - c*h)*(3/(5*(d*e - c*f)*(e + f*x)^(5/3)) + (d*(3/(2*(d*e - c *f)*(e + f*x)^(2/3)) + (d*(-((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))))/(d*e - c*f)))/(d*e - c*f)))/ (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 + 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]
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]
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 = 1.14 (sec) , antiderivative size = 495, normalized size of antiderivative = 1.00
| method | result | size |
| derivativedivides | \(\frac {3 \left (\frac {\ln \left (\left (f x +e \right )^{\frac {1}{3}}+\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a f -b e}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\left (f x +e \right )^{\frac {2}{3}}-\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}} \left (f x +e \right )^{\frac {1}{3}}+\left (\frac {a f -b e}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a f -b e}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (f x +e \right )^{\frac {1}{3}}}{\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a f -b e}{b}\right )^{\frac {2}{3}}}\right ) \left (a h -b g \right ) b^{2}}{\left (a d -b c \right ) \left (a f -b e \right )^{2}}-\frac {3 \left (-e h +f g \right )}{5 \left (c f -d e \right ) \left (a f -b e \right ) \left (f x +e \right )^{\frac {5}{3}}}-\frac {3 \left (a c h \,f^{2}-a d g \,f^{2}-b c \,f^{2} g -b d \,e^{2} h +2 b d e f g \right )}{2 \left (c f -d e \right )^{2} \left (a f -b e \right )^{2} \left (f x +e \right )^{\frac {2}{3}}}-\frac {3 \left (\frac {\ln \left (\left (f x +e \right )^{\frac {1}{3}}+\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}}\right )}{3 d \left (\frac {c f -d e}{d}\right )^{\frac {2}{3}}}-\frac {\ln \left (\left (f x +e \right )^{\frac {2}{3}}-\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}} \left (f x +e \right )^{\frac {1}{3}}+\left (\frac {c f -d e}{d}\right )^{\frac {2}{3}}\right )}{6 d \left (\frac {c f -d e}{d}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (f x +e \right )^{\frac {1}{3}}}{\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 d \left (\frac {c f -d e}{d}\right )^{\frac {2}{3}}}\right ) \left (c h -d g \right ) d^{2}}{\left (a d -b c \right ) \left (c f -d e \right )^{2}}\) | \(495\) |
| default | \(\frac {3 \left (\frac {\ln \left (\left (f x +e \right )^{\frac {1}{3}}+\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a f -b e}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\left (f x +e \right )^{\frac {2}{3}}-\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}} \left (f x +e \right )^{\frac {1}{3}}+\left (\frac {a f -b e}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a f -b e}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (f x +e \right )^{\frac {1}{3}}}{\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a f -b e}{b}\right )^{\frac {2}{3}}}\right ) \left (a h -b g \right ) b^{2}}{\left (a d -b c \right ) \left (a f -b e \right )^{2}}-\frac {3 \left (-e h +f g \right )}{5 \left (c f -d e \right ) \left (a f -b e \right ) \left (f x +e \right )^{\frac {5}{3}}}-\frac {3 \left (a c h \,f^{2}-a d g \,f^{2}-b c \,f^{2} g -b d \,e^{2} h +2 b d e f g \right )}{2 \left (c f -d e \right )^{2} \left (a f -b e \right )^{2} \left (f x +e \right )^{\frac {2}{3}}}-\frac {3 \left (\frac {\ln \left (\left (f x +e \right )^{\frac {1}{3}}+\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}}\right )}{3 d \left (\frac {c f -d e}{d}\right )^{\frac {2}{3}}}-\frac {\ln \left (\left (f x +e \right )^{\frac {2}{3}}-\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}} \left (f x +e \right )^{\frac {1}{3}}+\left (\frac {c f -d e}{d}\right )^{\frac {2}{3}}\right )}{6 d \left (\frac {c f -d e}{d}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (f x +e \right )^{\frac {1}{3}}}{\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 d \left (\frac {c f -d e}{d}\right )^{\frac {2}{3}}}\right ) \left (c h -d g \right ) d^{2}}{\left (a d -b c \right ) \left (c f -d e \right )^{2}}\) | \(495\) |
| pseudoelliptic | \(\frac {b \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 {2}{3}} \left (a d -b c \right ) \left (a f -b e \right )^{2}}-\frac {b \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 {2}{3}} \left (a d -b c \right ) \left (a f -b e \right )^{2}}+\frac {b \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 {2}{3}} \left (a d -b c \right ) \left (a f -b e \right )^{2}}+\frac {\frac {3 e h}{5}-\frac {3 f g}{5}}{\left (c f -d e \right ) \left (a f -b e \right ) \left (f x +e \right )^{\frac {5}{3}}}-\frac {3 \left (\left (-b c g +a \left (c h -d g \right )\right ) f^{2}+2 b d e f g -b d \,e^{2} h \right )}{2 \left (f x +e \right )^{\frac {2}{3}} \left (c f -d e \right )^{2} \left (a f -b e \right )^{2}}-\frac {d \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 (a d -b c \right ) \left (c f -d e \right )^{2}}+\frac {d \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 {2}{3}} \left (a d -b c \right ) \left (c f -d e \right )^{2}}-\frac {d \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 {2}{3}} \left (a d -b c \right ) \left (c f -d e \right )^{2}}\) | \(582\) |
Input:
int((h*x+g)/(b*x+a)/(d*x+c)/(f*x+e)^(8/3),x,method=_RETURNVERBOSE)
Output:
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)*b^2/(a*d-b*c)/(a*f-b*e)^2- 3/5*(-e*h+f*g)/(c*f-d*e)/(a*f-b*e)/(f*x+e)^(5/3)-3/2*(a*c*f^2*h-a*d*f^2*g- b*c*f^2*g-b*d*e^2*h+2*b*d*e*f*g)/(c*f-d*e)^2/(a*f-b*e)^2/(f*x+e)^(2/3)-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)*d^2/(a*d-b*c)/(c*f-d*e)^2
Timed out. \[ \int \frac {g+h x}{(a+b x) (c+d x) (e+f x)^{8/3}} \, dx=\text {Timed out} \] Input:
integrate((h*x+g)/(b*x+a)/(d*x+c)/(f*x+e)^(8/3),x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {g+h x}{(a+b x) (c+d x) (e+f x)^{8/3}} \, dx=\text {Timed out} \] Input:
integrate((h*x+g)/(b*x+a)/(d*x+c)/(f*x+e)**(8/3),x)
Output:
Timed out
Exception generated. \[ \int \frac {g+h x}{(a+b x) (c+d x) (e+f x)^{8/3}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((h*x+g)/(b*x+a)/(d*x+c)/(f*x+e)^(8/3),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
Leaf count of result is larger than twice the leaf count of optimal. 1239 vs. \(2 (425) = 850\).
Time = 3.49 (sec) , antiderivative size = 1239, normalized size of antiderivative = 2.50 \[ \int \frac {g+h x}{(a+b x) (c+d x) (e+f x)^{8/3}} \, dx=\text {Too large to display} \] Input:
integrate((h*x+g)/(b*x+a)/(d*x+c)/(f*x+e)^(8/3),x, algorithm="giac")
Output:
(b^3*g - a*b^2*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*e^3 - a*b^3*d*e^3 - 3*a*b^3*c*e^2*f + 3*a^2*b^2*d*e ^2*f + 3*a^2*b^2*c*e*f^2 - 3*a^3*b*d*e*f^2 - a^3*b*c*f^3 + a^4*d*f^3) - (d ^3*g - c*d^2*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^3*e^3 - a*d^4*e^3 - 3*b*c^2*d^2*e^2*f + 3*a*c*d^3*e^2 *f + 3*b*c^3*d*e*f^2 - 3*a*c^2*d^2*e*f^2 - b*c^4*f^3 + a*c^3*d*f^3) - (sqr t(3)*(b^3*e - a*b^2*f)^(1/3)*b^2*g - sqrt(3)*(b^3*e - a*b^2*f)^(1/3)*a*b*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^3 - 3*a*b^3*e^2*f + 3*a^2*b^2*e*f^2 - a^3*b*f^3)*c - (a*b^3*e^3 - 3*a^2*b^2*e^2*f + 3*a^3*b*e*f^2 - a^4*f^3)*d) - (sqrt(3)*(d^ 3*e - c*d^2*f)^(1/3)*d^2*g - sqrt(3)*(d^3*e - c*d^2*f)^(1/3)*c*d*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^3 - 3*c*d^3*e^2*f + 3*c^2*d^2*e*f^2 - c^3*d*f^3)*a - (c*d^3* e^3 - 3*c^2*d^2*e^2*f + 3*c^3*d*e*f^2 - c^4*f^3)*b) - 1/2*((b^3*e - a*b^2* f)^(1/3)*b^2*g - (b^3*e - a*b^2*f)^(1/3)*a*b*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^3 - 3*a *b^3*e^2*f + 3*a^2*b^2*e*f^2 - a^3*b*f^3)*c - (a*b^3*e^3 - 3*a^2*b^2*e^2*f + 3*a^3*b*e*f^2 - a^4*f^3)*d) - 1/2*((d^3*e - c*d^2*f)^(1/3)*d^2*g - (d^3 *e - c*d^2*f)^(1/3)*c*d*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^3 - 3*c*d^3*e^2*f + 3*c^2...
Timed out. \[ \int \frac {g+h x}{(a+b x) (c+d x) (e+f x)^{8/3}} \, dx=\text {Hanged} \] Input:
int((g + h*x)/((e + f*x)^(8/3)*(a + b*x)*(c + d*x)),x)
Output:
\text{Hanged}
Time = 2.12 (sec) , antiderivative size = 13315, normalized size of antiderivative = 26.90 \[ \int \frac {g+h x}{(a+b x) (c+d x) (e+f x)^{8/3}} \, dx =\text {Too large to display} \] Input:
int((h*x+g)/(b*x+a)/(d*x+c)/(f*x+e)^(8/3),x)
Output:
( - 10*d**(1/3)*(e + f*x)**(2/3)*(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**3*e*f**3*h - 10*d**(1/3)*(e + f*x)**(2/3)*(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**3*f**4*h*x + 30*d**(1/3)*(e + f*x)**(2/3)*(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**2*d*e**2*f**2*h + 30*d**(1/3)*(e + f*x)**(2/3)*(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**2*d*e*f**3*h *x - 30*d**(1/3)*(e + f*x)**(2/3)*(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*d**2*e**3*f*h - 30*d**(1/3)*(e + f*x)**(2/3)*(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*d**2*e**2*f **2*h*x + 10*d**(1/3)*(e + f*x)**(2/3)*(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**3*e**4*h + 10*d**(1/3)*(e + f*x)**(2/3)*(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**3*e**3...