Integrand size = 29, antiderivative size = 427 \[ \int \frac {g+h x}{(a+b x) (c+d x) (e+f x)^{5/3}} \, dx=-\frac {3 (f g-e h)}{2 (b e-a f) (d e-c f) (e+f x)^{2/3}}-\frac {\sqrt {3} b^{2/3} (b g-a h) \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} \sqrt [3]{e+f x}}{\sqrt [3]{b e-a f}}}{\sqrt {3}}\right )}{(b c-a d) (b e-a f)^{5/3}}+\frac {\sqrt {3} d^{2/3} (d g-c h) \arctan \left (\frac {1+\frac {2 \sqrt [3]{d} \sqrt [3]{e+f x}}{\sqrt [3]{d e-c f}}}{\sqrt {3}}\right )}{(b c-a d) (d e-c f)^{5/3}}-\frac {b^{2/3} (b g-a h) \log (a+b x)}{2 (b c-a d) (b e-a f)^{5/3}}+\frac {d^{2/3} (d g-c h) \log (c+d x)}{2 (b c-a d) (d e-c f)^{5/3}}+\frac {3 b^{2/3} (b g-a h) \log \left (\sqrt [3]{b e-a f}-\sqrt [3]{b} \sqrt [3]{e+f x}\right )}{2 (b c-a d) (b e-a f)^{5/3}}-\frac {3 d^{2/3} (d g-c h) \log \left (\sqrt [3]{d e-c f}-\sqrt [3]{d} \sqrt [3]{e+f x}\right )}{2 (b c-a d) (d e-c f)^{5/3}} \] Output:
1/2*(3*e*h-3*f*g)/(-a*f+b*e)/(-c*f+d*e)/(f*x+e)^(2/3)-3^(1/2)*b^(2/3)*(-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)^(5/3)+3^(1/2)*d^(2/3)*(-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)^(5/3)-1/ 2*b^(2/3)*(-a*h+b*g)*ln(b*x+a)/(-a*d+b*c)/(-a*f+b*e)^(5/3)+1/2*d^(2/3)*(-c *h+d*g)*ln(d*x+c)/(-a*d+b*c)/(-c*f+d*e)^(5/3)+3/2*b^(2/3)*(-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)^(5/3)-3/2*d^(2 /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)^(5/3)
Time = 10.34 (sec) , antiderivative size = 520, normalized size of antiderivative = 1.22 \[ \int \frac {g+h x}{(a+b x) (c+d x) (e+f x)^{5/3}} \, dx=\frac {1}{2} \left (\frac {3 (-f g+e h)}{(b e-a f) (d e-c f) (e+f x)^{2/3}}+\frac {2 \sqrt {3} b^{2/3} (b g-a h) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{e+f x}}{\sqrt [3]{-b e+a f}}}{\sqrt {3}}\right )}{(b c-a d) (-b e+a f)^{5/3}}-\frac {2 \sqrt {3} d^{2/3} (d g-c h) \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{e+f x}}{\sqrt [3]{-d e+c f}}}{\sqrt {3}}\right )}{(b c-a d) (-d e+c f)^{5/3}}+\frac {2 b^{2/3} (b g-a h) \log \left (\sqrt [3]{-b e+a f}+\sqrt [3]{b} \sqrt [3]{e+f x}\right )}{(-b c+a d) (-b e+a f)^{5/3}}+\frac {2 d^{2/3} (d g-c h) \log \left (\sqrt [3]{-d e+c f}+\sqrt [3]{d} \sqrt [3]{e+f x}\right )}{(b c-a d) (-d e+c f)^{5/3}}+\frac {b^{2/3} (b g-a h) \log \left ((-b e+a f)^{2/3}-\sqrt [3]{b} \sqrt [3]{-b e+a f} \sqrt [3]{e+f x}+b^{2/3} (e+f x)^{2/3}\right )}{(b c-a d) (-b e+a f)^{5/3}}+\frac {d^{2/3} (d g-c h) \log \left ((-d e+c f)^{2/3}-\sqrt [3]{d} \sqrt [3]{-d e+c f} \sqrt [3]{e+f x}+d^{2/3} (e+f x)^{2/3}\right )}{(-b c+a d) (-d e+c f)^{5/3}}\right ) \] Input:
Integrate[(g + h*x)/((a + b*x)*(c + d*x)*(e + f*x)^(5/3)),x]
Output:
((3*(-(f*g) + e*h))/((b*e - a*f)*(d*e - c*f)*(e + f*x)^(2/3)) + (2*Sqrt[3] *b^(2/3)*(b*g - a*h)*ArcTan[(1 - (2*b^(1/3)*(e + f*x)^(1/3))/(-(b*e) + a*f )^(1/3))/Sqrt[3]])/((b*c - a*d)*(-(b*e) + a*f)^(5/3)) - (2*Sqrt[3]*d^(2/3) *(d*g - c*h)*ArcTan[(1 - (2*d^(1/3)*(e + f*x)^(1/3))/(-(d*e) + c*f)^(1/3)) /Sqrt[3]])/((b*c - a*d)*(-(d*e) + c*f)^(5/3)) + (2*b^(2/3)*(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)^(5/3)) + (2*d^(2/3)*(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)^(5/3)) + (b^(2/3)*(b*g - a*h )*Log[(-(b*e) + a*f)^(2/3) - b^(1/3)*(-(b*e) + a*f)^(1/3)*(e + f*x)^(1/3) + b^(2/3)*(e + f*x)^(2/3)])/((b*c - a*d)*(-(b*e) + a*f)^(5/3)) + (d^(2/3)* (d*g - c*h)*Log[(-(d*e) + c*f)^(2/3) - d^(1/3)*(-(d*e) + c*f)^(1/3)*(e + f *x)^(1/3) + d^(2/3)*(e + f*x)^(2/3)])/((-(b*c) + a*d)*(-(d*e) + c*f)^(5/3) ))/2
Time = 0.46 (sec) , antiderivative size = 392, normalized size of antiderivative = 0.92, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {174, 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.
| \(\displaystyle \int \frac {g+h x}{(a+b x) (c+d x) (e+f x)^{5/3}} \, dx\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {(b g-a h) \int \frac {1}{(a+b x) (e+f x)^{5/3}}dx}{b c-a d}-\frac {(d g-c h) \int \frac {1}{(c+d x) (e+f x)^{5/3}}dx}{b c-a d}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {(b g-a h) \left (\frac {b \int \frac {1}{(a+b x) (e+f x)^{2/3}}dx}{b e-a f}+\frac {3}{2 (e+f x)^{2/3} (b e-a f)}\right )}{b c-a d}-\frac {(d g-c h) \left (\frac {d \int \frac {1}{(c+d x) (e+f x)^{2/3}}dx}{d e-c f}+\frac {3}{2 (e+f x)^{2/3} (d e-c f)}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 69 |
| \(\displaystyle \frac {(b g-a h) \left (\frac {b \left (-\frac {3 \int \frac {1}{\frac {(b e-a f)^{2/3}}{b^{2/3}}+\frac {\sqrt [3]{e+f x} \sqrt [3]{b e-a f}}{\sqrt [3]{b}}+(e+f x)^{2/3}}d\sqrt [3]{e+f x}}{2 b^{2/3} \sqrt [3]{b e-a f}}-\frac {3 \int \frac {1}{\frac {\sqrt [3]{b e-a f}}{\sqrt [3]{b}}-\sqrt [3]{e+f x}}d\sqrt [3]{e+f x}}{2 \sqrt [3]{b} (b e-a f)^{2/3}}-\frac {\log (a+b x)}{2 \sqrt [3]{b} (b e-a f)^{2/3}}\right )}{b e-a f}+\frac {3}{2 (e+f x)^{2/3} (b e-a f)}\right )}{b c-a d}-\frac {(d g-c h) \left (\frac {d \left (-\frac {3 \int \frac {1}{\frac {(d e-c f)^{2/3}}{d^{2/3}}+\frac {\sqrt [3]{e+f x} \sqrt [3]{d e-c f}}{\sqrt [3]{d}}+(e+f x)^{2/3}}d\sqrt [3]{e+f x}}{2 d^{2/3} \sqrt [3]{d e-c f}}-\frac {3 \int \frac {1}{\frac {\sqrt [3]{d e-c f}}{\sqrt [3]{d}}-\sqrt [3]{e+f x}}d\sqrt [3]{e+f x}}{2 \sqrt [3]{d} (d e-c f)^{2/3}}-\frac {\log (c+d x)}{2 \sqrt [3]{d} (d e-c f)^{2/3}}\right )}{d e-c f}+\frac {3}{2 (e+f x)^{2/3} (d e-c f)}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {(b g-a h) \left (\frac {b \left (-\frac {3 \int \frac {1}{\frac {(b e-a f)^{2/3}}{b^{2/3}}+\frac {\sqrt [3]{e+f x} \sqrt [3]{b e-a f}}{\sqrt [3]{b}}+(e+f x)^{2/3}}d\sqrt [3]{e+f x}}{2 b^{2/3} \sqrt [3]{b e-a f}}-\frac {\log (a+b x)}{2 \sqrt [3]{b} (b e-a f)^{2/3}}+\frac {3 \log \left (\sqrt [3]{b e-a f}-\sqrt [3]{b} \sqrt [3]{e+f x}\right )}{2 \sqrt [3]{b} (b e-a f)^{2/3}}\right )}{b e-a f}+\frac {3}{2 (e+f x)^{2/3} (b e-a f)}\right )}{b c-a d}-\frac {(d g-c h) \left (\frac {d \left (-\frac {3 \int \frac {1}{\frac {(d e-c f)^{2/3}}{d^{2/3}}+\frac {\sqrt [3]{e+f x} \sqrt [3]{d e-c f}}{\sqrt [3]{d}}+(e+f x)^{2/3}}d\sqrt [3]{e+f x}}{2 d^{2/3} \sqrt [3]{d e-c f}}-\frac {\log (c+d x)}{2 \sqrt [3]{d} (d e-c f)^{2/3}}+\frac {3 \log \left (\sqrt [3]{d e-c f}-\sqrt [3]{d} \sqrt [3]{e+f x}\right )}{2 \sqrt [3]{d} (d e-c f)^{2/3}}\right )}{d e-c f}+\frac {3}{2 (e+f x)^{2/3} (d e-c f)}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {(b g-a h) \left (\frac {b \left (\frac {3 \int \frac {1}{-(e+f x)^{2/3}-3}d\left (\frac {2 \sqrt [3]{b} \sqrt [3]{e+f x}}{\sqrt [3]{b e-a f}}+1\right )}{\sqrt [3]{b} (b e-a f)^{2/3}}-\frac {\log (a+b x)}{2 \sqrt [3]{b} (b e-a f)^{2/3}}+\frac {3 \log \left (\sqrt [3]{b e-a f}-\sqrt [3]{b} \sqrt [3]{e+f x}\right )}{2 \sqrt [3]{b} (b e-a f)^{2/3}}\right )}{b e-a f}+\frac {3}{2 (e+f x)^{2/3} (b e-a f)}\right )}{b c-a d}-\frac {(d g-c h) \left (\frac {d \left (\frac {3 \int \frac {1}{-(e+f x)^{2/3}-3}d\left (\frac {2 \sqrt [3]{d} \sqrt [3]{e+f x}}{\sqrt [3]{d e-c f}}+1\right )}{\sqrt [3]{d} (d e-c f)^{2/3}}-\frac {\log (c+d x)}{2 \sqrt [3]{d} (d e-c f)^{2/3}}+\frac {3 \log \left (\sqrt [3]{d e-c f}-\sqrt [3]{d} \sqrt [3]{e+f x}\right )}{2 \sqrt [3]{d} (d e-c f)^{2/3}}\right )}{d e-c f}+\frac {3}{2 (e+f x)^{2/3} (d e-c f)}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {(b g-a h) \left (\frac {b \left (-\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{b} \sqrt [3]{e+f x}}{\sqrt [3]{b e-a f}}+1}{\sqrt {3}}\right )}{\sqrt [3]{b} (b e-a f)^{2/3}}-\frac {\log (a+b x)}{2 \sqrt [3]{b} (b e-a f)^{2/3}}+\frac {3 \log \left (\sqrt [3]{b e-a f}-\sqrt [3]{b} \sqrt [3]{e+f x}\right )}{2 \sqrt [3]{b} (b e-a f)^{2/3}}\right )}{b e-a f}+\frac {3}{2 (e+f x)^{2/3} (b e-a f)}\right )}{b c-a d}-\frac {(d g-c h) \left (\frac {d \left (-\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{d} \sqrt [3]{e+f x}}{\sqrt [3]{d e-c f}}+1}{\sqrt {3}}\right )}{\sqrt [3]{d} (d e-c f)^{2/3}}-\frac {\log (c+d x)}{2 \sqrt [3]{d} (d e-c f)^{2/3}}+\frac {3 \log \left (\sqrt [3]{d e-c f}-\sqrt [3]{d} \sqrt [3]{e+f x}\right )}{2 \sqrt [3]{d} (d e-c f)^{2/3}}\right )}{d e-c f}+\frac {3}{2 (e+f x)^{2/3} (d e-c f)}\right )}{b c-a d}\) |
Input:
Int[(g + h*x)/((a + b*x)*(c + d*x)*(e + f*x)^(5/3)),x]
Output:
((b*g - a*h)*(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*c - a*d) - ((d*g - c*h)*(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)))/(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 = 0.78 (sec) , antiderivative size = 423, normalized size of antiderivative = 0.99
| method | result | size |
| derivativedivides | \(-\frac {3 \left (\frac {\ln \left (\left (f x +e \right )^{\frac {1}{3}}+\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a f -b e}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\left (f x +e \right )^{\frac {2}{3}}-\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}} \left (f x +e \right )^{\frac {1}{3}}+\left (\frac {a f -b e}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a f -b e}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (f x +e \right )^{\frac {1}{3}}}{\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a f -b e}{b}\right )^{\frac {2}{3}}}\right ) b \left (a h -b g \right )}{\left (a d -b c \right ) \left (a f -b e \right )}+\frac {3 \left (\frac {\ln \left (\left (f x +e \right )^{\frac {1}{3}}+\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}}\right )}{3 d \left (\frac {c f -d e}{d}\right )^{\frac {2}{3}}}-\frac {\ln \left (\left (f x +e \right )^{\frac {2}{3}}-\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}} \left (f x +e \right )^{\frac {1}{3}}+\left (\frac {c f -d e}{d}\right )^{\frac {2}{3}}\right )}{6 d \left (\frac {c f -d e}{d}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (f x +e \right )^{\frac {1}{3}}}{\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 d \left (\frac {c f -d e}{d}\right )^{\frac {2}{3}}}\right ) d \left (c h -d g \right )}{\left (a d -b c \right ) \left (c f -d e \right )}-\frac {3 \left (-e h +f g \right )}{2 \left (c f -d e \right ) \left (a f -b e \right ) \left (f x +e \right )^{\frac {2}{3}}}\) | \(423\) |
| default | \(-\frac {3 \left (\frac {\ln \left (\left (f x +e \right )^{\frac {1}{3}}+\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a f -b e}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\left (f x +e \right )^{\frac {2}{3}}-\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}} \left (f x +e \right )^{\frac {1}{3}}+\left (\frac {a f -b e}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a f -b e}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (f x +e \right )^{\frac {1}{3}}}{\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a f -b e}{b}\right )^{\frac {2}{3}}}\right ) b \left (a h -b g \right )}{\left (a d -b c \right ) \left (a f -b e \right )}+\frac {3 \left (\frac {\ln \left (\left (f x +e \right )^{\frac {1}{3}}+\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}}\right )}{3 d \left (\frac {c f -d e}{d}\right )^{\frac {2}{3}}}-\frac {\ln \left (\left (f x +e \right )^{\frac {2}{3}}-\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}} \left (f x +e \right )^{\frac {1}{3}}+\left (\frac {c f -d e}{d}\right )^{\frac {2}{3}}\right )}{6 d \left (\frac {c f -d e}{d}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (f x +e \right )^{\frac {1}{3}}}{\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 d \left (\frac {c f -d e}{d}\right )^{\frac {2}{3}}}\right ) d \left (c h -d g \right )}{\left (a d -b c \right ) \left (c f -d e \right )}-\frac {3 \left (-e h +f g \right )}{2 \left (c f -d e \right ) \left (a f -b e \right ) \left (f x +e \right )^{\frac {2}{3}}}\) | \(423\) |
| pseudoelliptic | \(-\frac {\ln \left (\left (f x +e \right )^{\frac {1}{3}}+\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}}\right ) \left (a h -b g \right )}{\left (\frac {a f -b e}{b}\right )^{\frac {2}{3}} \left (a f -b e \right ) \left (a d -b c \right )}+\frac {\ln \left (\left (f x +e \right )^{\frac {2}{3}}-\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}} \left (f x +e \right )^{\frac {1}{3}}+\left (\frac {a f -b e}{b}\right )^{\frac {2}{3}}\right ) \left (a h -b g \right )}{2 \left (\frac {a f -b e}{b}\right )^{\frac {2}{3}} \left (a f -b e \right ) \left (a d -b c \right )}-\frac {\sqrt {3}\, \arctan \left (\frac {2 \sqrt {3}\, \left (f x +e \right )^{\frac {1}{3}}}{3 \left (\frac {a f -b e}{b}\right )^{\frac {1}{3}}}-\frac {\sqrt {3}}{3}\right ) \left (a h -b g \right )}{\left (\frac {a f -b e}{b}\right )^{\frac {2}{3}} \left (a f -b e \right ) \left (a d -b c \right )}+\frac {\ln \left (\left (f x +e \right )^{\frac {1}{3}}+\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}}\right ) \left (c h -d g \right )}{\left (\frac {c f -d e}{d}\right )^{\frac {2}{3}} \left (a d -b c \right ) \left (c f -d e \right )}-\frac {\ln \left (\left (f x +e \right )^{\frac {2}{3}}-\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}} \left (f x +e \right )^{\frac {1}{3}}+\left (\frac {c f -d e}{d}\right )^{\frac {2}{3}}\right ) \left (c h -d g \right )}{2 \left (\frac {c f -d e}{d}\right )^{\frac {2}{3}} \left (a d -b c \right ) \left (c f -d e \right )}+\frac {\sqrt {3}\, \arctan \left (\frac {2 \sqrt {3}\, \left (f x +e \right )^{\frac {1}{3}}}{3 \left (\frac {c f -d e}{d}\right )^{\frac {1}{3}}}-\frac {\sqrt {3}}{3}\right ) \left (c h -d g \right )}{\left (\frac {c f -d e}{d}\right )^{\frac {2}{3}} \left (a d -b c \right ) \left (c f -d e \right )}+\frac {\frac {3 e h}{2}-\frac {3 f g}{2}}{\left (c f -d e \right ) \left (a f -b e \right ) \left (f x +e \right )^{\frac {2}{3}}}\) | \(511\) |
Input:
int((h*x+g)/(b*x+a)/(d*x+c)/(f*x+e)^(5/3),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)))*b*(a*h-b*g)/(a*d-b*c)/(a*f-b*e)+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)))*d*(c*h-d*g)/(a*d-b*c)/(c*f-d*e)-3/2*(-e *h+f*g)/(c*f-d*e)/(a*f-b*e)/(f*x+e)^(2/3)
Timed out. \[ \int \frac {g+h x}{(a+b x) (c+d x) (e+f x)^{5/3}} \, dx=\text {Timed out} \] Input:
integrate((h*x+g)/(b*x+a)/(d*x+c)/(f*x+e)^(5/3),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {g+h x}{(a+b x) (c+d x) (e+f x)^{5/3}} \, dx=\int \frac {g + h x}{\left (a + b x\right ) \left (c + d x\right ) \left (e + f x\right )^{\frac {5}{3}}}\, dx \] Input:
integrate((h*x+g)/(b*x+a)/(d*x+c)/(f*x+e)**(5/3),x)
Output:
Integral((g + h*x)/((a + b*x)*(c + d*x)*(e + f*x)**(5/3)), x)
Exception generated. \[ \int \frac {g+h x}{(a+b x) (c+d x) (e+f x)^{5/3}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((h*x+g)/(b*x+a)/(d*x+c)/(f*x+e)^(5/3),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. 838 vs. \(2 (361) = 722\).
Time = 0.79 (sec) , antiderivative size = 838, normalized size of antiderivative = 1.96 \[ \int \frac {g+h x}{(a+b x) (c+d x) (e+f x)^{5/3}} \, dx =\text {Too large to display} \] Input:
integrate((h*x+g)/(b*x+a)/(d*x+c)/(f*x+e)^(5/3),x, algorithm="giac")
Output:
(b^2*g - a*b*h)*((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 - c*d*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^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)^(1/3)*b*g - sqrt(3)*(b^3*e - a*b^2*f)^(1/3)*a*h)*arctan(1/3*sqrt(3)*( 2*(f*x + e)^(1/3) + ((b*e - a*f)/b)^(1/3))/((b*e - a*f)/b)^(1/3))/((b^3*e^ 2 - 2*a*b^2*e*f + a^2*b*f^2)*c - (a*b^2*e^2 - 2*a^2*b*e*f + a^3*f^2)*d) - (sqrt(3)*(d^3*e - c*d^2*f)^(1/3)*d*g - sqrt(3)*(d^3*e - c*d^2*f)^(1/3)*c*h )*arctan(1/3*sqrt(3)*(2*(f*x + e)^(1/3) + ((d*e - c*f)/d)^(1/3))/((d*e - c *f)/d)^(1/3))/((d^3*e^2 - 2*c*d^2*e*f + c^2*d*f^2)*a - (c*d^2*e^2 - 2*c^2* d*e*f + c^3*f^2)*b) - 1/2*((b^3*e - a*b^2*f)^(1/3)*b*g - (b^3*e - a*b^2*f) ^(1/3)*a*h)*log((f*x + e)^(2/3) + (f*x + e)^(1/3)*((b*e - a*f)/b)^(1/3) + ((b*e - a*f)/b)^(2/3))/((b^3*e^2 - 2*a*b^2*e*f + a^2*b*f^2)*c - (a*b^2*e^2 - 2*a^2*b*e*f + a^3*f^2)*d) - 1/2*((d^3*e - c*d^2*f)^(1/3)*d*g - (d^3*e - c*d^2*f)^(1/3)*c*h)*log((f*x + e)^(2/3) + (f*x + e)^(1/3)*((d*e - c*f)/d) ^(1/3) + ((d*e - c*f)/d)^(2/3))/((d^3*e^2 - 2*c*d^2*e*f + c^2*d*f^2)*a - ( c*d^2*e^2 - 2*c^2*d*e*f + c^3*f^2)*b) - 3/2*(f*g - e*h)/((b*d*e^2 - b*c*e* f - a*d*e*f + a*c*f^2)*(f*x + e)^(2/3))
Time = 18.76 (sec) , antiderivative size = 413671, normalized size of antiderivative = 968.78 \[ \int \frac {g+h x}{(a+b x) (c+d x) (e+f x)^{5/3}} \, dx=\text {Too large to display} \] Input:
int((g + h*x)/((e + f*x)^(5/3)*(a + b*x)*(c + d*x)),x)
Output:
log((e + f*x)^(1/3)*(486*b^14*d^14*e^14*f^4*g^4 - 3402*a*b^13*d^14*e^13*f^ 5*g^4 - 3402*b^14*c*d^13*e^13*f^5*g^4 + 243*a^6*b^8*c^8*d^6*f^18*g^4 + 243 *a^8*b^6*c^6*d^8*f^18*g^4 + 486*a^9*b^5*c^9*d^5*f^18*h^4 + 10449*a^2*b^12* d^14*e^12*f^6*g^4 - 18468*a^3*b^11*d^14*e^11*f^7*g^4 + 20655*a^4*b^10*d^14 *e^10*f^8*g^4 - 15066*a^5*b^9*d^14*e^9*f^9*g^4 + 7047*a^6*b^8*d^14*e^8*f^1 0*g^4 - 1944*a^7*b^7*d^14*e^7*f^11*g^4 + 243*a^8*b^6*d^14*e^6*f^12*g^4 + 1 0449*b^14*c^2*d^12*e^12*f^6*g^4 - 18468*b^14*c^3*d^11*e^11*f^7*g^4 + 20655 *b^14*c^4*d^10*e^10*f^8*g^4 - 15066*b^14*c^5*d^9*e^9*f^9*g^4 + 7047*b^14*c ^6*d^8*e^8*f^10*g^4 - 1944*b^14*c^7*d^7*e^7*f^11*g^4 + 243*b^14*c^8*d^6*e^ 6*f^12*g^4 - 69984*a*b^13*c^2*d^12*e^11*f^7*g^4 + 120528*a*b^13*c^3*d^11*e ^10*f^8*g^4 - 131220*a*b^13*c^4*d^10*e^9*f^9*g^4 + 93312*a*b^13*c^5*d^9*e^ 8*f^10*g^4 - 42768*a*b^13*c^6*d^8*e^7*f^11*g^4 + 11664*a*b^13*c^7*d^7*e^6* f^12*g^4 - 1458*a*b^13*c^8*d^6*e^5*f^13*g^4 - 69984*a^2*b^12*c*d^13*e^11*f ^7*g^4 + 120528*a^3*b^11*c*d^13*e^10*f^8*g^4 - 131220*a^4*b^10*c*d^13*e^9* f^9*g^4 + 93312*a^5*b^9*c*d^13*e^8*f^10*g^4 - 1458*a^5*b^9*c^8*d^6*e*f^17* g^4 - 42768*a^6*b^8*c*d^13*e^7*f^11*g^4 - 1944*a^6*b^8*c^7*d^7*e*f^17*g^4 + 11664*a^7*b^7*c*d^13*e^6*f^12*g^4 - 1944*a^7*b^7*c^6*d^8*e*f^17*g^4 - 14 58*a^8*b^6*c*d^13*e^5*f^13*g^4 - 1458*a^8*b^6*c^5*d^9*e*f^17*g^4 + 243*a*b ^13*c^3*d^11*e^14*f^4*h^4 - 1458*a*b^13*c^4*d^10*e^13*f^5*h^4 + 3645*a*b^1 3*c^5*d^9*e^12*f^6*h^4 - 4860*a*b^13*c^6*d^8*e^11*f^7*h^4 + 3645*a*b^13...
Time = 0.84 (sec) , antiderivative size = 4618, normalized size of antiderivative = 10.81 \[ \int \frac {g+h x}{(a+b x) (c+d x) (e+f x)^{5/3}} \, dx =\text {Too large to display} \] Input:
int((h*x+g)/(b*x+a)/(d*x+c)/(f*x+e)^(5/3),x)
Output:
(2*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*c**2*f**2*h - 4*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*c*d*e*f*h + 2*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)*sq rt(3) - 2*b**(1/3)*(e + f*x)**(1/6))/(b**(1/6)*(a*f - b*e)**(1/6)))*a*b*d* *2*e**2*h - 2*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)))*b**2*c**2*f**2*g + 4*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)))*b**2*c*d*e*f*g - 2*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)))*b**2*d**2*e**2*g + 2*d**(1/3)*(e + f*x)**(2/3)*(a*f - b*e)**(1/3)*s qrt(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*c**2*f**2*h - 4*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*c*d* e*f*h + 2*d**(1/3)*(e + f*x)**(2/3)*(a*f - b*e)**(1/3)*sqrt(3)*atan((b*...