Integrand size = 29, antiderivative size = 446 \[ \int \frac {(e+f x)^{4/3} (g+h x)}{(a+b x) (c+d x)} \, dx=-\frac {3 (a d f h-b (d f g+d e h-c f h)) \sqrt [3]{e+f x}}{b^2 d^2}+\frac {3 h (e+f x)^{4/3}}{4 b d}-\frac {\sqrt {3} (b e-a f)^{4/3} (b g-a h) \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} \sqrt [3]{e+f x}}{\sqrt [3]{b e-a f}}}{\sqrt {3}}\right )}{b^{7/3} (b c-a d)}+\frac {\sqrt {3} (d e-c f)^{4/3} (d g-c h) \arctan \left (\frac {1+\frac {2 \sqrt [3]{d} \sqrt [3]{e+f x}}{\sqrt [3]{d e-c f}}}{\sqrt {3}}\right )}{d^{7/3} (b c-a d)}-\frac {(b e-a f)^{4/3} (b g-a h) \log (a+b x)}{2 b^{7/3} (b c-a d)}+\frac {(d e-c f)^{4/3} (d g-c h) \log (c+d x)}{2 d^{7/3} (b c-a d)}+\frac {3 (b e-a f)^{4/3} (b g-a h) \log \left (\sqrt [3]{b e-a f}-\sqrt [3]{b} \sqrt [3]{e+f x}\right )}{2 b^{7/3} (b c-a d)}-\frac {3 (d e-c f)^{4/3} (d g-c h) \log \left (\sqrt [3]{d e-c f}-\sqrt [3]{d} \sqrt [3]{e+f x}\right )}{2 d^{7/3} (b c-a d)} \] Output:
-3*(a*d*f*h-b*(-c*f*h+d*e*h+d*f*g))*(f*x+e)^(1/3)/b^2/d^2+3/4*h*(f*x+e)^(4 /3)/b/d-3^(1/2)*(-a*f+b*e)^(4/3)*(-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^(7/3)/(-a*d+b*c)+3^(1/2)*(-c*f+d*e)^( 4/3)*(-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^(7/3)/(-a*d+b*c)-1/2*(-a*f+b*e)^(4/3)*(-a*h+b*g)*ln(b*x+a)/b^(7/3 )/(-a*d+b*c)+1/2*(-c*f+d*e)^(4/3)*(-c*h+d*g)*ln(d*x+c)/d^(7/3)/(-a*d+b*c)+ 3/2*(-a*f+b*e)^(4/3)*(-a*h+b*g)*ln((-a*f+b*e)^(1/3)-b^(1/3)*(f*x+e)^(1/3)) /b^(7/3)/(-a*d+b*c)-3/2*(-c*f+d*e)^(4/3)*(-c*h+d*g)*ln((-c*f+d*e)^(1/3)-d^ (1/3)*(f*x+e)^(1/3))/d^(7/3)/(-a*d+b*c)
Time = 2.03 (sec) , antiderivative size = 501, normalized size of antiderivative = 1.12 \[ \int \frac {(e+f x)^{4/3} (g+h x)}{(a+b x) (c+d x)} \, dx=\frac {3 \sqrt [3]{b} \sqrt [3]{d} (b c-a d) \sqrt [3]{e+f x} (-4 b c f h-4 a d f h+b d (4 f g+5 e h+f h x))-4 \sqrt {3} d^{7/3} (-b e+a f)^{4/3} (b g-a h) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \sqrt [3]{e+f x}}{\sqrt [3]{-b e+a f}}}{\sqrt {3}}\right )+4 \sqrt {3} b^{7/3} (-d e+c f)^{4/3} (d g-c h) \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{e+f x}}{\sqrt [3]{-d e+c f}}}{\sqrt {3}}\right )+4 d^{7/3} (-b e+a f)^{4/3} (b g-a h) \log \left (\sqrt [3]{-b e+a f}+\sqrt [3]{b} \sqrt [3]{e+f x}\right )-4 b^{7/3} (-d e+c f)^{4/3} (d g-c h) \log \left (\sqrt [3]{-d e+c f}+\sqrt [3]{d} \sqrt [3]{e+f x}\right )-2 d^{7/3} (-b e+a f)^{4/3} (b g-a h) \log \left ((-b e+a f)^{2/3}-\sqrt [3]{b} \sqrt [3]{-b e+a f} \sqrt [3]{e+f x}+b^{2/3} (e+f x)^{2/3}\right )+2 b^{7/3} (-d e+c f)^{4/3} (d g-c h) \log \left ((-d e+c f)^{2/3}-\sqrt [3]{d} \sqrt [3]{-d e+c f} \sqrt [3]{e+f x}+d^{2/3} (e+f x)^{2/3}\right )}{4 b^{7/3} d^{7/3} (b c-a d)} \] Input:
Integrate[((e + f*x)^(4/3)*(g + h*x))/((a + b*x)*(c + d*x)),x]
Output:
(3*b^(1/3)*d^(1/3)*(b*c - a*d)*(e + f*x)^(1/3)*(-4*b*c*f*h - 4*a*d*f*h + b *d*(4*f*g + 5*e*h + f*h*x)) - 4*Sqrt[3]*d^(7/3)*(-(b*e) + a*f)^(4/3)*(b*g - a*h)*ArcTan[(1 - (2*b^(1/3)*(e + f*x)^(1/3))/(-(b*e) + a*f)^(1/3))/Sqrt[ 3]] + 4*Sqrt[3]*b^(7/3)*(-(d*e) + c*f)^(4/3)*(d*g - c*h)*ArcTan[(1 - (2*d^ (1/3)*(e + f*x)^(1/3))/(-(d*e) + c*f)^(1/3))/Sqrt[3]] + 4*d^(7/3)*(-(b*e) + a*f)^(4/3)*(b*g - a*h)*Log[(-(b*e) + a*f)^(1/3) + b^(1/3)*(e + f*x)^(1/3 )] - 4*b^(7/3)*(-(d*e) + c*f)^(4/3)*(d*g - c*h)*Log[(-(d*e) + c*f)^(1/3) + d^(1/3)*(e + f*x)^(1/3)] - 2*d^(7/3)*(-(b*e) + a*f)^(4/3)*(b*g - a*h)*Log [(-(b*e) + a*f)^(2/3) - b^(1/3)*(-(b*e) + a*f)^(1/3)*(e + f*x)^(1/3) + b^( 2/3)*(e + f*x)^(2/3)] + 2*b^(7/3)*(-(d*e) + c*f)^(4/3)*(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)])/(4*b^(7/3)*d^(7/3)*(b*c - a*d))
Time = 0.53 (sec) , antiderivative size = 432, 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, 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.
| \(\displaystyle \int \frac {(e+f x)^{4/3} (g+h x)}{(a+b x) (c+d x)} \, dx\) |
| \(\Big \downarrow \) 174 |
| \(\displaystyle \frac {(b g-a h) \int \frac {(e+f x)^{4/3}}{a+b x}dx}{b c-a d}-\frac {(d g-c h) \int \frac {(e+f x)^{4/3}}{c+d x}dx}{b c-a d}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(b g-a h) \left (\frac {(b e-a f) \int \frac {\sqrt [3]{e+f x}}{a+b x}dx}{b}+\frac {3 (e+f x)^{4/3}}{4 b}\right )}{b c-a d}-\frac {(d g-c h) \left (\frac {(d e-c f) \int \frac {\sqrt [3]{e+f x}}{c+d x}dx}{d}+\frac {3 (e+f x)^{4/3}}{4 d}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {(b g-a h) \left (\frac {(b e-a f) \left (\frac {(b e-a f) \int \frac {1}{(a+b x) (e+f x)^{2/3}}dx}{b}+\frac {3 \sqrt [3]{e+f x}}{b}\right )}{b}+\frac {3 (e+f x)^{4/3}}{4 b}\right )}{b c-a d}-\frac {(d g-c h) \left (\frac {(d e-c f) \left (\frac {(d e-c f) \int \frac {1}{(c+d x) (e+f x)^{2/3}}dx}{d}+\frac {3 \sqrt [3]{e+f x}}{d}\right )}{d}+\frac {3 (e+f x)^{4/3}}{4 d}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 69 |
| \(\displaystyle \frac {(b g-a h) \left (\frac {(b e-a f) \left (\frac {(b e-a f) \left (-\frac {3 \int \frac {1}{\frac {(b e-a f)^{2/3}}{b^{2/3}}+\frac {\sqrt [3]{e+f x} \sqrt [3]{b e-a f}}{\sqrt [3]{b}}+(e+f x)^{2/3}}d\sqrt [3]{e+f x}}{2 b^{2/3} \sqrt [3]{b e-a f}}-\frac {3 \int \frac {1}{\frac {\sqrt [3]{b e-a f}}{\sqrt [3]{b}}-\sqrt [3]{e+f x}}d\sqrt [3]{e+f x}}{2 \sqrt [3]{b} (b e-a f)^{2/3}}-\frac {\log (a+b x)}{2 \sqrt [3]{b} (b e-a f)^{2/3}}\right )}{b}+\frac {3 \sqrt [3]{e+f x}}{b}\right )}{b}+\frac {3 (e+f x)^{4/3}}{4 b}\right )}{b c-a d}-\frac {(d g-c h) \left (\frac {(d e-c f) \left (\frac {(d e-c f) \left (-\frac {3 \int \frac {1}{\frac {(d e-c f)^{2/3}}{d^{2/3}}+\frac {\sqrt [3]{e+f x} \sqrt [3]{d e-c f}}{\sqrt [3]{d}}+(e+f x)^{2/3}}d\sqrt [3]{e+f x}}{2 d^{2/3} \sqrt [3]{d e-c f}}-\frac {3 \int \frac {1}{\frac {\sqrt [3]{d e-c f}}{\sqrt [3]{d}}-\sqrt [3]{e+f x}}d\sqrt [3]{e+f x}}{2 \sqrt [3]{d} (d e-c f)^{2/3}}-\frac {\log (c+d x)}{2 \sqrt [3]{d} (d e-c f)^{2/3}}\right )}{d}+\frac {3 \sqrt [3]{e+f x}}{d}\right )}{d}+\frac {3 (e+f x)^{4/3}}{4 d}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {(b g-a h) \left (\frac {(b e-a f) \left (\frac {(b e-a f) \left (-\frac {3 \int \frac {1}{\frac {(b e-a f)^{2/3}}{b^{2/3}}+\frac {\sqrt [3]{e+f x} \sqrt [3]{b e-a f}}{\sqrt [3]{b}}+(e+f x)^{2/3}}d\sqrt [3]{e+f x}}{2 b^{2/3} \sqrt [3]{b e-a f}}-\frac {\log (a+b x)}{2 \sqrt [3]{b} (b e-a f)^{2/3}}+\frac {3 \log \left (\sqrt [3]{b e-a f}-\sqrt [3]{b} \sqrt [3]{e+f x}\right )}{2 \sqrt [3]{b} (b e-a f)^{2/3}}\right )}{b}+\frac {3 \sqrt [3]{e+f x}}{b}\right )}{b}+\frac {3 (e+f x)^{4/3}}{4 b}\right )}{b c-a d}-\frac {(d g-c h) \left (\frac {(d e-c f) \left (\frac {(d e-c f) \left (-\frac {3 \int \frac {1}{\frac {(d e-c f)^{2/3}}{d^{2/3}}+\frac {\sqrt [3]{e+f x} \sqrt [3]{d e-c f}}{\sqrt [3]{d}}+(e+f x)^{2/3}}d\sqrt [3]{e+f x}}{2 d^{2/3} \sqrt [3]{d e-c f}}-\frac {\log (c+d x)}{2 \sqrt [3]{d} (d e-c f)^{2/3}}+\frac {3 \log \left (\sqrt [3]{d e-c f}-\sqrt [3]{d} \sqrt [3]{e+f x}\right )}{2 \sqrt [3]{d} (d e-c f)^{2/3}}\right )}{d}+\frac {3 \sqrt [3]{e+f x}}{d}\right )}{d}+\frac {3 (e+f x)^{4/3}}{4 d}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {(b g-a h) \left (\frac {(b e-a f) \left (\frac {(b e-a f) \left (\frac {3 \int \frac {1}{-(e+f x)^{2/3}-3}d\left (\frac {2 \sqrt [3]{b} \sqrt [3]{e+f x}}{\sqrt [3]{b e-a f}}+1\right )}{\sqrt [3]{b} (b e-a f)^{2/3}}-\frac {\log (a+b x)}{2 \sqrt [3]{b} (b e-a f)^{2/3}}+\frac {3 \log \left (\sqrt [3]{b e-a f}-\sqrt [3]{b} \sqrt [3]{e+f x}\right )}{2 \sqrt [3]{b} (b e-a f)^{2/3}}\right )}{b}+\frac {3 \sqrt [3]{e+f x}}{b}\right )}{b}+\frac {3 (e+f x)^{4/3}}{4 b}\right )}{b c-a d}-\frac {(d g-c h) \left (\frac {(d e-c f) \left (\frac {(d e-c f) \left (\frac {3 \int \frac {1}{-(e+f x)^{2/3}-3}d\left (\frac {2 \sqrt [3]{d} \sqrt [3]{e+f x}}{\sqrt [3]{d e-c f}}+1\right )}{\sqrt [3]{d} (d e-c f)^{2/3}}-\frac {\log (c+d x)}{2 \sqrt [3]{d} (d e-c f)^{2/3}}+\frac {3 \log \left (\sqrt [3]{d e-c f}-\sqrt [3]{d} \sqrt [3]{e+f x}\right )}{2 \sqrt [3]{d} (d e-c f)^{2/3}}\right )}{d}+\frac {3 \sqrt [3]{e+f x}}{d}\right )}{d}+\frac {3 (e+f x)^{4/3}}{4 d}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {(b g-a h) \left (\frac {(b e-a f) \left (\frac {(b e-a f) \left (-\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{b} \sqrt [3]{e+f x}}{\sqrt [3]{b e-a f}}+1}{\sqrt {3}}\right )}{\sqrt [3]{b} (b e-a f)^{2/3}}-\frac {\log (a+b x)}{2 \sqrt [3]{b} (b e-a f)^{2/3}}+\frac {3 \log \left (\sqrt [3]{b e-a f}-\sqrt [3]{b} \sqrt [3]{e+f x}\right )}{2 \sqrt [3]{b} (b e-a f)^{2/3}}\right )}{b}+\frac {3 \sqrt [3]{e+f x}}{b}\right )}{b}+\frac {3 (e+f x)^{4/3}}{4 b}\right )}{b c-a d}-\frac {(d g-c h) \left (\frac {(d e-c f) \left (\frac {(d e-c f) \left (-\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{d} \sqrt [3]{e+f x}}{\sqrt [3]{d e-c f}}+1}{\sqrt {3}}\right )}{\sqrt [3]{d} (d e-c f)^{2/3}}-\frac {\log (c+d x)}{2 \sqrt [3]{d} (d e-c f)^{2/3}}+\frac {3 \log \left (\sqrt [3]{d e-c f}-\sqrt [3]{d} \sqrt [3]{e+f x}\right )}{2 \sqrt [3]{d} (d e-c f)^{2/3}}\right )}{d}+\frac {3 \sqrt [3]{e+f x}}{d}\right )}{d}+\frac {3 (e+f x)^{4/3}}{4 d}\right )}{b c-a d}\) |
Input:
Int[((e + f*x)^(4/3)*(g + h*x))/((a + b*x)*(c + d*x)),x]
Output:
((b*g - a*h)*((3*(e + f*x)^(4/3))/(4*b) + ((b*e - a*f)*((3*(e + f*x)^(1/3) )/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^(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))/b))/(b*c - a*d) - ((d*g - c*h)* ((3*(e + f*x)^(4/3))/(4*d) + ((d*e - c*f)*((3*(e + f*x)^(1/3))/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^(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))/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_))^(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 = 1.29 (sec) , antiderivative size = 432, normalized size of antiderivative = 0.97
| method | result | size |
| pseudoelliptic | \(-\frac {-\left (-6 d \left (\frac {c f -d e}{d}\right )^{\frac {2}{3}} \left (\left (\left (\left (-\frac {h x}{4}-g \right ) f -\frac {5 e h}{4}\right ) b +a f h \right ) d +b c f h \right ) \left (a d -b c \right ) \left (f x +e \right )^{\frac {1}{3}}+\left (-2 \arctan \left (\frac {\sqrt {3}\, \left (2 \left (f x +e \right )^{\frac {1}{3}}-\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {c f -d e}{d}\right )^{\frac {1}{3}}}\right ) \sqrt {3}+\ln \left (\left (f x +e \right )^{\frac {2}{3}}-\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}} \left (f x +e \right )^{\frac {1}{3}}+\left (\frac {c f -d e}{d}\right )^{\frac {2}{3}}\right )-2 \ln \left (\left (f x +e \right )^{\frac {1}{3}}+\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}}\right )\right ) \left (c f -d e \right )^{2} \left (c h -d g \right ) b^{2}\right ) b \left (\frac {a f -b e}{b}\right )^{\frac {2}{3}}+\left (a f -b e \right )^{2} d^{3} \left (a h -b g \right ) \left (\frac {c f -d e}{d}\right )^{\frac {2}{3}} \left (-2 \arctan \left (\frac {\sqrt {3}\, \left (2 \left (f x +e \right )^{\frac {1}{3}}-\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a f -b e}{b}\right )^{\frac {1}{3}}}\right ) \sqrt {3}+\ln \left (\left (f x +e \right )^{\frac {2}{3}}-\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}} \left (f x +e \right )^{\frac {1}{3}}+\left (\frac {a f -b e}{b}\right )^{\frac {2}{3}}\right )-2 \ln \left (\left (f x +e \right )^{\frac {1}{3}}+\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}}\right )\right )}{2 \left (\frac {c f -d e}{d}\right )^{\frac {2}{3}} \left (\frac {a f -b e}{b}\right )^{\frac {2}{3}} b^{3} d^{3} \left (a d -b c \right )}\) | \(432\) |
| derivativedivides | \(-\frac {3 \left (-\frac {h \left (f x +e \right )^{\frac {4}{3}} b d}{4}+a d f h \left (f x +e \right )^{\frac {1}{3}}+b c f h \left (f x +e \right )^{\frac {1}{3}}-b d e h \left (f x +e \right )^{\frac {1}{3}}-b d f g \left (f x +e \right )^{\frac {1}{3}}\right )}{b^{2} d^{2}}+\frac {3 \left (\frac {\ln \left (\left (f x +e \right )^{\frac {1}{3}}+\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a f -b e}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\left (f x +e \right )^{\frac {2}{3}}-\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}} \left (f x +e \right )^{\frac {1}{3}}+\left (\frac {a f -b e}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a f -b e}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (f x +e \right )^{\frac {1}{3}}}{\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a f -b e}{b}\right )^{\frac {2}{3}}}\right ) \left (f^{2} a^{3} h -2 a^{2} b e f h -a^{2} b \,f^{2} g +a \,b^{2} e^{2} h +2 a \,b^{2} e f g -b^{3} e^{2} g \right )}{b^{2} \left (a d -b c \right )}+\frac {3 \left (\frac {\ln \left (\left (f x +e \right )^{\frac {1}{3}}+\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}}\right )}{3 d \left (\frac {c f -d e}{d}\right )^{\frac {2}{3}}}-\frac {\ln \left (\left (f x +e \right )^{\frac {2}{3}}-\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}} \left (f x +e \right )^{\frac {1}{3}}+\left (\frac {c f -d e}{d}\right )^{\frac {2}{3}}\right )}{6 d \left (\frac {c f -d e}{d}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (f x +e \right )^{\frac {1}{3}}}{\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 d \left (\frac {c f -d e}{d}\right )^{\frac {2}{3}}}\right ) \left (-c^{3} f^{2} h +2 c^{2} d e f h +c^{2} d \,f^{2} g -c \,d^{2} e^{2} h -2 c \,d^{2} e f g +d^{3} e^{2} g \right )}{d^{2} \left (a d -b c \right )}\) | \(535\) |
| default | \(-\frac {3 \left (-\frac {h \left (f x +e \right )^{\frac {4}{3}} b d}{4}+a d f h \left (f x +e \right )^{\frac {1}{3}}+b c f h \left (f x +e \right )^{\frac {1}{3}}-b d e h \left (f x +e \right )^{\frac {1}{3}}-b d f g \left (f x +e \right )^{\frac {1}{3}}\right )}{b^{2} d^{2}}+\frac {3 \left (\frac {\ln \left (\left (f x +e \right )^{\frac {1}{3}}+\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a f -b e}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (\left (f x +e \right )^{\frac {2}{3}}-\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}} \left (f x +e \right )^{\frac {1}{3}}+\left (\frac {a f -b e}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a f -b e}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (f x +e \right )^{\frac {1}{3}}}{\left (\frac {a f -b e}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a f -b e}{b}\right )^{\frac {2}{3}}}\right ) \left (f^{2} a^{3} h -2 a^{2} b e f h -a^{2} b \,f^{2} g +a \,b^{2} e^{2} h +2 a \,b^{2} e f g -b^{3} e^{2} g \right )}{b^{2} \left (a d -b c \right )}+\frac {3 \left (\frac {\ln \left (\left (f x +e \right )^{\frac {1}{3}}+\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}}\right )}{3 d \left (\frac {c f -d e}{d}\right )^{\frac {2}{3}}}-\frac {\ln \left (\left (f x +e \right )^{\frac {2}{3}}-\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}} \left (f x +e \right )^{\frac {1}{3}}+\left (\frac {c f -d e}{d}\right )^{\frac {2}{3}}\right )}{6 d \left (\frac {c f -d e}{d}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (f x +e \right )^{\frac {1}{3}}}{\left (\frac {c f -d e}{d}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 d \left (\frac {c f -d e}{d}\right )^{\frac {2}{3}}}\right ) \left (-c^{3} f^{2} h +2 c^{2} d e f h +c^{2} d \,f^{2} g -c \,d^{2} e^{2} h -2 c \,d^{2} e f g +d^{3} e^{2} g \right )}{d^{2} \left (a d -b c \right )}\) | \(535\) |
Input:
int((f*x+e)^(4/3)*(h*x+g)/(b*x+a)/(d*x+c),x,method=_RETURNVERBOSE)
Output:
-1/2/((c*f-d*e)/d)^(2/3)*(-(-6*d*((c*f-d*e)/d)^(2/3)*((((-1/4*h*x-g)*f-5/4 *e*h)*b+a*f*h)*d+b*c*f*h)*(a*d-b*c)*(f*x+e)^(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*(c*h-d*g)*b^2)*b*((a*f-b*e)/b)^ (2/3)+(a*f-b*e)^2*d^3*(a*h-b*g)*((c*f-d*e)/d)^(2/3)*(-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*f-b*e)/b)^(2/3)/b^3/d^3/(a*d-b*c)
Timed out. \[ \int \frac {(e+f x)^{4/3} (g+h x)}{(a+b x) (c+d x)} \, dx=\text {Timed out} \] Input:
integrate((f*x+e)^(4/3)*(h*x+g)/(b*x+a)/(d*x+c),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {(e+f x)^{4/3} (g+h x)}{(a+b x) (c+d x)} \, dx=\int \frac {\left (e + f x\right )^{\frac {4}{3}} \left (g + h x\right )}{\left (a + b x\right ) \left (c + d x\right )}\, dx \] Input:
integrate((f*x+e)**(4/3)*(h*x+g)/(b*x+a)/(d*x+c),x)
Output:
Integral((e + f*x)**(4/3)*(g + h*x)/((a + b*x)*(c + d*x)), x)
Exception generated. \[ \int \frac {(e+f x)^{4/3} (g+h x)}{(a+b x) (c+d x)} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((f*x+e)^(4/3)*(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. 857 vs. \(2 (378) = 756\).
Time = 0.40 (sec) , antiderivative size = 857, normalized size of antiderivative = 1.92 \[ \int \frac {(e+f x)^{4/3} (g+h x)}{(a+b x) (c+d x)} \, dx =\text {Too large to display} \] Input:
integrate((f*x+e)^(4/3)*(h*x+g)/(b*x+a)/(d*x+c),x, algorithm="giac")
Output:
(b^3*e^2*g - 2*a*b^2*e*f*g + a^2*b*f^2*g - a*b^2*e^2*h + 2*a^2*b*e*f*h - a ^3*f^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 - a*b^3*d*e - a*b^3*c*f + a^2*b^2*d*f) - (d^3*e^2*g - 2*c *d^2*e*f*g + c^2*d*f^2*g - c*d^2*e^2*h + 2*c^2*d*e*f*h - c^3*f^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 - a*d^4*e - b*c^2*d^2*f + a*c*d^3*f) - ((b^3*e - a*b^2*f)^(1/3)*(sqrt(3 )*b^2*e - sqrt(3)*a*b*f)*g - (b^3*e - a*b^2*f)^(1/3)*(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^4*c - a*b^3*d) + ((d^3*e - c*d^2*f)^(1/3)*(sq rt(3)*d^2*e - sqrt(3)*c*d*f)*g - (d^3*e - c*d^2*f)^(1/3)*(sqrt(3)*c*d*e - sqrt(3)*c^2*f)*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*c*d^3 - a*d^4) - 1/2*((b^3*e - a*b^2*f)^( 1/3)*(b^2*e - a*b*f)*g - (b^3*e - a*b^2*f)^(1/3)*(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^4*c - a*b^3*d) + 1/2*((d^3*e - c*d^2*f)^(1/3)*(d^2*e - c*d*f)*g - (d^3*e - c*d^2*f)^(1/3)*(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) + ((d*e - c*f)/d)^(2/3))/(b*c*d^3 - a*d^4) + 3/4*(4*(f*x + e)^(1/3)*b^3*d^3*f*g + (f*x + e)^(4/3)*b^3*d^3*h + 4*(f*x + e)^(1/3)*b^3*d^3*e*h - 4*(f*x + e)^(1/3)*b^3*c*d^2*f*h - 4*(f*x + e)^(1/3) *a*b^2*d^3*f*h)/(b^4*d^4)
Timed out. \[ \int \frac {(e+f x)^{4/3} (g+h x)}{(a+b x) (c+d x)} \, dx=\text {Hanged} \] Input:
int(((e + f*x)^(4/3)*(g + h*x))/((a + b*x)*(c + d*x)),x)
Output:
\text{Hanged}
Time = 0.70 (sec) , antiderivative size = 4185, normalized size of antiderivative = 9.38 \[ \int \frac {(e+f x)^{4/3} (g+h x)}{(a+b x) (c+d x)} \, dx =\text {Too large to display} \] Input:
int((f*x+e)^(4/3)*(h*x+g)/(b*x+a)/(d*x+c),x)
Output:
( - 4*d**(1/3)*(c*f - d*e)**(2/3)*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 + 8*d**(1/3)*(c*f - d*e)**(2/3)*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 + 4*d**(1/3)*(c*f - d*e)**(2/3)*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 - 4*d**(1/3)*(c*f - d*e)**(2/3 )*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 - 8*d**(1/3)*(c *f - d*e)**(2/3)*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 + 4*d**(1/3)*(c*f - d*e)**(2/3)*sqrt(3)*atan((b**(1/6)*(a*f - b*e)**(1/6)*s qrt(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 - 4*d**(1/3)*(c*f - d*e)**(2/3)*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 + 8*d**(1/3)*(c*f - d*e)**(2/3)*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 + 4*d**(1/3)*(c*f - d*e)**(2/3)*sqr t(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 - 4*d**(1/3)*(c*f...