Integrand size = 38, antiderivative size = 351 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{x \left (a+b x^3\right )^3} \, dx=\frac {x \left (a (b d-a g)+a (b e-a h) x-b (b c-a f) x^2\right )}{6 a^2 b \left (a+b x^3\right )^2}+\frac {x \left (a (5 b d+a g)+2 a (2 b e+a h) x-3 b (3 b c-a f) x^2\right )}{18 a^3 b \left (a+b x^3\right )}-\frac {\left (5 b^{4/3} d+2 \sqrt [3]{a} b e+a \sqrt [3]{b} g+a^{4/3} h\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{8/3} b^{5/3}}+\frac {c \log (x)}{a^3}+\frac {\left (\frac {5 b d+a g}{b^{2/3}}-\sqrt [3]{a} \left (2 e+\frac {a h}{b}\right )\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{8/3} b^{2/3}}-\frac {\left (\frac {5 b d+a g}{b^{2/3}}-\sqrt [3]{a} \left (2 e+\frac {a h}{b}\right )\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{8/3} b^{2/3}}-\frac {c \log \left (a+b x^3\right )}{3 a^3} \] Output:
1/6*x*(a*(-a*g+b*d)+a*(-a*h+b*e)*x-b*(-a*f+b*c)*x^2)/a^2/b/(b*x^3+a)^2+1/1 8*x*(a*(a*g+5*b*d)+2*a*(a*h+2*b*e)*x-3*b*(-a*f+3*b*c)*x^2)/a^3/b/(b*x^3+a) -1/27*(5*b^(4/3)*d+2*a^(1/3)*b*e+a*b^(1/3)*g+a^(4/3)*h)*arctan(1/3*(a^(1/3 )-2*b^(1/3)*x)*3^(1/2)/a^(1/3))*3^(1/2)/a^(8/3)/b^(5/3)+c*ln(x)/a^3+1/27*( (a*g+5*b*d)/b^(2/3)-a^(1/3)*(2*e+a*h/b))*ln(a^(1/3)+b^(1/3)*x)/a^(8/3)/b^( 2/3)-1/54*((a*g+5*b*d)/b^(2/3)-a^(1/3)*(2*e+a*h/b))*ln(a^(2/3)-a^(1/3)*b^( 1/3)*x+b^(2/3)*x^2)/a^(8/3)/b^(2/3)-1/3*c*ln(b*x^3+a)/a^3
Time = 0.20 (sec) , antiderivative size = 311, normalized size of antiderivative = 0.89 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{x \left (a+b x^3\right )^3} \, dx=\frac {\frac {3 a (6 b c+b x (5 d+4 e x)+a x (g+2 h x))}{b \left (a+b x^3\right )}-\frac {9 a^2 (-b (c+x (d+e x))+a (f+x (g+h x)))}{b \left (a+b x^3\right )^2}-\frac {2 \sqrt {3} \sqrt [3]{a} \left (5 b^{4/3} d+2 \sqrt [3]{a} b e+a \sqrt [3]{b} g+a^{4/3} h\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{b^{5/3}}+54 c \log (x)+\frac {2 \sqrt [3]{a} \left (5 b^{4/3} d-2 \sqrt [3]{a} b e+a \sqrt [3]{b} g-a^{4/3} h\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{5/3}}+\frac {\sqrt [3]{a} \left (-5 b^{4/3} d+2 \sqrt [3]{a} b e-a \sqrt [3]{b} g+a^{4/3} h\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{b^{5/3}}-18 c \log \left (a+b x^3\right )}{54 a^3} \] Input:
Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5)/(x*(a + b*x^3)^3),x]
Output:
((3*a*(6*b*c + b*x*(5*d + 4*e*x) + a*x*(g + 2*h*x)))/(b*(a + b*x^3)) - (9* a^2*(-(b*(c + x*(d + e*x))) + a*(f + x*(g + h*x))))/(b*(a + b*x^3)^2) - (2 *Sqrt[3]*a^(1/3)*(5*b^(4/3)*d + 2*a^(1/3)*b*e + a*b^(1/3)*g + a^(4/3)*h)*A rcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(5/3) + 54*c*Log[x] + (2*a^( 1/3)*(5*b^(4/3)*d - 2*a^(1/3)*b*e + a*b^(1/3)*g - a^(4/3)*h)*Log[a^(1/3) + b^(1/3)*x])/b^(5/3) + (a^(1/3)*(-5*b^(4/3)*d + 2*a^(1/3)*b*e - a*b^(1/3)* g + a^(4/3)*h)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(5/3) - 1 8*c*Log[a + b*x^3])/(54*a^3)
Time = 1.74 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.06, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2368, 25, 2368, 27, 2373, 2009}
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.
| \(\displaystyle \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{x \left (a+b x^3\right )^3} \, dx\) |
| \(\Big \downarrow \) 2368 |
| \(\displaystyle \frac {x \left (-b x^2 (b c-a f)+a (b d-a g)+a x (b e-a h)\right )}{6 a^2 b \left (a+b x^3\right )^2}-\frac {\int -\frac {-3 b^2 \left (\frac {b c}{a}-f\right ) x^3+2 b (2 b e+a h) x^2+b (5 b d+a g) x+6 b^2 c}{x \left (b x^3+a\right )^2}dx}{6 a b^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {-3 b^2 \left (\frac {b c}{a}-f\right ) x^3+2 b (2 b e+a h) x^2+b (5 b d+a g) x+6 b^2 c}{x \left (b x^3+a\right )^2}dx}{6 a b^2}+\frac {x \left (-b x^2 (b c-a f)+a (b d-a g)+a x (b e-a h)\right )}{6 a^2 b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 2368 |
| \(\displaystyle \frac {\frac {x \left (-3 b^2 x^2 (3 b c-a f)+a b (a g+5 b d)+2 a b x (a h+2 b e)\right )}{3 a^2 \left (a+b x^3\right )}-\frac {\int -\frac {2 \left (9 c b^3+(2 b e+a h) x^2 b^2+(5 b d+a g) x b^2\right )}{x \left (b x^3+a\right )}dx}{3 a b}}{6 a b^2}+\frac {x \left (-b x^2 (b c-a f)+a (b d-a g)+a x (b e-a h)\right )}{6 a^2 b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 \int \frac {9 c b^3+(2 b e+a h) x^2 b^2+(5 b d+a g) x b^2}{x \left (b x^3+a\right )}dx}{3 a b}+\frac {x \left (-3 b^2 x^2 (3 b c-a f)+a b (a g+5 b d)+2 a b x (a h+2 b e)\right )}{3 a^2 \left (a+b x^3\right )}}{6 a b^2}+\frac {x \left (-b x^2 (b c-a f)+a (b d-a g)+a x (b e-a h)\right )}{6 a^2 b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 2373 |
| \(\displaystyle \frac {\frac {2 \int \left (\frac {9 c b^3}{a x}+\frac {\left (-9 b^2 c x^2+a (2 b e+a h) x+a (5 b d+a g)\right ) b^2}{a \left (b x^3+a\right )}\right )dx}{3 a b}+\frac {x \left (-3 b^2 x^2 (3 b c-a f)+a b (a g+5 b d)+2 a b x (a h+2 b e)\right )}{3 a^2 \left (a+b x^3\right )}}{6 a b^2}+\frac {x \left (-b x^2 (b c-a f)+a (b d-a g)+a x (b e-a h)\right )}{6 a^2 b \left (a+b x^3\right )^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {x \left (-b x^2 (b c-a f)+a (b d-a g)+a x (b e-a h)\right )}{6 a^2 b \left (a+b x^3\right )^2}+\frac {\frac {2 \left (-\frac {b^{4/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (a^{4/3} h+2 \sqrt [3]{a} b e+a \sqrt [3]{b} g+5 b^{4/3} d\right )}{\sqrt {3} a^{2/3}}-\frac {b^{5/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-\frac {\sqrt [3]{a} (a h+2 b e)}{\sqrt [3]{b}}+a g+5 b d\right )}{6 a^{2/3}}+\frac {b^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (a g+5 b d)-\sqrt [3]{a} (a h+2 b e)\right )}{3 a^{2/3}}-\frac {3 b^3 c \log \left (a+b x^3\right )}{a}+\frac {9 b^3 c \log (x)}{a}\right )}{3 a b}+\frac {x \left (-3 b^2 x^2 (3 b c-a f)+a b (a g+5 b d)+2 a b x (a h+2 b e)\right )}{3 a^2 \left (a+b x^3\right )}}{6 a b^2}\) |
Input:
Int[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5)/(x*(a + b*x^3)^3),x]
Output:
(x*(a*(b*d - a*g) + a*(b*e - a*h)*x - b*(b*c - a*f)*x^2))/(6*a^2*b*(a + b* x^3)^2) + ((x*(a*b*(5*b*d + a*g) + 2*a*b*(2*b*e + a*h)*x - 3*b^2*(3*b*c - a*f)*x^2))/(3*a^2*(a + b*x^3)) + (2*(-((b^(4/3)*(5*b^(4/3)*d + 2*a^(1/3)*b *e + a*b^(1/3)*g + a^(4/3)*h)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1 /3))])/(Sqrt[3]*a^(2/3))) + (9*b^3*c*Log[x])/a + (b^(4/3)*(b^(1/3)*(5*b*d + a*g) - a^(1/3)*(2*b*e + a*h))*Log[a^(1/3) + b^(1/3)*x])/(3*a^(2/3)) - (b ^(5/3)*(5*b*d + a*g - (a^(1/3)*(2*b*e + a*h))/b^(1/3))*Log[a^(2/3) - a^(1/ 3)*b^(1/3)*x + b^(2/3)*x^2])/(6*a^(2/3)) - (3*b^3*c*Log[a + b*x^3])/a))/(3 *a*b))/(6*a*b^2)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Pq_)*(x_)^(m_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, Module[{Q = PolynomialQuotient[a*b^(Floor[(q - 1)/n] + 1)*x^ m*Pq, a + b*x^n, x], R = PolynomialRemainder[a*b^(Floor[(q - 1)/n] + 1)*x^m *Pq, a + b*x^n, x], i}, Simp[(-x)*R*((a + b*x^n)^(p + 1)/(a^2*n*(p + 1)*b^( Floor[(q - 1)/n] + 1))), x] + Simp[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)) Int[x^m*(a + b*x^n)^(p + 1)*ExpandToSum[(n*(p + 1)*Q)/x^m + Sum[((n*(p + 1) + i + 1)/a)*Coeff[R, x, i]*x^(i - m), {i, 0, n - 1}], x], x], x]]] /; F reeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1] && ILtQ[m, 0]
Int[((Pq_)*((c_.)*(x_))^(m_.))/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[E xpandIntegrand[(c*x)^m*(Pq/(a + b*x^n)), x], x] /; FreeQ[{a, b, c, m}, x] & & PolyQ[Pq, x] && IntegerQ[n] && !IGtQ[m, 0]
Time = 0.15 (sec) , antiderivative size = 339, normalized size of antiderivative = 0.97
| method | result | size |
| default | \(\frac {\frac {\left (\frac {1}{9} a^{2} h +\frac {2}{9} a b e \right ) x^{5}+\left (\frac {1}{18} a^{2} g +\frac {5}{18} d a b \right ) x^{4}+\frac {a b c \,x^{3}}{3}-\frac {a^{2} \left (a h -7 b e \right ) x^{2}}{18 b}-\frac {a^{2} \left (a g -4 b d \right ) x}{9 b}-\frac {a^{2} \left (a f -3 c b \right )}{6 b}}{\left (b \,x^{3}+a \right )^{2}}+\frac {\left (a^{2} g +5 d a b \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )+\left (a^{2} h +2 a b e \right ) \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )-3 b c \ln \left (b \,x^{3}+a \right )}{9 b}}{a^{3}}+\frac {c \ln \left (x \right )}{a^{3}}\) | \(339\) |
| risch | \(\frac {\frac {\left (a h +2 b e \right ) x^{5}}{9 a^{2}}+\frac {\left (a g +5 b d \right ) x^{4}}{18 a^{2}}+\frac {b c \,x^{3}}{3 a^{2}}-\frac {\left (a h -7 b e \right ) x^{2}}{18 a b}-\frac {\left (a g -4 b d \right ) x}{9 a b}-\frac {a f -3 c b}{6 a b}}{\left (b \,x^{3}+a \right )^{2}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{9} b^{5} \textit {\_Z}^{3}+27 a^{6} b^{5} c \,\textit {\_Z}^{2}+\left (3 a^{6} b^{2} g h +15 a^{5} b^{3} d h +6 a^{5} b^{3} e g +30 a^{4} b^{4} d e +243 a^{3} b^{5} c^{2}\right ) \textit {\_Z} +a^{5} h^{3}+6 a^{4} b e \,h^{2}-a^{4} b \,g^{3}+27 a^{3} b^{2} c g h -15 a^{3} b^{2} d \,g^{2}+12 a^{3} b^{2} e^{2} h +135 a^{2} b^{3} c d h +54 a^{2} b^{3} c e g -75 a^{2} b^{3} d^{2} g +8 a^{2} b^{3} e^{3}+270 a \,b^{4} c d e -125 a \,b^{4} d^{3}+729 b^{5} c^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{8} b^{5}-72 \textit {\_R}^{2} a^{5} b^{5} c +\left (-10 a^{5} b^{2} g h -50 a^{4} b^{3} d h -20 a^{4} b^{3} e g -100 a^{3} b^{4} d e -324 a^{2} b^{5} c^{2}\right ) \textit {\_R} -3 a^{4} h^{3}-18 a^{3} b e \,h^{2}+3 a^{3} b \,g^{3}-54 a^{2} b^{2} c g h +45 a^{2} b^{2} d \,g^{2}-36 a^{2} b^{2} e^{2} h -270 a \,b^{3} c d h -108 a \,b^{3} c e g +225 a \,b^{3} d^{2} g -24 a \,b^{3} e^{3}-540 b^{4} c d e +375 b^{4} d^{3}\right ) x +\left (a^{7} b^{3} h +2 b^{4} e \,a^{6}\right ) \textit {\_R}^{2}+\left (-g^{2} a^{5} b^{2}-18 b^{3} c h \,a^{4}-10 b^{3} d g \,a^{4}-36 b^{4} c e \,a^{3}-25 a^{3} b^{4} d^{2}\right ) \textit {\_R} +27 a^{2} b^{2} c \,g^{2}-243 a \,b^{3} c^{2} h +270 a \,b^{3} c d g -486 b^{4} c^{2} e +675 b^{4} c \,d^{2}\right )\right )}{27}+\frac {c \ln \left (-x \right )}{a^{3}}\) | \(656\) |
Input:
int((h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/x/(b*x^3+a)^3,x,method=_RETURNVERBOSE)
Output:
1/a^3*(((1/9*a^2*h+2/9*a*b*e)*x^5+(1/18*a^2*g+5/18*d*a*b)*x^4+1/3*a*b*c*x^ 3-1/18*a^2*(a*h-7*b*e)/b*x^2-1/9*a^2*(a*g-4*b*d)/b*x-1/6*a^2*(a*f-3*b*c)/b )/(b*x^3+a)^2+1/9/b*((a^2*g+5*a*b*d)*(1/3/b/(a/b)^(2/3)*ln(x+(a/b)^(1/3))- 1/6/b/(a/b)^(2/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))+1/3/b/(a/b)^(2/3)*3^(1 /2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1)))+(a^2*h+2*a*b*e)*(-1/3/b/(a/b) ^(1/3)*ln(x+(a/b)^(1/3))+1/6/b/(a/b)^(1/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3 ))+1/3*3^(1/2)/b/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1)))-3*b* c*ln(b*x^3+a)))+c*ln(x)/a^3
Result contains complex when optimal does not.
Time = 20.30 (sec) , antiderivative size = 12815, normalized size of antiderivative = 36.51 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{x \left (a+b x^3\right )^3} \, dx=\text {Too large to display} \] Input:
integrate((h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/x/(b*x^3+a)^3,x, algorithm="fric as")
Output:
Too large to include
Timed out. \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{x \left (a+b x^3\right )^3} \, dx=\text {Timed out} \] Input:
integrate((h*x**5+g*x**4+f*x**3+e*x**2+d*x+c)/x/(b*x**3+a)**3,x)
Output:
Timed out
Time = 0.12 (sec) , antiderivative size = 368, normalized size of antiderivative = 1.05 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{x \left (a+b x^3\right )^3} \, dx=\frac {6 \, b^{2} c x^{3} + 2 \, {\left (2 \, b^{2} e + a b h\right )} x^{5} + {\left (5 \, b^{2} d + a b g\right )} x^{4} + 9 \, a b c - 3 \, a^{2} f + {\left (7 \, a b e - a^{2} h\right )} x^{2} + 2 \, {\left (4 \, a b d - a^{2} g\right )} x}{18 \, {\left (a^{2} b^{3} x^{6} + 2 \, a^{3} b^{2} x^{3} + a^{4} b\right )}} + \frac {c \log \left (x\right )}{a^{3}} + \frac {\sqrt {3} {\left (2 \, a b e \left (\frac {a}{b}\right )^{\frac {2}{3}} + a^{2} h \left (\frac {a}{b}\right )^{\frac {2}{3}} + 5 \, a b d \left (\frac {a}{b}\right )^{\frac {1}{3}} + a^{2} g \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{4} b} - \frac {{\left (18 \, b^{2} c \left (\frac {a}{b}\right )^{\frac {2}{3}} - 2 \, a b e \left (\frac {a}{b}\right )^{\frac {1}{3}} - a^{2} h \left (\frac {a}{b}\right )^{\frac {1}{3}} + 5 \, a b d + a^{2} g\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a^{3} b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (9 \, b^{2} c \left (\frac {a}{b}\right )^{\frac {2}{3}} + 2 \, a b e \left (\frac {a}{b}\right )^{\frac {1}{3}} + a^{2} h \left (\frac {a}{b}\right )^{\frac {1}{3}} - 5 \, a b d - a^{2} g\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a^{3} b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \] Input:
integrate((h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/x/(b*x^3+a)^3,x, algorithm="maxi ma")
Output:
1/18*(6*b^2*c*x^3 + 2*(2*b^2*e + a*b*h)*x^5 + (5*b^2*d + a*b*g)*x^4 + 9*a* b*c - 3*a^2*f + (7*a*b*e - a^2*h)*x^2 + 2*(4*a*b*d - a^2*g)*x)/(a^2*b^3*x^ 6 + 2*a^3*b^2*x^3 + a^4*b) + c*log(x)/a^3 + 1/27*sqrt(3)*(2*a*b*e*(a/b)^(2 /3) + a^2*h*(a/b)^(2/3) + 5*a*b*d*(a/b)^(1/3) + a^2*g*(a/b)^(1/3))*arctan( 1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a^4*b) - 1/54*(18*b^2*c*(a/b )^(2/3) - 2*a*b*e*(a/b)^(1/3) - a^2*h*(a/b)^(1/3) + 5*a*b*d + a^2*g)*log(x ^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a^3*b^2*(a/b)^(2/3)) - 1/27*(9*b^2*c*(a /b)^(2/3) + 2*a*b*e*(a/b)^(1/3) + a^2*h*(a/b)^(1/3) - 5*a*b*d - a^2*g)*log (x + (a/b)^(1/3))/(a^3*b^2*(a/b)^(2/3))
Time = 0.14 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.06 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{x \left (a+b x^3\right )^3} \, dx=-\frac {c \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{3}} + \frac {c \log \left ({\left | x \right |}\right )}{a^{3}} - \frac {\sqrt {3} {\left (5 \, b^{2} d + a b g - 2 \, \left (-a b^{2}\right )^{\frac {1}{3}} b e - \left (-a b^{2}\right )^{\frac {1}{3}} a h\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{2} b} - \frac {{\left (5 \, b^{2} d + a b g + 2 \, \left (-a b^{2}\right )^{\frac {1}{3}} b e + \left (-a b^{2}\right )^{\frac {1}{3}} a h\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{2} b} + \frac {6 \, a b^{2} c x^{3} + 2 \, {\left (2 \, a b^{2} e + a^{2} b h\right )} x^{5} + {\left (5 \, a b^{2} d + a^{2} b g\right )} x^{4} + 9 \, a^{2} b c - 3 \, a^{3} f + {\left (7 \, a^{2} b e - a^{3} h\right )} x^{2} + 2 \, {\left (4 \, a^{2} b d - a^{3} g\right )} x}{18 \, {\left (b x^{3} + a\right )}^{2} a^{3} b} - \frac {{\left (2 \, a^{4} b^{3} e \left (-\frac {a}{b}\right )^{\frac {1}{3}} + a^{5} b^{2} h \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 5 \, a^{4} b^{3} d + a^{5} b^{2} g\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{7} b^{3}} \] Input:
integrate((h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/x/(b*x^3+a)^3,x, algorithm="giac ")
Output:
-1/3*c*log(abs(b*x^3 + a))/a^3 + c*log(abs(x))/a^3 - 1/27*sqrt(3)*(5*b^2*d + a*b*g - 2*(-a*b^2)^(1/3)*b*e - (-a*b^2)^(1/3)*a*h)*arctan(1/3*sqrt(3)*( 2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/((-a*b^2)^(2/3)*a^2*b) - 1/54*(5*b^2*d + a*b*g + 2*(-a*b^2)^(1/3)*b*e + (-a*b^2)^(1/3)*a*h)*log(x^2 + x*(-a/b)^(1/ 3) + (-a/b)^(2/3))/((-a*b^2)^(2/3)*a^2*b) + 1/18*(6*a*b^2*c*x^3 + 2*(2*a*b ^2*e + a^2*b*h)*x^5 + (5*a*b^2*d + a^2*b*g)*x^4 + 9*a^2*b*c - 3*a^3*f + (7 *a^2*b*e - a^3*h)*x^2 + 2*(4*a^2*b*d - a^3*g)*x)/((b*x^3 + a)^2*a^3*b) - 1 /27*(2*a^4*b^3*e*(-a/b)^(1/3) + a^5*b^2*h*(-a/b)^(1/3) + 5*a^4*b^3*d + a^5 *b^2*g)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^7*b^3)
Time = 6.88 (sec) , antiderivative size = 1716, normalized size of antiderivative = 4.89 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{x \left (a+b x^3\right )^3} \, dx=\text {Too large to display} \] Input:
int((c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5)/(x*(a + b*x^3)^3),x)
Output:
((3*b*c - a*f)/(6*a*b) + (x^4*(5*b*d + a*g))/(18*a^2) + (x^5*(2*b*e + a*h) )/(9*a^2) + (x*(4*b*d - a*g))/(9*a*b) + (x^2*(7*b*e - a*h))/(18*a*b) + (b* c*x^3)/(3*a^2))/(a^2 + b^2*x^6 + 2*a*b*x^3) + symsum(log((c*(25*b^2*d^2 + a^2*g^2 - 18*b^2*c*e - 9*a*b*c*h + 10*a*b*d*g))/(81*a^6) - (root(19683*a^9 *b^5*z^3 + 19683*a^6*b^5*c*z^2 + 81*a^6*b^2*g*h*z + 405*a^5*b^3*d*h*z + 16 2*a^5*b^3*e*g*z + 810*a^4*b^4*d*e*z + 6561*a^3*b^5*c^2*z + 270*a*b^4*c*d*e + 27*a^3*b^2*c*g*h + 135*a^2*b^3*c*d*h + 54*a^2*b^3*c*e*g + 6*a^4*b*e*h^2 + 12*a^3*b^2*e^2*h - 75*a^2*b^3*d^2*g - 15*a^3*b^2*d*g^2 + 8*a^2*b^3*e^3 - a^4*b*g^3 - 125*a*b^4*d^3 + 729*b^5*c^3 + a^5*h^3, z, k)*(a^3*g^2 + 25*a *b^2*d^2 + 324*b^3*c^2*x + 2916*root(19683*a^9*b^5*z^3 + 19683*a^6*b^5*c*z ^2 + 81*a^6*b^2*g*h*z + 405*a^5*b^3*d*h*z + 162*a^5*b^3*e*g*z + 810*a^4*b^ 4*d*e*z + 6561*a^3*b^5*c^2*z + 270*a*b^4*c*d*e + 27*a^3*b^2*c*g*h + 135*a^ 2*b^3*c*d*h + 54*a^2*b^3*c*e*g + 6*a^4*b*e*h^2 + 12*a^3*b^2*e^2*h - 75*a^2 *b^3*d^2*g - 15*a^3*b^2*d*g^2 + 8*a^2*b^3*e^3 - a^4*b*g^3 - 125*a*b^4*d^3 + 729*b^5*c^3 + a^5*h^3, z, k)^2*a^6*b^3*x - 27*root(19683*a^9*b^5*z^3 + 1 9683*a^6*b^5*c*z^2 + 81*a^6*b^2*g*h*z + 405*a^5*b^3*d*h*z + 162*a^5*b^3*e* g*z + 810*a^4*b^4*d*e*z + 6561*a^3*b^5*c^2*z + 270*a*b^4*c*d*e + 27*a^3*b^ 2*c*g*h + 135*a^2*b^3*c*d*h + 54*a^2*b^3*c*e*g + 6*a^4*b*e*h^2 + 12*a^3*b^ 2*e^2*h - 75*a^2*b^3*d^2*g - 15*a^3*b^2*d*g^2 + 8*a^2*b^3*e^3 - a^4*b*g^3 - 125*a*b^4*d^3 + 729*b^5*c^3 + a^5*h^3, z, k)*a^5*b*h + 36*a*b^2*c*e +...
Time = 0.19 (sec) , antiderivative size = 1340, normalized size of antiderivative = 3.82 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{x \left (a+b x^3\right )^3} \, dx =\text {Too large to display} \] Input:
int((h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/x/(b*x^3+a)^3,x)
Output:
( - 2*b**(1/3)*a**(2/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*s qrt(3)))*a**3*g - 10*b**(1/3)*a**(2/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3) *x)/(a**(1/3)*sqrt(3)))*a**2*b*d - 4*b**(1/3)*a**(2/3)*sqrt(3)*atan((a**(1 /3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a**2*b*g*x**3 - 20*b**(1/3)*a**(2/ 3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a*b**2*d*x** 3 - 2*b**(1/3)*a**(2/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*s qrt(3)))*a*b**2*g*x**6 - 10*b**(1/3)*a**(2/3)*sqrt(3)*atan((a**(1/3) - 2*b **(1/3)*x)/(a**(1/3)*sqrt(3)))*b**3*d*x**6 - 2*sqrt(3)*atan((a**(1/3) - 2* b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a**4*h - 4*sqrt(3)*atan((a**(1/3) - 2*b**( 1/3)*x)/(a**(1/3)*sqrt(3)))*a**3*b*e - 4*sqrt(3)*atan((a**(1/3) - 2*b**(1/ 3)*x)/(a**(1/3)*sqrt(3)))*a**3*b*h*x**3 - 8*sqrt(3)*atan((a**(1/3) - 2*b** (1/3)*x)/(a**(1/3)*sqrt(3)))*a**2*b**2*e*x**3 - 2*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a**2*b**2*h*x**6 - 4*sqrt(3)*atan((a**( 1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a*b**3*e*x**6 - b**(1/3)*a**(2/3) *log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a**3*g - 5*b**(1/3)*a **(2/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a**2*b*d - 2*b **(1/3)*a**(2/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a**2* b*g*x**3 - 10*b**(1/3)*a**(2/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2 /3)*x**2)*a*b**2*d*x**3 - b**(1/3)*a**(2/3)*log(a**(2/3) - b**(1/3)*a**(1/ 3)*x + b**(2/3)*x**2)*a*b**2*g*x**6 - 5*b**(1/3)*a**(2/3)*log(a**(2/3) ...