Integrand size = 23, antiderivative size = 340 \[ \int \frac {c+d x+e x^2}{x^4 \left (a+b x^3\right )^4} \, dx=-\frac {c}{3 a^4 x^3}-\frac {d}{2 a^4 x^2}-\frac {e}{a^4 x}-\frac {x \left (b d+b e x-\frac {b^2 c x^2}{a}\right )}{9 a^2 \left (a+b x^3\right )^3}-\frac {x \left (17 b d+16 b e x-\frac {24 b^2 c x^2}{a}\right )}{54 a^3 \left (a+b x^3\right )^2}-\frac {x \left (139 b d+118 b e x-\frac {234 b^2 c x^2}{a}\right )}{162 a^4 \left (a+b x^3\right )}+\frac {20 \sqrt [3]{b} \left (11 \sqrt [3]{b} d+7 \sqrt [3]{a} e\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{81 \sqrt {3} a^{14/3}}-\frac {4 b c \log (x)}{a^5}-\frac {20 \sqrt [3]{b} \left (11 \sqrt [3]{b} d-7 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{14/3}}+\frac {10 \sqrt [3]{b} \left (11 \sqrt [3]{b} d-7 \sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{243 a^{14/3}}+\frac {4 b c \log \left (a+b x^3\right )}{3 a^5} \]
-1/3*c/a^4/x^3-1/2*d/a^4/x^2-e/a^4/x-1/9*x*(b*d+b*e*x-b^2*c*x^2/a)/a^2/(b* x^3+a)^3-1/54*x*(17*b*d+16*b*e*x-24*b^2*c*x^2/a)/a^3/(b*x^3+a)^2-1/162*x*( 139*b*d+118*b*e*x-234*b^2*c*x^2/a)/a^4/(b*x^3+a)-4*b*c*ln(x)/a^5-20/243*b^ (1/3)*(11*b^(1/3)*d-7*a^(1/3)*e)*ln(a^(1/3)+b^(1/3)*x)/a^(14/3)+10/243*b^( 1/3)*(11*b^(1/3)*d-7*a^(1/3)*e)*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/ a^(14/3)+4/3*b*c*ln(b*x^3+a)/a^5+20/243*b^(1/3)*(11*b^(1/3)*d+7*a^(1/3)*e) *arctan(1/3*(a^(1/3)-2*b^(1/3)*x)/a^(1/3)*3^(1/2))/a^(14/3)*3^(1/2)
Time = 0.50 (sec) , antiderivative size = 284, normalized size of antiderivative = 0.84 \[ \int \frac {c+d x+e x^2}{x^4 \left (a+b x^3\right )^4} \, dx=-\frac {\frac {162 a c}{x^3}+\frac {243 a d}{x^2}+\frac {486 a e}{x}+\frac {54 a^3 b (c+x (d+e x))}{\left (a+b x^3\right )^3}+\frac {9 a^2 b (18 c+x (17 d+16 e x))}{\left (a+b x^3\right )^2}+\frac {3 a b (162 c+x (139 d+118 e x))}{a+b x^3}-40 \sqrt {3} \sqrt [3]{a} \sqrt [3]{b} \left (11 \sqrt [3]{b} d+7 \sqrt [3]{a} e\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )+1944 b c \log (x)+40 \sqrt [3]{b} \left (11 \sqrt [3]{a} \sqrt [3]{b} d-7 a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-20 \sqrt [3]{b} \left (11 \sqrt [3]{a} \sqrt [3]{b} d-7 a^{2/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-648 b c \log \left (a+b x^3\right )}{486 a^5} \]
-1/486*((162*a*c)/x^3 + (243*a*d)/x^2 + (486*a*e)/x + (54*a^3*b*(c + x*(d + e*x)))/(a + b*x^3)^3 + (9*a^2*b*(18*c + x*(17*d + 16*e*x)))/(a + b*x^3)^ 2 + (3*a*b*(162*c + x*(139*d + 118*e*x)))/(a + b*x^3) - 40*Sqrt[3]*a^(1/3) *b^(1/3)*(11*b^(1/3)*d + 7*a^(1/3)*e)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/S qrt[3]] + 1944*b*c*Log[x] + 40*b^(1/3)*(11*a^(1/3)*b^(1/3)*d - 7*a^(2/3)*e )*Log[a^(1/3) + b^(1/3)*x] - 20*b^(1/3)*(11*a^(1/3)*b^(1/3)*d - 7*a^(2/3)* e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] - 648*b*c*Log[a + b*x^3] )/a^5
Time = 1.15 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.14, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {2368, 25, 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.
| \(\displaystyle \int \frac {c+d x+e x^2}{x^4 \left (a+b x^3\right )^4} \, dx\) |
| \(\Big \downarrow \) 2368 |
| \(\displaystyle -\frac {\int -\frac {\frac {6 b^3 c x^6}{a^2}-\frac {7 b^2 e x^5}{a}-\frac {8 b^2 d x^4}{a}-\frac {9 b^2 c x^3}{a}+9 b e x^2+9 b d x+9 b c}{x^4 \left (b x^3+a\right )^3}dx}{9 a b}-\frac {x \left (-\frac {b^2 c x^2}{a}+b d+b e x\right )}{9 a^2 \left (a+b x^3\right )^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\frac {6 b^3 c x^6}{a^2}-\frac {7 b^2 e x^5}{a}-\frac {8 b^2 d x^4}{a}-\frac {9 b^2 c x^3}{a}+9 b e x^2+9 b d x+9 b c}{x^4 \left (b x^3+a\right )^3}dx}{9 a b}-\frac {x \left (-\frac {b^2 c x^2}{a}+b d+b e x\right )}{9 a^2 \left (a+b x^3\right )^3}\) |
| \(\Big \downarrow \) 2368 |
| \(\displaystyle \frac {-\frac {\int -\frac {\frac {72 b^5 c x^6}{a^2}-\frac {64 b^4 e x^5}{a}-\frac {85 b^4 d x^4}{a}-\frac {108 b^4 c x^3}{a}+54 b^3 e x^2+54 b^3 d x+54 b^3 c}{x^4 \left (b x^3+a\right )^2}dx}{6 a b^2}-\frac {x \left (-\frac {24 b^3 c x^2}{a}+17 b^2 d+16 b^2 e x\right )}{6 a^2 \left (a+b x^3\right )^2}}{9 a b}-\frac {x \left (-\frac {b^2 c x^2}{a}+b d+b e x\right )}{9 a^2 \left (a+b x^3\right )^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {\frac {72 b^5 c x^6}{a^2}-\frac {64 b^4 e x^5}{a}-\frac {85 b^4 d x^4}{a}-\frac {108 b^4 c x^3}{a}+54 b^3 e x^2+54 b^3 d x+54 b^3 c}{x^4 \left (b x^3+a\right )^2}dx}{6 a b^2}-\frac {x \left (-\frac {24 b^3 c x^2}{a}+17 b^2 d+16 b^2 e x\right )}{6 a^2 \left (a+b x^3\right )^2}}{9 a b}-\frac {x \left (-\frac {b^2 c x^2}{a}+b d+b e x\right )}{9 a^2 \left (a+b x^3\right )^3}\) |
| \(\Big \downarrow \) 2368 |
| \(\displaystyle \frac {\frac {-\frac {\int -\frac {2 \left (-\frac {59 e x^5 b^6}{a}-\frac {139 d x^4 b^6}{a}-\frac {243 c x^3 b^6}{a}+81 e x^2 b^5+81 c b^5+81 d x b^5\right )}{x^4 \left (b x^3+a\right )}dx}{3 a b^2}-\frac {x \left (-\frac {234 b^5 c x^2}{a}+139 b^4 d+118 b^4 e x\right )}{3 a^2 \left (a+b x^3\right )}}{6 a b^2}-\frac {x \left (-\frac {24 b^3 c x^2}{a}+17 b^2 d+16 b^2 e x\right )}{6 a^2 \left (a+b x^3\right )^2}}{9 a b}-\frac {x \left (-\frac {b^2 c x^2}{a}+b d+b e x\right )}{9 a^2 \left (a+b x^3\right )^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {2 \int \frac {-\frac {59 e x^5 b^6}{a}-\frac {139 d x^4 b^6}{a}-\frac {243 c x^3 b^6}{a}+81 e x^2 b^5+81 c b^5+81 d x b^5}{x^4 \left (b x^3+a\right )}dx}{3 a b^2}-\frac {x \left (-\frac {234 b^5 c x^2}{a}+139 b^4 d+118 b^4 e x\right )}{3 a^2 \left (a+b x^3\right )}}{6 a b^2}-\frac {x \left (-\frac {24 b^3 c x^2}{a}+17 b^2 d+16 b^2 e x\right )}{6 a^2 \left (a+b x^3\right )^2}}{9 a b}-\frac {x \left (-\frac {b^2 c x^2}{a}+b d+b e x\right )}{9 a^2 \left (a+b x^3\right )^3}\) |
| \(\Big \downarrow \) 2373 |
| \(\displaystyle \frac {\frac {\frac {2 \int \left (-\frac {324 c b^6}{a^2 x}-\frac {4 \left (-81 b c x^2+35 a e x+55 a d\right ) b^6}{a^2 \left (b x^3+a\right )}+\frac {81 e b^5}{a x^2}+\frac {81 d b^5}{a x^3}+\frac {81 c b^5}{a x^4}\right )dx}{3 a b^2}-\frac {x \left (-\frac {234 b^5 c x^2}{a}+139 b^4 d+118 b^4 e x\right )}{3 a^2 \left (a+b x^3\right )}}{6 a b^2}-\frac {x \left (-\frac {24 b^3 c x^2}{a}+17 b^2 d+16 b^2 e x\right )}{6 a^2 \left (a+b x^3\right )^2}}{9 a b}-\frac {x \left (-\frac {b^2 c x^2}{a}+b d+b e x\right )}{9 a^2 \left (a+b x^3\right )^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\frac {2 \left (\frac {20 b^{16/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (7 \sqrt [3]{a} e+11 \sqrt [3]{b} d\right )}{\sqrt {3} a^{5/3}}+\frac {10 b^{16/3} \left (11 \sqrt [3]{b} d-7 \sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{3 a^{5/3}}-\frac {20 b^{16/3} \left (11 \sqrt [3]{b} d-7 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{5/3}}+\frac {108 b^6 c \log \left (a+b x^3\right )}{a^2}-\frac {324 b^6 c \log (x)}{a^2}-\frac {27 b^5 c}{a x^3}-\frac {81 b^5 d}{2 a x^2}-\frac {81 b^5 e}{a x}\right )}{3 a b^2}-\frac {x \left (-\frac {234 b^5 c x^2}{a}+139 b^4 d+118 b^4 e x\right )}{3 a^2 \left (a+b x^3\right )}}{6 a b^2}-\frac {x \left (-\frac {24 b^3 c x^2}{a}+17 b^2 d+16 b^2 e x\right )}{6 a^2 \left (a+b x^3\right )^2}}{9 a b}-\frac {x \left (-\frac {b^2 c x^2}{a}+b d+b e x\right )}{9 a^2 \left (a+b x^3\right )^3}\) |
-1/9*(x*(b*d + b*e*x - (b^2*c*x^2)/a))/(a^2*(a + b*x^3)^3) + (-1/6*(x*(17* b^2*d + 16*b^2*e*x - (24*b^3*c*x^2)/a))/(a^2*(a + b*x^3)^2) + (-1/3*(x*(13 9*b^4*d + 118*b^4*e*x - (234*b^5*c*x^2)/a))/(a^2*(a + b*x^3)) + (2*((-27*b ^5*c)/(a*x^3) - (81*b^5*d)/(2*a*x^2) - (81*b^5*e)/(a*x) + (20*b^(16/3)*(11 *b^(1/3)*d + 7*a^(1/3)*e)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3)) ])/(Sqrt[3]*a^(5/3)) - (324*b^6*c*Log[x])/a^2 - (20*b^(16/3)*(11*b^(1/3)*d - 7*a^(1/3)*e)*Log[a^(1/3) + b^(1/3)*x])/(3*a^(5/3)) + (10*b^(16/3)*(11*b ^(1/3)*d - 7*a^(1/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(3 *a^(5/3)) + (108*b^6*c*Log[a + b*x^3])/a^2))/(3*a*b^2))/(6*a*b^2))/(9*a*b)
3.4.64.3.1 Defintions of rubi rules used
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 = 1.56 (sec) , antiderivative size = 336, normalized size of antiderivative = 0.99
| method | result | size |
| default | \(-\frac {c}{3 a^{4} x^{3}}-\frac {d}{2 a^{4} x^{2}}-\frac {e}{a^{4} x}-\frac {4 b c \ln \left (x \right )}{a^{5}}-\frac {b \left (\frac {\frac {59}{81} a \,b^{2} e \,x^{8}+\frac {139}{162} a \,b^{2} d \,x^{7}+a \,b^{2} c \,x^{6}+\frac {142}{81} a^{2} b e \,x^{5}+\frac {329}{162} a^{2} b d \,x^{4}+\frac {7}{3} a^{2} x^{3} b c +\frac {92}{81} a^{3} e \,x^{2}+\frac {104}{81} a^{3} d x +\frac {13}{9} c \,a^{3}}{\left (b \,x^{3}+a \right )^{3}}+\frac {220 a d \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{81}+\frac {140 a e \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{81}-\frac {4 c \ln \left (b \,x^{3}+a \right )}{3}\right )}{a^{5}}\) | \(336\) |
| risch | \(\frac {-\frac {140 b^{3} e \,x^{11}}{81 a^{4}}-\frac {110 b^{3} d \,x^{10}}{81 a^{4}}-\frac {4 b^{3} c \,x^{9}}{3 a^{4}}-\frac {385 e \,b^{2} x^{8}}{81 a^{3}}-\frac {286 d \,b^{2} x^{7}}{81 a^{3}}-\frac {10 c \,b^{2} x^{6}}{3 a^{3}}-\frac {335 b e \,x^{5}}{81 a^{2}}-\frac {451 b d \,x^{4}}{162 a^{2}}-\frac {22 b c \,x^{3}}{9 a^{2}}-\frac {e \,x^{2}}{a}-\frac {x d}{2 a}-\frac {c}{3 a}}{x^{3} \left (b \,x^{3}+a \right )^{3}}+\frac {4 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{15} \textit {\_Z}^{3}-243 a^{10} b c \,\textit {\_Z}^{2}+\left (5775 a^{6} b d e +19683 a^{5} b^{2} c^{2}\right ) \textit {\_Z} -42875 a^{2} b \,e^{3}-467775 a \,b^{2} c d e +166375 a \,b^{2} d^{3}-531441 b^{3} c^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{14}+648 a^{9} b c \,\textit {\_R}^{2}+\left (-19250 a^{5} b d e -26244 a^{4} b^{2} c^{2}\right ) \textit {\_R} +128625 a b \,e^{3}+935550 b^{2} d c e -499125 b^{2} d^{3}\right ) x -35 \textit {\_R}^{2} a^{10} e +\left (-5670 a^{5} b c e -3025 a^{5} b \,d^{2}\right ) \textit {\_R} +688905 b^{2} c^{2} e -735075 b^{2} c \,d^{2}\right )\right )}{243}-\frac {4 b c \ln \left (x \right )}{a^{5}}\) | \(347\) |
-1/3*c/a^4/x^3-1/2*d/a^4/x^2-e/a^4/x-4*b*c*ln(x)/a^5-1/a^5*b*((59/81*a*b^2 *e*x^8+139/162*a*b^2*d*x^7+a*b^2*c*x^6+142/81*a^2*b*e*x^5+329/162*a^2*b*d* x^4+7/3*a^2*x^3*b*c+92/81*a^3*e*x^2+104/81*a^3*d*x+13/9*c*a^3)/(b*x^3+a)^3 +220/81*a*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)))+140/81*a*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)*a rctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1)))-4/3*c*ln(b*x^3+a))
Result contains complex when optimal does not.
Time = 1.44 (sec) , antiderivative size = 5670, normalized size of antiderivative = 16.68 \[ \int \frac {c+d x+e x^2}{x^4 \left (a+b x^3\right )^4} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {c+d x+e x^2}{x^4 \left (a+b x^3\right )^4} \, dx=\text {Timed out} \]
Time = 0.31 (sec) , antiderivative size = 330, normalized size of antiderivative = 0.97 \[ \int \frac {c+d x+e x^2}{x^4 \left (a+b x^3\right )^4} \, dx=-\frac {280 \, b^{3} e x^{11} + 220 \, b^{3} d x^{10} + 216 \, b^{3} c x^{9} + 770 \, a b^{2} e x^{8} + 572 \, a b^{2} d x^{7} + 540 \, a b^{2} c x^{6} + 670 \, a^{2} b e x^{5} + 451 \, a^{2} b d x^{4} + 396 \, a^{2} b c x^{3} + 162 \, a^{3} e x^{2} + 81 \, a^{3} d x + 54 \, a^{3} c}{162 \, {\left (a^{4} b^{3} x^{12} + 3 \, a^{5} b^{2} x^{9} + 3 \, a^{6} b x^{6} + a^{7} x^{3}\right )}} - \frac {4 \, b c \log \left (x\right )}{a^{5}} - \frac {20 \, \sqrt {3} {\left (7 \, a e \left (\frac {a}{b}\right )^{\frac {2}{3}} + 11 \, a d \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} b \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{243 \, a^{6}} + \frac {2 \, {\left (162 \, b c \left (\frac {a}{b}\right )^{\frac {2}{3}} - 35 \, a e \left (\frac {a}{b}\right )^{\frac {1}{3}} + 55 \, a d\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{243 \, a^{5} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {4 \, {\left (81 \, b c \left (\frac {a}{b}\right )^{\frac {2}{3}} + 35 \, a e \left (\frac {a}{b}\right )^{\frac {1}{3}} - 55 \, a d\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{243 \, a^{5} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
-1/162*(280*b^3*e*x^11 + 220*b^3*d*x^10 + 216*b^3*c*x^9 + 770*a*b^2*e*x^8 + 572*a*b^2*d*x^7 + 540*a*b^2*c*x^6 + 670*a^2*b*e*x^5 + 451*a^2*b*d*x^4 + 396*a^2*b*c*x^3 + 162*a^3*e*x^2 + 81*a^3*d*x + 54*a^3*c)/(a^4*b^3*x^12 + 3 *a^5*b^2*x^9 + 3*a^6*b*x^6 + a^7*x^3) - 4*b*c*log(x)/a^5 - 20/243*sqrt(3)* (7*a*e*(a/b)^(2/3) + 11*a*d*(a/b)^(1/3))*b*arctan(1/3*sqrt(3)*(2*x - (a/b) ^(1/3))/(a/b)^(1/3))/a^6 + 2/243*(162*b*c*(a/b)^(2/3) - 35*a*e*(a/b)^(1/3) + 55*a*d)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a^5*(a/b)^(2/3)) + 4/24 3*(81*b*c*(a/b)^(2/3) + 35*a*e*(a/b)^(1/3) - 55*a*d)*log(x + (a/b)^(1/3))/ (a^5*(a/b)^(2/3))
Time = 0.28 (sec) , antiderivative size = 326, normalized size of antiderivative = 0.96 \[ \int \frac {c+d x+e x^2}{x^4 \left (a+b x^3\right )^4} \, dx=\frac {4 \, b c \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{5}} - \frac {4 \, b c \log \left ({\left | x \right |}\right )}{a^{5}} - \frac {20 \, \sqrt {3} {\left (11 \, \left (-a b^{2}\right )^{\frac {1}{3}} b d - 7 \, \left (-a b^{2}\right )^{\frac {2}{3}} e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{243 \, a^{5} b} - \frac {10 \, {\left (11 \, \left (-a b^{2}\right )^{\frac {1}{3}} b d + 7 \, \left (-a b^{2}\right )^{\frac {2}{3}} e\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{243 \, a^{5} b} - \frac {280 \, b^{3} e x^{11} + 220 \, b^{3} d x^{10} + 216 \, b^{3} c x^{9} + 770 \, a b^{2} e x^{8} + 572 \, a b^{2} d x^{7} + 540 \, a b^{2} c x^{6} + 670 \, a^{2} b e x^{5} + 451 \, a^{2} b d x^{4} + 396 \, a^{2} b c x^{3} + 162 \, a^{3} e x^{2} + 81 \, a^{3} d x + 54 \, a^{3} c}{162 \, {\left (b x^{4} + a x\right )}^{3} a^{4}} + \frac {20 \, {\left (7 \, a^{6} b^{2} e \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 11 \, a^{6} b^{2} d\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{243 \, a^{11} b} \]
4/3*b*c*log(abs(b*x^3 + a))/a^5 - 4*b*c*log(abs(x))/a^5 - 20/243*sqrt(3)*( 11*(-a*b^2)^(1/3)*b*d - 7*(-a*b^2)^(2/3)*e)*arctan(1/3*sqrt(3)*(2*x + (-a/ b)^(1/3))/(-a/b)^(1/3))/(a^5*b) - 10/243*(11*(-a*b^2)^(1/3)*b*d + 7*(-a*b^ 2)^(2/3)*e)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^5*b) - 1/162*(280* b^3*e*x^11 + 220*b^3*d*x^10 + 216*b^3*c*x^9 + 770*a*b^2*e*x^8 + 572*a*b^2* d*x^7 + 540*a*b^2*c*x^6 + 670*a^2*b*e*x^5 + 451*a^2*b*d*x^4 + 396*a^2*b*c* x^3 + 162*a^3*e*x^2 + 81*a^3*d*x + 54*a^3*c)/((b*x^4 + a*x)^3*a^4) + 20/24 3*(7*a^6*b^2*e*(-a/b)^(1/3) + 11*a^6*b^2*d)*(-a/b)^(1/3)*log(abs(x - (-a/b )^(1/3)))/(a^11*b)
Time = 12.23 (sec) , antiderivative size = 918, normalized size of antiderivative = 2.70 \[ \int \frac {c+d x+e x^2}{x^4 \left (a+b x^3\right )^4} \, dx=\left (\sum _{k=1}^3\ln \left (-\frac {b^3\,\left ({\mathrm {root}\left (14348907\,a^{15}\,z^3-57395628\,a^{10}\,b\,c\,z^2+22453200\,a^6\,b\,d\,e\,z+76527504\,a^5\,b^2\,c^2\,z-29937600\,a\,b^2\,c\,d\,e-2744000\,a^2\,b\,e^3+10648000\,a\,b^2\,d^3-34012224\,b^3\,c^3,z,k\right )}^2\,a^{10}\,e\,688905+3920400\,b^2\,c\,d^2-3674160\,b^2\,c^2\,e+{\mathrm {root}\left (14348907\,a^{15}\,z^3-57395628\,a^{10}\,b\,c\,z^2+22453200\,a^6\,b\,d\,e\,z+76527504\,a^5\,b^2\,c^2\,z-29937600\,a\,b^2\,c\,d\,e-2744000\,a^2\,b\,e^3+10648000\,a\,b^2\,d^3-34012224\,b^3\,c^3,z,k\right )}^3\,a^{14}\,x\,4782969+2662000\,b^2\,d^3\,x-686000\,a\,b\,e^3\,x+\mathrm {root}\left (14348907\,a^{15}\,z^3-57395628\,a^{10}\,b\,c\,z^2+22453200\,a^6\,b\,d\,e\,z+76527504\,a^5\,b^2\,c^2\,z-29937600\,a\,b^2\,c\,d\,e-2744000\,a^2\,b\,e^3+10648000\,a\,b^2\,d^3-34012224\,b^3\,c^3,z,k\right )\,a^5\,b\,d^2\,980100-{\mathrm {root}\left (14348907\,a^{15}\,z^3-57395628\,a^{10}\,b\,c\,z^2+22453200\,a^6\,b\,d\,e\,z+76527504\,a^5\,b^2\,c^2\,z-29937600\,a\,b^2\,c\,d\,e-2744000\,a^2\,b\,e^3+10648000\,a\,b^2\,d^3-34012224\,b^3\,c^3,z,k\right )}^2\,a^9\,b\,c\,x\,12754584+\mathrm {root}\left (14348907\,a^{15}\,z^3-57395628\,a^{10}\,b\,c\,z^2+22453200\,a^6\,b\,d\,e\,z+76527504\,a^5\,b^2\,c^2\,z-29937600\,a\,b^2\,c\,d\,e-2744000\,a^2\,b\,e^3+10648000\,a\,b^2\,d^3-34012224\,b^3\,c^3,z,k\right )\,a^4\,b^2\,c^2\,x\,8503056+\mathrm {root}\left (14348907\,a^{15}\,z^3-57395628\,a^{10}\,b\,c\,z^2+22453200\,a^6\,b\,d\,e\,z+76527504\,a^5\,b^2\,c^2\,z-29937600\,a\,b^2\,c\,d\,e-2744000\,a^2\,b\,e^3+10648000\,a\,b^2\,d^3-34012224\,b^3\,c^3,z,k\right )\,a^5\,b\,c\,e\,1837080-4989600\,b^2\,c\,d\,e\,x+\mathrm {root}\left (14348907\,a^{15}\,z^3-57395628\,a^{10}\,b\,c\,z^2+22453200\,a^6\,b\,d\,e\,z+76527504\,a^5\,b^2\,c^2\,z-29937600\,a\,b^2\,c\,d\,e-2744000\,a^2\,b\,e^3+10648000\,a\,b^2\,d^3-34012224\,b^3\,c^3,z,k\right )\,a^5\,b\,d\,e\,x\,6237000\right )\,4}{a^{12}\,531441}\right )\,\mathrm {root}\left (14348907\,a^{15}\,z^3-57395628\,a^{10}\,b\,c\,z^2+22453200\,a^6\,b\,d\,e\,z+76527504\,a^5\,b^2\,c^2\,z-29937600\,a\,b^2\,c\,d\,e-2744000\,a^2\,b\,e^3+10648000\,a\,b^2\,d^3-34012224\,b^3\,c^3,z,k\right )\right )-\frac {\frac {c}{3\,a}+\frac {e\,x^2}{a}+\frac {d\,x}{2\,a}+\frac {10\,b^2\,c\,x^6}{3\,a^3}+\frac {4\,b^3\,c\,x^9}{3\,a^4}+\frac {286\,b^2\,d\,x^7}{81\,a^3}+\frac {110\,b^3\,d\,x^{10}}{81\,a^4}+\frac {385\,b^2\,e\,x^8}{81\,a^3}+\frac {140\,b^3\,e\,x^{11}}{81\,a^4}+\frac {22\,b\,c\,x^3}{9\,a^2}+\frac {451\,b\,d\,x^4}{162\,a^2}+\frac {335\,b\,e\,x^5}{81\,a^2}}{a^3\,x^3+3\,a^2\,b\,x^6+3\,a\,b^2\,x^9+b^3\,x^{12}}-\frac {4\,b\,c\,\ln \left (x\right )}{a^5} \]
symsum(log(-(4*b^3*(688905*root(14348907*a^15*z^3 - 57395628*a^10*b*c*z^2 + 22453200*a^6*b*d*e*z + 76527504*a^5*b^2*c^2*z - 29937600*a*b^2*c*d*e - 2 744000*a^2*b*e^3 + 10648000*a*b^2*d^3 - 34012224*b^3*c^3, z, k)^2*a^10*e + 3920400*b^2*c*d^2 - 3674160*b^2*c^2*e + 4782969*root(14348907*a^15*z^3 - 57395628*a^10*b*c*z^2 + 22453200*a^6*b*d*e*z + 76527504*a^5*b^2*c^2*z - 29 937600*a*b^2*c*d*e - 2744000*a^2*b*e^3 + 10648000*a*b^2*d^3 - 34012224*b^3 *c^3, z, k)^3*a^14*x + 2662000*b^2*d^3*x - 686000*a*b*e^3*x + 980100*root( 14348907*a^15*z^3 - 57395628*a^10*b*c*z^2 + 22453200*a^6*b*d*e*z + 7652750 4*a^5*b^2*c^2*z - 29937600*a*b^2*c*d*e - 2744000*a^2*b*e^3 + 10648000*a*b^ 2*d^3 - 34012224*b^3*c^3, z, k)*a^5*b*d^2 - 12754584*root(14348907*a^15*z^ 3 - 57395628*a^10*b*c*z^2 + 22453200*a^6*b*d*e*z + 76527504*a^5*b^2*c^2*z - 29937600*a*b^2*c*d*e - 2744000*a^2*b*e^3 + 10648000*a*b^2*d^3 - 34012224 *b^3*c^3, z, k)^2*a^9*b*c*x + 8503056*root(14348907*a^15*z^3 - 57395628*a^ 10*b*c*z^2 + 22453200*a^6*b*d*e*z + 76527504*a^5*b^2*c^2*z - 29937600*a*b^ 2*c*d*e - 2744000*a^2*b*e^3 + 10648000*a*b^2*d^3 - 34012224*b^3*c^3, z, k) *a^4*b^2*c^2*x + 1837080*root(14348907*a^15*z^3 - 57395628*a^10*b*c*z^2 + 22453200*a^6*b*d*e*z + 76527504*a^5*b^2*c^2*z - 29937600*a*b^2*c*d*e - 274 4000*a^2*b*e^3 + 10648000*a*b^2*d^3 - 34012224*b^3*c^3, z, k)*a^5*b*c*e - 4989600*b^2*c*d*e*x + 6237000*root(14348907*a^15*z^3 - 57395628*a^10*b*c*z ^2 + 22453200*a^6*b*d*e*z + 76527504*a^5*b^2*c^2*z - 29937600*a*b^2*c*d...