Integrand size = 23, antiderivative size = 301 \[ \int \frac {c+d x+e x^2}{x^2 \left (a+b x^3\right )^4} \, dx=-\frac {c}{a^4 x}+\frac {x \left (a e-b c x-b d x^2\right )}{9 a^2 \left (a+b x^3\right )^3}+\frac {x \left (8 a e-16 b c x-15 b d x^2\right )}{54 a^3 \left (a+b x^3\right )^2}+\frac {x \left (40 a e-118 b c x-99 b d x^2\right )}{162 a^4 \left (a+b x^3\right )}+\frac {20 \left (7 b^{2/3} c-2 a^{2/3} e\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{81 \sqrt {3} a^{13/3} \sqrt [3]{b}}+\frac {d \log (x)}{a^4}+\frac {20 \left (7 b^{2/3} c+2 a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{13/3} \sqrt [3]{b}}-\frac {10 \left (7 b^{2/3} c+2 a^{2/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{243 a^{13/3} \sqrt [3]{b}}-\frac {d \log \left (a+b x^3\right )}{3 a^4} \] Output:
-c/a^4/x+1/9*x*(-b*d*x^2-b*c*x+a*e)/a^2/(b*x^3+a)^3+1/54*x*(-15*b*d*x^2-16 *b*c*x+8*a*e)/a^3/(b*x^3+a)^2+1/162*x*(-99*b*d*x^2-118*b*c*x+40*a*e)/a^4/( b*x^3+a)+20/243*(7*b^(2/3)*c-2*a^(2/3)*e)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x) *3^(1/2)/a^(1/3))*3^(1/2)/a^(13/3)/b^(1/3)+d*ln(x)/a^4+20/243*(7*b^(2/3)*c +2*a^(2/3)*e)*ln(a^(1/3)+b^(1/3)*x)/a^(13/3)/b^(1/3)-10/243*(7*b^(2/3)*c+2 *a^(2/3)*e)*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(13/3)/b^(1/3)-1/3 *d*ln(b*x^3+a)/a^4
Time = 0.21 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.93 \[ \int \frac {c+d x+e x^2}{x^2 \left (a+b x^3\right )^4} \, dx=\frac {-\frac {486 a c}{x}+\frac {9 a^2 \left (9 a d+8 a e x-16 b c x^2\right )}{\left (a+b x^3\right )^2}+\frac {6 a \left (27 a d+20 a e x-59 b c x^2\right )}{a+b x^3}+\frac {54 a^3 \left (-b c x^2+a (d+e x)\right )}{\left (a+b x^3\right )^3}-\frac {40 \sqrt {3} a^{2/3} \left (-7 b^{2/3} c+2 a^{2/3} e\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}+486 a d \log (x)+\frac {40 \left (7 a^{2/3} b^{2/3} c+2 a^{4/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}-\frac {20 \left (7 a^{2/3} b^{2/3} c+2 a^{4/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [3]{b}}-162 a d \log \left (a+b x^3\right )}{486 a^5} \] Input:
Integrate[(c + d*x + e*x^2)/(x^2*(a + b*x^3)^4),x]
Output:
((-486*a*c)/x + (9*a^2*(9*a*d + 8*a*e*x - 16*b*c*x^2))/(a + b*x^3)^2 + (6* a*(27*a*d + 20*a*e*x - 59*b*c*x^2))/(a + b*x^3) + (54*a^3*(-(b*c*x^2) + a* (d + e*x)))/(a + b*x^3)^3 - (40*Sqrt[3]*a^(2/3)*(-7*b^(2/3)*c + 2*a^(2/3)* e)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(1/3) + 486*a*d*Log[x] + (40*(7*a^(2/3)*b^(2/3)*c + 2*a^(4/3)*e)*Log[a^(1/3) + b^(1/3)*x])/b^(1/3) - (20*(7*a^(2/3)*b^(2/3)*c + 2*a^(4/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(1/3) - 162*a*d*Log[a + b*x^3])/(486*a^5)
Time = 1.65 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.17, 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.
| \(\displaystyle \int \frac {c+d x+e x^2}{x^2 \left (a+b x^3\right )^4} \, dx\) |
| \(\Big \downarrow \) 2368 |
| \(\displaystyle \frac {x \left (a e-b c x-b d x^2\right )}{9 a^2 \left (a+b x^3\right )^3}-\frac {\int -\frac {-\frac {6 b^2 d x^4}{a}-\frac {7 b^2 c x^3}{a}+8 b e x^2+9 b d x+9 b c}{x^2 \left (b x^3+a\right )^3}dx}{9 a b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {-\frac {6 b^2 d x^4}{a}-\frac {7 b^2 c x^3}{a}+8 b e x^2+9 b d x+9 b c}{x^2 \left (b x^3+a\right )^3}dx}{9 a b}+\frac {x \left (a e-b c x-b d x^2\right )}{9 a^2 \left (a+b x^3\right )^3}\) |
| \(\Big \downarrow \) 2368 |
| \(\displaystyle \frac {\frac {x \left (8 a b e-16 b^2 c x-15 b^2 d x^2\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac {\int -\frac {-\frac {45 d x^4 b^4}{a}-\frac {64 c x^3 b^4}{a}+40 e x^2 b^3+54 c b^3+54 d x b^3}{x^2 \left (b x^3+a\right )^2}dx}{6 a b^2}}{9 a b}+\frac {x \left (a e-b c x-b d x^2\right )}{9 a^2 \left (a+b x^3\right )^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {-\frac {45 d x^4 b^4}{a}-\frac {64 c x^3 b^4}{a}+40 e x^2 b^3+54 c b^3+54 d x b^3}{x^2 \left (b x^3+a\right )^2}dx}{6 a b^2}+\frac {x \left (8 a b e-16 b^2 c x-15 b^2 d x^2\right )}{6 a^2 \left (a+b x^3\right )^2}}{9 a b}+\frac {x \left (a e-b c x-b d x^2\right )}{9 a^2 \left (a+b x^3\right )^3}\) |
| \(\Big \downarrow \) 2368 |
| \(\displaystyle \frac {\frac {\frac {x \left (40 a b^3 e-118 b^4 c x-99 b^4 d x^2\right )}{3 a^2 \left (a+b x^3\right )}-\frac {\int -\frac {2 \left (-\frac {59 c x^3 b^6}{a}+40 e x^2 b^5+81 c b^5+81 d x b^5\right )}{x^2 \left (b x^3+a\right )}dx}{3 a b^2}}{6 a b^2}+\frac {x \left (8 a b e-16 b^2 c x-15 b^2 d x^2\right )}{6 a^2 \left (a+b x^3\right )^2}}{9 a b}+\frac {x \left (a e-b c x-b d x^2\right )}{9 a^2 \left (a+b x^3\right )^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {2 \int \frac {-\frac {59 c x^3 b^6}{a}+40 e x^2 b^5+81 c b^5+81 d x b^5}{x^2 \left (b x^3+a\right )}dx}{3 a b^2}+\frac {x \left (40 a b^3 e-118 b^4 c x-99 b^4 d x^2\right )}{3 a^2 \left (a+b x^3\right )}}{6 a b^2}+\frac {x \left (8 a b e-16 b^2 c x-15 b^2 d x^2\right )}{6 a^2 \left (a+b x^3\right )^2}}{9 a b}+\frac {x \left (a e-b c x-b d x^2\right )}{9 a^2 \left (a+b x^3\right )^3}\) |
| \(\Big \downarrow \) 2373 |
| \(\displaystyle \frac {\frac {\frac {2 \int \left (\frac {81 d b^5}{a x}+\frac {\left (-81 b d x^2-140 b c x+40 a e\right ) b^5}{a \left (b x^3+a\right )}+\frac {81 c b^5}{a x^2}\right )dx}{3 a b^2}+\frac {x \left (40 a b^3 e-118 b^4 c x-99 b^4 d x^2\right )}{3 a^2 \left (a+b x^3\right )}}{6 a b^2}+\frac {x \left (8 a b e-16 b^2 c x-15 b^2 d x^2\right )}{6 a^2 \left (a+b x^3\right )^2}}{9 a b}+\frac {x \left (a e-b c x-b d x^2\right )}{9 a^2 \left (a+b x^3\right )^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {x \left (a e-b c x-b d x^2\right )}{9 a^2 \left (a+b x^3\right )^3}+\frac {\frac {x \left (8 a b e-16 b^2 c x-15 b^2 d x^2\right )}{6 a^2 \left (a+b x^3\right )^2}+\frac {\frac {2 \left (\frac {20 b^{14/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (7 b^{2/3} c-2 a^{2/3} e\right )}{\sqrt {3} a^{4/3}}-\frac {10 b^{14/3} \left (2 a^{2/3} e+7 b^{2/3} c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{3 a^{4/3}}+\frac {20 b^{14/3} \left (2 a^{2/3} e+7 b^{2/3} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{4/3}}-\frac {81 b^5 c}{a x}-\frac {27 b^5 d \log \left (a+b x^3\right )}{a}+\frac {81 b^5 d \log (x)}{a}\right )}{3 a b^2}+\frac {x \left (40 a b^3 e-118 b^4 c x-99 b^4 d x^2\right )}{3 a^2 \left (a+b x^3\right )}}{6 a b^2}}{9 a b}\) |
Input:
Int[(c + d*x + e*x^2)/(x^2*(a + b*x^3)^4),x]
Output:
(x*(a*e - b*c*x - b*d*x^2))/(9*a^2*(a + b*x^3)^3) + ((x*(8*a*b*e - 16*b^2* c*x - 15*b^2*d*x^2))/(6*a^2*(a + b*x^3)^2) + ((x*(40*a*b^3*e - 118*b^4*c*x - 99*b^4*d*x^2))/(3*a^2*(a + b*x^3)) + (2*((-81*b^5*c)/(a*x) + (20*b^(14/ 3)*(7*b^(2/3)*c - 2*a^(2/3)*e)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^( 1/3))])/(Sqrt[3]*a^(4/3)) + (81*b^5*d*Log[x])/a + (20*b^(14/3)*(7*b^(2/3)* c + 2*a^(2/3)*e)*Log[a^(1/3) + b^(1/3)*x])/(3*a^(4/3)) - (10*b^(14/3)*(7*b ^(2/3)*c + 2*a^(2/3)*e)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(3 *a^(4/3)) - (27*b^5*d*Log[a + b*x^3])/a))/(3*a*b^2))/(6*a*b^2))/(9*a*b)
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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.14 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.04
| method | result | size |
| risch | \(\frac {-\frac {140 b^{3} c \,x^{9}}{81 a^{4}}+\frac {20 b^{2} e \,x^{8}}{81 a^{3}}+\frac {d \,b^{2} x^{7}}{3 a^{3}}-\frac {385 b^{2} c \,x^{6}}{81 a^{3}}+\frac {52 e b \,x^{5}}{81 a^{2}}+\frac {5 d b \,x^{4}}{6 a^{2}}-\frac {335 b c \,x^{3}}{81 a^{2}}+\frac {41 e \,x^{2}}{81 a}+\frac {11 d x}{18 a}-\frac {c}{a}}{x \left (b \,x^{3}+a \right )^{3}}+\frac {d \ln \left (x \right )}{a^{4}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{13} b \,\textit {\_Z}^{3}+243 a^{9} b d \,\textit {\_Z}^{2}+\left (-16800 a^{5} b c e +19683 a^{5} b \,d^{2}\right ) \textit {\_Z} -64000 a^{2} e^{3}-1360800 a b c d e +531441 a b \,d^{3}-2744000 c^{3} b^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (-\textit {\_R}^{3} a^{13} b -162 \textit {\_R}^{2} a^{9} b d +\left (14000 a^{5} b c e -6561 a^{5} b \,d^{2}\right ) \textit {\_R} +48000 a^{2} e^{3}+680400 a b c d e +2058000 c^{3} b^{2}\right ) x -35 a^{9} b c \,\textit {\_R}^{2}+\left (-400 a^{6} e^{2}+5670 a^{5} b c d \right ) \textit {\_R} +97200 a^{2} d \,e^{2}+688905 a b c \,d^{2}\right )\right )}{243}\) | \(313\) |
| default | \(\frac {\frac {-\frac {59}{81} b^{3} c \,x^{8}+\frac {20}{81} a \,b^{2} e \,x^{7}+\frac {1}{3} a \,b^{2} d \,x^{6}-\frac {142}{81} a \,b^{2} c \,x^{5}+\frac {52}{81} a^{2} b e \,x^{4}+\frac {5}{6} a^{2} b d \,x^{3}-\frac {92}{81} a^{2} b c \,x^{2}+\frac {41}{81} a^{3} e x +\frac {11}{18} a^{3} d}{\left (b \,x^{3}+a \right )^{3}}+\frac {40 a e \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{81}-\frac {140 c b \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{81}-\frac {d \ln \left (b \,x^{3}+a \right )}{3}}{a^{4}}-\frac {c}{a^{4} x}+\frac {d \ln \left (x \right )}{a^{4}}\) | \(315\) |
Input:
int((e*x^2+d*x+c)/x^2/(b*x^3+a)^4,x,method=_RETURNVERBOSE)
Output:
(-140/81/a^4*b^3*c*x^9+20/81/a^3*b^2*e*x^8+1/3/a^3*d*b^2*x^7-385/81/a^3*b^ 2*c*x^6+52/81/a^2*e*b*x^5+5/6/a^2*d*b*x^4-335/81*b/a^2*c*x^3+41/81*e/a*x^2 +11/18*d/a*x-c/a)/x/(b*x^3+a)^3+d*ln(x)/a^4+1/243*sum(_R*ln((-_R^3*a^13*b- 162*_R^2*a^9*b*d+(14000*a^5*b*c*e-6561*a^5*b*d^2)*_R+48000*a^2*e^3+680400* a*b*c*d*e+2058000*c^3*b^2)*x-35*a^9*b*c*_R^2+(-400*a^6*e^2+5670*a^5*b*c*d) *_R+97200*a^2*d*e^2+688905*a*b*c*d^2),_R=RootOf(a^13*b*_Z^3+243*a^9*b*d*_Z ^2+(-16800*a^5*b*c*e+19683*a^5*b*d^2)*_Z-64000*a^2*e^3-1360800*a*b*c*d*e+5 31441*a*b*d^3-2744000*c^3*b^2))
Result contains complex when optimal does not.
Time = 1.00 (sec) , antiderivative size = 5250, normalized size of antiderivative = 17.44 \[ \int \frac {c+d x+e x^2}{x^2 \left (a+b x^3\right )^4} \, dx=\text {Too large to display} \] Input:
integrate((e*x^2+d*x+c)/x^2/(b*x^3+a)^4,x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {c+d x+e x^2}{x^2 \left (a+b x^3\right )^4} \, dx=\text {Timed out} \] Input:
integrate((e*x**2+d*x+c)/x**2/(b*x**3+a)**4,x)
Output:
Timed out
Time = 0.13 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.04 \[ \int \frac {c+d x+e x^2}{x^2 \left (a+b x^3\right )^4} \, dx=-\frac {280 \, b^{3} c x^{9} - 40 \, a b^{2} e x^{8} - 54 \, a b^{2} d x^{7} + 770 \, a b^{2} c x^{6} - 104 \, a^{2} b e x^{5} - 135 \, a^{2} b d x^{4} + 670 \, a^{2} b c x^{3} - 82 \, a^{3} e x^{2} - 99 \, a^{3} d x + 162 \, a^{3} c}{162 \, {\left (a^{4} b^{3} x^{10} + 3 \, a^{5} b^{2} x^{7} + 3 \, a^{6} b x^{4} + a^{7} x\right )}} + \frac {d \log \left (x\right )}{a^{4}} - \frac {20 \, \sqrt {3} {\left (7 \, b c \left (\frac {a}{b}\right )^{\frac {2}{3}} - 2 \, a e \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{243 \, a^{5}} - \frac {{\left (81 \, b d \left (\frac {a}{b}\right )^{\frac {2}{3}} + 70 \, b c \left (\frac {a}{b}\right )^{\frac {1}{3}} + 20 \, a e\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{243 \, a^{4} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (81 \, b d \left (\frac {a}{b}\right )^{\frac {2}{3}} - 140 \, b c \left (\frac {a}{b}\right )^{\frac {1}{3}} - 40 \, a e\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{243 \, a^{4} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} \] Input:
integrate((e*x^2+d*x+c)/x^2/(b*x^3+a)^4,x, algorithm="maxima")
Output:
-1/162*(280*b^3*c*x^9 - 40*a*b^2*e*x^8 - 54*a*b^2*d*x^7 + 770*a*b^2*c*x^6 - 104*a^2*b*e*x^5 - 135*a^2*b*d*x^4 + 670*a^2*b*c*x^3 - 82*a^3*e*x^2 - 99* a^3*d*x + 162*a^3*c)/(a^4*b^3*x^10 + 3*a^5*b^2*x^7 + 3*a^6*b*x^4 + a^7*x) + d*log(x)/a^4 - 20/243*sqrt(3)*(7*b*c*(a/b)^(2/3) - 2*a*e*(a/b)^(1/3))*ar ctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/a^5 - 1/243*(81*b*d*(a/b )^(2/3) + 70*b*c*(a/b)^(1/3) + 20*a*e)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/ 3))/(a^4*b*(a/b)^(2/3)) - 1/243*(81*b*d*(a/b)^(2/3) - 140*b*c*(a/b)^(1/3) - 40*a*e)*log(x + (a/b)^(1/3))/(a^4*b*(a/b)^(2/3))
Time = 0.13 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.01 \[ \int \frac {c+d x+e x^2}{x^2 \left (a+b x^3\right )^4} \, dx=-\frac {d \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{4}} + \frac {d \log \left ({\left | x \right |}\right )}{a^{4}} + \frac {20 \, \sqrt {3} {\left (2 \, \left (-a b^{2}\right )^{\frac {1}{3}} a e + 7 \, \left (-a b^{2}\right )^{\frac {2}{3}} c\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{243 \, a^{5} b} + \frac {10 \, {\left (2 \, \left (-a b^{2}\right )^{\frac {1}{3}} a e - 7 \, \left (-a b^{2}\right )^{\frac {2}{3}} c\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{243 \, a^{5} b} - \frac {280 \, b^{3} c x^{9} - 40 \, a b^{2} e x^{8} - 54 \, a b^{2} d x^{7} + 770 \, a b^{2} c x^{6} - 104 \, a^{2} b e x^{5} - 135 \, a^{2} b d x^{4} + 670 \, a^{2} b c x^{3} - 82 \, a^{3} e x^{2} - 99 \, a^{3} d x + 162 \, a^{3} c}{162 \, {\left (b x^{3} + a\right )}^{3} a^{4} x} + \frac {20 \, {\left (7 \, a^{4} b^{2} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, a^{5} b e\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{243 \, a^{9} b} \] Input:
integrate((e*x^2+d*x+c)/x^2/(b*x^3+a)^4,x, algorithm="giac")
Output:
-1/3*d*log(abs(b*x^3 + a))/a^4 + d*log(abs(x))/a^4 + 20/243*sqrt(3)*(2*(-a *b^2)^(1/3)*a*e + 7*(-a*b^2)^(2/3)*c)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/ 3))/(-a/b)^(1/3))/(a^5*b) + 10/243*(2*(-a*b^2)^(1/3)*a*e - 7*(-a*b^2)^(2/3 )*c)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^5*b) - 1/162*(280*b^3*c*x ^9 - 40*a*b^2*e*x^8 - 54*a*b^2*d*x^7 + 770*a*b^2*c*x^6 - 104*a^2*b*e*x^5 - 135*a^2*b*d*x^4 + 670*a^2*b*c*x^3 - 82*a^3*e*x^2 - 99*a^3*d*x + 162*a^3*c )/((b*x^3 + a)^3*a^4*x) + 20/243*(7*a^4*b^2*c*(-a/b)^(1/3) - 2*a^5*b*e)*(- a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^9*b)
Time = 6.43 (sec) , antiderivative size = 840, normalized size of antiderivative = 2.79 \[ \int \frac {c+d x+e x^2}{x^2 \left (a+b x^3\right )^4} \, dx =\text {Too large to display} \] Input:
int((c + d*x + e*x^2)/(x^2*(a + b*x^3)^4),x)
Output:
((41*e*x^2)/(81*a) - c/a + (11*d*x)/(18*a) - (385*b^2*c*x^6)/(81*a^3) - (1 40*b^3*c*x^9)/(81*a^4) + (b^2*d*x^7)/(3*a^3) + (20*b^2*e*x^8)/(81*a^3) - ( 335*b*c*x^3)/(81*a^2) + (5*b*d*x^4)/(6*a^2) + (52*b*e*x^5)/(81*a^2))/(a^3* x + b^3*x^10 + 3*a^2*b*x^4 + 3*a*b^2*x^7) + symsum(log((4*b^2*(32400*a^2*d *e^2 - 32400*root(14348907*a^13*b*z^3 + 14348907*a^9*b*d*z^2 - 4082400*a^5 *b*c*e*z + 4782969*a^5*b*d^2*z - 1360800*a*b*c*d*e + 531441*a*b*d^3 - 6400 0*a^2*e^3 - 2744000*b^2*c^3, z, k)*a^6*e^2 + 686000*b^2*c^3*x + 16000*a^2* e^3*x + 229635*a*b*c*d^2 - 688905*root(14348907*a^13*b*z^3 + 14348907*a^9* b*d*z^2 - 4082400*a^5*b*c*e*z + 4782969*a^5*b*d^2*z - 1360800*a*b*c*d*e + 531441*a*b*d^3 - 64000*a^2*e^3 - 2744000*b^2*c^3, z, k)^2*a^9*b*c - 478296 9*root(14348907*a^13*b*z^3 + 14348907*a^9*b*d*z^2 - 4082400*a^5*b*c*e*z + 4782969*a^5*b*d^2*z - 1360800*a*b*c*d*e + 531441*a*b*d^3 - 64000*a^2*e^3 - 2744000*b^2*c^3, z, k)^3*a^13*b*x - 531441*root(14348907*a^13*b*z^3 + 143 48907*a^9*b*d*z^2 - 4082400*a^5*b*c*e*z + 4782969*a^5*b*d^2*z - 1360800*a* b*c*d*e + 531441*a*b*d^3 - 64000*a^2*e^3 - 2744000*b^2*c^3, z, k)*a^5*b*d^ 2*x - 3188646*root(14348907*a^13*b*z^3 + 14348907*a^9*b*d*z^2 - 4082400*a^ 5*b*c*e*z + 4782969*a^5*b*d^2*z - 1360800*a*b*c*d*e + 531441*a*b*d^3 - 640 00*a^2*e^3 - 2744000*b^2*c^3, z, k)^2*a^9*b*d*x + 459270*root(14348907*a^1 3*b*z^3 + 14348907*a^9*b*d*z^2 - 4082400*a^5*b*c*e*z + 4782969*a^5*b*d^2*z - 1360800*a*b*c*d*e + 531441*a*b*d^3 - 64000*a^2*e^3 - 2744000*b^2*c^3...
Time = 0.16 (sec) , antiderivative size = 1099, normalized size of antiderivative = 3.65 \[ \int \frac {c+d x+e x^2}{x^2 \left (a+b x^3\right )^4} \, dx =\text {Too large to display} \] Input:
int((e*x^2+d*x+c)/x^2/(b*x^3+a)^4,x)
Output:
( - 80*b**(1/3)*a**(2/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)* sqrt(3)))*a**3*e*x - 240*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*e*x**4 - 240*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*e*x**7 - 80*b**( 1/3)*a**(2/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*b **3*e*x**10 + 280*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3) ))*a**3*b*c*x + 840*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt( 3)))*a**2*b**2*c*x**4 + 840*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/ 3)*sqrt(3)))*a*b**3*c*x**7 + 280*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a **(1/3)*sqrt(3)))*b**4*c*x**10 - 40*b**(1/3)*a**(2/3)*log(a**(2/3) - b**(1 /3)*a**(1/3)*x + b**(2/3)*x**2)*a**3*e*x - 120*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*e*x**4 - 120*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*e*x**7 - 40*b**(1/3)*a**(2/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)* b**3*e*x**10 + 80*b**(1/3)*a**(2/3)*log(a**(1/3) + b**(1/3)*x)*a**3*e*x + 240*b**(1/3)*a**(2/3)*log(a**(1/3) + b**(1/3)*x)*a**2*b*e*x**4 + 240*b**(1 /3)*a**(2/3)*log(a**(1/3) + b**(1/3)*x)*a*b**2*e*x**7 + 80*b**(1/3)*a**(2/ 3)*log(a**(1/3) + b**(1/3)*x)*b**3*e*x**10 - 162*b**(2/3)*a**(1/3)*log(a** (2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a**3*d*x - 486*b**(2/3)*a**(1 /3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a**2*b*d*x**4 -...