\(\int \frac {c+d x+e x^2}{x^4 (a+b x^3)^3} \, dx\) [357]

Optimal result

Integrand size = 23, antiderivative size = 298 \[ \int \frac {c+d x+e x^2}{x^4 \left (a+b x^3\right )^3} \, dx=-\frac {c}{3 a^3 x^3}-\frac {d}{2 a^3 x^2}-\frac {e}{a^3 x}-\frac {x \left (b d+b e x-\frac {b^2 c x^2}{a}\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac {x \left (11 b d+10 b e x-\frac {15 b^2 c x^2}{a}\right )}{18 a^3 \left (a+b x^3\right )}+\frac {2 \sqrt [3]{b} \left (10 \sqrt [3]{b} d+7 \sqrt [3]{a} e\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{11/3}}-\frac {3 b c \log (x)}{a^4}-\frac {2 \sqrt [3]{b} \left (10 \sqrt [3]{b} d-7 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3}}+\frac {\sqrt [3]{b} \left (10 \sqrt [3]{b} d-7 \sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 a^{11/3}}+\frac {b c \log \left (a+b x^3\right )}{a^4} \]

[Out]

-1/3*c/a^3/x^3-1/2*d/a^3/x^2-e/a^3/x-1/6*x*(b*d+b*e*x-b^2*c*x^2/a)/a^2/(b*x^3+a)^2-1/18*x*(11*b*d+10*b*e*x-15*
b^2*c*x^2/a)/a^3/(b*x^3+a)-3*b*c*ln(x)/a^4-2/27*b^(1/3)*(10*b^(1/3)*d-7*a^(1/3)*e)*ln(a^(1/3)+b^(1/3)*x)/a^(11
/3)+1/27*b^(1/3)*(10*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^(11/3)+b*c*ln(b*x^3+a)
/a^4+2/27*b^(1/3)*(10*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^(11/3)*3^(1/2
)

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {1843, 1848, 1885, 1874, 31, 648, 631, 210, 642, 266} \[ \int \frac {c+d x+e x^2}{x^4 \left (a+b x^3\right )^3} \, dx=\frac {2 \sqrt [3]{b} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (7 \sqrt [3]{a} e+10 \sqrt [3]{b} d\right )}{9 \sqrt {3} a^{11/3}}+\frac {\sqrt [3]{b} \left (10 \sqrt [3]{b} d-7 \sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 a^{11/3}}-\frac {2 \sqrt [3]{b} \left (10 \sqrt [3]{b} d-7 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3}}+\frac {b c \log \left (a+b x^3\right )}{a^4}-\frac {3 b c \log (x)}{a^4}-\frac {x \left (-\frac {15 b^2 c x^2}{a}+11 b d+10 b e x\right )}{18 a^3 \left (a+b x^3\right )}-\frac {c}{3 a^3 x^3}-\frac {d}{2 a^3 x^2}-\frac {e}{a^3 x}-\frac {x \left (-\frac {b^2 c x^2}{a}+b d+b e x\right )}{6 a^2 \left (a+b x^3\right )^2} \]

[In]

Int[(c + d*x + e*x^2)/(x^4*(a + b*x^3)^3),x]

[Out]

-1/3*c/(a^3*x^3) - d/(2*a^3*x^2) - e/(a^3*x) - (x*(b*d + b*e*x - (b^2*c*x^2)/a))/(6*a^2*(a + b*x^3)^2) - (x*(1
1*b*d + 10*b*e*x - (15*b^2*c*x^2)/a))/(18*a^3*(a + b*x^3)) + (2*b^(1/3)*(10*b^(1/3)*d + 7*a^(1/3)*e)*ArcTan[(a
^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(11/3)) - (3*b*c*Log[x])/a^4 - (2*b^(1/3)*(10*b^(1/3)*d
- 7*a^(1/3)*e)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(11/3)) + (b^(1/3)*(10*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])/(27*a^(11/3)) + (b*c*Log[a + b*x^3])/a^4

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 210

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])

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[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])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 1843

Int[(Pq_)*(x_)^(m_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, Module[{Q = Polynomi
alQuotient[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}, Dist[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), Int[x^m*(a + b*x^n)^(p + 1)*Expand
ToSum[(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] + S
imp[(-x)*R*((a + b*x^n)^(p + 1)/(a^2*n*(p + 1)*b^(Floor[(q - 1)/n] + 1))), x]]] /; FreeQ[{a, b}, x] && PolyQ[P
q, x] && IGtQ[n, 0] && LtQ[p, -1] && ILtQ[m, 0]

Rule 1848

Int[((Pq_)*((c_.)*(x_))^(m_.))/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(Pq/(a + b*x
^n)), x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IntegerQ[n] &&  !IGtQ[m, 0]

Rule 1874

Int[((A_) + (B_.)*(x_))/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{r = Numerator[Rt[a/b, 3]], s = Denominator[R
t[a/b, 3]]}, Dist[(-r)*((B*r - A*s)/(3*a*s)), Int[1/(r + s*x), x], x] + Dist[r/(3*a*s), Int[(r*(B*r + 2*A*s) +
s*(B*r - A*s)*x)/(r^2 - r*s*x + s^2*x^2), x], x]] /; FreeQ[{a, b, A, B}, x] && NeQ[a*B^3 - b*A^3, 0] && PosQ[
a/b]

Rule 1885

Int[(P2_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x,
2]}, Int[(A + B*x)/(a + b*x^3), x] + Dist[C, Int[x^2/(a + b*x^3), x], x] /; EqQ[a*B^3 - b*A^3, 0] ||  !Ration
alQ[a/b]] /; FreeQ[{a, b}, x] && PolyQ[P2, x, 2]

Rubi steps \begin{align*} \text {integral}& = -\frac {x \left (b d+b e x-\frac {b^2 c x^2}{a}\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac {\int \frac {-6 b c-6 b d x-6 b e x^2+\frac {6 b^2 c x^3}{a}+\frac {5 b^2 d x^4}{a}+\frac {4 b^2 e x^5}{a}-\frac {3 b^3 c x^6}{a^2}}{x^4 \left (a+b x^3\right )^2} \, dx}{6 a b} \\ & = -\frac {x \left (b d+b e x-\frac {b^2 c x^2}{a}\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac {x \left (11 b d+10 b e x-\frac {15 b^2 c x^2}{a}\right )}{18 a^3 \left (a+b x^3\right )}+\frac {\int \frac {18 b^3 c+18 b^3 d x+18 b^3 e x^2-\frac {36 b^4 c x^3}{a}-\frac {22 b^4 d x^4}{a}-\frac {10 b^4 e x^5}{a}}{x^4 \left (a+b x^3\right )} \, dx}{18 a^2 b^3} \\ & = -\frac {x \left (b d+b e x-\frac {b^2 c x^2}{a}\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac {x \left (11 b d+10 b e x-\frac {15 b^2 c x^2}{a}\right )}{18 a^3 \left (a+b x^3\right )}+\frac {\int \left (\frac {18 b^3 c}{a x^4}+\frac {18 b^3 d}{a x^3}+\frac {18 b^3 e}{a x^2}-\frac {54 b^4 c}{a^2 x}-\frac {2 b^4 \left (20 a d+14 a e x-27 b c x^2\right )}{a^2 \left (a+b x^3\right )}\right ) \, dx}{18 a^2 b^3} \\ & = -\frac {c}{3 a^3 x^3}-\frac {d}{2 a^3 x^2}-\frac {e}{a^3 x}-\frac {x \left (b d+b e x-\frac {b^2 c x^2}{a}\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac {x \left (11 b d+10 b e x-\frac {15 b^2 c x^2}{a}\right )}{18 a^3 \left (a+b x^3\right )}-\frac {3 b c \log (x)}{a^4}-\frac {b \int \frac {20 a d+14 a e x-27 b c x^2}{a+b x^3} \, dx}{9 a^4} \\ & = -\frac {c}{3 a^3 x^3}-\frac {d}{2 a^3 x^2}-\frac {e}{a^3 x}-\frac {x \left (b d+b e x-\frac {b^2 c x^2}{a}\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac {x \left (11 b d+10 b e x-\frac {15 b^2 c x^2}{a}\right )}{18 a^3 \left (a+b x^3\right )}-\frac {3 b c \log (x)}{a^4}-\frac {b \int \frac {20 a d+14 a e x}{a+b x^3} \, dx}{9 a^4}+\frac {\left (3 b^2 c\right ) \int \frac {x^2}{a+b x^3} \, dx}{a^4} \\ & = -\frac {c}{3 a^3 x^3}-\frac {d}{2 a^3 x^2}-\frac {e}{a^3 x}-\frac {x \left (b d+b e x-\frac {b^2 c x^2}{a}\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac {x \left (11 b d+10 b e x-\frac {15 b^2 c x^2}{a}\right )}{18 a^3 \left (a+b x^3\right )}-\frac {3 b c \log (x)}{a^4}+\frac {b c \log \left (a+b x^3\right )}{a^4}-\frac {b^{2/3} \int \frac {\sqrt [3]{a} \left (40 a \sqrt [3]{b} d+14 a^{4/3} e\right )+\sqrt [3]{b} \left (-20 a \sqrt [3]{b} d+14 a^{4/3} e\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{27 a^{14/3}}-\frac {\left (2 b \left (10 d-\frac {7 \sqrt [3]{a} e}{\sqrt [3]{b}}\right )\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{11/3}} \\ & = -\frac {c}{3 a^3 x^3}-\frac {d}{2 a^3 x^2}-\frac {e}{a^3 x}-\frac {x \left (b d+b e x-\frac {b^2 c x^2}{a}\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac {x \left (11 b d+10 b e x-\frac {15 b^2 c x^2}{a}\right )}{18 a^3 \left (a+b x^3\right )}-\frac {3 b c \log (x)}{a^4}-\frac {2 \sqrt [3]{b} \left (10 \sqrt [3]{b} d-7 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3}}+\frac {b c \log \left (a+b x^3\right )}{a^4}+\frac {\left (\sqrt [3]{b} \left (10 \sqrt [3]{b} d-7 \sqrt [3]{a} e\right )\right ) \int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{27 a^{11/3}}-\frac {\left (b^{2/3} \left (10 \sqrt [3]{b} d+7 \sqrt [3]{a} e\right )\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{9 a^{10/3}} \\ & = -\frac {c}{3 a^3 x^3}-\frac {d}{2 a^3 x^2}-\frac {e}{a^3 x}-\frac {x \left (b d+b e x-\frac {b^2 c x^2}{a}\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac {x \left (11 b d+10 b e x-\frac {15 b^2 c x^2}{a}\right )}{18 a^3 \left (a+b x^3\right )}-\frac {3 b c \log (x)}{a^4}-\frac {2 \sqrt [3]{b} \left (10 \sqrt [3]{b} d-7 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3}}+\frac {\sqrt [3]{b} \left (10 \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 )}{27 a^{11/3}}+\frac {b c \log \left (a+b x^3\right )}{a^4}-\frac {\left (2 \sqrt [3]{b} \left (10 \sqrt [3]{b} d+7 \sqrt [3]{a} e\right )\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{9 a^{11/3}} \\ & = -\frac {c}{3 a^3 x^3}-\frac {d}{2 a^3 x^2}-\frac {e}{a^3 x}-\frac {x \left (b d+b e x-\frac {b^2 c x^2}{a}\right )}{6 a^2 \left (a+b x^3\right )^2}-\frac {x \left (11 b d+10 b e x-\frac {15 b^2 c x^2}{a}\right )}{18 a^3 \left (a+b x^3\right )}+\frac {2 \sqrt [3]{b} \left (10 \sqrt [3]{b} d+7 \sqrt [3]{a} e\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{11/3}}-\frac {3 b c \log (x)}{a^4}-\frac {2 \sqrt [3]{b} \left (10 \sqrt [3]{b} d-7 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3}}+\frac {\sqrt [3]{b} \left (10 \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 )}{27 a^{11/3}}+\frac {b c \log \left (a+b x^3\right )}{a^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.86 \[ \int \frac {c+d x+e x^2}{x^4 \left (a+b x^3\right )^3} \, dx=-\frac {\frac {18 a c}{x^3}+\frac {27 a d}{x^2}+\frac {54 a e}{x}+\frac {9 a^2 b (c+x (d+e x))}{\left (a+b x^3\right )^2}+\frac {3 a b (12 c+x (11 d+10 e x))}{a+b x^3}-4 \sqrt {3} \sqrt [3]{a} \sqrt [3]{b} \left (10 \sqrt [3]{b} d+7 \sqrt [3]{a} e\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )+162 b c \log (x)+4 \sqrt [3]{b} \left (10 \sqrt [3]{a} \sqrt [3]{b} d-7 a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-2 \sqrt [3]{b} \left (10 \sqrt [3]{a} \sqrt [3]{b} d-7 a^{2/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-54 b c \log \left (a+b x^3\right )}{54 a^4} \]

[In]

Integrate[(c + d*x + e*x^2)/(x^4*(a + b*x^3)^3),x]

[Out]

-1/54*((18*a*c)/x^3 + (27*a*d)/x^2 + (54*a*e)/x + (9*a^2*b*(c + x*(d + e*x)))/(a + b*x^3)^2 + (3*a*b*(12*c + x
*(11*d + 10*e*x)))/(a + b*x^3) - 4*Sqrt[3]*a^(1/3)*b^(1/3)*(10*b^(1/3)*d + 7*a^(1/3)*e)*ArcTan[(1 - (2*b^(1/3)
*x)/a^(1/3))/Sqrt[3]] + 162*b*c*Log[x] + 4*b^(1/3)*(10*a^(1/3)*b^(1/3)*d - 7*a^(2/3)*e)*Log[a^(1/3) + b^(1/3)*
x] - 2*b^(1/3)*(10*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] - 54*b*c*Lo
g[a + b*x^3])/a^4

Maple [A] (verified)

Time = 1.54 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.01

method	result	size
default	\(-\frac {c}{3 a^{3} x^{3}}-\frac {d}{2 a^{3} x^{2}}-\frac {e}{a^{3} x}-\frac {3 b c \ln \left (x \right )}{a^{4}}-\frac {b \left (\frac {\frac {5}{9} a b e \,x^{5}+\frac {11}{18} a b d \,x^{4}+\frac {2}{3} a b c \,x^{3}+\frac {13}{18} a^{2} e \,x^{2}+\frac {7}{9} a^{2} d x +\frac {5}{6} a^{2} c}{\left (b \,x^{3}+a \right )^{2}}+\frac {20 a d \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{9}+\frac {14 a e \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9}-c \ln \left (b \,x^{3}+a \right )\right )}{a^{4}}\)	\(301\)
risch	\(\frac {-\frac {14 e \,b^{2} x^{8}}{9 a^{3}}-\frac {10 d \,b^{2} x^{7}}{9 a^{3}}-\frac {c \,b^{2} x^{6}}{a^{3}}-\frac {49 b e \,x^{5}}{18 a^{2}}-\frac {16 b d \,x^{4}}{9 a^{2}}-\frac {3 b c \,x^{3}}{2 a^{2}}-\frac {e \,x^{2}}{a}-\frac {x d}{2 a}-\frac {c}{3 a}}{x^{3} \left (b \,x^{3}+a \right )^{2}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{12} \textit {\_Z}^{3}-81 a^{8} b c \,\textit {\_Z}^{2}+\left (840 a^{5} b d e +2187 a^{4} b^{2} c^{2}\right ) \textit {\_Z} -2744 a^{2} b \,e^{3}-22680 a \,b^{2} c d e +8000 a \,b^{2} d^{3}-19683 b^{3} c^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (-2 \textit {\_R}^{3} a^{11}+108 a^{7} b c \,\textit {\_R}^{2}+\left (-1400 a^{4} b d e -1458 a^{3} b^{2} c^{2}\right ) \textit {\_R} +4116 a b \,e^{3}+22680 b^{2} d c e -12000 b^{2} d^{3}\right ) x -7 \textit {\_R}^{2} a^{8} e +\left (-378 a^{4} b c e -200 a^{4} b \,d^{2}\right ) \textit {\_R} +15309 b^{2} c^{2} e -16200 b^{2} c \,d^{2}\right )\right )}{27}-\frac {3 b c \ln \left (x \right )}{a^{4}}\)	\(311\)

[In]

int((e*x^2+d*x+c)/x^4/(b*x^3+a)^3,x,method=_RETURNVERBOSE)

[Out]

-1/3*c/a^3/x^3-1/2*d/a^3/x^2-e/a^3/x-3*b*c*ln(x)/a^4-1/a^4*b*((5/9*a*b*e*x^5+11/18*a*b*d*x^4+2/3*a*b*c*x^3+13/
18*a^2*e*x^2+7/9*a^2*d*x+5/6*a^2*c)/(b*x^3+a)^2+20/9*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)))+14/9*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)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1)))-c*ln(b*x^3+a))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.77 (sec) , antiderivative size = 5550, normalized size of antiderivative = 18.62 \[ \int \frac {c+d x+e x^2}{x^4 \left (a+b x^3\right )^3} \, dx=\text {Too large to display} \]

[In]

integrate((e*x^2+d*x+c)/x^4/(b*x^3+a)^3,x, algorithm="fricas")

[Out]

Too large to include

Sympy [F(-1)]

Timed out. \[ \int \frac {c+d x+e x^2}{x^4 \left (a+b x^3\right )^3} \, dx=\text {Timed out} \]

[In]

integrate((e*x**2+d*x+c)/x**4/(b*x**3+a)**3,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 283, normalized size of antiderivative = 0.95 \[ \int \frac {c+d x+e x^2}{x^4 \left (a+b x^3\right )^3} \, dx=-\frac {28 \, b^{2} e x^{8} + 20 \, b^{2} d x^{7} + 18 \, b^{2} c x^{6} + 49 \, a b e x^{5} + 32 \, a b d x^{4} + 27 \, a b c x^{3} + 18 \, a^{2} e x^{2} + 9 \, a^{2} d x + 6 \, a^{2} c}{18 \, {\left (a^{3} b^{2} x^{9} + 2 \, a^{4} b x^{6} + a^{5} x^{3}\right )}} - \frac {3 \, b c \log \left (x\right )}{a^{4}} - \frac {2 \, \sqrt {3} {\left (7 \, a e \left (\frac {a}{b}\right )^{\frac {2}{3}} + 10 \, a d \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} b \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{5}} + \frac {{\left (27 \, b c \left (\frac {a}{b}\right )^{\frac {2}{3}} - 7 \, a e \left (\frac {a}{b}\right )^{\frac {1}{3}} + 10 \, a d\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{27 \, a^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (27 \, b c \left (\frac {a}{b}\right )^{\frac {2}{3}} + 14 \, a e \left (\frac {a}{b}\right )^{\frac {1}{3}} - 20 \, a d\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]

[In]

integrate((e*x^2+d*x+c)/x^4/(b*x^3+a)^3,x, algorithm="maxima")

[Out]

-1/18*(28*b^2*e*x^8 + 20*b^2*d*x^7 + 18*b^2*c*x^6 + 49*a*b*e*x^5 + 32*a*b*d*x^4 + 27*a*b*c*x^3 + 18*a^2*e*x^2
+ 9*a^2*d*x + 6*a^2*c)/(a^3*b^2*x^9 + 2*a^4*b*x^6 + a^5*x^3) - 3*b*c*log(x)/a^4 - 2/27*sqrt(3)*(7*a*e*(a/b)^(2
/3) + 10*a*d*(a/b)^(1/3))*b*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/a^5 + 1/27*(27*b*c*(a/b)^(2/3)
- 7*a*e*(a/b)^(1/3) + 10*a*d)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a^4*(a/b)^(2/3)) + 1/27*(27*b*c*(a/b)^(
2/3) + 14*a*e*(a/b)^(1/3) - 20*a*d)*log(x + (a/b)^(1/3))/(a^4*(a/b)^(2/3))

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.00 \[ \int \frac {c+d x+e x^2}{x^4 \left (a+b x^3\right )^3} \, dx=\frac {b c \log \left ({\left | b x^{3} + a \right |}\right )}{a^{4}} - \frac {3 \, b c \log \left ({\left | x \right |}\right )}{a^{4}} - \frac {2 \, \sqrt {3} {\left (10 \, \left (-a b^{2}\right )^{\frac {1}{3}} b d - 7 \, \left (-a b^{2}\right )^{\frac {2}{3}} e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{4} b} - \frac {{\left (10 \, \left (-a b^{2}\right )^{\frac {1}{3}} b d + 7 \, \left (-a b^{2}\right )^{\frac {2}{3}} e\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{27 \, a^{4} b} + \frac {2 \, {\left (7 \, a^{5} b^{2} e \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 10 \, a^{5} b^{2} d\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{9} b} - \frac {28 \, a b^{2} e x^{8} + 20 \, a b^{2} d x^{7} + 18 \, a b^{2} c x^{6} + 49 \, a^{2} b e x^{5} + 32 \, a^{2} b d x^{4} + 27 \, a^{2} b c x^{3} + 18 \, a^{3} e x^{2} + 9 \, a^{3} d x + 6 \, a^{3} c}{18 \, {\left (b x^{3} + a\right )}^{2} a^{4} x^{3}} \]

[In]

integrate((e*x^2+d*x+c)/x^4/(b*x^3+a)^3,x, algorithm="giac")

[Out]

b*c*log(abs(b*x^3 + a))/a^4 - 3*b*c*log(abs(x))/a^4 - 2/27*sqrt(3)*(10*(-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^4*b) - 1/27*(10*(-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^4*b) + 2/27*(7*a^5*b^2*e*(-a/b)^(1/3) + 10*a^5*b^2*d)*(-a/b)
^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^9*b) - 1/18*(28*a*b^2*e*x^8 + 20*a*b^2*d*x^7 + 18*a*b^2*c*x^6 + 49*a^2*b*
e*x^5 + 32*a^2*b*d*x^4 + 27*a^2*b*c*x^3 + 18*a^3*e*x^2 + 9*a^3*d*x + 6*a^3*c)/((b*x^3 + a)^2*a^4*x^3)

Mupad [B] (verification not implemented)

Time = 9.79 (sec) , antiderivative size = 870, normalized size of antiderivative = 2.92 \[ \int \frac {c+d x+e x^2}{x^4 \left (a+b x^3\right )^3} \, dx=\left (\sum _{k=1}^3\ln \left (-\frac {b^3\,\left ({\mathrm {root}\left (19683\,a^{12}\,z^3-59049\,a^8\,b\,c\,z^2+22680\,a^5\,b\,d\,e\,z+59049\,a^4\,b^2\,c^2\,z-22680\,a\,b^2\,c\,d\,e-2744\,a^2\,b\,e^3+8000\,a\,b^2\,d^3-19683\,b^3\,c^3,z,k\right )}^2\,a^8\,e\,1701+5400\,b^2\,c\,d^2-5103\,b^2\,c^2\,e+{\mathrm {root}\left (19683\,a^{12}\,z^3-59049\,a^8\,b\,c\,z^2+22680\,a^5\,b\,d\,e\,z+59049\,a^4\,b^2\,c^2\,z-22680\,a\,b^2\,c\,d\,e-2744\,a^2\,b\,e^3+8000\,a\,b^2\,d^3-19683\,b^3\,c^3,z,k\right )}^3\,a^{11}\,x\,13122+4000\,b^2\,d^3\,x-1372\,a\,b\,e^3\,x+\mathrm {root}\left (19683\,a^{12}\,z^3-59049\,a^8\,b\,c\,z^2+22680\,a^5\,b\,d\,e\,z+59049\,a^4\,b^2\,c^2\,z-22680\,a\,b^2\,c\,d\,e-2744\,a^2\,b\,e^3+8000\,a\,b^2\,d^3-19683\,b^3\,c^3,z,k\right )\,a^4\,b\,d^2\,1800-{\mathrm {root}\left (19683\,a^{12}\,z^3-59049\,a^8\,b\,c\,z^2+22680\,a^5\,b\,d\,e\,z+59049\,a^4\,b^2\,c^2\,z-22680\,a\,b^2\,c\,d\,e-2744\,a^2\,b\,e^3+8000\,a\,b^2\,d^3-19683\,b^3\,c^3,z,k\right )}^2\,a^7\,b\,c\,x\,26244+\mathrm {root}\left (19683\,a^{12}\,z^3-59049\,a^8\,b\,c\,z^2+22680\,a^5\,b\,d\,e\,z+59049\,a^4\,b^2\,c^2\,z-22680\,a\,b^2\,c\,d\,e-2744\,a^2\,b\,e^3+8000\,a\,b^2\,d^3-19683\,b^3\,c^3,z,k\right )\,a^3\,b^2\,c^2\,x\,13122+\mathrm {root}\left (19683\,a^{12}\,z^3-59049\,a^8\,b\,c\,z^2+22680\,a^5\,b\,d\,e\,z+59049\,a^4\,b^2\,c^2\,z-22680\,a\,b^2\,c\,d\,e-2744\,a^2\,b\,e^3+8000\,a\,b^2\,d^3-19683\,b^3\,c^3,z,k\right )\,a^4\,b\,c\,e\,3402-7560\,b^2\,c\,d\,e\,x+\mathrm {root}\left (19683\,a^{12}\,z^3-59049\,a^8\,b\,c\,z^2+22680\,a^5\,b\,d\,e\,z+59049\,a^4\,b^2\,c^2\,z-22680\,a\,b^2\,c\,d\,e-2744\,a^2\,b\,e^3+8000\,a\,b^2\,d^3-19683\,b^3\,c^3,z,k\right )\,a^4\,b\,d\,e\,x\,12600\right )\,2}{a^9\,729}\right )\,\mathrm {root}\left (19683\,a^{12}\,z^3-59049\,a^8\,b\,c\,z^2+22680\,a^5\,b\,d\,e\,z+59049\,a^4\,b^2\,c^2\,z-22680\,a\,b^2\,c\,d\,e-2744\,a^2\,b\,e^3+8000\,a\,b^2\,d^3-19683\,b^3\,c^3,z,k\right )\right )-\frac {\frac {c}{3\,a}+\frac {e\,x^2}{a}+\frac {d\,x}{2\,a}+\frac {b^2\,c\,x^6}{a^3}+\frac {10\,b^2\,d\,x^7}{9\,a^3}+\frac {14\,b^2\,e\,x^8}{9\,a^3}+\frac {3\,b\,c\,x^3}{2\,a^2}+\frac {16\,b\,d\,x^4}{9\,a^2}+\frac {49\,b\,e\,x^5}{18\,a^2}}{a^2\,x^3+2\,a\,b\,x^6+b^2\,x^9}-\frac {3\,b\,c\,\ln \left (x\right )}{a^4} \]

[In]

int((c + d*x + e*x^2)/(x^4*(a + b*x^3)^3),x)

[Out]

symsum(log(-(2*b^3*(1701*root(19683*a^12*z^3 - 59049*a^8*b*c*z^2 + 22680*a^5*b*d*e*z + 59049*a^4*b^2*c^2*z - 2
2680*a*b^2*c*d*e - 2744*a^2*b*e^3 + 8000*a*b^2*d^3 - 19683*b^3*c^3, z, k)^2*a^8*e + 5400*b^2*c*d^2 - 5103*b^2*
c^2*e + 13122*root(19683*a^12*z^3 - 59049*a^8*b*c*z^2 + 22680*a^5*b*d*e*z + 59049*a^4*b^2*c^2*z - 22680*a*b^2*
c*d*e - 2744*a^2*b*e^3 + 8000*a*b^2*d^3 - 19683*b^3*c^3, z, k)^3*a^11*x + 4000*b^2*d^3*x - 1372*a*b*e^3*x + 18
00*root(19683*a^12*z^3 - 59049*a^8*b*c*z^2 + 22680*a^5*b*d*e*z + 59049*a^4*b^2*c^2*z - 22680*a*b^2*c*d*e - 274
4*a^2*b*e^3 + 8000*a*b^2*d^3 - 19683*b^3*c^3, z, k)*a^4*b*d^2 - 26244*root(19683*a^12*z^3 - 59049*a^8*b*c*z^2
+ 22680*a^5*b*d*e*z + 59049*a^4*b^2*c^2*z - 22680*a*b^2*c*d*e - 2744*a^2*b*e^3 + 8000*a*b^2*d^3 - 19683*b^3*c^
3, z, k)^2*a^7*b*c*x + 13122*root(19683*a^12*z^3 - 59049*a^8*b*c*z^2 + 22680*a^5*b*d*e*z + 59049*a^4*b^2*c^2*z
- 22680*a*b^2*c*d*e - 2744*a^2*b*e^3 + 8000*a*b^2*d^3 - 19683*b^3*c^3, z, k)*a^3*b^2*c^2*x + 3402*root(19683*
a^12*z^3 - 59049*a^8*b*c*z^2 + 22680*a^5*b*d*e*z + 59049*a^4*b^2*c^2*z - 22680*a*b^2*c*d*e - 2744*a^2*b*e^3 +
8000*a*b^2*d^3 - 19683*b^3*c^3, z, k)*a^4*b*c*e - 7560*b^2*c*d*e*x + 12600*root(19683*a^12*z^3 - 59049*a^8*b*c
*z^2 + 22680*a^5*b*d*e*z + 59049*a^4*b^2*c^2*z - 22680*a*b^2*c*d*e - 2744*a^2*b*e^3 + 8000*a*b^2*d^3 - 19683*b
^3*c^3, z, k)*a^4*b*d*e*x))/(729*a^9))*root(19683*a^12*z^3 - 59049*a^8*b*c*z^2 + 22680*a^5*b*d*e*z + 59049*a^4
*b^2*c^2*z - 22680*a*b^2*c*d*e - 2744*a^2*b*e^3 + 8000*a*b^2*d^3 - 19683*b^3*c^3, z, k), k, 1, 3) - (c/(3*a) +
(e*x^2)/a + (d*x)/(2*a) + (b^2*c*x^6)/a^3 + (10*b^2*d*x^7)/(9*a^3) + (14*b^2*e*x^8)/(9*a^3) + (3*b*c*x^3)/(2*
a^2) + (16*b*d*x^4)/(9*a^2) + (49*b*e*x^5)/(18*a^2))/(a^2*x^3 + b^2*x^9 + 2*a*b*x^6) - (3*b*c*log(x))/a^4