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

Optimal. Leaf size=340 \[ \frac {10 \sqrt [3]{b} \left (11 \sqrt [3]{b} d-7 \sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{243 a^{14/3}}-\frac {20 \sqrt [3]{b} \left (11 \sqrt [3]{b} d-7 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{14/3}}+\frac {20 \sqrt [3]{b} \left (7 \sqrt [3]{a} e+11 \sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{81 \sqrt {3} a^{14/3}}+\frac {4 b c \log \left (a+b x^3\right )}{3 a^5}-\frac {4 b c \log (x)}{a^5}-\frac {x \left (-\frac {234 b^2 c x^2}{a}+139 b d+118 b e x\right )}{162 a^4 \left (a+b x^3\right )}-\frac {c}{3 a^4 x^3}-\frac {d}{2 a^4 x^2}-\frac {e}{a^4 x}-\frac {x \left (-\frac {24 b^2 c x^2}{a}+17 b d+16 b e x\right )}{54 a^3 \left (a+b x^3\right )^2}-\frac {x \left (-\frac {b^2 c x^2}{a}+b d+b e x\right )}{9 a^2 \left (a+b x^3\right )^3} \]

[Out]

-1/3*c/a^4/x^3-1/2*d/a^4/x^2-e/a^4/x-1/9*x*(b*d+b*x*e-b^2*c*x^2/a)/a^2/(b*x^3+a)^3-1/54*x*(17*b*d+16*b*x*e-24*
b^2*c*x^2/a)/a^3/(b*x^3+a)^2-1/162*x*(139*b*d+118*b*x*e-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)

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Rubi [A]  time = 0.77, antiderivative size = 340, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {1829, 1834, 1871, 1860, 31, 634, 617, 204, 628, 260} \[ -\frac {x \left (-\frac {234 b^2 c x^2}{a}+139 b d+118 b e x\right )}{162 a^4 \left (a+b x^3\right )}-\frac {x \left (-\frac {24 b^2 c x^2}{a}+17 b d+16 b e x\right )}{54 a^3 \left (a+b x^3\right )^2}-\frac {x \left (-\frac {b^2 c x^2}{a}+b d+b e x\right )}{9 a^2 \left (a+b x^3\right )^3}+\frac {10 \sqrt [3]{b} \left (11 \sqrt [3]{b} d-7 \sqrt [3]{a} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{243 a^{14/3}}+\frac {4 b c \log \left (a+b x^3\right )}{3 a^5}-\frac {4 b c \log (x)}{a^5}-\frac {20 \sqrt [3]{b} \left (11 \sqrt [3]{b} d-7 \sqrt [3]{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{243 a^{14/3}}+\frac {20 \sqrt [3]{b} \left (7 \sqrt [3]{a} e+11 \sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{81 \sqrt {3} a^{14/3}}-\frac {c}{3 a^4 x^3}-\frac {d}{2 a^4 x^2}-\frac {e}{a^4 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-c/(3*a^4*x^3) - d/(2*a^4*x^2) - e/(a^4*x) - (x*(b*d + b*e*x - (b^2*c*x^2)/a))/(9*a^2*(a + b*x^3)^3) - (x*(17*
b*d + 16*b*e*x - (24*b^2*c*x^2)/a))/(54*a^3*(a + b*x^3)^2) - (x*(139*b*d + 118*b*e*x - (234*b^2*c*x^2)/a))/(16
2*a^4*(a + b*x^3)) + (20*b^(1/3)*(11*b^(1/3)*d + 7*a^(1/3)*e)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))
])/(81*Sqrt[3]*a^(14/3)) - (4*b*c*Log[x])/a^5 - (20*b^(1/3)*(11*b^(1/3)*d - 7*a^(1/3)*e)*Log[a^(1/3) + b^(1/3)
*x])/(243*a^(14/3)) + (10*b^(1/3)*(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])
/(243*a^(14/3)) + (4*b*c*Log[a + b*x^3])/(3*a^5)

Rule 31

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

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 260

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 617

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 628

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 634

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 1829

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)*Coeff[R, x, i]*x^(i - m))/a, {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[Pq,
x] && IGtQ[n, 0] && LtQ[p, -1] && ILtQ[m, 0]

Rule 1834

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 1860

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 1871

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*} \int \frac {c+d x+e x^2}{x^4 \left (a+b x^3\right )^4} \, dx &=-\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 {\int \frac {-9 b c-9 b d x-9 b e x^2+\frac {9 b^2 c x^3}{a}+\frac {8 b^2 d x^4}{a}+\frac {7 b^2 e x^5}{a}-\frac {6 b^3 c x^6}{a^2}}{x^4 \left (a+b x^3\right )^3} \, dx}{9 a b}\\ &=-\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 {\int \frac {54 b^3 c+54 b^3 d x+54 b^3 e x^2-\frac {108 b^4 c x^3}{a}-\frac {85 b^4 d x^4}{a}-\frac {64 b^4 e x^5}{a}+\frac {72 b^5 c x^6}{a^2}}{x^4 \left (a+b x^3\right )^2} \, dx}{54 a^2 b^3}\\ &=-\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 {\int \frac {-162 b^5 c-162 b^5 d x-162 b^5 e x^2+\frac {486 b^6 c x^3}{a}+\frac {278 b^6 d x^4}{a}+\frac {118 b^6 e x^5}{a}}{x^4 \left (a+b x^3\right )} \, dx}{162 a^3 b^5}\\ &=-\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 {\int \left (-\frac {162 b^5 c}{a x^4}-\frac {162 b^5 d}{a x^3}-\frac {162 b^5 e}{a x^2}+\frac {648 b^6 c}{a^2 x}+\frac {8 b^6 \left (55 a d+35 a e x-81 b c x^2\right )}{a^2 \left (a+b x^3\right )}\right ) \, dx}{162 a^3 b^5}\\ &=-\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 {4 b c \log (x)}{a^5}-\frac {(4 b) \int \frac {55 a d+35 a e x-81 b c x^2}{a+b x^3} \, dx}{81 a^5}\\ &=-\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 {4 b c \log (x)}{a^5}-\frac {(4 b) \int \frac {55 a d+35 a e x}{a+b x^3} \, dx}{81 a^5}+\frac {\left (4 b^2 c\right ) \int \frac {x^2}{a+b x^3} \, dx}{a^5}\\ &=-\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 {4 b c \log (x)}{a^5}+\frac {4 b c \log \left (a+b x^3\right )}{3 a^5}-\frac {\left (4 b^{2/3}\right ) \int \frac {\sqrt [3]{a} \left (110 a \sqrt [3]{b} d+35 a^{4/3} e\right )+\sqrt [3]{b} \left (-55 a \sqrt [3]{b} d+35 a^{4/3} e\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{243 a^{17/3}}-\frac {\left (20 b \left (11 d-\frac {7 \sqrt [3]{a} e}{\sqrt [3]{b}}\right )\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{243 a^{14/3}}\\ &=-\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 {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 {4 b c \log \left (a+b x^3\right )}{3 a^5}+\frac {\left (10 \sqrt [3]{b} \left (11 \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}{243 a^{14/3}}-\frac {\left (10 b^{2/3} \left (11 \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}{81 a^{13/3}}\\ &=-\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 {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}-\frac {\left (20 \sqrt [3]{b} \left (11 \sqrt [3]{b} d+7 \sqrt [3]{a} e\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{81 a^{14/3}}\\ &=-\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 ) \tan ^{-1}\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}\\ \end {align*}

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Mathematica [A]  time = 0.60, size = 284, normalized size = 0.84 \[ -\frac {-20 \sqrt [3]{b} \left (11 \sqrt [3]{a} \sqrt [3]{b} d-7 a^{2/3} e\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+40 \sqrt [3]{b} \left (11 \sqrt [3]{a} \sqrt [3]{b} d-7 a^{2/3} e\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+\frac {54 a^3 b (c+x (d+e x))}{\left (a+b x^3\right )^3}+\frac {9 a^2 b (18 c+x (17 d+16 e x))}{\left (a+b x^3\right )^2}+\frac {3 a b (162 c+x (139 d+118 e x))}{a+b x^3}-648 b c \log \left (a+b x^3\right )-40 \sqrt {3} \sqrt [3]{a} \sqrt [3]{b} \left (7 \sqrt [3]{a} e+11 \sqrt [3]{b} d\right ) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )+\frac {162 a c}{x^3}+\frac {243 a d}{x^2}+\frac {486 a e}{x}+1944 b c \log (x)}{486 a^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-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*(1
8*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))/Sqrt[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

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fricas [C]  time = 3.39, size = 5670, normalized size = 16.68 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/486*(840*a*b^3*e*x^11 + 660*a*b^3*d*x^10 + 648*a*b^3*c*x^9 + 2310*a^2*b^2*e*x^8 + 1716*a^2*b^2*d*x^7 + 1620
*a^2*b^2*c*x^6 + 2010*a^3*b*e*x^5 + 1353*a^3*b*d*x^4 + 1188*a^3*b*c*x^3 + 486*a^4*e*x^2 + 243*a^4*d*x + 162*a^
4*c + 2*(a^5*b^3*x^12 + 3*a^6*b^2*x^9 + 3*a^7*b*x^6 + a^8*x^3)*(4^(2/3)*(-I*sqrt(3) + 1)*(6561*b^2*c^2/a^10 -
(6561*b^2*c^2 + 1925*a*b*d*e)/a^10)/(1062882*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343*a*e^3)*b/a^14 - 243*(6561*b^
2*c^2 + 1925*a*b*d*e)*b*c/a^15 + (531441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3 - 1701*c*d*e)*a*b^2)/a^15)^(
1/3) + 4^(1/3)*(I*sqrt(3) + 1)*(1062882*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343*a*e^3)*b/a^14 - 243*(6561*b^2*c^2
+ 1925*a*b*d*e)*b*c/a^15 + (531441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3 - 1701*c*d*e)*a*b^2)/a^15)^(1/3)
- 324*b*c/a^5)*log(7*(4^(2/3)*(-I*sqrt(3) + 1)*(6561*b^2*c^2/a^10 - (6561*b^2*c^2 + 1925*a*b*d*e)/a^10)/(10628
82*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343*a*e^3)*b/a^14 - 243*(6561*b^2*c^2 + 1925*a*b*d*e)*b*c/a^15 + (531441*b
^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3 - 1701*c*d*e)*a*b^2)/a^15)^(1/3) + 4^(1/3)*(I*sqrt(3) + 1)*(1062882*b^
3*c^3/a^15 + 125*(1331*b*d^3 + 343*a*e^3)*b/a^14 - 243*(6561*b^2*c^2 + 1925*a*b*d*e)*b*c/a^15 + (531441*b^3*c^
3 + 42875*a^2*b*e^3 - 275*(605*d^3 - 1701*c*d*e)*a*b^2)/a^15)^(1/3) - 324*b*c/a^5)^2*a^10*e + 784080*b^2*c*d^2
+ 734832*b^2*c^2*e + 431200*a*b*d*e^2 + 4*(605*a^5*b*d^2 + 1134*a^5*b*c*e)*(4^(2/3)*(-I*sqrt(3) + 1)*(6561*b^
2*c^2/a^10 - (6561*b^2*c^2 + 1925*a*b*d*e)/a^10)/(1062882*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343*a*e^3)*b/a^14 -
243*(6561*b^2*c^2 + 1925*a*b*d*e)*b*c/a^15 + (531441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3 - 1701*c*d*e)*a
*b^2)/a^15)^(1/3) + 4^(1/3)*(I*sqrt(3) + 1)*(1062882*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343*a*e^3)*b/a^14 - 243*
(6561*b^2*c^2 + 1925*a*b*d*e)*b*c/a^15 + (531441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3 - 1701*c*d*e)*a*b^2)
/a^15)^(1/3) - 324*b*c/a^5) + 400*(1331*b^2*d^3 + 343*a*b*e^3)*x) - (972*b^4*c*x^12 + 2916*a*b^3*c*x^9 + 2916*
a^2*b^2*c*x^6 + 972*a^3*b*c*x^3 + (a^5*b^3*x^12 + 3*a^6*b^2*x^9 + 3*a^7*b*x^6 + a^8*x^3)*(4^(2/3)*(-I*sqrt(3)
+ 1)*(6561*b^2*c^2/a^10 - (6561*b^2*c^2 + 1925*a*b*d*e)/a^10)/(1062882*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343*a*
e^3)*b/a^14 - 243*(6561*b^2*c^2 + 1925*a*b*d*e)*b*c/a^15 + (531441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3 -
1701*c*d*e)*a*b^2)/a^15)^(1/3) + 4^(1/3)*(I*sqrt(3) + 1)*(1062882*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343*a*e^3)*
b/a^14 - 243*(6561*b^2*c^2 + 1925*a*b*d*e)*b*c/a^15 + (531441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3 - 1701*
c*d*e)*a*b^2)/a^15)^(1/3) - 324*b*c/a^5) + 3*sqrt(1/3)*(a^5*b^3*x^12 + 3*a^6*b^2*x^9 + 3*a^7*b*x^6 + a^8*x^3)*
sqrt(-((4^(2/3)*(-I*sqrt(3) + 1)*(6561*b^2*c^2/a^10 - (6561*b^2*c^2 + 1925*a*b*d*e)/a^10)/(1062882*b^3*c^3/a^1
5 + 125*(1331*b*d^3 + 343*a*e^3)*b/a^14 - 243*(6561*b^2*c^2 + 1925*a*b*d*e)*b*c/a^15 + (531441*b^3*c^3 + 42875
*a^2*b*e^3 - 275*(605*d^3 - 1701*c*d*e)*a*b^2)/a^15)^(1/3) + 4^(1/3)*(I*sqrt(3) + 1)*(1062882*b^3*c^3/a^15 + 1
25*(1331*b*d^3 + 343*a*e^3)*b/a^14 - 243*(6561*b^2*c^2 + 1925*a*b*d*e)*b*c/a^15 + (531441*b^3*c^3 + 42875*a^2*
b*e^3 - 275*(605*d^3 - 1701*c*d*e)*a*b^2)/a^15)^(1/3) - 324*b*c/a^5)^2*a^10 + 648*(4^(2/3)*(-I*sqrt(3) + 1)*(6
561*b^2*c^2/a^10 - (6561*b^2*c^2 + 1925*a*b*d*e)/a^10)/(1062882*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343*a*e^3)*b/
a^14 - 243*(6561*b^2*c^2 + 1925*a*b*d*e)*b*c/a^15 + (531441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3 - 1701*c*
d*e)*a*b^2)/a^15)^(1/3) + 4^(1/3)*(I*sqrt(3) + 1)*(1062882*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343*a*e^3)*b/a^14
- 243*(6561*b^2*c^2 + 1925*a*b*d*e)*b*c/a^15 + (531441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3 - 1701*c*d*e)*
a*b^2)/a^15)^(1/3) - 324*b*c/a^5)*a^5*b*c + 104976*b^2*c^2 + 123200*a*b*d*e)/a^10))*log(-7*(4^(2/3)*(-I*sqrt(3
) + 1)*(6561*b^2*c^2/a^10 - (6561*b^2*c^2 + 1925*a*b*d*e)/a^10)/(1062882*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343*
a*e^3)*b/a^14 - 243*(6561*b^2*c^2 + 1925*a*b*d*e)*b*c/a^15 + (531441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3
- 1701*c*d*e)*a*b^2)/a^15)^(1/3) + 4^(1/3)*(I*sqrt(3) + 1)*(1062882*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343*a*e^3
)*b/a^14 - 243*(6561*b^2*c^2 + 1925*a*b*d*e)*b*c/a^15 + (531441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3 - 170
1*c*d*e)*a*b^2)/a^15)^(1/3) - 324*b*c/a^5)^2*a^10*e - 784080*b^2*c*d^2 - 734832*b^2*c^2*e - 431200*a*b*d*e^2 -
4*(605*a^5*b*d^2 + 1134*a^5*b*c*e)*(4^(2/3)*(-I*sqrt(3) + 1)*(6561*b^2*c^2/a^10 - (6561*b^2*c^2 + 1925*a*b*d*
e)/a^10)/(1062882*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343*a*e^3)*b/a^14 - 243*(6561*b^2*c^2 + 1925*a*b*d*e)*b*c/a
^15 + (531441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3 - 1701*c*d*e)*a*b^2)/a^15)^(1/3) + 4^(1/3)*(I*sqrt(3) +
1)*(1062882*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343*a*e^3)*b/a^14 - 243*(6561*b^2*c^2 + 1925*a*b*d*e)*b*c/a^15 +
(531441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3 - 1701*c*d*e)*a*b^2)/a^15)^(1/3) - 324*b*c/a^5) + 800*(1331*
b^2*d^3 + 343*a*b*e^3)*x + 3*sqrt(1/3)*(7*(4^(2/3)*(-I*sqrt(3) + 1)*(6561*b^2*c^2/a^10 - (6561*b^2*c^2 + 1925*
a*b*d*e)/a^10)/(1062882*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343*a*e^3)*b/a^14 - 243*(6561*b^2*c^2 + 1925*a*b*d*e)
*b*c/a^15 + (531441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3 - 1701*c*d*e)*a*b^2)/a^15)^(1/3) + 4^(1/3)*(I*sqr
t(3) + 1)*(1062882*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343*a*e^3)*b/a^14 - 243*(6561*b^2*c^2 + 1925*a*b*d*e)*b*c/
a^15 + (531441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3 - 1701*c*d*e)*a*b^2)/a^15)^(1/3) - 324*b*c/a^5)*a^10*e
- 2420*a^5*b*d^2 + 2268*a^5*b*c*e)*sqrt(-((4^(2/3)*(-I*sqrt(3) + 1)*(6561*b^2*c^2/a^10 - (6561*b^2*c^2 + 1925
*a*b*d*e)/a^10)/(1062882*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343*a*e^3)*b/a^14 - 243*(6561*b^2*c^2 + 1925*a*b*d*e
)*b*c/a^15 + (531441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3 - 1701*c*d*e)*a*b^2)/a^15)^(1/3) + 4^(1/3)*(I*sq
rt(3) + 1)*(1062882*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343*a*e^3)*b/a^14 - 243*(6561*b^2*c^2 + 1925*a*b*d*e)*b*c
/a^15 + (531441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3 - 1701*c*d*e)*a*b^2)/a^15)^(1/3) - 324*b*c/a^5)^2*a^1
0 + 648*(4^(2/3)*(-I*sqrt(3) + 1)*(6561*b^2*c^2/a^10 - (6561*b^2*c^2 + 1925*a*b*d*e)/a^10)/(1062882*b^3*c^3/a^
15 + 125*(1331*b*d^3 + 343*a*e^3)*b/a^14 - 243*(6561*b^2*c^2 + 1925*a*b*d*e)*b*c/a^15 + (531441*b^3*c^3 + 4287
5*a^2*b*e^3 - 275*(605*d^3 - 1701*c*d*e)*a*b^2)/a^15)^(1/3) + 4^(1/3)*(I*sqrt(3) + 1)*(1062882*b^3*c^3/a^15 +
125*(1331*b*d^3 + 343*a*e^3)*b/a^14 - 243*(6561*b^2*c^2 + 1925*a*b*d*e)*b*c/a^15 + (531441*b^3*c^3 + 42875*a^2
*b*e^3 - 275*(605*d^3 - 1701*c*d*e)*a*b^2)/a^15)^(1/3) - 324*b*c/a^5)*a^5*b*c + 104976*b^2*c^2 + 123200*a*b*d*
e)/a^10)) - (972*b^4*c*x^12 + 2916*a*b^3*c*x^9 + 2916*a^2*b^2*c*x^6 + 972*a^3*b*c*x^3 + (a^5*b^3*x^12 + 3*a^6*
b^2*x^9 + 3*a^7*b*x^6 + a^8*x^3)*(4^(2/3)*(-I*sqrt(3) + 1)*(6561*b^2*c^2/a^10 - (6561*b^2*c^2 + 1925*a*b*d*e)/
a^10)/(1062882*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343*a*e^3)*b/a^14 - 243*(6561*b^2*c^2 + 1925*a*b*d*e)*b*c/a^15
+ (531441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3 - 1701*c*d*e)*a*b^2)/a^15)^(1/3) + 4^(1/3)*(I*sqrt(3) + 1)
*(1062882*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343*a*e^3)*b/a^14 - 243*(6561*b^2*c^2 + 1925*a*b*d*e)*b*c/a^15 + (5
31441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3 - 1701*c*d*e)*a*b^2)/a^15)^(1/3) - 324*b*c/a^5) - 3*sqrt(1/3)*(
a^5*b^3*x^12 + 3*a^6*b^2*x^9 + 3*a^7*b*x^6 + a^8*x^3)*sqrt(-((4^(2/3)*(-I*sqrt(3) + 1)*(6561*b^2*c^2/a^10 - (6
561*b^2*c^2 + 1925*a*b*d*e)/a^10)/(1062882*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343*a*e^3)*b/a^14 - 243*(6561*b^2*
c^2 + 1925*a*b*d*e)*b*c/a^15 + (531441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3 - 1701*c*d*e)*a*b^2)/a^15)^(1/
3) + 4^(1/3)*(I*sqrt(3) + 1)*(1062882*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343*a*e^3)*b/a^14 - 243*(6561*b^2*c^2 +
1925*a*b*d*e)*b*c/a^15 + (531441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3 - 1701*c*d*e)*a*b^2)/a^15)^(1/3) -
324*b*c/a^5)^2*a^10 + 648*(4^(2/3)*(-I*sqrt(3) + 1)*(6561*b^2*c^2/a^10 - (6561*b^2*c^2 + 1925*a*b*d*e)/a^10)/(
1062882*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343*a*e^3)*b/a^14 - 243*(6561*b^2*c^2 + 1925*a*b*d*e)*b*c/a^15 + (531
441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3 - 1701*c*d*e)*a*b^2)/a^15)^(1/3) + 4^(1/3)*(I*sqrt(3) + 1)*(10628
82*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343*a*e^3)*b/a^14 - 243*(6561*b^2*c^2 + 1925*a*b*d*e)*b*c/a^15 + (531441*b
^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3 - 1701*c*d*e)*a*b^2)/a^15)^(1/3) - 324*b*c/a^5)*a^5*b*c + 104976*b^2*c
^2 + 123200*a*b*d*e)/a^10))*log(-7*(4^(2/3)*(-I*sqrt(3) + 1)*(6561*b^2*c^2/a^10 - (6561*b^2*c^2 + 1925*a*b*d*e
)/a^10)/(1062882*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343*a*e^3)*b/a^14 - 243*(6561*b^2*c^2 + 1925*a*b*d*e)*b*c/a^
15 + (531441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3 - 1701*c*d*e)*a*b^2)/a^15)^(1/3) + 4^(1/3)*(I*sqrt(3) +
1)*(1062882*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343*a*e^3)*b/a^14 - 243*(6561*b^2*c^2 + 1925*a*b*d*e)*b*c/a^15 +
(531441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3 - 1701*c*d*e)*a*b^2)/a^15)^(1/3) - 324*b*c/a^5)^2*a^10*e - 78
4080*b^2*c*d^2 - 734832*b^2*c^2*e - 431200*a*b*d*e^2 - 4*(605*a^5*b*d^2 + 1134*a^5*b*c*e)*(4^(2/3)*(-I*sqrt(3)
+ 1)*(6561*b^2*c^2/a^10 - (6561*b^2*c^2 + 1925*a*b*d*e)/a^10)/(1062882*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343*a
*e^3)*b/a^14 - 243*(6561*b^2*c^2 + 1925*a*b*d*e)*b*c/a^15 + (531441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3 -
1701*c*d*e)*a*b^2)/a^15)^(1/3) + 4^(1/3)*(I*sqrt(3) + 1)*(1062882*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343*a*e^3)
*b/a^14 - 243*(6561*b^2*c^2 + 1925*a*b*d*e)*b*c/a^15 + (531441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3 - 1701
*c*d*e)*a*b^2)/a^15)^(1/3) - 324*b*c/a^5) + 800*(1331*b^2*d^3 + 343*a*b*e^3)*x - 3*sqrt(1/3)*(7*(4^(2/3)*(-I*s
qrt(3) + 1)*(6561*b^2*c^2/a^10 - (6561*b^2*c^2 + 1925*a*b*d*e)/a^10)/(1062882*b^3*c^3/a^15 + 125*(1331*b*d^3 +
343*a*e^3)*b/a^14 - 243*(6561*b^2*c^2 + 1925*a*b*d*e)*b*c/a^15 + (531441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(605
*d^3 - 1701*c*d*e)*a*b^2)/a^15)^(1/3) + 4^(1/3)*(I*sqrt(3) + 1)*(1062882*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343*
a*e^3)*b/a^14 - 243*(6561*b^2*c^2 + 1925*a*b*d*e)*b*c/a^15 + (531441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3
- 1701*c*d*e)*a*b^2)/a^15)^(1/3) - 324*b*c/a^5)*a^10*e - 2420*a^5*b*d^2 + 2268*a^5*b*c*e)*sqrt(-((4^(2/3)*(-I*
sqrt(3) + 1)*(6561*b^2*c^2/a^10 - (6561*b^2*c^2 + 1925*a*b*d*e)/a^10)/(1062882*b^3*c^3/a^15 + 125*(1331*b*d^3
+ 343*a*e^3)*b/a^14 - 243*(6561*b^2*c^2 + 1925*a*b*d*e)*b*c/a^15 + (531441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(60
5*d^3 - 1701*c*d*e)*a*b^2)/a^15)^(1/3) + 4^(1/3)*(I*sqrt(3) + 1)*(1062882*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343
*a*e^3)*b/a^14 - 243*(6561*b^2*c^2 + 1925*a*b*d*e)*b*c/a^15 + (531441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3
- 1701*c*d*e)*a*b^2)/a^15)^(1/3) - 324*b*c/a^5)^2*a^10 + 648*(4^(2/3)*(-I*sqrt(3) + 1)*(6561*b^2*c^2/a^10 - (
6561*b^2*c^2 + 1925*a*b*d*e)/a^10)/(1062882*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343*a*e^3)*b/a^14 - 243*(6561*b^2
*c^2 + 1925*a*b*d*e)*b*c/a^15 + (531441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3 - 1701*c*d*e)*a*b^2)/a^15)^(1
/3) + 4^(1/3)*(I*sqrt(3) + 1)*(1062882*b^3*c^3/a^15 + 125*(1331*b*d^3 + 343*a*e^3)*b/a^14 - 243*(6561*b^2*c^2
+ 1925*a*b*d*e)*b*c/a^15 + (531441*b^3*c^3 + 42875*a^2*b*e^3 - 275*(605*d^3 - 1701*c*d*e)*a*b^2)/a^15)^(1/3) -
324*b*c/a^5)*a^5*b*c + 104976*b^2*c^2 + 123200*a*b*d*e)/a^10)) + 1944*(b^4*c*x^12 + 3*a*b^3*c*x^9 + 3*a^2*b^2
*c*x^6 + a^3*b*c*x^3)*log(x))/(a^5*b^3*x^12 + 3*a^6*b^2*x^9 + 3*a^7*b*x^6 + a^8*x^3)

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giac [A]  time = 0.20, size = 333, normalized size = 0.98 \[ \frac {4 \, b c \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{5}} - \frac {4 \, b c \log \left ({\left | x \right |}\right )}{a^{5}} - \frac {20 \, \sqrt {3} {\left (11 \, \left (-a b^{2}\right )^{\frac {1}{3}} b d - 7 \, \left (-a b^{2}\right )^{\frac {2}{3}} e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{243 \, a^{5} b} - \frac {10 \, {\left (11 \, \left (-a b^{2}\right )^{\frac {1}{3}} b d + 7 \, \left (-a b^{2}\right )^{\frac {2}{3}} e\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{243 \, a^{5} b} - \frac {280 \, b^{3} x^{11} e + 220 \, b^{3} d x^{10} + 216 \, b^{3} c x^{9} + 770 \, a b^{2} x^{8} e + 572 \, a b^{2} d x^{7} + 540 \, a b^{2} c x^{6} + 670 \, a^{2} b x^{5} e + 451 \, a^{2} b d x^{4} + 396 \, a^{2} b c x^{3} + 162 \, a^{3} x^{2} e + 81 \, a^{3} d x + 54 \, a^{3} c}{162 \, {\left (b x^{4} + a x\right )}^{3} a^{4}} + \frac {20 \, {\left (7 \, a^{6} b^{2} \left (-\frac {a}{b}\right )^{\frac {1}{3}} e + 11 \, a^{6} b^{2} d\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{243 \, a^{11} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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*x^11*e + 220*b^3*d*x^10 + 216*
b^3*c*x^9 + 770*a*b^2*x^8*e + 572*a*b^2*d*x^7 + 540*a*b^2*c*x^6 + 670*a^2*b*x^5*e + 451*a^2*b*d*x^4 + 396*a^2*
b*c*x^3 + 162*a^3*x^2*e + 81*a^3*d*x + 54*a^3*c)/((b*x^4 + a*x)^3*a^4) + 20/243*(7*a^6*b^2*(-a/b)^(1/3)*e + 11
*a^6*b^2*d)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^11*b)

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maple [A]  time = 0.07, size = 415, normalized size = 1.22 \[ -\frac {59 b^{3} e \,x^{8}}{81 \left (b \,x^{3}+a \right )^{3} a^{4}}-\frac {139 b^{3} d \,x^{7}}{162 \left (b \,x^{3}+a \right )^{3} a^{4}}-\frac {b^{3} c \,x^{6}}{\left (b \,x^{3}+a \right )^{3} a^{4}}-\frac {142 b^{2} e \,x^{5}}{81 \left (b \,x^{3}+a \right )^{3} a^{3}}-\frac {329 b^{2} d \,x^{4}}{162 \left (b \,x^{3}+a \right )^{3} a^{3}}-\frac {7 b^{2} c \,x^{3}}{3 \left (b \,x^{3}+a \right )^{3} a^{3}}-\frac {92 b e \,x^{2}}{81 \left (b \,x^{3}+a \right )^{3} a^{2}}-\frac {104 b d x}{81 \left (b \,x^{3}+a \right )^{3} a^{2}}-\frac {13 b c}{9 \left (b \,x^{3}+a \right )^{3} a^{2}}-\frac {220 \sqrt {3}\, d \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{243 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{4}}-\frac {220 d \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{243 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{4}}+\frac {110 d \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{243 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{4}}-\frac {140 \sqrt {3}\, e \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{243 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{4}}+\frac {140 e \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{243 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{4}}-\frac {70 e \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{243 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{4}}-\frac {4 b c \ln \relax (x )}{a^{5}}+\frac {4 b c \ln \left (b \,x^{3}+a \right )}{3 a^{5}}-\frac {e}{a^{4} x}-\frac {d}{2 a^{4} x^{2}}-\frac {c}{3 a^{4} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-59/81/a^4*b^3/(b*x^3+a)^3*e*x^8-139/162/a^4*b^3/(b*x^3+a)^3*d*x^7-1/a^4*b^3/(b*x^3+a)^3*c*x^6-142/81/a^3*b^2/
(b*x^3+a)^3*e*x^5-329/162/a^3*b^2/(b*x^3+a)^3*d*x^4-7/3/a^3*b^2/(b*x^3+a)^3*c*x^3-92/81/a^2*b/(b*x^3+a)^3*e*x^
2-104/81/a^2*b/(b*x^3+a)^3*d*x-13/9/a^2*b/(b*x^3+a)^3*c-220/243/a^4*d/(a/b)^(2/3)*ln(x+(a/b)^(1/3))+110/243/a^
4*d/(a/b)^(2/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))-220/243/a^4*d/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b
)^(1/3)*x-1))+140/243/a^4*e/(a/b)^(1/3)*ln(x+(a/b)^(1/3))-70/243/a^4*e/(a/b)^(1/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^
(2/3))-140/243/a^4*e*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+4/3*b*c*ln(b*x^3+a)/a^5-1/3*c
/a^4/x^3-1/2*d/a^4/x^2-e/a^4/x-4*b*c*ln(x)/a^5

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maxima [A]  time = 3.08, size = 330, normalized size = 0.97 \[ -\frac {280 \, b^{3} e x^{11} + 220 \, b^{3} d x^{10} + 216 \, b^{3} c x^{9} + 770 \, a b^{2} e x^{8} + 572 \, a b^{2} d x^{7} + 540 \, a b^{2} c x^{6} + 670 \, a^{2} b e x^{5} + 451 \, a^{2} b d x^{4} + 396 \, a^{2} b c x^{3} + 162 \, a^{3} e x^{2} + 81 \, a^{3} d x + 54 \, a^{3} c}{162 \, {\left (a^{4} b^{3} x^{12} + 3 \, a^{5} b^{2} x^{9} + 3 \, a^{6} b x^{6} + a^{7} x^{3}\right )}} - \frac {4 \, b c \log \relax (x)}{a^{5}} - \frac {20 \, \sqrt {3} {\left (7 \, a e \left (\frac {a}{b}\right )^{\frac {2}{3}} + 11 \, a d \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} b \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{243 \, a^{6}} + \frac {2 \, {\left (162 \, b c \left (\frac {a}{b}\right )^{\frac {2}{3}} - 35 \, a e \left (\frac {a}{b}\right )^{\frac {1}{3}} + 55 \, a d\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{243 \, a^{5} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {4 \, {\left (81 \, b c \left (\frac {a}{b}\right )^{\frac {2}{3}} + 35 \, a e \left (\frac {a}{b}\right )^{\frac {1}{3}} - 55 \, a d\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{243 \, a^{5} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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)) + 4/243*(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))

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mupad [B]  time = 0.52, size = 918, normalized size = 2.70 \[ \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 \relax (x)}{a^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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 - 2744000*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 - 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)^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^1
0*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 + 1064800
0*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 + 22
453200*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 + 765
27504*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*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*e - 2744000*a^2*b*e^3 + 10648000*a*b^2*d^3 - 34012224*b^3*c^3,
z, k)*a^5*b*d*e*x))/(531441*a^12))*root(14348907*a^15*z^3 - 57395628*a^10*b*c*z^2 + 22453200*a^6*b*d*e*z + 76
527504*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)
, k, 1, 3) - (c/(3*a) + (e*x^2)/a + (d*x)/(2*a) + (10*b^2*c*x^6)/(3*a^3) + (4*b^3*c*x^9)/(3*a^4) + (286*b^2*d*
x^7)/(81*a^3) + (110*b^3*d*x^10)/(81*a^4) + (385*b^2*e*x^8)/(81*a^3) + (140*b^3*e*x^11)/(81*a^4) + (22*b*c*x^3
)/(9*a^2) + (451*b*d*x^4)/(162*a^2) + (335*b*e*x^5)/(81*a^2))/(a^3*x^3 + b^3*x^12 + 3*a^2*b*x^6 + 3*a*b^2*x^9)
- (4*b*c*log(x))/a^5

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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