3.298 \(\int \frac {c+d x^3+e x^6+f x^9}{x^6 (a+b x^3)^3} \, dx\)

Optimal. Leaf size=316 \[ \frac {3 b c-a d}{2 a^4 x^2}-\frac {c}{5 a^3 x^5}+\frac {x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^3 b \left (a+b x^3\right )^2}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 f+5 a^2 b e-20 a b^2 d+44 b^3 c\right )}{54 a^{14/3} b^{4/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 f+5 a^2 b e-20 a b^2 d+44 b^3 c\right )}{27 a^{14/3} b^{4/3}}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (a^3 f+5 a^2 b e-20 a b^2 d+44 b^3 c\right )}{9 \sqrt {3} a^{14/3} b^{4/3}}+\frac {x \left (a^3 f+5 a^2 b e-11 a b^2 d+17 b^3 c\right )}{18 a^4 b \left (a+b x^3\right )} \]

[Out]

-1/5*c/a^3/x^5+1/2*(-a*d+3*b*c)/a^4/x^2+1/6*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*x/a^3/b/(b*x^3+a)^2+1/18*(a^3*f+5*a
^2*b*e-11*a*b^2*d+17*b^3*c)*x/a^4/b/(b*x^3+a)+1/27*(a^3*f+5*a^2*b*e-20*a*b^2*d+44*b^3*c)*ln(a^(1/3)+b^(1/3)*x)
/a^(14/3)/b^(4/3)-1/54*(a^3*f+5*a^2*b*e-20*a*b^2*d+44*b^3*c)*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(14/3
)/b^(4/3)-1/27*(a^3*f+5*a^2*b*e-20*a*b^2*d+44*b^3*c)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)/a^(1/3)*3^(1/2))/a^(14/3
)/b^(4/3)*3^(1/2)

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Rubi [A]  time = 0.37, antiderivative size = 316, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1829, 1484, 1488, 200, 31, 634, 617, 204, 628} \[ \frac {x \left (5 a^2 b e+a^3 f-11 a b^2 d+17 b^3 c\right )}{18 a^4 b \left (a+b x^3\right )}+\frac {x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 a^3 b \left (a+b x^3\right )^2}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (5 a^2 b e+a^3 f-20 a b^2 d+44 b^3 c\right )}{54 a^{14/3} b^{4/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (5 a^2 b e+a^3 f-20 a b^2 d+44 b^3 c\right )}{27 a^{14/3} b^{4/3}}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (5 a^2 b e+a^3 f-20 a b^2 d+44 b^3 c\right )}{9 \sqrt {3} a^{14/3} b^{4/3}}+\frac {3 b c-a d}{2 a^4 x^2}-\frac {c}{5 a^3 x^5} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^3 + e*x^6 + f*x^9)/(x^6*(a + b*x^3)^3),x]

[Out]

-c/(5*a^3*x^5) + (3*b*c - a*d)/(2*a^4*x^2) + ((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/(6*a^3*b*(a + b*x^3)^2) +
((17*b^3*c - 11*a*b^2*d + 5*a^2*b*e + a^3*f)*x)/(18*a^4*b*(a + b*x^3)) - ((44*b^3*c - 20*a*b^2*d + 5*a^2*b*e
+ a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(14/3)*b^(4/3)) + ((44*b^3*c - 20*a*b
^2*d + 5*a^2*b*e + a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(27*a^(14/3)*b^(4/3)) - ((44*b^3*c - 20*a*b^2*d + 5*a^2*b*
e + a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(14/3)*b^(4/3))

Rule 31

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

Rule 200

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], 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 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 1484

Int[(x_)^(m_)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_), x_Symbol] :> S
imp[((-d)^((m - Mod[m, n])/n - 1)*(c*d^2 - b*d*e + a*e^2)^p*x^(Mod[m, n] + 1)*(d + e*x^n)^(q + 1))/(n*e^(2*p +
(m - Mod[m, n])/n)*(q + 1)), x] + Dist[(-d)^((m - Mod[m, n])/n - 1)/(n*e^(2*p)*(q + 1)), Int[x^m*(d + e*x^n)^
(q + 1)*ExpandToSum[Together[(1*(n*(-d)^(-((m - Mod[m, n])/n) + 1)*e^(2*p)*(q + 1)*(a + b*x^n + c*x^(2*n))^p -
((c*d^2 - b*d*e + a*e^2)^p/(e^((m - Mod[m, n])/n)*x^(m - Mod[m, n])))*(d*(Mod[m, n] + 1) + e*(Mod[m, n] + n*(
q + 1) + 1)*x^n)))/(d + e*x^n)], x], x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0
] && IGtQ[n, 0] && IGtQ[p, 0] && ILtQ[q, -1] && ILtQ[m, 0]

Rule 1488

Int[((f_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Sy
mbol] :> Int[ExpandIntegrand[(f*x)^m*(d + e*x^n)^q*(a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, e,
f, m, q}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] && IGtQ[p, 0]

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]

Rubi steps

\begin {align*} \int \frac {c+d x^3+e x^6+f x^9}{x^6 \left (a+b x^3\right )^3} \, dx &=\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^3 b \left (a+b x^3\right )^2}-\frac {\int \frac {-6 b^3 c+6 b^3 \left (\frac {b c}{a}-d\right ) x^3-b^2 \left (\frac {5 b^3 c}{a^2}-\frac {5 b^2 d}{a}+5 b e+a f\right ) x^6}{x^6 \left (a+b x^3\right )^2} \, dx}{6 a b^3}\\ &=\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^3 b \left (a+b x^3\right )^2}+\frac {\left (17 b^3 c-11 a b^2 d+5 a^2 b e+a^3 f\right ) x}{18 a^4 b \left (a+b x^3\right )}-\frac {\int \frac {-18 a^2 b^5 c+18 a b^5 (2 b c-a d) x^3-2 b^4 \left (17 b^3 c-11 a b^2 d+5 a^2 b e+a^3 f\right ) x^6}{x^6 \left (a+b x^3\right )} \, dx}{18 a^4 b^5}\\ &=\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^3 b \left (a+b x^3\right )^2}+\frac {\left (17 b^3 c-11 a b^2 d+5 a^2 b e+a^3 f\right ) x}{18 a^4 b \left (a+b x^3\right )}-\frac {\int \left (-\frac {18 a b^5 c}{x^6}+\frac {18 b^5 (3 b c-a d)}{x^3}-\frac {2 b^4 \left (44 b^3 c-20 a b^2 d+5 a^2 b e+a^3 f\right )}{a+b x^3}\right ) \, dx}{18 a^4 b^5}\\ &=-\frac {c}{5 a^3 x^5}+\frac {3 b c-a d}{2 a^4 x^2}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^3 b \left (a+b x^3\right )^2}+\frac {\left (17 b^3 c-11 a b^2 d+5 a^2 b e+a^3 f\right ) x}{18 a^4 b \left (a+b x^3\right )}+\frac {\left (44 b^3 c-20 a b^2 d+5 a^2 b e+a^3 f\right ) \int \frac {1}{a+b x^3} \, dx}{9 a^4 b}\\ &=-\frac {c}{5 a^3 x^5}+\frac {3 b c-a d}{2 a^4 x^2}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^3 b \left (a+b x^3\right )^2}+\frac {\left (17 b^3 c-11 a b^2 d+5 a^2 b e+a^3 f\right ) x}{18 a^4 b \left (a+b x^3\right )}+\frac {\left (44 b^3 c-20 a b^2 d+5 a^2 b e+a^3 f\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{14/3} b}+\frac {\left (44 b^3 c-20 a b^2 d+5 a^2 b e+a^3 f\right ) \int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{27 a^{14/3} b}\\ &=-\frac {c}{5 a^3 x^5}+\frac {3 b c-a d}{2 a^4 x^2}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^3 b \left (a+b x^3\right )^2}+\frac {\left (17 b^3 c-11 a b^2 d+5 a^2 b e+a^3 f\right ) x}{18 a^4 b \left (a+b x^3\right )}+\frac {\left (44 b^3 c-20 a b^2 d+5 a^2 b e+a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{14/3} b^{4/3}}-\frac {\left (44 b^3 c-20 a b^2 d+5 a^2 b e+a^3 f\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}{54 a^{14/3} b^{4/3}}+\frac {\left (44 b^3 c-20 a b^2 d+5 a^2 b e+a^3 f\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 a^{13/3} b}\\ &=-\frac {c}{5 a^3 x^5}+\frac {3 b c-a d}{2 a^4 x^2}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^3 b \left (a+b x^3\right )^2}+\frac {\left (17 b^3 c-11 a b^2 d+5 a^2 b e+a^3 f\right ) x}{18 a^4 b \left (a+b x^3\right )}+\frac {\left (44 b^3 c-20 a b^2 d+5 a^2 b e+a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{14/3} b^{4/3}}-\frac {\left (44 b^3 c-20 a b^2 d+5 a^2 b e+a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{14/3} b^{4/3}}+\frac {\left (44 b^3 c-20 a b^2 d+5 a^2 b e+a^3 f\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{9 a^{14/3} b^{4/3}}\\ &=-\frac {c}{5 a^3 x^5}+\frac {3 b c-a d}{2 a^4 x^2}+\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{6 a^3 b \left (a+b x^3\right )^2}+\frac {\left (17 b^3 c-11 a b^2 d+5 a^2 b e+a^3 f\right ) x}{18 a^4 b \left (a+b x^3\right )}-\frac {\left (44 b^3 c-20 a b^2 d+5 a^2 b e+a^3 f\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{14/3} b^{4/3}}+\frac {\left (44 b^3 c-20 a b^2 d+5 a^2 b e+a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{14/3} b^{4/3}}-\frac {\left (44 b^3 c-20 a b^2 d+5 a^2 b e+a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{14/3} b^{4/3}}\\ \end {align*}

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Mathematica [A]  time = 0.28, size = 299, normalized size = 0.95 \[ \frac {-\frac {135 a^{2/3} (a d-3 b c)}{x^2}-\frac {54 a^{5/3} c}{x^5}+\frac {10 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 f+5 a^2 b e-20 a b^2 d+44 b^3 c\right )}{b^{4/3}}-\frac {10 \sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right ) \left (a^3 f+5 a^2 b e-20 a b^2 d+44 b^3 c\right )}{b^{4/3}}-\frac {45 a^{5/3} x \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{b \left (a+b x^3\right )^2}+\frac {15 a^{2/3} x \left (a^3 f+5 a^2 b e-11 a b^2 d+17 b^3 c\right )}{b \left (a+b x^3\right )}-\frac {5 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 f+5 a^2 b e-20 a b^2 d+44 b^3 c\right )}{b^{4/3}}}{270 a^{14/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^6*(a + b*x^3)^3),x]

[Out]

((-54*a^(5/3)*c)/x^5 - (135*a^(2/3)*(-3*b*c + a*d))/x^2 - (45*a^(5/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*x
)/(b*(a + b*x^3)^2) + (15*a^(2/3)*(17*b^3*c - 11*a*b^2*d + 5*a^2*b*e + a^3*f)*x)/(b*(a + b*x^3)) - (10*Sqrt[3]
*(44*b^3*c - 20*a*b^2*d + 5*a^2*b*e + a^3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(4/3) + (10*(44*b^
3*c - 20*a*b^2*d + 5*a^2*b*e + a^3*f)*Log[a^(1/3) + b^(1/3)*x])/b^(4/3) - (5*(44*b^3*c - 20*a*b^2*d + 5*a^2*b*
e + a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(4/3))/(270*a^(14/3))

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fricas [B]  time = 0.66, size = 1247, normalized size = 3.95 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^9+e*x^6+d*x^3+c)/x^6/(b*x^3+a)^3,x, algorithm="fricas")

[Out]

[1/270*(15*(44*a^2*b^5*c - 20*a^3*b^4*d + 5*a^4*b^3*e + a^5*b^2*f)*x^9 - 54*a^5*b^2*c + 6*(176*a^3*b^4*c - 80*
a^4*b^3*d + 20*a^5*b^2*e - 5*a^6*b*f)*x^6 + 27*(11*a^4*b^3*c - 5*a^5*b^2*d)*x^3 + 15*sqrt(1/3)*((44*a*b^6*c -
20*a^2*b^5*d + 5*a^3*b^4*e + a^4*b^3*f)*x^11 + 2*(44*a^2*b^5*c - 20*a^3*b^4*d + 5*a^4*b^3*e + a^5*b^2*f)*x^8 +
(44*a^3*b^4*c - 20*a^4*b^3*d + 5*a^5*b^2*e + a^6*b*f)*x^5)*sqrt(-(a^2*b)^(1/3)/b)*log((2*a*b*x^3 - 3*(a^2*b)^
(1/3)*a*x - a^2 + 3*sqrt(1/3)*(2*a*b*x^2 + (a^2*b)^(2/3)*x - (a^2*b)^(1/3)*a)*sqrt(-(a^2*b)^(1/3)/b))/(b*x^3 +
a)) - 5*((44*b^5*c - 20*a*b^4*d + 5*a^2*b^3*e + a^3*b^2*f)*x^11 + 2*(44*a*b^4*c - 20*a^2*b^3*d + 5*a^3*b^2*e
+ a^4*b*f)*x^8 + (44*a^2*b^3*c - 20*a^3*b^2*d + 5*a^4*b*e + a^5*f)*x^5)*(a^2*b)^(2/3)*log(a*b*x^2 - (a^2*b)^(2
/3)*x + (a^2*b)^(1/3)*a) + 10*((44*b^5*c - 20*a*b^4*d + 5*a^2*b^3*e + a^3*b^2*f)*x^11 + 2*(44*a*b^4*c - 20*a^2
*b^3*d + 5*a^3*b^2*e + a^4*b*f)*x^8 + (44*a^2*b^3*c - 20*a^3*b^2*d + 5*a^4*b*e + a^5*f)*x^5)*(a^2*b)^(2/3)*log
(a*b*x + (a^2*b)^(2/3)))/(a^6*b^4*x^11 + 2*a^7*b^3*x^8 + a^8*b^2*x^5), 1/270*(15*(44*a^2*b^5*c - 20*a^3*b^4*d
+ 5*a^4*b^3*e + a^5*b^2*f)*x^9 - 54*a^5*b^2*c + 6*(176*a^3*b^4*c - 80*a^4*b^3*d + 20*a^5*b^2*e - 5*a^6*b*f)*x^
6 + 27*(11*a^4*b^3*c - 5*a^5*b^2*d)*x^3 + 30*sqrt(1/3)*((44*a*b^6*c - 20*a^2*b^5*d + 5*a^3*b^4*e + a^4*b^3*f)*
x^11 + 2*(44*a^2*b^5*c - 20*a^3*b^4*d + 5*a^4*b^3*e + a^5*b^2*f)*x^8 + (44*a^3*b^4*c - 20*a^4*b^3*d + 5*a^5*b^
2*e + a^6*b*f)*x^5)*sqrt((a^2*b)^(1/3)/b)*arctan(sqrt(1/3)*(2*(a^2*b)^(2/3)*x - (a^2*b)^(1/3)*a)*sqrt((a^2*b)^
(1/3)/b)/a^2) - 5*((44*b^5*c - 20*a*b^4*d + 5*a^2*b^3*e + a^3*b^2*f)*x^11 + 2*(44*a*b^4*c - 20*a^2*b^3*d + 5*a
^3*b^2*e + a^4*b*f)*x^8 + (44*a^2*b^3*c - 20*a^3*b^2*d + 5*a^4*b*e + a^5*f)*x^5)*(a^2*b)^(2/3)*log(a*b*x^2 - (
a^2*b)^(2/3)*x + (a^2*b)^(1/3)*a) + 10*((44*b^5*c - 20*a*b^4*d + 5*a^2*b^3*e + a^3*b^2*f)*x^11 + 2*(44*a*b^4*c
- 20*a^2*b^3*d + 5*a^3*b^2*e + a^4*b*f)*x^8 + (44*a^2*b^3*c - 20*a^3*b^2*d + 5*a^4*b*e + a^5*f)*x^5)*(a^2*b)^
(2/3)*log(a*b*x + (a^2*b)^(2/3)))/(a^6*b^4*x^11 + 2*a^7*b^3*x^8 + a^8*b^2*x^5)]

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giac [A]  time = 0.23, size = 310, normalized size = 0.98 \[ -\frac {\sqrt {3} {\left (44 \, b^{3} c - 20 \, a b^{2} d + a^{3} f + 5 \, a^{2} b e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{4}} - \frac {{\left (44 \, b^{3} c - 20 \, a b^{2} d + a^{3} f + 5 \, a^{2} b e\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{4}} - \frac {{\left (44 \, b^{3} c - 20 \, a b^{2} d + a^{3} f + 5 \, a^{2} b e\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{5} b} + \frac {17 \, b^{4} c x^{4} - 11 \, a b^{3} d x^{4} + a^{3} b f x^{4} + 5 \, a^{2} b^{2} x^{4} e + 20 \, a b^{3} c x - 14 \, a^{2} b^{2} d x - 2 \, a^{4} f x + 8 \, a^{3} b x e}{18 \, {\left (b x^{3} + a\right )}^{2} a^{4} b} + \frac {15 \, b c x^{3} - 5 \, a d x^{3} - 2 \, a c}{10 \, a^{4} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^9+e*x^6+d*x^3+c)/x^6/(b*x^3+a)^3,x, algorithm="giac")

[Out]

-1/27*sqrt(3)*(44*b^3*c - 20*a*b^2*d + a^3*f + 5*a^2*b*e)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3)
)/((-a*b^2)^(2/3)*a^4) - 1/54*(44*b^3*c - 20*a*b^2*d + a^3*f + 5*a^2*b*e)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2
/3))/((-a*b^2)^(2/3)*a^4) - 1/27*(44*b^3*c - 20*a*b^2*d + a^3*f + 5*a^2*b*e)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(
1/3)))/(a^5*b) + 1/18*(17*b^4*c*x^4 - 11*a*b^3*d*x^4 + a^3*b*f*x^4 + 5*a^2*b^2*x^4*e + 20*a*b^3*c*x - 14*a^2*b
^2*d*x - 2*a^4*f*x + 8*a^3*b*x*e)/((b*x^3 + a)^2*a^4*b) + 1/10*(15*b*c*x^3 - 5*a*d*x^3 - 2*a*c)/(a^4*x^5)

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maple [B]  time = 0.06, size = 566, normalized size = 1.79 \[ \frac {f \,x^{4}}{18 \left (b \,x^{3}+a \right )^{2} a}+\frac {5 b e \,x^{4}}{18 \left (b \,x^{3}+a \right )^{2} a^{2}}-\frac {11 b^{2} d \,x^{4}}{18 \left (b \,x^{3}+a \right )^{2} a^{3}}+\frac {17 b^{3} c \,x^{4}}{18 \left (b \,x^{3}+a \right )^{2} a^{4}}+\frac {4 e x}{9 \left (b \,x^{3}+a \right )^{2} a}-\frac {7 b d x}{9 \left (b \,x^{3}+a \right )^{2} a^{2}}+\frac {10 b^{2} c x}{9 \left (b \,x^{3}+a \right )^{2} a^{3}}-\frac {f x}{9 \left (b \,x^{3}+a \right )^{2} b}+\frac {\sqrt {3}\, f \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} a \,b^{2}}+\frac {f \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} a \,b^{2}}-\frac {f \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \left (\frac {a}{b}\right )^{\frac {2}{3}} a \,b^{2}}+\frac {5 \sqrt {3}\, e \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{2} b}+\frac {5 e \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{2} b}-\frac {5 e \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{2} b}-\frac {20 \sqrt {3}\, d \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{3}}-\frac {20 d \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{3}}+\frac {10 d \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{3}}+\frac {44 \sqrt {3}\, b c \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{4}}+\frac {44 b c \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{4}}-\frac {22 b c \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{4}}-\frac {d}{2 a^{3} x^{2}}+\frac {3 b c}{2 a^{4} x^{2}}-\frac {c}{5 a^{3} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x^9+e*x^6+d*x^3+c)/x^6/(b*x^3+a)^3,x)

[Out]

1/18/a/(b*x^3+a)^2*x^4*f+5/18/a^2/(b*x^3+a)^2*x^4*b*e-11/18/a^3/(b*x^3+a)^2*x^4*b^2*d+17/18/a^4/(b*x^3+a)^2*x^
4*b^3*c-1/9/(b*x^3+a)^2/b*x*f+4/9/a/(b*x^3+a)^2*x*e-7/9/a^2/(b*x^3+a)^2*b*x*d+10/9/a^3/(b*x^3+a)^2*x*b^2*c+1/2
7/a/b^2/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*f+5/27/a^2/b/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*e-20/27/a^3/(a/b)^(2/3)*ln(x+
(a/b)^(1/3))*d+44/27/a^4*b/(a/b)^(2/3)*ln(x+(a/b)^(1/3))*c-1/54/a/b^2/(a/b)^(2/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(
2/3))*f-5/54/a^2/b/(a/b)^(2/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))*e+10/27/a^3/(a/b)^(2/3)*ln(x^2-(a/b)^(1/3)*x+
(a/b)^(2/3))*d-22/27/a^4*b/(a/b)^(2/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))*c+1/27/a/b^2/(a/b)^(2/3)*3^(1/2)*arct
an(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*f+5/27/a^2/b/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*e
-20/27/a^3/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*d+44/27/a^4*b/(a/b)^(2/3)*3^(1/2)*arcta
n(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*c-1/5/a^3*c/x^5-1/2*d/a^3/x^2+3/2/a^4/x^2*b*c

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maxima [A]  time = 3.05, size = 318, normalized size = 1.01 \[ \frac {5 \, {\left (44 \, b^{4} c - 20 \, a b^{3} d + 5 \, a^{2} b^{2} e + a^{3} b f\right )} x^{9} + 2 \, {\left (176 \, a b^{3} c - 80 \, a^{2} b^{2} d + 20 \, a^{3} b e - 5 \, a^{4} f\right )} x^{6} - 18 \, a^{3} b c + 9 \, {\left (11 \, a^{2} b^{2} c - 5 \, a^{3} b d\right )} x^{3}}{90 \, {\left (a^{4} b^{3} x^{11} + 2 \, a^{5} b^{2} x^{8} + a^{6} b x^{5}\right )}} + \frac {\sqrt {3} {\left (44 \, b^{3} c - 20 \, a b^{2} d + 5 \, a^{2} b e + a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, a^{4} b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (44 \, b^{3} c - 20 \, a b^{2} d + 5 \, a^{2} b e + a^{3} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{54 \, a^{4} b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (44 \, b^{3} c - 20 \, a b^{2} d + 5 \, a^{2} b e + a^{3} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a^{4} b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^9+e*x^6+d*x^3+c)/x^6/(b*x^3+a)^3,x, algorithm="maxima")

[Out]

1/90*(5*(44*b^4*c - 20*a*b^3*d + 5*a^2*b^2*e + a^3*b*f)*x^9 + 2*(176*a*b^3*c - 80*a^2*b^2*d + 20*a^3*b*e - 5*a
^4*f)*x^6 - 18*a^3*b*c + 9*(11*a^2*b^2*c - 5*a^3*b*d)*x^3)/(a^4*b^3*x^11 + 2*a^5*b^2*x^8 + a^6*b*x^5) + 1/27*s
qrt(3)*(44*b^3*c - 20*a*b^2*d + 5*a^2*b*e + a^3*f)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a^4*b^
2*(a/b)^(2/3)) - 1/54*(44*b^3*c - 20*a*b^2*d + 5*a^2*b*e + a^3*f)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a^4*
b^2*(a/b)^(2/3)) + 1/27*(44*b^3*c - 20*a*b^2*d + 5*a^2*b*e + a^3*f)*log(x + (a/b)^(1/3))/(a^4*b^2*(a/b)^(2/3))

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mupad [B]  time = 5.20, size = 293, normalized size = 0.93 \[ \frac {\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (f\,a^3+5\,e\,a^2\,b-20\,d\,a\,b^2+44\,c\,b^3\right )}{27\,a^{14/3}\,b^{4/3}}-\frac {\frac {c}{5\,a}-\frac {x^9\,\left (f\,a^3+5\,e\,a^2\,b-20\,d\,a\,b^2+44\,c\,b^3\right )}{18\,a^4}+\frac {x^3\,\left (5\,a\,d-11\,b\,c\right )}{10\,a^2}-\frac {x^6\,\left (-5\,f\,a^3+20\,e\,a^2\,b-80\,d\,a\,b^2+176\,c\,b^3\right )}{45\,a^3\,b}}{a^2\,x^5+2\,a\,b\,x^8+b^2\,x^{11}}+\frac {\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (f\,a^3+5\,e\,a^2\,b-20\,d\,a\,b^2+44\,c\,b^3\right )}{27\,a^{14/3}\,b^{4/3}}-\frac {\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (f\,a^3+5\,e\,a^2\,b-20\,d\,a\,b^2+44\,c\,b^3\right )}{27\,a^{14/3}\,b^{4/3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x^3 + e*x^6 + f*x^9)/(x^6*(a + b*x^3)^3),x)

[Out]

(log(b^(1/3)*x + a^(1/3))*(44*b^3*c + a^3*f - 20*a*b^2*d + 5*a^2*b*e))/(27*a^(14/3)*b^(4/3)) - (c/(5*a) - (x^9
*(44*b^3*c + a^3*f - 20*a*b^2*d + 5*a^2*b*e))/(18*a^4) + (x^3*(5*a*d - 11*b*c))/(10*a^2) - (x^6*(176*b^3*c - 5
*a^3*f - 80*a*b^2*d + 20*a^2*b*e))/(45*a^3*b))/(a^2*x^5 + b^2*x^11 + 2*a*b*x^8) + (log(3^(1/2)*a^(1/3)*1i + 2*
b^(1/3)*x - a^(1/3))*((3^(1/2)*1i)/2 - 1/2)*(44*b^3*c + a^3*f - 20*a*b^2*d + 5*a^2*b*e))/(27*a^(14/3)*b^(4/3))
- (log(3^(1/2)*a^(1/3)*1i - 2*b^(1/3)*x + a^(1/3))*((3^(1/2)*1i)/2 + 1/2)*(44*b^3*c + a^3*f - 20*a*b^2*d + 5*
a^2*b*e))/(27*a^(14/3)*b^(4/3))

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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((f*x**9+e*x**6+d*x**3+c)/x**6/(b*x**3+a)**3,x)

[Out]

Timed out

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