3.275 \(\int \frac {c+d x^3+e x^6+f x^9}{x^{14} (a+b x^3)^2} \, dx\)

Optimal. Leaf size=375 \[ \frac {2 b c-a d}{10 a^3 x^{10}}-\frac {c}{13 a^2 x^{13}}-\frac {a^2 e-2 a b d+3 b^2 c}{7 a^4 x^7}-\frac {b^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-7 a^3 f+10 a^2 b e-13 a b^2 d+16 b^3 c\right )}{18 a^{19/3}}+\frac {b^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-7 a^3 f+10 a^2 b e-13 a b^2 d+16 b^3 c\right )}{9 a^{19/3}}+\frac {b^{4/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (-7 a^3 f+10 a^2 b e-13 a b^2 d+16 b^3 c\right )}{3 \sqrt {3} a^{19/3}}-\frac {b^2 x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^6 \left (a+b x^3\right )}-\frac {b \left (-2 a^3 f+3 a^2 b e-4 a b^2 d+5 b^3 c\right )}{a^6 x}+\frac {a^3 (-f)+2 a^2 b e-3 a b^2 d+4 b^3 c}{4 a^5 x^4} \]

[Out]

-1/13*c/a^2/x^13+1/10*(-a*d+2*b*c)/a^3/x^10+1/7*(-a^2*e+2*a*b*d-3*b^2*c)/a^4/x^7+1/4*(-a^3*f+2*a^2*b*e-3*a*b^2
*d+4*b^3*c)/a^5/x^4-b*(-2*a^3*f+3*a^2*b*e-4*a*b^2*d+5*b^3*c)/a^6/x-1/3*b^2*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*x^2/
a^6/(b*x^3+a)+1/9*b^(4/3)*(-7*a^3*f+10*a^2*b*e-13*a*b^2*d+16*b^3*c)*ln(a^(1/3)+b^(1/3)*x)/a^(19/3)-1/18*b^(4/3
)*(-7*a^3*f+10*a^2*b*e-13*a*b^2*d+16*b^3*c)*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(19/3)+1/9*b^(4/3)*(-7
*a^3*f+10*a^2*b*e-13*a*b^2*d+16*b^3*c)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)/a^(1/3)*3^(1/2))/a^(19/3)*3^(1/2)

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Rubi [A]  time = 0.53, antiderivative size = 375, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {1829, 1834, 292, 31, 634, 617, 204, 628} \[ -\frac {b^2 x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^6 \left (a+b x^3\right )}+\frac {2 a^2 b e+a^3 (-f)-3 a b^2 d+4 b^3 c}{4 a^5 x^4}-\frac {b^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (10 a^2 b e-7 a^3 f-13 a b^2 d+16 b^3 c\right )}{18 a^{19/3}}-\frac {b \left (3 a^2 b e-2 a^3 f-4 a b^2 d+5 b^3 c\right )}{a^6 x}+\frac {b^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (10 a^2 b e-7 a^3 f-13 a b^2 d+16 b^3 c\right )}{9 a^{19/3}}+\frac {b^{4/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (10 a^2 b e-7 a^3 f-13 a b^2 d+16 b^3 c\right )}{3 \sqrt {3} a^{19/3}}-\frac {a^2 e-2 a b d+3 b^2 c}{7 a^4 x^7}+\frac {2 b c-a d}{10 a^3 x^{10}}-\frac {c}{13 a^2 x^{13}} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^3 + e*x^6 + f*x^9)/(x^14*(a + b*x^3)^2),x]

[Out]

-c/(13*a^2*x^13) + (2*b*c - a*d)/(10*a^3*x^10) - (3*b^2*c - 2*a*b*d + a^2*e)/(7*a^4*x^7) + (4*b^3*c - 3*a*b^2*
d + 2*a^2*b*e - a^3*f)/(4*a^5*x^4) - (b*(5*b^3*c - 4*a*b^2*d + 3*a^2*b*e - 2*a^3*f))/(a^6*x) - (b^2*(b^3*c - a
*b^2*d + a^2*b*e - a^3*f)*x^2)/(3*a^6*(a + b*x^3)) + (b^(4/3)*(16*b^3*c - 13*a*b^2*d + 10*a^2*b*e - 7*a^3*f)*A
rcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]*a^(19/3)) + (b^(4/3)*(16*b^3*c - 13*a*b^2*d + 10*
a^2*b*e - 7*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(9*a^(19/3)) - (b^(4/3)*(16*b^3*c - 13*a*b^2*d + 10*a^2*b*e - 7*a
^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*a^(19/3))

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 292

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

Rubi steps

\begin {align*} \int \frac {c+d x^3+e x^6+f x^9}{x^{14} \left (a+b x^3\right )^2} \, dx &=-\frac {b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^6 \left (a+b x^3\right )}-\frac {\int \frac {-3 b^3 c+3 b^3 \left (\frac {b c}{a}-d\right ) x^3-\frac {3 b^3 \left (b^2 c-a b d+a^2 e\right ) x^6}{a^2}+\frac {3 b^3 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^9}{a^3}-\frac {3 b^4 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^{12}}{a^4}+\frac {b^5 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^{15}}{a^5}}{x^{14} \left (a+b x^3\right )} \, dx}{3 a b^3}\\ &=-\frac {b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^6 \left (a+b x^3\right )}-\frac {\int \left (-\frac {3 b^3 c}{a x^{14}}-\frac {3 b^3 (-2 b c+a d)}{a^2 x^{11}}-\frac {3 b^3 \left (3 b^2 c-2 a b d+a^2 e\right )}{a^3 x^8}-\frac {3 b^3 \left (-4 b^3 c+3 a b^2 d-2 a^2 b e+a^3 f\right )}{a^4 x^5}+\frac {3 b^4 \left (-5 b^3 c+4 a b^2 d-3 a^2 b e+2 a^3 f\right )}{a^5 x^2}-\frac {b^5 \left (-16 b^3 c+13 a b^2 d-10 a^2 b e+7 a^3 f\right ) x}{a^5 \left (a+b x^3\right )}\right ) \, dx}{3 a b^3}\\ &=-\frac {c}{13 a^2 x^{13}}+\frac {2 b c-a d}{10 a^3 x^{10}}-\frac {3 b^2 c-2 a b d+a^2 e}{7 a^4 x^7}+\frac {4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f}{4 a^5 x^4}-\frac {b \left (5 b^3 c-4 a b^2 d+3 a^2 b e-2 a^3 f\right )}{a^6 x}-\frac {b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^6 \left (a+b x^3\right )}-\frac {\left (b^2 \left (16 b^3 c-13 a b^2 d+10 a^2 b e-7 a^3 f\right )\right ) \int \frac {x}{a+b x^3} \, dx}{3 a^6}\\ &=-\frac {c}{13 a^2 x^{13}}+\frac {2 b c-a d}{10 a^3 x^{10}}-\frac {3 b^2 c-2 a b d+a^2 e}{7 a^4 x^7}+\frac {4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f}{4 a^5 x^4}-\frac {b \left (5 b^3 c-4 a b^2 d+3 a^2 b e-2 a^3 f\right )}{a^6 x}-\frac {b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^6 \left (a+b x^3\right )}+\frac {\left (b^{5/3} \left (16 b^3 c-13 a b^2 d+10 a^2 b e-7 a^3 f\right )\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{19/3}}-\frac {\left (b^{5/3} \left (16 b^3 c-13 a b^2 d+10 a^2 b e-7 a^3 f\right )\right ) \int \frac {\sqrt [3]{a}+\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{9 a^{19/3}}\\ &=-\frac {c}{13 a^2 x^{13}}+\frac {2 b c-a d}{10 a^3 x^{10}}-\frac {3 b^2 c-2 a b d+a^2 e}{7 a^4 x^7}+\frac {4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f}{4 a^5 x^4}-\frac {b \left (5 b^3 c-4 a b^2 d+3 a^2 b e-2 a^3 f\right )}{a^6 x}-\frac {b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^6 \left (a+b x^3\right )}+\frac {b^{4/3} \left (16 b^3 c-13 a b^2 d+10 a^2 b e-7 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{19/3}}-\frac {\left (b^{4/3} \left (16 b^3 c-13 a b^2 d+10 a^2 b e-7 a^3 f\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}{18 a^{19/3}}-\frac {\left (b^{5/3} \left (16 b^3 c-13 a b^2 d+10 a^2 b e-7 a^3 f\right )\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 a^6}\\ &=-\frac {c}{13 a^2 x^{13}}+\frac {2 b c-a d}{10 a^3 x^{10}}-\frac {3 b^2 c-2 a b d+a^2 e}{7 a^4 x^7}+\frac {4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f}{4 a^5 x^4}-\frac {b \left (5 b^3 c-4 a b^2 d+3 a^2 b e-2 a^3 f\right )}{a^6 x}-\frac {b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^6 \left (a+b x^3\right )}+\frac {b^{4/3} \left (16 b^3 c-13 a b^2 d+10 a^2 b e-7 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{19/3}}-\frac {b^{4/3} \left (16 b^3 c-13 a b^2 d+10 a^2 b e-7 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{19/3}}-\frac {\left (b^{4/3} \left (16 b^3 c-13 a b^2 d+10 a^2 b e-7 a^3 f\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 a^{19/3}}\\ &=-\frac {c}{13 a^2 x^{13}}+\frac {2 b c-a d}{10 a^3 x^{10}}-\frac {3 b^2 c-2 a b d+a^2 e}{7 a^4 x^7}+\frac {4 b^3 c-3 a b^2 d+2 a^2 b e-a^3 f}{4 a^5 x^4}-\frac {b \left (5 b^3 c-4 a b^2 d+3 a^2 b e-2 a^3 f\right )}{a^6 x}-\frac {b^2 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{3 a^6 \left (a+b x^3\right )}+\frac {b^{4/3} \left (16 b^3 c-13 a b^2 d+10 a^2 b e-7 a^3 f\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{19/3}}+\frac {b^{4/3} \left (16 b^3 c-13 a b^2 d+10 a^2 b e-7 a^3 f\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{19/3}}-\frac {b^{4/3} \left (16 b^3 c-13 a b^2 d+10 a^2 b e-7 a^3 f\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{19/3}}\\ \end {align*}

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Mathematica [A]  time = 0.41, size = 370, normalized size = 0.99 \[ \frac {2 b c-a d}{10 a^3 x^{10}}-\frac {c}{13 a^2 x^{13}}-\frac {a^2 e-2 a b d+3 b^2 c}{7 a^4 x^7}+\frac {b^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (7 a^3 f-10 a^2 b e+13 a b^2 d-16 b^3 c\right )}{18 a^{19/3}}+\frac {b^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (-7 a^3 f+10 a^2 b e-13 a b^2 d+16 b^3 c\right )}{9 a^{19/3}}+\frac {b^{4/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right ) \left (-7 a^3 f+10 a^2 b e-13 a b^2 d+16 b^3 c\right )}{3 \sqrt {3} a^{19/3}}+\frac {b^2 x^2 \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{3 a^6 \left (a+b x^3\right )}+\frac {b \left (2 a^3 f-3 a^2 b e+4 a b^2 d-5 b^3 c\right )}{a^6 x}+\frac {a^3 (-f)+2 a^2 b e-3 a b^2 d+4 b^3 c}{4 a^5 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x^3 + e*x^6 + f*x^9)/(x^14*(a + b*x^3)^2),x]

[Out]

-1/13*c/(a^2*x^13) + (2*b*c - a*d)/(10*a^3*x^10) - (3*b^2*c - 2*a*b*d + a^2*e)/(7*a^4*x^7) + (4*b^3*c - 3*a*b^
2*d + 2*a^2*b*e - a^3*f)/(4*a^5*x^4) + (b*(-5*b^3*c + 4*a*b^2*d - 3*a^2*b*e + 2*a^3*f))/(a^6*x) + (b^2*(-(b^3*
c) + a*b^2*d - a^2*b*e + a^3*f)*x^2)/(3*a^6*(a + b*x^3)) + (b^(4/3)*(16*b^3*c - 13*a*b^2*d + 10*a^2*b*e - 7*a^
3*f)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/(3*Sqrt[3]*a^(19/3)) + (b^(4/3)*(16*b^3*c - 13*a*b^2*d + 10*
a^2*b*e - 7*a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(9*a^(19/3)) + (b^(4/3)*(-16*b^3*c + 13*a*b^2*d - 10*a^2*b*e + 7*
a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(18*a^(19/3))

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fricas [A]  time = 0.81, size = 507, normalized size = 1.35 \[ -\frac {5460 \, {\left (16 \, b^{5} c - 13 \, a b^{4} d + 10 \, a^{2} b^{3} e - 7 \, a^{3} b^{2} f\right )} x^{15} + 4095 \, {\left (16 \, a b^{4} c - 13 \, a^{2} b^{3} d + 10 \, a^{3} b^{2} e - 7 \, a^{4} b f\right )} x^{12} - 585 \, {\left (16 \, a^{2} b^{3} c - 13 \, a^{3} b^{2} d + 10 \, a^{4} b e - 7 \, a^{5} f\right )} x^{9} + 234 \, {\left (16 \, a^{3} b^{2} c - 13 \, a^{4} b d + 10 \, a^{5} e\right )} x^{6} + 1260 \, a^{5} c - 126 \, {\left (16 \, a^{4} b c - 13 \, a^{5} d\right )} x^{3} + 1820 \, \sqrt {3} {\left ({\left (16 \, b^{5} c - 13 \, a b^{4} d + 10 \, a^{2} b^{3} e - 7 \, a^{3} b^{2} f\right )} x^{16} + {\left (16 \, a b^{4} c - 13 \, a^{2} b^{3} d + 10 \, a^{3} b^{2} e - 7 \, a^{4} b f\right )} x^{13}\right )} \left (-\frac {b}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {2}{3} \, \sqrt {3} x \left (-\frac {b}{a}\right )^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) - 910 \, {\left ({\left (16 \, b^{5} c - 13 \, a b^{4} d + 10 \, a^{2} b^{3} e - 7 \, a^{3} b^{2} f\right )} x^{16} + {\left (16 \, a b^{4} c - 13 \, a^{2} b^{3} d + 10 \, a^{3} b^{2} e - 7 \, a^{4} b f\right )} x^{13}\right )} \left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x^{2} - a x \left (-\frac {b}{a}\right )^{\frac {2}{3}} - a \left (-\frac {b}{a}\right )^{\frac {1}{3}}\right ) + 1820 \, {\left ({\left (16 \, b^{5} c - 13 \, a b^{4} d + 10 \, a^{2} b^{3} e - 7 \, a^{3} b^{2} f\right )} x^{16} + {\left (16 \, a b^{4} c - 13 \, a^{2} b^{3} d + 10 \, a^{3} b^{2} e - 7 \, a^{4} b f\right )} x^{13}\right )} \left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x + a \left (-\frac {b}{a}\right )^{\frac {2}{3}}\right )}{16380 \, {\left (a^{6} b x^{16} + a^{7} x^{13}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^9+e*x^6+d*x^3+c)/x^14/(b*x^3+a)^2,x, algorithm="fricas")

[Out]

-1/16380*(5460*(16*b^5*c - 13*a*b^4*d + 10*a^2*b^3*e - 7*a^3*b^2*f)*x^15 + 4095*(16*a*b^4*c - 13*a^2*b^3*d + 1
0*a^3*b^2*e - 7*a^4*b*f)*x^12 - 585*(16*a^2*b^3*c - 13*a^3*b^2*d + 10*a^4*b*e - 7*a^5*f)*x^9 + 234*(16*a^3*b^2
*c - 13*a^4*b*d + 10*a^5*e)*x^6 + 1260*a^5*c - 126*(16*a^4*b*c - 13*a^5*d)*x^3 + 1820*sqrt(3)*((16*b^5*c - 13*
a*b^4*d + 10*a^2*b^3*e - 7*a^3*b^2*f)*x^16 + (16*a*b^4*c - 13*a^2*b^3*d + 10*a^3*b^2*e - 7*a^4*b*f)*x^13)*(-b/
a)^(1/3)*arctan(2/3*sqrt(3)*x*(-b/a)^(1/3) + 1/3*sqrt(3)) - 910*((16*b^5*c - 13*a*b^4*d + 10*a^2*b^3*e - 7*a^3
*b^2*f)*x^16 + (16*a*b^4*c - 13*a^2*b^3*d + 10*a^3*b^2*e - 7*a^4*b*f)*x^13)*(-b/a)^(1/3)*log(b*x^2 - a*x*(-b/a
)^(2/3) - a*(-b/a)^(1/3)) + 1820*((16*b^5*c - 13*a*b^4*d + 10*a^2*b^3*e - 7*a^3*b^2*f)*x^16 + (16*a*b^4*c - 13
*a^2*b^3*d + 10*a^3*b^2*e - 7*a^4*b*f)*x^13)*(-b/a)^(1/3)*log(b*x + a*(-b/a)^(2/3)))/(a^6*b*x^16 + a^7*x^13)

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giac [A]  time = 0.21, size = 482, normalized size = 1.29 \[ \frac {\sqrt {3} {\left (16 \, \left (-a b^{2}\right )^{\frac {2}{3}} b^{3} c - 13 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b^{2} d - 7 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{3} f + 10 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{2} b e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{7}} + \frac {{\left (16 \, b^{5} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 13 \, a b^{4} d \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 7 \, a^{3} b^{2} f \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 10 \, a^{2} b^{3} \left (-\frac {a}{b}\right )^{\frac {1}{3}} e\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a^{7}} - \frac {{\left (16 \, \left (-a b^{2}\right )^{\frac {2}{3}} b^{3} c - 13 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b^{2} d - 7 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{3} f + 10 \, \left (-a b^{2}\right )^{\frac {2}{3}} a^{2} b e\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, a^{7}} - \frac {b^{5} c x^{2} - a b^{4} d x^{2} - a^{3} b^{2} f x^{2} + a^{2} b^{3} x^{2} e}{3 \, {\left (b x^{3} + a\right )} a^{6}} - \frac {9100 \, b^{4} c x^{12} - 7280 \, a b^{3} d x^{12} - 3640 \, a^{3} b f x^{12} + 5460 \, a^{2} b^{2} x^{12} e - 1820 \, a b^{3} c x^{9} + 1365 \, a^{2} b^{2} d x^{9} + 455 \, a^{4} f x^{9} - 910 \, a^{3} b x^{9} e + 780 \, a^{2} b^{2} c x^{6} - 520 \, a^{3} b d x^{6} + 260 \, a^{4} x^{6} e - 364 \, a^{3} b c x^{3} + 182 \, a^{4} d x^{3} + 140 \, a^{4} c}{1820 \, a^{6} x^{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^9+e*x^6+d*x^3+c)/x^14/(b*x^3+a)^2,x, algorithm="giac")

[Out]

1/9*sqrt(3)*(16*(-a*b^2)^(2/3)*b^3*c - 13*(-a*b^2)^(2/3)*a*b^2*d - 7*(-a*b^2)^(2/3)*a^3*f + 10*(-a*b^2)^(2/3)*
a^2*b*e)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/a^7 + 1/9*(16*b^5*c*(-a/b)^(1/3) - 13*a*b^4*d*(
-a/b)^(1/3) - 7*a^3*b^2*f*(-a/b)^(1/3) + 10*a^2*b^3*(-a/b)^(1/3)*e)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/a^
7 - 1/18*(16*(-a*b^2)^(2/3)*b^3*c - 13*(-a*b^2)^(2/3)*a*b^2*d - 7*(-a*b^2)^(2/3)*a^3*f + 10*(-a*b^2)^(2/3)*a^2
*b*e)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/a^7 - 1/3*(b^5*c*x^2 - a*b^4*d*x^2 - a^3*b^2*f*x^2 + a^2*b^3*x^
2*e)/((b*x^3 + a)*a^6) - 1/1820*(9100*b^4*c*x^12 - 7280*a*b^3*d*x^12 - 3640*a^3*b*f*x^12 + 5460*a^2*b^2*x^12*e
- 1820*a*b^3*c*x^9 + 1365*a^2*b^2*d*x^9 + 455*a^4*f*x^9 - 910*a^3*b*x^9*e + 780*a^2*b^2*c*x^6 - 520*a^3*b*d*x
^6 + 260*a^4*x^6*e - 364*a^3*b*c*x^3 + 182*a^4*d*x^3 + 140*a^4*c)/(a^6*x^13)

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maple [A]  time = 0.06, size = 631, normalized size = 1.68 \[ \frac {b^{2} f \,x^{2}}{3 \left (b \,x^{3}+a \right ) a^{3}}-\frac {b^{3} e \,x^{2}}{3 \left (b \,x^{3}+a \right ) a^{4}}+\frac {b^{4} d \,x^{2}}{3 \left (b \,x^{3}+a \right ) a^{5}}-\frac {b^{5} c \,x^{2}}{3 \left (b \,x^{3}+a \right ) a^{6}}+\frac {7 \sqrt {3}\, b f \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{3}}-\frac {7 b f \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{3}}+\frac {7 b f \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{3}}-\frac {10 \sqrt {3}\, b^{2} e \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{4}}+\frac {10 b^{2} e \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{4}}-\frac {5 b^{2} e \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{4}}+\frac {13 \sqrt {3}\, b^{3} d \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{5}}-\frac {13 b^{3} d \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{5}}+\frac {13 b^{3} d \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{5}}-\frac {16 \sqrt {3}\, b^{4} c \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{6}}+\frac {16 b^{4} c \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{6}}-\frac {8 b^{4} c \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{6}}+\frac {2 b f}{a^{3} x}-\frac {3 b^{2} e}{a^{4} x}+\frac {4 b^{3} d}{a^{5} x}-\frac {5 b^{4} c}{a^{6} x}-\frac {f}{4 a^{2} x^{4}}+\frac {b e}{2 a^{3} x^{4}}-\frac {3 b^{2} d}{4 a^{4} x^{4}}+\frac {b^{3} c}{a^{5} x^{4}}-\frac {e}{7 a^{2} x^{7}}+\frac {2 b d}{7 a^{3} x^{7}}-\frac {3 b^{2} c}{7 a^{4} x^{7}}-\frac {d}{10 a^{2} x^{10}}+\frac {b c}{5 a^{3} x^{10}}-\frac {c}{13 a^{2} x^{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x^9+e*x^6+d*x^3+c)/x^14/(b*x^3+a)^2,x)

[Out]

-3/4/a^4/x^4*b^2*d+1/a^5/x^4*b^3*c+2*b/a^3/x*f-3*b^2/a^4/x*e+4*b^3/a^5/x*d-5*b^4/a^6/x*c+1/5/a^3/x^10*b*c+2/7/
a^3/x^7*b*d-3/7/a^4/x^7*b^2*c+1/2/a^3/x^4*b*e-1/10/a^2/x^10*d+1/3*b^2/a^3*x^2/(b*x^3+a)*f-1/3*b^3/a^4*x^2/(b*x
^3+a)*e+7/9*b/a^3*f*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-10/9*b^2/a^4*e*3^(1/2)/(a/b)^(
1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))+13/9*b^3/a^5*d*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1
/3)*x-1))-16/9*b^4/a^6*c*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))-1/7/a^2/x^7*e-1/4/a^2/x^4
*f+1/3*b^4/a^5*x^2/(b*x^3+a)*d-1/3*b^5/a^6*x^2/(b*x^3+a)*c+10/9*b^2/a^4*e/(a/b)^(1/3)*ln(x+(a/b)^(1/3))-5/9*b^
2/a^4*e/(a/b)^(1/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))-13/9*b^3/a^5*d/(a/b)^(1/3)*ln(x+(a/b)^(1/3))+13/18*b^3/a
^5*d/(a/b)^(1/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))+16/9*b^4/a^6*c/(a/b)^(1/3)*ln(x+(a/b)^(1/3))-8/9*b^4/a^6*c/
(a/b)^(1/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))-7/9*b/a^3*f/(a/b)^(1/3)*ln(x+(a/b)^(1/3))+7/18*b/a^3*f/(a/b)^(1/
3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))-1/13*c/a^2/x^13

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maxima [A]  time = 2.97, size = 374, normalized size = 1.00 \[ -\frac {1820 \, {\left (16 \, b^{5} c - 13 \, a b^{4} d + 10 \, a^{2} b^{3} e - 7 \, a^{3} b^{2} f\right )} x^{15} + 1365 \, {\left (16 \, a b^{4} c - 13 \, a^{2} b^{3} d + 10 \, a^{3} b^{2} e - 7 \, a^{4} b f\right )} x^{12} - 195 \, {\left (16 \, a^{2} b^{3} c - 13 \, a^{3} b^{2} d + 10 \, a^{4} b e - 7 \, a^{5} f\right )} x^{9} + 78 \, {\left (16 \, a^{3} b^{2} c - 13 \, a^{4} b d + 10 \, a^{5} e\right )} x^{6} + 420 \, a^{5} c - 42 \, {\left (16 \, a^{4} b c - 13 \, a^{5} d\right )} x^{3}}{5460 \, {\left (a^{6} b x^{16} + a^{7} x^{13}\right )}} - \frac {\sqrt {3} {\left (16 \, b^{4} c - 13 \, a b^{3} d + 10 \, a^{2} b^{2} e - 7 \, a^{3} b f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{6} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {{\left (16 \, b^{4} c - 13 \, a b^{3} d + 10 \, a^{2} b^{2} e - 7 \, a^{3} b f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, a^{6} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {{\left (16 \, b^{4} c - 13 \, a b^{3} d + 10 \, a^{2} b^{2} e - 7 \, a^{3} b f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, a^{6} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^9+e*x^6+d*x^3+c)/x^14/(b*x^3+a)^2,x, algorithm="maxima")

[Out]

-1/5460*(1820*(16*b^5*c - 13*a*b^4*d + 10*a^2*b^3*e - 7*a^3*b^2*f)*x^15 + 1365*(16*a*b^4*c - 13*a^2*b^3*d + 10
*a^3*b^2*e - 7*a^4*b*f)*x^12 - 195*(16*a^2*b^3*c - 13*a^3*b^2*d + 10*a^4*b*e - 7*a^5*f)*x^9 + 78*(16*a^3*b^2*c
- 13*a^4*b*d + 10*a^5*e)*x^6 + 420*a^5*c - 42*(16*a^4*b*c - 13*a^5*d)*x^3)/(a^6*b*x^16 + a^7*x^13) - 1/9*sqrt
(3)*(16*b^4*c - 13*a*b^3*d + 10*a^2*b^2*e - 7*a^3*b*f)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a^
6*(a/b)^(1/3)) - 1/18*(16*b^4*c - 13*a*b^3*d + 10*a^2*b^2*e - 7*a^3*b*f)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3)
)/(a^6*(a/b)^(1/3)) + 1/9*(16*b^4*c - 13*a*b^3*d + 10*a^2*b^2*e - 7*a^3*b*f)*log(x + (a/b)^(1/3))/(a^6*(a/b)^(
1/3))

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mupad [B]  time = 5.12, size = 348, normalized size = 0.93 \[ \frac {b^{4/3}\,\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (-7\,f\,a^3+10\,e\,a^2\,b-13\,d\,a\,b^2+16\,c\,b^3\right )}{9\,a^{19/3}}-\frac {\frac {c}{13\,a}-\frac {x^9\,\left (-7\,f\,a^3+10\,e\,a^2\,b-13\,d\,a\,b^2+16\,c\,b^3\right )}{28\,a^4}+\frac {x^3\,\left (13\,a\,d-16\,b\,c\right )}{130\,a^2}+\frac {x^6\,\left (10\,e\,a^2-13\,d\,a\,b+16\,c\,b^2\right )}{70\,a^3}+\frac {b\,x^{12}\,\left (-7\,f\,a^3+10\,e\,a^2\,b-13\,d\,a\,b^2+16\,c\,b^3\right )}{4\,a^5}+\frac {b^2\,x^{15}\,\left (-7\,f\,a^3+10\,e\,a^2\,b-13\,d\,a\,b^2+16\,c\,b^3\right )}{3\,a^6}}{b\,x^{16}+a\,x^{13}}-\frac {b^{4/3}\,\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-7\,f\,a^3+10\,e\,a^2\,b-13\,d\,a\,b^2+16\,c\,b^3\right )}{9\,a^{19/3}}+\frac {b^{4/3}\,\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-7\,f\,a^3+10\,e\,a^2\,b-13\,d\,a\,b^2+16\,c\,b^3\right )}{9\,a^{19/3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x^3 + e*x^6 + f*x^9)/(x^14*(a + b*x^3)^2),x)

[Out]

(b^(4/3)*log(b^(1/3)*x + a^(1/3))*(16*b^3*c - 7*a^3*f - 13*a*b^2*d + 10*a^2*b*e))/(9*a^(19/3)) - (c/(13*a) - (
x^9*(16*b^3*c - 7*a^3*f - 13*a*b^2*d + 10*a^2*b*e))/(28*a^4) + (x^3*(13*a*d - 16*b*c))/(130*a^2) + (x^6*(16*b^
2*c + 10*a^2*e - 13*a*b*d))/(70*a^3) + (b*x^12*(16*b^3*c - 7*a^3*f - 13*a*b^2*d + 10*a^2*b*e))/(4*a^5) + (b^2*
x^15*(16*b^3*c - 7*a^3*f - 13*a*b^2*d + 10*a^2*b*e))/(3*a^6))/(a*x^13 + b*x^16) - (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)*(16*b^3*c - 7*a^3*f - 13*a*b^2*d + 10*a^2*b*e))/(9*a^(19/3
)) + (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)*(16*b^3*c - 7*a^3*f - 13*
a*b^2*d + 10*a^2*b*e))/(9*a^(19/3))

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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**14/(b*x**3+a)**2,x)

[Out]

Timed out

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