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

Optimal. Leaf size=348 \[ \frac {x^{10} \left (a^2 f-a b e+b^2 d\right )}{10 b^3}+\frac {a^2 x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{b^6}-\frac {a x^4 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{4 b^5}+\frac {x^7 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{7 b^4}+\frac {a^{7/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^{19/3}}-\frac {a^{7/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^{19/3}}+\frac {a^{7/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\sqrt {3} b^{19/3}}+\frac {x^{13} (b e-a f)}{13 b^2}+\frac {f x^{16}}{16 b} \]

[Out]

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

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Rubi [A]  time = 0.33, antiderivative size = 348, 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 = {1836, 1488, 200, 31, 634, 617, 204, 628} \[ \frac {x^7 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{7 b^4}-\frac {a x^4 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{4 b^5}+\frac {a^{7/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 b^{19/3}}+\frac {a^2 x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{b^6}-\frac {a^{7/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^{19/3}}+\frac {a^{7/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{\sqrt {3} b^{19/3}}+\frac {x^{10} \left (a^2 f-a b e+b^2 d\right )}{10 b^3}+\frac {x^{13} (b e-a f)}{13 b^2}+\frac {f x^{16}}{16 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x)/b^6 - (a*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^4)/(4*b^5) + ((b^3*
c - a*b^2*d + a^2*b*e - a^3*f)*x^7)/(7*b^4) + ((b^2*d - a*b*e + a^2*f)*x^10)/(10*b^3) + ((b*e - a*f)*x^13)/(13
*b^2) + (f*x^16)/(16*b) + (a^(7/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]
*a^(1/3))])/(Sqrt[3]*b^(19/3)) - (a^(7/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(3*b^(
19/3)) + (a^(7/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*b^(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 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 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 1836

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, With[{Pqq =
Coeff[Pq, x, q]}, Dist[1/(b*(m + q + n*p + 1)), Int[(c*x)^m*ExpandToSum[b*(m + q + n*p + 1)*(Pq - Pqq*x^q) - a
*Pqq*(m + q - n + 1)*x^(q - n), x]*(a + b*x^n)^p, x], x] + Simp[(Pqq*(c*x)^(m + q - n + 1)*(a + b*x^n)^(p + 1)
)/(b*c^(q - n + 1)*(m + q + n*p + 1)), x]] /; NeQ[m + q + n*p + 1, 0] && q - n >= 0 && (IntegerQ[2*p] || Integ
erQ[p + (q + 1)/(2*n)])] /; FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && IGtQ[n, 0]

Rubi steps

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

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Mathematica [A]  time = 0.09, size = 351, normalized size = 1.01 \[ \frac {x^{10} \left (a^2 f-a b e+b^2 d\right )}{10 b^3}-\frac {a^2 x \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{b^6}+\frac {a x^4 \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{4 b^5}+\frac {x^7 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{7 b^4}-\frac {a^{7/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{6 b^{19/3}}+\frac {a^{7/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{3 b^{19/3}}+\frac {a^{7/3} \tan ^{-1}\left (\frac {2 \sqrt [3]{b} x-\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{\sqrt {3} b^{19/3}}+\frac {x^{13} (b e-a f)}{13 b^2}+\frac {f x^{16}}{16 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.64, size = 342, normalized size = 0.98 \[ \frac {1365 \, b^{5} f x^{16} + 1680 \, {\left (b^{5} e - a b^{4} f\right )} x^{13} + 2184 \, {\left (b^{5} d - a b^{4} e + a^{2} b^{3} f\right )} x^{10} + 3120 \, {\left (b^{5} c - a b^{4} d + a^{2} b^{3} e - a^{3} b^{2} f\right )} x^{7} - 5460 \, {\left (a b^{4} c - a^{2} b^{3} d + a^{3} b^{2} e - a^{4} b f\right )} x^{4} - 7280 \, \sqrt {3} {\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} b x \left (\frac {a}{b}\right )^{\frac {2}{3}} - \sqrt {3} a}{3 \, a}\right ) + 3640 \, {\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) - 7280 \, {\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) + 21840 \, {\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f\right )} x}{21840 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/21840*(1365*b^5*f*x^16 + 1680*(b^5*e - a*b^4*f)*x^13 + 2184*(b^5*d - a*b^4*e + a^2*b^3*f)*x^10 + 3120*(b^5*c
- a*b^4*d + a^2*b^3*e - a^3*b^2*f)*x^7 - 5460*(a*b^4*c - a^2*b^3*d + a^3*b^2*e - a^4*b*f)*x^4 - 7280*sqrt(3)*
(a^2*b^3*c - a^3*b^2*d + a^4*b*e - a^5*f)*(a/b)^(1/3)*arctan(1/3*(2*sqrt(3)*b*x*(a/b)^(2/3) - sqrt(3)*a)/a) +
3640*(a^2*b^3*c - a^3*b^2*d + a^4*b*e - a^5*f)*(a/b)^(1/3)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3)) - 7280*(a^2*
b^3*c - a^3*b^2*d + a^4*b*e - a^5*f)*(a/b)^(1/3)*log(x + (a/b)^(1/3)) + 21840*(a^2*b^3*c - a^3*b^2*d + a^4*b*e
- a^5*f)*x)/b^6

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giac [A]  time = 0.18, size = 454, normalized size = 1.30 \[ -\frac {\sqrt {3} {\left (\left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b^{3} c - \left (-a b^{2}\right )^{\frac {1}{3}} a^{3} b^{2} d - \left (-a b^{2}\right )^{\frac {1}{3}} a^{5} f + \left (-a b^{2}\right )^{\frac {1}{3}} a^{4} b e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, b^{7}} - \frac {{\left (\left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b^{3} c - \left (-a b^{2}\right )^{\frac {1}{3}} a^{3} b^{2} d - \left (-a b^{2}\right )^{\frac {1}{3}} a^{5} f + \left (-a b^{2}\right )^{\frac {1}{3}} a^{4} b e\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, b^{7}} + \frac {{\left (a^{3} b^{13} c - a^{4} b^{12} d - a^{6} b^{10} f + a^{5} b^{11} e\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a b^{16}} + \frac {455 \, b^{15} f x^{16} - 560 \, a b^{14} f x^{13} + 560 \, b^{15} x^{13} e + 728 \, b^{15} d x^{10} + 728 \, a^{2} b^{13} f x^{10} - 728 \, a b^{14} x^{10} e + 1040 \, b^{15} c x^{7} - 1040 \, a b^{14} d x^{7} - 1040 \, a^{3} b^{12} f x^{7} + 1040 \, a^{2} b^{13} x^{7} e - 1820 \, a b^{14} c x^{4} + 1820 \, a^{2} b^{13} d x^{4} + 1820 \, a^{4} b^{11} f x^{4} - 1820 \, a^{3} b^{12} x^{4} e + 7280 \, a^{2} b^{13} c x - 7280 \, a^{3} b^{12} d x - 7280 \, a^{5} b^{10} f x + 7280 \, a^{4} b^{11} x e}{7280 \, b^{16}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*sqrt(3)*((-a*b^2)^(1/3)*a^2*b^3*c - (-a*b^2)^(1/3)*a^3*b^2*d - (-a*b^2)^(1/3)*a^5*f + (-a*b^2)^(1/3)*a^4*
b*e)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/b^7 - 1/6*((-a*b^2)^(1/3)*a^2*b^3*c - (-a*b^2)^(1/3
)*a^3*b^2*d - (-a*b^2)^(1/3)*a^5*f + (-a*b^2)^(1/3)*a^4*b*e)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/b^7 + 1/
3*(a^3*b^13*c - a^4*b^12*d - a^6*b^10*f + a^5*b^11*e)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a*b^16) + 1/728
0*(455*b^15*f*x^16 - 560*a*b^14*f*x^13 + 560*b^15*x^13*e + 728*b^15*d*x^10 + 728*a^2*b^13*f*x^10 - 728*a*b^14*
x^10*e + 1040*b^15*c*x^7 - 1040*a*b^14*d*x^7 - 1040*a^3*b^12*f*x^7 + 1040*a^2*b^13*x^7*e - 1820*a*b^14*c*x^4 +
1820*a^2*b^13*d*x^4 + 1820*a^4*b^11*f*x^4 - 1820*a^3*b^12*x^4*e + 7280*a^2*b^13*c*x - 7280*a^3*b^12*d*x - 728
0*a^5*b^10*f*x + 7280*a^4*b^11*x*e)/b^16

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maple [A]  time = 0.05, size = 592, normalized size = 1.70 \[ \frac {f \,x^{16}}{16 b}-\frac {a f \,x^{13}}{13 b^{2}}+\frac {e \,x^{13}}{13 b}+\frac {a^{2} f \,x^{10}}{10 b^{3}}-\frac {a e \,x^{10}}{10 b^{2}}+\frac {d \,x^{10}}{10 b}-\frac {a^{3} f \,x^{7}}{7 b^{4}}+\frac {a^{2} e \,x^{7}}{7 b^{3}}-\frac {a d \,x^{7}}{7 b^{2}}+\frac {c \,x^{7}}{7 b}+\frac {a^{4} f \,x^{4}}{4 b^{5}}-\frac {a^{3} e \,x^{4}}{4 b^{4}}+\frac {a^{2} d \,x^{4}}{4 b^{3}}-\frac {a c \,x^{4}}{4 b^{2}}+\frac {\sqrt {3}\, a^{6} f \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{7}}+\frac {a^{6} f \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{7}}-\frac {a^{6} f \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{7}}-\frac {\sqrt {3}\, a^{5} e \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{6}}-\frac {a^{5} e \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{6}}+\frac {a^{5} e \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{6}}-\frac {a^{5} f x}{b^{6}}+\frac {\sqrt {3}\, a^{4} d \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{5}}+\frac {a^{4} d \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{5}}-\frac {a^{4} d \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{5}}+\frac {a^{4} e x}{b^{5}}-\frac {\sqrt {3}\, a^{3} c \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{4}}-\frac {a^{3} c \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{4}}+\frac {a^{3} c \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{4}}-\frac {a^{3} d x}{b^{4}}+\frac {a^{2} c x}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 3.01, size = 351, normalized size = 1.01 \[ \frac {455 \, b^{5} f x^{16} + 560 \, {\left (b^{5} e - a b^{4} f\right )} x^{13} + 728 \, {\left (b^{5} d - a b^{4} e + a^{2} b^{3} f\right )} x^{10} + 1040 \, {\left (b^{5} c - a b^{4} d + a^{2} b^{3} e - a^{3} b^{2} f\right )} x^{7} - 1820 \, {\left (a b^{4} c - a^{2} b^{3} d + a^{3} b^{2} e - a^{4} b f\right )} x^{4} + 7280 \, {\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f\right )} x}{7280 \, b^{6}} - \frac {\sqrt {3} {\left (a^{3} b^{3} c - a^{4} b^{2} d + a^{5} b e - a^{6} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, b^{7} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (a^{3} b^{3} c - a^{4} b^{2} d + a^{5} b e - a^{6} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, b^{7} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (a^{3} b^{3} c - a^{4} b^{2} d + a^{5} b e - a^{6} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, b^{7} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7280*(455*b^5*f*x^16 + 560*(b^5*e - a*b^4*f)*x^13 + 728*(b^5*d - a*b^4*e + a^2*b^3*f)*x^10 + 1040*(b^5*c - a
*b^4*d + a^2*b^3*e - a^3*b^2*f)*x^7 - 1820*(a*b^4*c - a^2*b^3*d + a^3*b^2*e - a^4*b*f)*x^4 + 7280*(a^2*b^3*c -
a^3*b^2*d + a^4*b*e - a^5*f)*x)/b^6 - 1/3*sqrt(3)*(a^3*b^3*c - a^4*b^2*d + a^5*b*e - a^6*f)*arctan(1/3*sqrt(3
)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(b^7*(a/b)^(2/3)) + 1/6*(a^3*b^3*c - a^4*b^2*d + a^5*b*e - a^6*f)*log(x^2 -
x*(a/b)^(1/3) + (a/b)^(2/3))/(b^7*(a/b)^(2/3)) - 1/3*(a^3*b^3*c - a^4*b^2*d + a^5*b*e - a^6*f)*log(x + (a/b)^
(1/3))/(b^7*(a/b)^(2/3))

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mupad [B]  time = 0.31, size = 358, normalized size = 1.03 \[ x^{13}\,\left (\frac {e}{13\,b}-\frac {a\,f}{13\,b^2}\right )+x^{10}\,\left (\frac {d}{10\,b}-\frac {a\,\left (\frac {e}{b}-\frac {a\,f}{b^2}\right )}{10\,b}\right )+x^7\,\left (\frac {c}{7\,b}-\frac {a\,\left (\frac {d}{b}-\frac {a\,\left (\frac {e}{b}-\frac {a\,f}{b^2}\right )}{b}\right )}{7\,b}\right )+\frac {f\,x^{16}}{16\,b}-\frac {a^{7/3}\,\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,b^{19/3}}+\frac {a^2\,x\,\left (\frac {c}{b}-\frac {a\,\left (\frac {d}{b}-\frac {a\,\left (\frac {e}{b}-\frac {a\,f}{b^2}\right )}{b}\right )}{b}\right )}{b^2}-\frac {a\,x^4\,\left (\frac {c}{b}-\frac {a\,\left (\frac {d}{b}-\frac {a\,\left (\frac {e}{b}-\frac {a\,f}{b^2}\right )}{b}\right )}{b}\right )}{4\,b}-\frac {a^{7/3}\,\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,b^{19/3}}+\frac {a^{7/3}\,\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,b^{19/3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^13*(e/(13*b) - (a*f)/(13*b^2)) + x^10*(d/(10*b) - (a*(e/b - (a*f)/b^2))/(10*b)) + x^7*(c/(7*b) - (a*(d/b - (
a*(e/b - (a*f)/b^2))/b))/(7*b)) + (f*x^16)/(16*b) - (a^(7/3)*log(b^(1/3)*x + a^(1/3))*(b^3*c - a^3*f - a*b^2*d
+ a^2*b*e))/(3*b^(19/3)) + (a^2*x*(c/b - (a*(d/b - (a*(e/b - (a*f)/b^2))/b))/b))/b^2 - (a*x^4*(c/b - (a*(d/b
- (a*(e/b - (a*f)/b^2))/b))/b))/(4*b) - (a^(7/3)*log(3^(1/2)*a^(1/3)*1i + 2*b^(1/3)*x - a^(1/3))*((3^(1/2)*1i)
/2 - 1/2)*(b^3*c - a^3*f - a*b^2*d + a^2*b*e))/(3*b^(19/3)) + (a^(7/3)*log(3^(1/2)*a^(1/3)*1i - 2*b^(1/3)*x +
a^(1/3))*((3^(1/2)*1i)/2 + 1/2)*(b^3*c - a^3*f - a*b^2*d + a^2*b*e))/(3*b^(19/3))

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sympy [A]  time = 4.35, size = 469, normalized size = 1.35 \[ x^{13} \left (- \frac {a f}{13 b^{2}} + \frac {e}{13 b}\right ) + x^{10} \left (\frac {a^{2} f}{10 b^{3}} - \frac {a e}{10 b^{2}} + \frac {d}{10 b}\right ) + x^{7} \left (- \frac {a^{3} f}{7 b^{4}} + \frac {a^{2} e}{7 b^{3}} - \frac {a d}{7 b^{2}} + \frac {c}{7 b}\right ) + x^{4} \left (\frac {a^{4} f}{4 b^{5}} - \frac {a^{3} e}{4 b^{4}} + \frac {a^{2} d}{4 b^{3}} - \frac {a c}{4 b^{2}}\right ) + x \left (- \frac {a^{5} f}{b^{6}} + \frac {a^{4} e}{b^{5}} - \frac {a^{3} d}{b^{4}} + \frac {a^{2} c}{b^{3}}\right ) + \operatorname {RootSum} {\left (27 t^{3} b^{19} - a^{16} f^{3} + 3 a^{15} b e f^{2} - 3 a^{14} b^{2} d f^{2} - 3 a^{14} b^{2} e^{2} f + 3 a^{13} b^{3} c f^{2} + 6 a^{13} b^{3} d e f + a^{13} b^{3} e^{3} - 6 a^{12} b^{4} c e f - 3 a^{12} b^{4} d^{2} f - 3 a^{12} b^{4} d e^{2} + 6 a^{11} b^{5} c d f + 3 a^{11} b^{5} c e^{2} + 3 a^{11} b^{5} d^{2} e - 3 a^{10} b^{6} c^{2} f - 6 a^{10} b^{6} c d e - a^{10} b^{6} d^{3} + 3 a^{9} b^{7} c^{2} e + 3 a^{9} b^{7} c d^{2} - 3 a^{8} b^{8} c^{2} d + a^{7} b^{9} c^{3}, \left (t \mapsto t \log {\left (\frac {3 t b^{6}}{a^{5} f - a^{4} b e + a^{3} b^{2} d - a^{2} b^{3} c} + x \right )} \right )\right )} + \frac {f x^{16}}{16 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**13*(-a*f/(13*b**2) + e/(13*b)) + x**10*(a**2*f/(10*b**3) - a*e/(10*b**2) + d/(10*b)) + x**7*(-a**3*f/(7*b**
4) + a**2*e/(7*b**3) - a*d/(7*b**2) + c/(7*b)) + x**4*(a**4*f/(4*b**5) - a**3*e/(4*b**4) + a**2*d/(4*b**3) - a
*c/(4*b**2)) + x*(-a**5*f/b**6 + a**4*e/b**5 - a**3*d/b**4 + a**2*c/b**3) + RootSum(27*_t**3*b**19 - a**16*f**
3 + 3*a**15*b*e*f**2 - 3*a**14*b**2*d*f**2 - 3*a**14*b**2*e**2*f + 3*a**13*b**3*c*f**2 + 6*a**13*b**3*d*e*f +
a**13*b**3*e**3 - 6*a**12*b**4*c*e*f - 3*a**12*b**4*d**2*f - 3*a**12*b**4*d*e**2 + 6*a**11*b**5*c*d*f + 3*a**1
1*b**5*c*e**2 + 3*a**11*b**5*d**2*e - 3*a**10*b**6*c**2*f - 6*a**10*b**6*c*d*e - a**10*b**6*d**3 + 3*a**9*b**7
*c**2*e + 3*a**9*b**7*c*d**2 - 3*a**8*b**8*c**2*d + a**7*b**9*c**3, Lambda(_t, _t*log(3*_t*b**6/(a**5*f - a**4
*b*e + a**3*b**2*d - a**2*b**3*c) + x))) + f*x**16/(16*b)

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