3.236 \(\int \frac {x^4 (c+d x^3+e x^6+f x^9)}{a+b x^3} \, dx\)

Optimal. Leaf size=279 \[ \frac {x^5 \left (a^2 f-a b e+b^2 d\right )}{5 b^3}+\frac {x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{2 b^4}-\frac {a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^{14/3}}+\frac {a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^{14/3}}+\frac {a^{2/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\sqrt {3} b^{14/3}}+\frac {x^8 (b e-a f)}{8 b^2}+\frac {f x^{11}}{11 b} \]

[Out]

1/2*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*x^2/b^4+1/5*(a^2*f-a*b*e+b^2*d)*x^5/b^3+1/8*(-a*f+b*e)*x^8/b^2+1/11*f*x^11/
b+1/3*a^(2/3)*(-a^3*f+a^2*b*e-a*b^2*d+b^3*c)*ln(a^(1/3)+b^(1/3)*x)/b^(14/3)-1/6*a^(2/3)*(-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^(14/3)+1/3*a^(2/3)*(-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^(14/3)*3^(1/2)

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Rubi [A]  time = 0.27, antiderivative size = 279, 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, 292, 31, 634, 617, 204, 628} \[ \frac {x^2 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{2 b^4}-\frac {a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 b^{14/3}}+\frac {a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^{14/3}}+\frac {a^{2/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{\sqrt {3} b^{14/3}}+\frac {x^5 \left (a^2 f-a b e+b^2 d\right )}{5 b^3}+\frac {x^8 (b e-a f)}{8 b^2}+\frac {f x^{11}}{11 b} \]

Antiderivative was successfully verified.

[In]

Int[(x^4*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3),x]

[Out]

((b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/(2*b^4) + ((b^2*d - a*b*e + a^2*f)*x^5)/(5*b^3) + ((b*e - a*f)*x^8)/
(8*b^2) + (f*x^11)/(11*b) + (a^(2/3)*(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^(14/3)) + (a^(2/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*Log[a^(1/3) + b^(1/3)*x])/(3*b
^(14/3)) - (a^(2/3)*(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^(
14/3))

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 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^4 \left (c+d x^3+e x^6+f x^9\right )}{a+b x^3} \, dx &=\frac {f x^{11}}{11 b}+\frac {\int \frac {x^4 \left (11 b c+11 b d x^3+11 (b e-a f) x^6\right )}{a+b x^3} \, dx}{11 b}\\ &=\frac {f x^{11}}{11 b}+\frac {\int \left (\frac {11 \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x}{b^3}+\frac {11 \left (b^2 d-a b e+a^2 f\right ) x^4}{b^2}+\frac {11 (b e-a f) x^7}{b}+\frac {11 \left (-a b^3 c+a^2 b^2 d-a^3 b e+a^4 f\right ) x}{b^3 \left (a+b x^3\right )}\right ) \, dx}{11 b}\\ &=\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{2 b^4}+\frac {\left (b^2 d-a b e+a^2 f\right ) x^5}{5 b^3}+\frac {(b e-a f) x^8}{8 b^2}+\frac {f x^{11}}{11 b}-\frac {\left (a \left (b^3 c-a b^2 d+a^2 b e-a^3 f\right )\right ) \int \frac {x}{a+b x^3} \, dx}{b^4}\\ &=\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{2 b^4}+\frac {\left (b^2 d-a b e+a^2 f\right ) x^5}{5 b^3}+\frac {(b e-a f) x^8}{8 b^2}+\frac {f x^{11}}{11 b}+\frac {\left (a^{2/3} \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^{13/3}}-\frac {\left (a^{2/3} \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} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{3 b^{13/3}}\\ &=\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{2 b^4}+\frac {\left (b^2 d-a b e+a^2 f\right ) x^5}{5 b^3}+\frac {(b e-a f) x^8}{8 b^2}+\frac {f x^{11}}{11 b}+\frac {a^{2/3} \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^{14/3}}-\frac {\left (a^{2/3} \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^{14/3}}-\frac {\left (a \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^{13/3}}\\ &=\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{2 b^4}+\frac {\left (b^2 d-a b e+a^2 f\right ) x^5}{5 b^3}+\frac {(b e-a f) x^8}{8 b^2}+\frac {f x^{11}}{11 b}+\frac {a^{2/3} \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^{14/3}}-\frac {a^{2/3} \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^{14/3}}-\frac {\left (a^{2/3} \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^{14/3}}\\ &=\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) x^2}{2 b^4}+\frac {\left (b^2 d-a b e+a^2 f\right ) x^5}{5 b^3}+\frac {(b e-a f) x^8}{8 b^2}+\frac {f x^{11}}{11 b}+\frac {a^{2/3} \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^{14/3}}+\frac {a^{2/3} \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^{14/3}}-\frac {a^{2/3} \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^{14/3}}\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 266, normalized size = 0.95 \[ \frac {264 b^{5/3} x^5 \left (a^2 f-a b e+b^2 d\right )+660 b^{2/3} x^2 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )-440 a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )-440 \sqrt {3} a^{2/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )+220 a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )+165 b^{8/3} x^8 (b e-a f)+120 b^{11/3} f x^{11}}{1320 b^{14/3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^4*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3),x]

[Out]

(660*b^(2/3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x^2 + 264*b^(5/3)*(b^2*d - a*b*e + a^2*f)*x^5 + 165*b^(8/3)*(
b*e - a*f)*x^8 + 120*b^(11/3)*f*x^11 - 440*Sqrt[3]*a^(2/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*ArcTan[(1 -
(2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] - 440*a^(2/3)*(-(b^3*c) + a*b^2*d - a^2*b*e + a^3*f)*Log[a^(1/3) + b^(1/3)*x]
+ 220*a^(2/3)*(-(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])/(1320*b^(
14/3))

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fricas [A]  time = 0.69, size = 281, normalized size = 1.01 \[ \frac {120 \, b^{3} f x^{11} + 165 \, {\left (b^{3} e - a b^{2} f\right )} x^{8} + 264 \, {\left (b^{3} d - a b^{2} e + a^{2} b f\right )} x^{5} + 660 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{2} - 440 \, \sqrt {3} {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} b x \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} + \sqrt {3} a}{3 \, a}\right ) + 220 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \log \left (a x^{2} - b x \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {2}{3}} - a \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}}\right ) - 440 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \log \left (a x + b \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {2}{3}}\right )}{1320 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a),x, algorithm="fricas")

[Out]

1/1320*(120*b^3*f*x^11 + 165*(b^3*e - a*b^2*f)*x^8 + 264*(b^3*d - a*b^2*e + a^2*b*f)*x^5 + 660*(b^3*c - a*b^2*
d + a^2*b*e - a^3*f)*x^2 - 440*sqrt(3)*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*(-a^2/b^2)^(1/3)*arctan(1/3*(2*sqrt
(3)*b*x*(-a^2/b^2)^(1/3) + sqrt(3)*a)/a) + 220*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*(-a^2/b^2)^(1/3)*log(a*x^2
- b*x*(-a^2/b^2)^(2/3) - a*(-a^2/b^2)^(1/3)) - 440*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*(-a^2/b^2)^(1/3)*log(a*
x + b*(-a^2/b^2)^(2/3)))/b^4

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giac [A]  time = 0.18, size = 386, normalized size = 1.38 \[ \frac {\sqrt {3} {\left (\left (-a b^{2}\right )^{\frac {2}{3}} b^{3} c - \left (-a b^{2}\right )^{\frac {2}{3}} a b^{2} d - \left (-a b^{2}\right )^{\frac {2}{3}} a^{3} f + \left (-a b^{2}\right )^{\frac {2}{3}} a^{2} b e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, b^{6}} - \frac {{\left (\left (-a b^{2}\right )^{\frac {2}{3}} b^{3} c - \left (-a b^{2}\right )^{\frac {2}{3}} a b^{2} d - \left (-a b^{2}\right )^{\frac {2}{3}} a^{3} f + \left (-a b^{2}\right )^{\frac {2}{3}} a^{2} b e\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, b^{6}} + \frac {{\left (a b^{10} c \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a^{2} b^{9} d \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a^{4} b^{7} f \left (-\frac {a}{b}\right )^{\frac {1}{3}} + a^{3} b^{8} \left (-\frac {a}{b}\right )^{\frac {1}{3}} e\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a b^{11}} + \frac {40 \, b^{10} f x^{11} - 55 \, a b^{9} f x^{8} + 55 \, b^{10} x^{8} e + 88 \, b^{10} d x^{5} + 88 \, a^{2} b^{8} f x^{5} - 88 \, a b^{9} x^{5} e + 220 \, b^{10} c x^{2} - 220 \, a b^{9} d x^{2} - 220 \, a^{3} b^{7} f x^{2} + 220 \, a^{2} b^{8} x^{2} e}{440 \, b^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(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)^(2/3)*b^3*c - (-a*b^2)^(2/3)*a*b^2*d - (-a*b^2)^(2/3)*a^3*f + (-a*b^2)^(2/3)*a^2*b*e)*ar
ctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/b^6 - 1/6*((-a*b^2)^(2/3)*b^3*c - (-a*b^2)^(2/3)*a*b^2*d -
(-a*b^2)^(2/3)*a^3*f + (-a*b^2)^(2/3)*a^2*b*e)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/b^6 + 1/3*(a*b^10*c*(
-a/b)^(1/3) - a^2*b^9*d*(-a/b)^(1/3) - a^4*b^7*f*(-a/b)^(1/3) + a^3*b^8*(-a/b)^(1/3)*e)*(-a/b)^(1/3)*log(abs(x
- (-a/b)^(1/3)))/(a*b^11) + 1/440*(40*b^10*f*x^11 - 55*a*b^9*f*x^8 + 55*b^10*x^8*e + 88*b^10*d*x^5 + 88*a^2*b
^8*f*x^5 - 88*a*b^9*x^5*e + 220*b^10*c*x^2 - 220*a*b^9*d*x^2 - 220*a^3*b^7*f*x^2 + 220*a^2*b^8*x^2*e)/b^11

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maple [B]  time = 0.05, size = 502, normalized size = 1.80 \[ \frac {f \,x^{11}}{11 b}-\frac {a f \,x^{8}}{8 b^{2}}+\frac {e \,x^{8}}{8 b}+\frac {a^{2} f \,x^{5}}{5 b^{3}}-\frac {a e \,x^{5}}{5 b^{2}}+\frac {d \,x^{5}}{5 b}-\frac {a^{3} f \,x^{2}}{2 b^{4}}+\frac {a^{2} e \,x^{2}}{2 b^{3}}-\frac {a d \,x^{2}}{2 b^{2}}+\frac {c \,x^{2}}{2 b}+\frac {\sqrt {3}\, a^{4} 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 {1}{3}} b^{5}}-\frac {a^{4} f \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{5}}+\frac {a^{4} 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 {1}{3}} b^{5}}-\frac {\sqrt {3}\, a^{3} 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 {1}{3}} b^{4}}+\frac {a^{3} e \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{4}}-\frac {a^{3} 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 {1}{3}} b^{4}}+\frac {\sqrt {3}\, a^{2} 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 {1}{3}} b^{3}}-\frac {a^{2} d \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{3}}+\frac {a^{2} 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 {1}{3}} b^{3}}-\frac {\sqrt {3}\, a 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 {1}{3}} b^{2}}+\frac {a c \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{2}}-\frac {a 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 {1}{3}} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a),x)

[Out]

1/11/b*f*x^11-1/8/b^2*x^8*a*f+1/8/b*x^8*e+1/5/b^3*x^5*a^2*f-1/5/b^2*x^5*a*e+1/5/b*x^5*d-1/2/b^4*x^2*a^3*f+1/2/
b^3*x^2*a^2*e-1/2/b^2*x^2*a*d+1/2/b*x^2*c-1/3*a^4/b^5/(a/b)^(1/3)*ln(x+(a/b)^(1/3))*f+1/3*a^3/b^4/(a/b)^(1/3)*
ln(x+(a/b)^(1/3))*e-1/3*a^2/b^3/(a/b)^(1/3)*ln(x+(a/b)^(1/3))*d+1/3*a/b^2/(a/b)^(1/3)*ln(x+(a/b)^(1/3))*c+1/6*
a^4/b^5/(a/b)^(1/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))*f-1/6*a^3/b^4/(a/b)^(1/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/
3))*e+1/6*a^2/b^3/(a/b)^(1/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))*d-1/6*a/b^2/(a/b)^(1/3)*ln(x^2-(a/b)^(1/3)*x+(
a/b)^(2/3))*c+1/3*a^4/b^5*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*f-1/3*a^3/b^4*3^(1/2)/(a
/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*e+1/3*a^2/b^3*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b
)^(1/3)*x-1))*d-1/3*a/b^2*3^(1/2)/(a/b)^(1/3)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))*c

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maxima [A]  time = 3.02, size = 269, normalized size = 0.96 \[ -\frac {\sqrt {3} {\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, b^{5} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {40 \, b^{3} f x^{11} + 55 \, {\left (b^{3} e - a b^{2} f\right )} x^{8} + 88 \, {\left (b^{3} d - a b^{2} e + a^{2} b f\right )} x^{5} + 220 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{2}}{440 \, b^{4}} - \frac {{\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, b^{5} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {{\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, b^{5} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(f*x^9+e*x^6+d*x^3+c)/(b*x^3+a),x, algorithm="maxima")

[Out]

-1/3*sqrt(3)*(a*b^3*c - a^2*b^2*d + a^3*b*e - a^4*f)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(b^5*
(a/b)^(1/3)) + 1/440*(40*b^3*f*x^11 + 55*(b^3*e - a*b^2*f)*x^8 + 88*(b^3*d - a*b^2*e + a^2*b*f)*x^5 + 220*(b^3
*c - a*b^2*d + a^2*b*e - a^3*f)*x^2)/b^4 - 1/6*(a*b^3*c - a^2*b^2*d + a^3*b*e - a^4*f)*log(x^2 - x*(a/b)^(1/3)
+ (a/b)^(2/3))/(b^5*(a/b)^(1/3)) + 1/3*(a*b^3*c - a^2*b^2*d + a^3*b*e - a^4*f)*log(x + (a/b)^(1/3))/(b^5*(a/b
)^(1/3))

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mupad [B]  time = 5.15, size = 267, normalized size = 0.96 \[ x^8\,\left (\frac {e}{8\,b}-\frac {a\,f}{8\,b^2}\right )+x^5\,\left (\frac {d}{5\,b}-\frac {a\,\left (\frac {e}{b}-\frac {a\,f}{b^2}\right )}{5\,b}\right )+x^2\,\left (\frac {c}{2\,b}-\frac {a\,\left (\frac {d}{b}-\frac {a\,\left (\frac {e}{b}-\frac {a\,f}{b^2}\right )}{b}\right )}{2\,b}\right )+\frac {f\,x^{11}}{11\,b}+\frac {a^{2/3}\,\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,b^{14/3}}-\frac {a^{2/3}\,\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,b^{14/3}}+\frac {a^{2/3}\,\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,b^{14/3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4*(c + d*x^3 + e*x^6 + f*x^9))/(a + b*x^3),x)

[Out]

x^8*(e/(8*b) - (a*f)/(8*b^2)) + x^5*(d/(5*b) - (a*(e/b - (a*f)/b^2))/(5*b)) + x^2*(c/(2*b) - (a*(d/b - (a*(e/b
- (a*f)/b^2))/b))/(2*b)) + (f*x^11)/(11*b) + (a^(2/3)*log(b^(1/3)*x + a^(1/3))*(b^3*c - a^3*f - a*b^2*d + a^2
*b*e))/(3*b^(14/3)) - (a^(2/3)*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^(14/3)) + (a^(2/3)*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^(14/3))

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sympy [A]  time = 2.48, size = 469, normalized size = 1.68 \[ x^{8} \left (- \frac {a f}{8 b^{2}} + \frac {e}{8 b}\right ) + x^{5} \left (\frac {a^{2} f}{5 b^{3}} - \frac {a e}{5 b^{2}} + \frac {d}{5 b}\right ) + x^{2} \left (- \frac {a^{3} f}{2 b^{4}} + \frac {a^{2} e}{2 b^{3}} - \frac {a d}{2 b^{2}} + \frac {c}{2 b}\right ) + \operatorname {RootSum} {\left (27 t^{3} b^{14} + a^{11} f^{3} - 3 a^{10} b e f^{2} + 3 a^{9} b^{2} d f^{2} + 3 a^{9} b^{2} e^{2} f - 3 a^{8} b^{3} c f^{2} - 6 a^{8} b^{3} d e f - a^{8} b^{3} e^{3} + 6 a^{7} b^{4} c e f + 3 a^{7} b^{4} d^{2} f + 3 a^{7} b^{4} d e^{2} - 6 a^{6} b^{5} c d f - 3 a^{6} b^{5} c e^{2} - 3 a^{6} b^{5} d^{2} e + 3 a^{5} b^{6} c^{2} f + 6 a^{5} b^{6} c d e + a^{5} b^{6} d^{3} - 3 a^{4} b^{7} c^{2} e - 3 a^{4} b^{7} c d^{2} + 3 a^{3} b^{8} c^{2} d - a^{2} b^{9} c^{3}, \left (t \mapsto t \log {\left (\frac {9 t^{2} b^{9}}{a^{7} f^{2} - 2 a^{6} b e f + 2 a^{5} b^{2} d f + a^{5} b^{2} e^{2} - 2 a^{4} b^{3} c f - 2 a^{4} b^{3} d e + 2 a^{3} b^{4} c e + a^{3} b^{4} d^{2} - 2 a^{2} b^{5} c d + a b^{6} c^{2}} + x \right )} \right )\right )} + \frac {f x^{11}}{11 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(f*x**9+e*x**6+d*x**3+c)/(b*x**3+a),x)

[Out]

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

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