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

Optimal. Leaf size=312 \[ \frac {x^7 \left (a^2 f-a b e+b^2 d\right )}{7 b^3}-\frac {a x \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{b^5}+\frac {x^4 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{4 b^4}-\frac {a^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 b^{16/3}}+\frac {a^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 b^{16/3}}-\frac {a^{4/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\sqrt {3} b^{16/3}}+\frac {x^{10} (b e-a f)}{10 b^2}+\frac {f x^{13}}{13 b} \]

[Out]

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

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Rubi [A]  time = 0.30, antiderivative size = 312, 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} \[ \frac {x^4 \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{4 b^4}-\frac {a^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 b^{16/3}}-\frac {a x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{b^5}+\frac {a^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 b^{16/3}}-\frac {a^{4/3} \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{\sqrt {3} b^{16/3}}+\frac {x^7 \left (a^2 f-a b e+b^2 d\right )}{7 b^3}+\frac {x^{10} (b e-a f)}{10 b^2}+\frac {f x^{13}}{13 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]  time = 0.11, size = 306, normalized size = 0.98 \[ \frac {x^7 \left (a^2 f-a b e+b^2 d\right )}{7 b^3}+\frac {a x \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{b^5}+\frac {x^4 \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{4 b^4}+\frac {a^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{6 b^{16/3}}-\frac {a^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{3 b^{16/3}}+\frac {a^{4/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{\sqrt {3} b^{16/3}}+\frac {x^{10} (b e-a f)}{10 b^2}+\frac {f x^{13}}{13 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.72, size = 304, normalized size = 0.97 \[ \frac {420 \, b^{4} f x^{13} + 546 \, {\left (b^{4} e - a b^{3} f\right )} x^{10} + 780 \, {\left (b^{4} d - a b^{3} e + a^{2} b^{2} f\right )} x^{7} + 1365 \, {\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} x^{4} - 1820 \, \sqrt {3} {\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} b x \left (-\frac {a}{b}\right )^{\frac {2}{3}} - \sqrt {3} a}{3 \, a}\right ) + 910 \, {\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right ) - 1820 \, {\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x - \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right ) - 5460 \, {\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} x}{5460 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5460*(420*b^4*f*x^13 + 546*(b^4*e - a*b^3*f)*x^10 + 780*(b^4*d - a*b^3*e + a^2*b^2*f)*x^7 + 1365*(b^4*c - a*
b^3*d + a^2*b^2*e - a^3*b*f)*x^4 - 1820*sqrt(3)*(a*b^3*c - a^2*b^2*d + a^3*b*e - a^4*f)*(-a/b)^(1/3)*arctan(1/
3*(2*sqrt(3)*b*x*(-a/b)^(2/3) - sqrt(3)*a)/a) + 910*(a*b^3*c - a^2*b^2*d + a^3*b*e - a^4*f)*(-a/b)^(1/3)*log(x
^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3)) - 1820*(a*b^3*c - a^2*b^2*d + a^3*b*e - a^4*f)*(-a/b)^(1/3)*log(x - (-a/b)
^(1/3)) - 5460*(a*b^3*c - a^2*b^2*d + a^3*b*e - a^4*f)*x)/b^5

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giac [A]  time = 0.18, size = 401, normalized size = 1.29 \[ \frac {\sqrt {3} {\left (\left (-a b^{2}\right )^{\frac {1}{3}} a b^{3} c - \left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b^{2} d - \left (-a b^{2}\right )^{\frac {1}{3}} a^{4} f + \left (-a b^{2}\right )^{\frac {1}{3}} a^{3} b e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, b^{6}} + \frac {{\left (\left (-a b^{2}\right )^{\frac {1}{3}} a b^{3} c - \left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b^{2} d - \left (-a b^{2}\right )^{\frac {1}{3}} a^{4} f + \left (-a b^{2}\right )^{\frac {1}{3}} a^{3} b e\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, b^{6}} - \frac {{\left (a^{2} b^{11} c - a^{3} b^{10} d - a^{5} b^{8} f + a^{4} b^{9} e\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a b^{13}} + \frac {140 \, b^{12} f x^{13} - 182 \, a b^{11} f x^{10} + 182 \, b^{12} x^{10} e + 260 \, b^{12} d x^{7} + 260 \, a^{2} b^{10} f x^{7} - 260 \, a b^{11} x^{7} e + 455 \, b^{12} c x^{4} - 455 \, a b^{11} d x^{4} - 455 \, a^{3} b^{9} f x^{4} + 455 \, a^{2} b^{10} x^{4} e - 1820 \, a b^{11} c x + 1820 \, a^{2} b^{10} d x + 1820 \, a^{4} b^{8} f x - 1820 \, a^{3} b^{9} x e}{1820 \, b^{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(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*b^3*c - (-a*b^2)^(1/3)*a^2*b^2*d - (-a*b^2)^(1/3)*a^4*f + (-a*b^2)^(1/3)*a^3*b*e
)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/b^6 + 1/6*((-a*b^2)^(1/3)*a*b^3*c - (-a*b^2)^(1/3)*a^2
*b^2*d - (-a*b^2)^(1/3)*a^4*f + (-a*b^2)^(1/3)*a^3*b*e)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/b^6 - 1/3*(a^
2*b^11*c - a^3*b^10*d - a^5*b^8*f + a^4*b^9*e)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a*b^13) + 1/1820*(140*
b^12*f*x^13 - 182*a*b^11*f*x^10 + 182*b^12*x^10*e + 260*b^12*d*x^7 + 260*a^2*b^10*f*x^7 - 260*a*b^11*x^7*e + 4
55*b^12*c*x^4 - 455*a*b^11*d*x^4 - 455*a^3*b^9*f*x^4 + 455*a^2*b^10*x^4*e - 1820*a*b^11*c*x + 1820*a^2*b^10*d*
x + 1820*a^4*b^8*f*x - 1820*a^3*b^9*x*e)/b^13

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maple [B]  time = 0.04, size = 544, normalized size = 1.74 \[ \frac {f \,x^{13}}{13 b}-\frac {a f \,x^{10}}{10 b^{2}}+\frac {e \,x^{10}}{10 b}+\frac {a^{2} f \,x^{7}}{7 b^{3}}-\frac {a e \,x^{7}}{7 b^{2}}+\frac {d \,x^{7}}{7 b}-\frac {a^{3} f \,x^{4}}{4 b^{4}}+\frac {a^{2} e \,x^{4}}{4 b^{3}}-\frac {a d \,x^{4}}{4 b^{2}}+\frac {c \,x^{4}}{4 b}-\frac {\sqrt {3}\, a^{5} 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^{6}}-\frac {a^{5} f \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} 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^{6}}+\frac {\sqrt {3}\, a^{4} 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^{5}}+\frac {a^{4} e \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} 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^{5}}+\frac {a^{4} f x}{b^{5}}-\frac {\sqrt {3}\, a^{3} 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^{4}}-\frac {a^{3} d \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} 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^{4}}-\frac {a^{3} e x}{b^{4}}+\frac {\sqrt {3}\, a^{2} 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^{3}}+\frac {a^{2} c \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{3}}-\frac {a^{2} 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^{3}}+\frac {a^{2} d x}{b^{3}}-\frac {a c x}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 2.97, size = 311, normalized size = 1.00 \[ \frac {140 \, b^{4} f x^{13} + 182 \, {\left (b^{4} e - a b^{3} f\right )} x^{10} + 260 \, {\left (b^{4} d - a b^{3} e + a^{2} b^{2} f\right )} x^{7} + 455 \, {\left (b^{4} c - a b^{3} d + a^{2} b^{2} e - a^{3} b f\right )} x^{4} - 1820 \, {\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e - a^{4} f\right )} x}{1820 \, b^{5}} + \frac {\sqrt {3} {\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, b^{6} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, b^{6} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (a^{2} b^{3} c - a^{3} b^{2} d + a^{4} b e - a^{5} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, b^{6} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1820*(140*b^4*f*x^13 + 182*(b^4*e - a*b^3*f)*x^10 + 260*(b^4*d - a*b^3*e + a^2*b^2*f)*x^7 + 455*(b^4*c - a*b
^3*d + a^2*b^2*e - a^3*b*f)*x^4 - 1820*(a*b^3*c - a^2*b^2*d + a^3*b*e - a^4*f)*x)/b^5 + 1/3*sqrt(3)*(a^2*b^3*c
- a^3*b^2*d + a^4*b*e - a^5*f)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(b^6*(a/b)^(2/3)) - 1/6*(a
^2*b^3*c - a^3*b^2*d + a^4*b*e - a^5*f)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(b^6*(a/b)^(2/3)) + 1/3*(a^2*b^
3*c - a^3*b^2*d + a^4*b*e - a^5*f)*log(x + (a/b)^(1/3))/(b^6*(a/b)^(2/3))

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mupad [B]  time = 5.19, size = 311, normalized size = 1.00 \[ x^{10}\,\left (\frac {e}{10\,b}-\frac {a\,f}{10\,b^2}\right )+x^7\,\left (\frac {d}{7\,b}-\frac {a\,\left (\frac {e}{b}-\frac {a\,f}{b^2}\right )}{7\,b}\right )+x^4\,\left (\frac {c}{4\,b}-\frac {a\,\left (\frac {d}{b}-\frac {a\,\left (\frac {e}{b}-\frac {a\,f}{b^2}\right )}{b}\right )}{4\,b}\right )+\frac {f\,x^{13}}{13\,b}+\frac {a^{4/3}\,\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,b^{16/3}}-\frac {a\,x\,\left (\frac {c}{b}-\frac {a\,\left (\frac {d}{b}-\frac {a\,\left (\frac {e}{b}-\frac {a\,f}{b^2}\right )}{b}\right )}{b}\right )}{b}+\frac {a^{4/3}\,\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,b^{16/3}}-\frac {a^{4/3}\,\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,b^{16/3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 3.24, size = 423, normalized size = 1.36 \[ x^{10} \left (- \frac {a f}{10 b^{2}} + \frac {e}{10 b}\right ) + x^{7} \left (\frac {a^{2} f}{7 b^{3}} - \frac {a e}{7 b^{2}} + \frac {d}{7 b}\right ) + x^{4} \left (- \frac {a^{3} f}{4 b^{4}} + \frac {a^{2} e}{4 b^{3}} - \frac {a d}{4 b^{2}} + \frac {c}{4 b}\right ) + x \left (\frac {a^{4} f}{b^{5}} - \frac {a^{3} e}{b^{4}} + \frac {a^{2} d}{b^{3}} - \frac {a c}{b^{2}}\right ) + \operatorname {RootSum} {\left (27 t^{3} b^{16} + a^{13} f^{3} - 3 a^{12} b e f^{2} + 3 a^{11} b^{2} d f^{2} + 3 a^{11} b^{2} e^{2} f - 3 a^{10} b^{3} c f^{2} - 6 a^{10} b^{3} d e f - a^{10} b^{3} e^{3} + 6 a^{9} b^{4} c e f + 3 a^{9} b^{4} d^{2} f + 3 a^{9} b^{4} d e^{2} - 6 a^{8} b^{5} c d f - 3 a^{8} b^{5} c e^{2} - 3 a^{8} b^{5} d^{2} e + 3 a^{7} b^{6} c^{2} f + 6 a^{7} b^{6} c d e + a^{7} b^{6} d^{3} - 3 a^{6} b^{7} c^{2} e - 3 a^{6} b^{7} c d^{2} + 3 a^{5} b^{8} c^{2} d - a^{4} b^{9} c^{3}, \left (t \mapsto t \log {\left (- \frac {3 t b^{5}}{a^{4} f - a^{3} b e + a^{2} b^{2} d - a b^{3} c} + x \right )} \right )\right )} + \frac {f x^{13}}{13 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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