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

Optimal. Leaf size=240 \[ \frac {x \left (a^2 f-a b e+b^2 d\right )}{b^3}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{6 a^{2/3} b^{10/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{3 a^{2/3} b^{10/3}}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{\sqrt {3} a^{2/3} b^{10/3}}+\frac {x^4 (b e-a f)}{4 b^2}+\frac {f x^7}{7 b} \]

[Out]

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

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Rubi [A]  time = 0.15, antiderivative size = 240, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {1887, 200, 31, 634, 617, 204, 628} \[ -\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 a^{2/3} b^{10/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^{2/3} b^{10/3}}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{\sqrt {3} a^{2/3} b^{10/3}}+\frac {x \left (a^2 f-a b e+b^2 d\right )}{b^3}+\frac {x^4 (b e-a f)}{4 b^2}+\frac {f x^7}{7 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 1887

Int[(Pq_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[Pq/(a + b*x^n), x], x] /; FreeQ[{a, b}, x
] && PolyQ[Pq, x] && IntegerQ[n]

Rubi steps

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

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Mathematica [A]  time = 0.17, size = 229, normalized size = 0.95 \[ \frac {84 \sqrt [3]{b} x \left (a^2 f-a b e+b^2 d\right )+\frac {28 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{a^{2/3}}+\frac {28 \sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{a^{2/3}}+\frac {14 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{a^{2/3}}+21 b^{4/3} x^4 (b e-a f)+12 b^{7/3} f x^7}{84 b^{10/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(84*b^(1/3)*(b^2*d - a*b*e + a^2*f)*x + 21*b^(4/3)*(b*e - a*f)*x^4 + 12*b^(7/3)*f*x^7 + (28*Sqrt[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]])/a^(2/3) + (28*(b^3*c - a*b^2*d + a^2
*b*e - a^3*f)*Log[a^(1/3) + b^(1/3)*x])/a^(2/3) + (14*(-(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])/a^(2/3))/(84*b^(10/3))

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fricas [A]  time = 0.54, size = 600, normalized size = 2.50 \[ \left [\frac {12 \, a^{2} b^{3} f x^{7} + 21 \, {\left (a^{2} b^{3} e - a^{3} b^{2} f\right )} x^{4} - 42 \, \sqrt {\frac {1}{3}} {\left (a b^{4} c - a^{2} b^{3} d + a^{3} b^{2} e - a^{4} b f\right )} \sqrt {\frac {\left (-a^{2} b\right )^{\frac {1}{3}}}{b}} \log \left (\frac {2 \, a b x^{3} + 3 \, \left (-a^{2} b\right )^{\frac {1}{3}} a x - a^{2} - 3 \, \sqrt {\frac {1}{3}} {\left (2 \, a b x^{2} + \left (-a^{2} b\right )^{\frac {2}{3}} x + \left (-a^{2} b\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (-a^{2} b\right )^{\frac {1}{3}}}{b}}}{b x^{3} + a}\right ) - 14 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (-a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{2} - \left (-a^{2} b\right )^{\frac {2}{3}} x - \left (-a^{2} b\right )^{\frac {1}{3}} a\right ) + 28 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (-a^{2} b\right )^{\frac {2}{3}} \log \left (a b x + \left (-a^{2} b\right )^{\frac {2}{3}}\right ) + 84 \, {\left (a^{2} b^{3} d - a^{3} b^{2} e + a^{4} b f\right )} x}{84 \, a^{2} b^{4}}, \frac {12 \, a^{2} b^{3} f x^{7} + 21 \, {\left (a^{2} b^{3} e - a^{3} b^{2} f\right )} x^{4} + 84 \, \sqrt {\frac {1}{3}} {\left (a b^{4} c - a^{2} b^{3} d + a^{3} b^{2} e - a^{4} b f\right )} \sqrt {-\frac {\left (-a^{2} b\right )^{\frac {1}{3}}}{b}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (-a^{2} b\right )^{\frac {2}{3}} x + \left (-a^{2} b\right )^{\frac {1}{3}} a\right )} \sqrt {-\frac {\left (-a^{2} b\right )^{\frac {1}{3}}}{b}}}{a^{2}}\right ) - 14 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (-a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{2} - \left (-a^{2} b\right )^{\frac {2}{3}} x - \left (-a^{2} b\right )^{\frac {1}{3}} a\right ) + 28 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (-a^{2} b\right )^{\frac {2}{3}} \log \left (a b x + \left (-a^{2} b\right )^{\frac {2}{3}}\right ) + 84 \, {\left (a^{2} b^{3} d - a^{3} b^{2} e + a^{4} b f\right )} x}{84 \, a^{2} b^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/84*(12*a^2*b^3*f*x^7 + 21*(a^2*b^3*e - a^3*b^2*f)*x^4 - 42*sqrt(1/3)*(a*b^4*c - a^2*b^3*d + a^3*b^2*e - a^4
*b*f)*sqrt((-a^2*b)^(1/3)/b)*log((2*a*b*x^3 + 3*(-a^2*b)^(1/3)*a*x - a^2 - 3*sqrt(1/3)*(2*a*b*x^2 + (-a^2*b)^(
2/3)*x + (-a^2*b)^(1/3)*a)*sqrt((-a^2*b)^(1/3)/b))/(b*x^3 + a)) - 14*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*(-a^2
*b)^(2/3)*log(a*b*x^2 - (-a^2*b)^(2/3)*x - (-a^2*b)^(1/3)*a) + 28*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*(-a^2*b)
^(2/3)*log(a*b*x + (-a^2*b)^(2/3)) + 84*(a^2*b^3*d - a^3*b^2*e + a^4*b*f)*x)/(a^2*b^4), 1/84*(12*a^2*b^3*f*x^7
+ 21*(a^2*b^3*e - a^3*b^2*f)*x^4 + 84*sqrt(1/3)*(a*b^4*c - a^2*b^3*d + a^3*b^2*e - a^4*b*f)*sqrt(-(-a^2*b)^(1
/3)/b)*arctan(sqrt(1/3)*(2*(-a^2*b)^(2/3)*x + (-a^2*b)^(1/3)*a)*sqrt(-(-a^2*b)^(1/3)/b)/a^2) - 14*(b^3*c - a*b
^2*d + a^2*b*e - a^3*f)*(-a^2*b)^(2/3)*log(a*b*x^2 - (-a^2*b)^(2/3)*x - (-a^2*b)^(1/3)*a) + 28*(b^3*c - a*b^2*
d + a^2*b*e - a^3*f)*(-a^2*b)^(2/3)*log(a*b*x + (-a^2*b)^(2/3)) + 84*(a^2*b^3*d - a^3*b^2*e + a^4*b*f)*x)/(a^2
*b^4)]

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giac [A]  time = 0.19, size = 253, normalized size = 1.05 \[ -\frac {\sqrt {3} {\left (b^{3} c - a b^{2} d - a^{3} f + a^{2} b e\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, \left (-a b^{2}\right )^{\frac {2}{3}} b^{2}} - \frac {{\left (b^{3} c - a b^{2} d - a^{3} f + a^{2} b e\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, \left (-a b^{2}\right )^{\frac {2}{3}} b^{2}} - \frac {{\left (b^{7} c - a b^{6} d - a^{3} b^{4} f + a^{2} b^{5} e\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a b^{7}} + \frac {4 \, b^{6} f x^{7} - 7 \, a b^{5} f x^{4} + 7 \, b^{6} x^{4} e + 28 \, b^{6} d x + 28 \, a^{2} b^{4} f x - 28 \, a b^{5} x e}{28 \, b^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*sqrt(3)*(b^3*c - a*b^2*d - a^3*f + a^2*b*e)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/((-a*b^
2)^(2/3)*b^2) - 1/6*(b^3*c - a*b^2*d - a^3*f + a^2*b*e)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/((-a*b^2)^(2/
3)*b^2) - 1/3*(b^7*c - a*b^6*d - a^3*b^4*f + a^2*b^5*e)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a*b^7) + 1/28
*(4*b^6*f*x^7 - 7*a*b^5*f*x^4 + 7*b^6*x^4*e + 28*b^6*d*x + 28*a^2*b^4*f*x - 28*a*b^5*x*e)/b^7

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maple [B]  time = 0.04, size = 442, normalized size = 1.84 \[ \frac {f \,x^{7}}{7 b}-\frac {a f \,x^{4}}{4 b^{2}}+\frac {e \,x^{4}}{4 b}-\frac {\sqrt {3}\, a^{3} 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^{4}}-\frac {a^{3} f \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} 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^{4}}+\frac {\sqrt {3}\, a^{2} 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^{3}}+\frac {a^{2} e \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} 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^{3}}+\frac {a^{2} f x}{b^{3}}-\frac {\sqrt {3}\, a 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^{2}}-\frac {a d \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{2}}+\frac {a 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^{2}}-\frac {a e x}{b^{2}}+\frac {\sqrt {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}+\frac {c \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} b}-\frac {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}+\frac {d x}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 3.01, size = 223, normalized size = 0.93 \[ \frac {4 \, b^{2} f x^{7} + 7 \, {\left (b^{2} e - a b f\right )} x^{4} + 28 \, {\left (b^{2} d - a b e + a^{2} f\right )} x}{28 \, b^{3}} + \frac {\sqrt {3} {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/28*(4*b^2*f*x^7 + 7*(b^2*e - a*b*f)*x^4 + 28*(b^2*d - a*b*e + a^2*f)*x)/b^3 + 1/3*sqrt(3)*(b^3*c - a*b^2*d +
a^2*b*e - a^3*f)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(b^4*(a/b)^(2/3)) - 1/6*(b^3*c - a*b^2*d
+ a^2*b*e - a^3*f)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(b^4*(a/b)^(2/3)) + 1/3*(b^3*c - a*b^2*d + a^2*b*e
- a^3*f)*log(x + (a/b)^(1/3))/(b^4*(a/b)^(2/3))

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mupad [B]  time = 5.17, size = 222, normalized size = 0.92 \[ x^4\,\left (\frac {e}{4\,b}-\frac {a\,f}{4\,b^2}\right )+x\,\left (\frac {d}{b}-\frac {a\,\left (\frac {e}{b}-\frac {a\,f}{b^2}\right )}{b}\right )+\frac {f\,x^7}{7\,b}+\frac {\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,a^{2/3}\,b^{10/3}}+\frac {\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,a^{2/3}\,b^{10/3}}-\frac {\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{3\,a^{2/3}\,b^{10/3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 3.41, size = 342, normalized size = 1.42 \[ x^{4} \left (- \frac {a f}{4 b^{2}} + \frac {e}{4 b}\right ) + x \left (\frac {a^{2} f}{b^{3}} - \frac {a e}{b^{2}} + \frac {d}{b}\right ) + \operatorname {RootSum} {\left (27 t^{3} a^{2} b^{10} + a^{9} f^{3} - 3 a^{8} b e f^{2} + 3 a^{7} b^{2} d f^{2} + 3 a^{7} b^{2} e^{2} f - 3 a^{6} b^{3} c f^{2} - 6 a^{6} b^{3} d e f - a^{6} b^{3} e^{3} + 6 a^{5} b^{4} c e f + 3 a^{5} b^{4} d^{2} f + 3 a^{5} b^{4} d e^{2} - 6 a^{4} b^{5} c d f - 3 a^{4} b^{5} c e^{2} - 3 a^{4} b^{5} d^{2} e + 3 a^{3} b^{6} c^{2} f + 6 a^{3} b^{6} c d e + a^{3} b^{6} d^{3} - 3 a^{2} b^{7} c^{2} e - 3 a^{2} b^{7} c d^{2} + 3 a b^{8} c^{2} d - b^{9} c^{3}, \left (t \mapsto t \log {\left (- \frac {3 t a b^{3}}{a^{3} f - a^{2} b e + a b^{2} d - b^{3} c} + x \right )} \right )\right )} + \frac {f x^{7}}{7 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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