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

Optimal. Leaf size=227 \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 a^{4/3} b^{8/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^{4/3} b^{8/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{\sqrt{3} a^{4/3} b^{8/3}}+\frac{x^2 (b e-a f)}{2 b^2}-\frac{c}{a x}+\frac{f x^5}{5 b} \]

[Out]

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

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Rubi [A]  time = 0.192612, antiderivative size = 227, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.233, Rules used = {1834, 292, 31, 634, 617, 204, 628} \[ -\frac{\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{6 a^{4/3} b^{8/3}}+\frac{\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{3 a^{4/3} b^{8/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{\sqrt{3} a^{4/3} b^{8/3}}+\frac{x^2 (b e-a f)}{2 b^2}-\frac{c}{a x}+\frac{f x^5}{5 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

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 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

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 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 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 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]

Rubi steps

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

Mathematica [A]  time = 0.128406, size = 224, normalized size = 0.99 \[ \frac{-5 x \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )+10 x \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )+10 \sqrt{3} x \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right ) \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )+15 a^{4/3} b^{2/3} x^3 (b e-a f)+6 a^{4/3} b^{5/3} f x^6-30 \sqrt [3]{a} b^{8/3} c}{30 a^{4/3} b^{8/3} x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-30*a^(1/3)*b^(8/3)*c + 15*a^(4/3)*b^(2/3)*(b*e - a*f)*x^3 + 6*a^(4/3)*b^(5/3)*f*x^6 + 10*Sqrt[3]*(b^3*c - a*
b^2*d + a^2*b*e - a^3*f)*x*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]] + 10*(b^3*c - a*b^2*d + a^2*b*e - a^3*f
)*x*Log[a^(1/3) + b^(1/3)*x] - 5*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*x*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/
3)*x^2])/(30*a^(4/3)*b^(8/3)*x)

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Maple [B]  time = 0.004, size = 419, normalized size = 1.9 \begin{align*}{\frac{f{x}^{5}}{5\,b}}-{\frac{a{x}^{2}f}{2\,{b}^{2}}}+{\frac{e{x}^{2}}{2\,b}}-{\frac{{a}^{2}f}{3\,{b}^{3}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{ae}{3\,{b}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{d}{3\,b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{c}{3\,a}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{{a}^{2}f}{6\,{b}^{3}}\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{ae}{6\,{b}^{2}}\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{d}{6\,b}\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{c}{6\,a}\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{{a}^{2}\sqrt{3}f}{3\,{b}^{3}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{a\sqrt{3}e}{3\,{b}^{2}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{d\sqrt{3}}{3\,b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{c\sqrt{3}}{3\,a}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-{\frac{c}{ax}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.44421, size = 1260, normalized size = 5.55 \begin{align*} \left [\frac{6 \, a^{2} b^{3} f x^{6} - 30 \, a b^{4} c + 15 \,{\left (a^{2} b^{3} e - a^{3} b^{2} f\right )} x^{3} - 15 \, \sqrt{\frac{1}{3}}{\left (a b^{4} c - a^{2} b^{3} d + a^{3} b^{2} e - a^{4} b f\right )} x \sqrt{\frac{\left (-a b^{2}\right )^{\frac{1}{3}}}{a}} \log \left (\frac{2 \, b^{2} x^{3} - a b + 3 \, \sqrt{\frac{1}{3}}{\left (a b x + 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} x^{2} + \left (-a b^{2}\right )^{\frac{1}{3}} a\right )} \sqrt{\frac{\left (-a b^{2}\right )^{\frac{1}{3}}}{a}} - 3 \, \left (-a b^{2}\right )^{\frac{2}{3}} x}{b x^{3} + a}\right ) - 5 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (-a b^{2}\right )^{\frac{2}{3}} x \log \left (b^{2} x^{2} + \left (-a b^{2}\right )^{\frac{1}{3}} b x + \left (-a b^{2}\right )^{\frac{2}{3}}\right ) + 10 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (-a b^{2}\right )^{\frac{2}{3}} x \log \left (b x - \left (-a b^{2}\right )^{\frac{1}{3}}\right )}{30 \, a^{2} b^{4} x}, \frac{6 \, a^{2} b^{3} f x^{6} - 30 \, a b^{4} c + 15 \,{\left (a^{2} b^{3} e - a^{3} b^{2} f\right )} x^{3} - 30 \, \sqrt{\frac{1}{3}}{\left (a b^{4} c - a^{2} b^{3} d + a^{3} b^{2} e - a^{4} b f\right )} x \sqrt{-\frac{\left (-a b^{2}\right )^{\frac{1}{3}}}{a}} \arctan \left (\frac{\sqrt{\frac{1}{3}}{\left (2 \, b x + \left (-a b^{2}\right )^{\frac{1}{3}}\right )} \sqrt{-\frac{\left (-a b^{2}\right )^{\frac{1}{3}}}{a}}}{b}\right ) - 5 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (-a b^{2}\right )^{\frac{2}{3}} x \log \left (b^{2} x^{2} + \left (-a b^{2}\right )^{\frac{1}{3}} b x + \left (-a b^{2}\right )^{\frac{2}{3}}\right ) + 10 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \left (-a b^{2}\right )^{\frac{2}{3}} x \log \left (b x - \left (-a b^{2}\right )^{\frac{1}{3}}\right )}{30 \, a^{2} b^{4} x}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/30*(6*a^2*b^3*f*x^6 - 30*a*b^4*c + 15*(a^2*b^3*e - a^3*b^2*f)*x^3 - 15*sqrt(1/3)*(a*b^4*c - a^2*b^3*d + a^3
*b^2*e - a^4*b*f)*x*sqrt((-a*b^2)^(1/3)/a)*log((2*b^2*x^3 - a*b + 3*sqrt(1/3)*(a*b*x + 2*(-a*b^2)^(2/3)*x^2 +
(-a*b^2)^(1/3)*a)*sqrt((-a*b^2)^(1/3)/a) - 3*(-a*b^2)^(2/3)*x)/(b*x^3 + a)) - 5*(b^3*c - a*b^2*d + a^2*b*e - a
^3*f)*(-a*b^2)^(2/3)*x*log(b^2*x^2 + (-a*b^2)^(1/3)*b*x + (-a*b^2)^(2/3)) + 10*(b^3*c - a*b^2*d + a^2*b*e - a^
3*f)*(-a*b^2)^(2/3)*x*log(b*x - (-a*b^2)^(1/3)))/(a^2*b^4*x), 1/30*(6*a^2*b^3*f*x^6 - 30*a*b^4*c + 15*(a^2*b^3
*e - a^3*b^2*f)*x^3 - 30*sqrt(1/3)*(a*b^4*c - a^2*b^3*d + a^3*b^2*e - a^4*b*f)*x*sqrt(-(-a*b^2)^(1/3)/a)*arcta
n(sqrt(1/3)*(2*b*x + (-a*b^2)^(1/3))*sqrt(-(-a*b^2)^(1/3)/a)/b) - 5*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*(-a*b^
2)^(2/3)*x*log(b^2*x^2 + (-a*b^2)^(1/3)*b*x + (-a*b^2)^(2/3)) + 10*(b^3*c - a*b^2*d + a^2*b*e - a^3*f)*(-a*b^2
)^(2/3)*x*log(b*x - (-a*b^2)^(1/3)))/(a^2*b^4*x)]

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Sympy [A]  time = 2.51496, size = 406, normalized size = 1.79 \begin{align*} \operatorname{RootSum}{\left (27 t^{3} a^{4} b^{8} + 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{9 t^{2} a^{3} b^{5}}{a^{6} f^{2} - 2 a^{5} b e f + 2 a^{4} b^{2} d f + a^{4} b^{2} e^{2} - 2 a^{3} b^{3} c f - 2 a^{3} b^{3} d e + 2 a^{2} b^{4} c e + a^{2} b^{4} d^{2} - 2 a b^{5} c d + b^{6} c^{2}} + x \right )} \right )\right )} + \frac{f x^{5}}{5 b} - \frac{x^{2} \left (a f - b e\right )}{2 b^{2}} - \frac{c}{a x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.09501, size = 428, normalized size = 1.89 \begin{align*} -\frac{c}{a x} + \frac{{\left (b^{3} c \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a b^{2} d \left (-\frac{a}{b}\right )^{\frac{1}{3}} - a^{3} f \left (-\frac{a}{b}\right )^{\frac{1}{3}} + a^{2} b \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^{2} b^{2}} + \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 \, a^{2} b^{4}} + \frac{2 \, b^{4} f x^{5} - 5 \, a b^{3} f x^{2} + 5 \, b^{4} x^{2} e}{10 \, b^{5}} - \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 \, a^{2} b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-c/(a*x) + 1/3*(b^3*c*(-a/b)^(1/3) - a*b^2*d*(-a/b)^(1/3) - a^3*f*(-a/b)^(1/3) + a^2*b*(-a/b)^(1/3)*e)*(-a/b)^
(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^2*b^2) + 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)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(a^2*b^4) + 1
/10*(2*b^4*f*x^5 - 5*a*b^3*f*x^2 + 5*b^4*x^2*e)/b^5 - 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))/(a^2*b^4)