3.5.14 \(\int \frac {x^2 (c+d x+e x^2+f x^3+g x^4+h x^5)}{(a+b x^3)^2} \, dx\) [414]

Optimal. Leaf size=290 \[ \frac {4 g x}{3 b^2}+\frac {5 h x^2}{6 b^2}-\frac {c+d x+e x^2+f x^3+g x^4+h x^5}{3 b \left (a+b x^3\right )}-\frac {\left (b^{4/3} d+2 \sqrt [3]{a} b e-4 a \sqrt [3]{b} g-5 a^{4/3} h\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{2/3} b^{8/3}}+\frac {\left (\sqrt [3]{b} (b d-4 a g)-\sqrt [3]{a} (2 b e-5 a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{2/3} b^{8/3}}-\frac {\left (\sqrt [3]{b} (b d-4 a g)-\sqrt [3]{a} (2 b e-5 a h)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{2/3} b^{8/3}}+\frac {f \log \left (a+b x^3\right )}{3 b^2} \]

[Out]

4/3*g*x/b^2+5/6*h*x^2/b^2+1/3*(-h*x^5-g*x^4-f*x^3-e*x^2-d*x-c)/b/(b*x^3+a)+1/9*(b^(1/3)*(-4*a*g+b*d)-a^(1/3)*(
-5*a*h+2*b*e))*ln(a^(1/3)+b^(1/3)*x)/a^(2/3)/b^(8/3)-1/18*(b^(1/3)*(-4*a*g+b*d)-a^(1/3)*(-5*a*h+2*b*e))*ln(a^(
2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(2/3)/b^(8/3)+1/3*f*ln(b*x^3+a)/b^2-1/9*(b^(4/3)*d+2*a^(1/3)*b*e-4*a*b^(
1/3)*g-5*a^(4/3)*h)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)/a^(1/3)*3^(1/2))/a^(2/3)/b^(8/3)*3^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.33, antiderivative size = 288, normalized size of antiderivative = 0.99, number of steps used = 11, number of rules used = 10, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {1837, 1901, 1885, 1874, 31, 648, 631, 210, 642, 266} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (-5 a^{4/3} h+2 \sqrt [3]{a} b e-4 a \sqrt [3]{b} g+b^{4/3} d\right )}{3 \sqrt {3} a^{2/3} b^{8/3}}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (-\frac {\sqrt [3]{a} (2 b e-5 a h)}{\sqrt [3]{b}}-4 a g+b d\right )}{18 a^{2/3} b^{7/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (b d-4 a g)-\sqrt [3]{a} (2 b e-5 a h)\right )}{9 a^{2/3} b^{8/3}}+\frac {f \log \left (a+b x^3\right )}{3 b^2}-\frac {c+d x+e x^2+f x^3+g x^4+h x^5}{3 b \left (a+b x^3\right )}+\frac {4 g x}{3 b^2}+\frac {5 h x^2}{6 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^2*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5))/(a + b*x^3)^2,x]

[Out]

(4*g*x)/(3*b^2) + (5*h*x^2)/(6*b^2) - (c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5)/(3*b*(a + b*x^3)) - ((b^(4/3)*
d + 2*a^(1/3)*b*e - 4*a*b^(1/3)*g - 5*a^(4/3)*h)*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(3*Sqrt[3]
*a^(2/3)*b^(8/3)) + ((b^(1/3)*(b*d - 4*a*g) - a^(1/3)*(2*b*e - 5*a*h))*Log[a^(1/3) + b^(1/3)*x])/(9*a^(2/3)*b^
(8/3)) - ((b*d - 4*a*g - (a^(1/3)*(2*b*e - 5*a*h))/b^(1/3))*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(1
8*a^(2/3)*b^(7/3)) + (f*Log[a + b*x^3])/(3*b^2)

Rule 31

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

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 266

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

Rule 631

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 642

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 648

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 1837

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[Pq*((a + b*x^n)^(p + 1)/(b*n*(p + 1))),
x] - Dist[1/(b*n*(p + 1)), Int[D[Pq, x]*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, m, n}, x] && PolyQ[Pq, x]
&& EqQ[m - n + 1, 0] && LtQ[p, -1]

Rule 1874

Int[((A_) + (B_.)*(x_))/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{r = Numerator[Rt[a/b, 3]], s = Denominator[R
t[a/b, 3]]}, Dist[(-r)*((B*r - A*s)/(3*a*s)), Int[1/(r + s*x), x], x] + Dist[r/(3*a*s), Int[(r*(B*r + 2*A*s) +
s*(B*r - A*s)*x)/(r^2 - r*s*x + s^2*x^2), x], x]] /; FreeQ[{a, b, A, B}, x] && NeQ[a*B^3 - b*A^3, 0] && PosQ[
a/b]

Rule 1885

Int[(P2_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x,
2]}, Int[(A + B*x)/(a + b*x^3), x] + Dist[C, Int[x^2/(a + b*x^3), x], x] /; EqQ[a*B^3 - b*A^3, 0] ||  !Ration
alQ[a/b]] /; FreeQ[{a, b}, x] && PolyQ[P2, x, 2]

Rule 1901

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 {x^2 \left (c+d x+e x^2+f x^3+g x^4+h x^5\right )}{\left (a+b x^3\right )^2} \, dx &=-\frac {c+d x+e x^2+f x^3+g x^4+h x^5}{3 b \left (a+b x^3\right )}+\frac {\int \frac {d+2 e x+3 f x^2+4 g x^3+5 h x^4}{a+b x^3} \, dx}{3 b}\\ &=-\frac {c+d x+e x^2+f x^3+g x^4+h x^5}{3 b \left (a+b x^3\right )}+\frac {\int \left (\frac {4 g}{b}+\frac {5 h x}{b}+\frac {b d-4 a g+(2 b e-5 a h) x+3 b f x^2}{b \left (a+b x^3\right )}\right ) \, dx}{3 b}\\ &=\frac {4 g x}{3 b^2}+\frac {5 h x^2}{6 b^2}-\frac {c+d x+e x^2+f x^3+g x^4+h x^5}{3 b \left (a+b x^3\right )}+\frac {\int \frac {b d-4 a g+(2 b e-5 a h) x+3 b f x^2}{a+b x^3} \, dx}{3 b^2}\\ &=\frac {4 g x}{3 b^2}+\frac {5 h x^2}{6 b^2}-\frac {c+d x+e x^2+f x^3+g x^4+h x^5}{3 b \left (a+b x^3\right )}+\frac {\int \frac {b d-4 a g+(2 b e-5 a h) x}{a+b x^3} \, dx}{3 b^2}+\frac {f \int \frac {x^2}{a+b x^3} \, dx}{b}\\ &=\frac {4 g x}{3 b^2}+\frac {5 h x^2}{6 b^2}-\frac {c+d x+e x^2+f x^3+g x^4+h x^5}{3 b \left (a+b x^3\right )}+\frac {f \log \left (a+b x^3\right )}{3 b^2}+\frac {\int \frac {\sqrt [3]{a} \left (2 \sqrt [3]{b} (b d-4 a g)+\sqrt [3]{a} (2 b e-5 a h)\right )+\sqrt [3]{b} \left (-\sqrt [3]{b} (b d-4 a g)+\sqrt [3]{a} (2 b e-5 a h)\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{9 a^{2/3} b^{7/3}}+\frac {\left (b d-4 a g-\frac {\sqrt [3]{a} (2 b e-5 a h)}{\sqrt [3]{b}}\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{2/3} b^2}\\ &=\frac {4 g x}{3 b^2}+\frac {5 h x^2}{6 b^2}-\frac {c+d x+e x^2+f x^3+g x^4+h x^5}{3 b \left (a+b x^3\right )}+\frac {\left (b d-4 a g-\frac {\sqrt [3]{a} (2 b e-5 a h)}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{2/3} b^{7/3}}+\frac {f \log \left (a+b x^3\right )}{3 b^2}+\frac {\left (b^{4/3} d+2 \sqrt [3]{a} b e-4 a \sqrt [3]{b} g-5 a^{4/3} h\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 \sqrt [3]{a} b^{7/3}}-\frac {\left (b d-4 a g-\frac {\sqrt [3]{a} (2 b e-5 a h)}{\sqrt [3]{b}}\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}{18 a^{2/3} b^{7/3}}\\ &=\frac {4 g x}{3 b^2}+\frac {5 h x^2}{6 b^2}-\frac {c+d x+e x^2+f x^3+g x^4+h x^5}{3 b \left (a+b x^3\right )}+\frac {\left (b d-4 a g-\frac {\sqrt [3]{a} (2 b e-5 a h)}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{2/3} b^{7/3}}-\frac {\left (b d-4 a g-\frac {\sqrt [3]{a} (2 b e-5 a h)}{\sqrt [3]{b}}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{2/3} b^{7/3}}+\frac {f \log \left (a+b x^3\right )}{3 b^2}+\frac {\left (b^{4/3} d+2 \sqrt [3]{a} b e-4 a \sqrt [3]{b} g-5 a^{4/3} h\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 a^{2/3} b^{8/3}}\\ &=\frac {4 g x}{3 b^2}+\frac {5 h x^2}{6 b^2}-\frac {c+d x+e x^2+f x^3+g x^4+h x^5}{3 b \left (a+b x^3\right )}-\frac {\left (b^{4/3} d+2 \sqrt [3]{a} b e-4 a \sqrt [3]{b} g-5 a^{4/3} h\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{2/3} b^{8/3}}+\frac {\left (b d-4 a g-\frac {\sqrt [3]{a} (2 b e-5 a h)}{\sqrt [3]{b}}\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{2/3} b^{7/3}}-\frac {\left (b d-4 a g-\frac {\sqrt [3]{a} (2 b e-5 a h)}{\sqrt [3]{b}}\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{2/3} b^{7/3}}+\frac {f \log \left (a+b x^3\right )}{3 b^2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.13, size = 280, normalized size = 0.97 \begin {gather*} \frac {18 b^{2/3} g x+9 b^{2/3} h x^2-\frac {6 b^{2/3} (b (c+x (d+e x))-a (f+x (g+h x)))}{a+b x^3}+\frac {2 \sqrt {3} \left (-b^{4/3} d-2 \sqrt [3]{a} b e+4 a \sqrt [3]{b} g+5 a^{4/3} h\right ) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{2/3}}+\frac {2 \left (b^{4/3} d-2 \sqrt [3]{a} b e-4 a \sqrt [3]{b} g+5 a^{4/3} h\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{2/3}}-\frac {\left (b^{4/3} d-2 \sqrt [3]{a} b e-4 a \sqrt [3]{b} g+5 a^{4/3} h\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{2/3}}+6 b^{2/3} f \log \left (a+b x^3\right )}{18 b^{8/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^2*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5))/(a + b*x^3)^2,x]

[Out]

(18*b^(2/3)*g*x + 9*b^(2/3)*h*x^2 - (6*b^(2/3)*(b*(c + x*(d + e*x)) - a*(f + x*(g + h*x))))/(a + b*x^3) + (2*S
qrt[3]*(-(b^(4/3)*d) - 2*a^(1/3)*b*e + 4*a*b^(1/3)*g + 5*a^(4/3)*h)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]
])/a^(2/3) + (2*(b^(4/3)*d - 2*a^(1/3)*b*e - 4*a*b^(1/3)*g + 5*a^(4/3)*h)*Log[a^(1/3) + b^(1/3)*x])/a^(2/3) -
((b^(4/3)*d - 2*a^(1/3)*b*e - 4*a*b^(1/3)*g + 5*a^(4/3)*h)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(
2/3) + 6*b^(2/3)*f*Log[a + b*x^3])/(18*b^(8/3))

________________________________________________________________________________________

Maple [A]
time = 0.41, size = 278, normalized size = 0.96

method	result	size
risch	\(\frac {h \,x^{2}}{2 b^{2}}+\frac {g x}{b^{2}}+\frac {\left (\frac {a h}{3}-\frac {b e}{3}\right ) x^{2}+\left (\frac {a g}{3}-\frac {b d}{3}\right ) x +\frac {a f}{3}-\frac {b c}{3}}{b^{2} \left (b \,x^{3}+a \right )}+\frac {\munderset {\textit {\_R} =\RootOf \left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\left (3 b f \,\textit {\_R}^{2}+\left (-5 a h +2 b e \right ) \textit {\_R} -4 a g +b d \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{9 b^{3}}\)	\(115\)
default	\(\frac {\frac {1}{2} h \,x^{2}+g x}{b^{2}}-\frac {\frac {\left (-\frac {a h}{3}+\frac {b e}{3}\right ) x^{2}+\left (-\frac {a g}{3}+\frac {b d}{3}\right ) x -\frac {a f}{3}+\frac {b c}{3}}{b \,x^{3}+a}+\frac {\left (4 a g -b d \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{3}+\frac {\left (5 a h -2 b e \right ) \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3}-\frac {f \ln \left (b \,x^{3}+a \right )}{3}}{b^{2}}\)	\(278\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^3+a)^2,x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.53, size = 289, normalized size = 1.00 \begin {gather*} \frac {{\left (a h - b e\right )} x^{2} - b c + a f - {\left (b d - a g\right )} x}{3 \, {\left (b^{3} x^{3} + a b^{2}\right )}} - \frac {\sqrt {3} {\left (5 \, a h \left (\frac {a}{b}\right )^{\frac {2}{3}} - 2 \, b \left (\frac {a}{b}\right )^{\frac {2}{3}} e - b d \left (\frac {a}{b}\right )^{\frac {1}{3}} + 4 \, a g \left (\frac {a}{b}\right )^{\frac {1}{3}}\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 )}{9 \, a b^{2}} + \frac {h x^{2} + 2 \, g x}{2 \, b^{2}} + \frac {{\left (6 \, b f \left (\frac {a}{b}\right )^{\frac {2}{3}} - 5 \, a h \left (\frac {a}{b}\right )^{\frac {1}{3}} + 2 \, b \left (\frac {a}{b}\right )^{\frac {1}{3}} e - b d + 4 \, a g\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (3 \, b f \left (\frac {a}{b}\right )^{\frac {2}{3}} + 5 \, a h \left (\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, b \left (\frac {a}{b}\right )^{\frac {1}{3}} e + b d - 4 \, a g\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^3+a)^2,x, algorithm="maxima")

[Out]

1/3*((a*h - b*e)*x^2 - b*c + a*f - (b*d - a*g)*x)/(b^3*x^3 + a*b^2) - 1/9*sqrt(3)*(5*a*h*(a/b)^(2/3) - 2*b*(a/
b)^(2/3)*e - b*d*(a/b)^(1/3) + 4*a*g*(a/b)^(1/3))*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a*b^2)
+ 1/2*(h*x^2 + 2*g*x)/b^2 + 1/18*(6*b*f*(a/b)^(2/3) - 5*a*h*(a/b)^(1/3) + 2*b*(a/b)^(1/3)*e - b*d + 4*a*g)*log
(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(b^3*(a/b)^(2/3)) + 1/9*(3*b*f*(a/b)^(2/3) + 5*a*h*(a/b)^(1/3) - 2*b*(a/b)
^(1/3)*e + b*d - 4*a*g)*log(x + (a/b)^(1/3))/(b^3*(a/b)^(2/3))

________________________________________________________________________________________

Fricas [C] Result contains complex when optimal does not.
time = 1.76, size = 12153, normalized size = 41.91 \begin {gather*} \text {too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^3+a)^2,x, algorithm="fricas")

[Out]

1/36*(18*b*h*x^5 + 36*b*g*x^4 - 6*(2*b*e - 5*a*h)*x^2 - 2*(b^3*x^3 + a*b^2)*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*(9
*f^2/b^4 - (2*b^2*d*e + 20*a^2*g*h + (9*f^2 - 8*e*g - 5*d*h)*a*b)/(a*b^5))/(54*f^3/b^6 - 9*(2*b^2*d*e + 20*a^2
*g*h + (9*f^2 - 8*e*g - 5*d*h)*a*b)*f/(a*b^7) - (b^4*d^3 + 8*a*b^3*e^3 - 12*a*b^3*d^2*g + 48*a^2*b^2*d*g^2 - 6
4*a^3*b*g^3 - 60*a^2*b^2*e^2*h + 150*a^3*b*e*h^2 - 125*a^4*h^3)/(a^2*b^8) + (b^4*d^3 + 125*a^4*h^3 - 2*(32*g^3
- 90*f*g*h + 75*e*h^2)*a^3*b + 3*(9*f^3 - 24*e*f*g + 20*e^2*h + (16*g^2 - 15*f*h)*d)*a^2*b^2 - 2*(4*e^3 - 9*d
*e*f + 6*d^2*g)*a*b^3)/(a^2*b^8))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(54*f^3/b^6 - 9*(2*b^2*d*e + 20*a^2*g*h
+ (9*f^2 - 8*e*g - 5*d*h)*a*b)*f/(a*b^7) - (b^4*d^3 + 8*a*b^3*e^3 - 12*a*b^3*d^2*g + 48*a^2*b^2*d*g^2 - 64*a^3
*b*g^3 - 60*a^2*b^2*e^2*h + 150*a^3*b*e*h^2 - 125*a^4*h^3)/(a^2*b^8) + (b^4*d^3 + 125*a^4*h^3 - 2*(32*g^3 - 90
*f*g*h + 75*e*h^2)*a^3*b + 3*(9*f^3 - 24*e*f*g + 20*e^2*h + (16*g^2 - 15*f*h)*d)*a^2*b^2 - 2*(4*e^3 - 9*d*e*f
+ 6*d^2*g) ...

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(h*x**5+g*x**4+f*x**3+e*x**2+d*x+c)/(b*x**3+a)**2,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]
time = 0.54, size = 307, normalized size = 1.06 \begin {gather*} \frac {f \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{2}} - \frac {\sqrt {3} {\left (b^{2} d - 4 \, a b g + 5 \, \left (-a b^{2}\right )^{\frac {1}{3}} a h - 2 \, \left (-a b^{2}\right )^{\frac {1}{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 )}{9 \, \left (-a b^{2}\right )^{\frac {2}{3}} b^{2}} - \frac {{\left (b^{2} d - 4 \, a b g - 5 \, \left (-a b^{2}\right )^{\frac {1}{3}} a h + 2 \, \left (-a b^{2}\right )^{\frac {1}{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 )}{18 \, \left (-a b^{2}\right )^{\frac {2}{3}} b^{2}} + \frac {{\left (a h - b e\right )} x^{2} - b c + a f - {\left (b d - a g\right )} x}{3 \, {\left (b x^{3} + a\right )} b^{2}} + \frac {b^{2} h x^{2} + 2 \, b^{2} g x}{2 \, b^{4}} + \frac {{\left (5 \, a b^{3} h \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, b^{4} \left (-\frac {a}{b}\right )^{\frac {1}{3}} e - b^{4} d + 4 \, a b^{3} g\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a b^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(b*x^3+a)^2,x, algorithm="giac")

[Out]

1/3*f*log(abs(b*x^3 + a))/b^2 - 1/9*sqrt(3)*(b^2*d - 4*a*b*g + 5*(-a*b^2)^(1/3)*a*h - 2*(-a*b^2)^(1/3)*b*e)*ar
ctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/((-a*b^2)^(2/3)*b^2) - 1/18*(b^2*d - 4*a*b*g - 5*(-a*b^2)^
(1/3)*a*h + 2*(-a*b^2)^(1/3)*b*e)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/((-a*b^2)^(2/3)*b^2) + 1/3*((a*h -
b*e)*x^2 - b*c + a*f - (b*d - a*g)*x)/((b*x^3 + a)*b^2) + 1/2*(b^2*h*x^2 + 2*b^2*g*x)/b^4 + 1/9*(5*a*b^3*h*(-a
/b)^(1/3) - 2*b^4*(-a/b)^(1/3)*e - b^4*d + 4*a*b^3*g)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a*b^5)

________________________________________________________________________________________

Mupad [B]
time = 0.14, size = 816, normalized size = 2.81 \begin {gather*} \left (\sum _{k=1}^3\ln \left (\frac {9\,a\,b\,f^2+2\,b^2\,d\,e+20\,a^2\,g\,h-5\,a\,b\,d\,h-8\,a\,b\,e\,g}{9\,b^3}+\mathrm {root}\left (729\,a^2\,b^8\,z^3-729\,a^2\,b^6\,f\,z^2+54\,a\,b^5\,d\,e\,z+540\,a^3\,b^3\,g\,h\,z-216\,a^2\,b^4\,e\,g\,z-135\,a^2\,b^4\,d\,h\,z+243\,a^2\,b^4\,f^2\,z-180\,a^3\,b\,f\,g\,h-18\,a\,b^3\,d\,e\,f+72\,a^2\,b^2\,e\,f\,g+45\,a^2\,b^2\,d\,f\,h+150\,a^3\,b\,e\,h^2+12\,a\,b^3\,d^2\,g-60\,a^2\,b^2\,e^2\,h-48\,a^2\,b^2\,d\,g^2-27\,a^2\,b^2\,f^3+64\,a^3\,b\,g^3+8\,a\,b^3\,e^3-125\,a^4\,h^3-b^4\,d^3,z,k\right )\,\left (-6\,a\,f+\frac {x\,\left (9\,b^4\,d-36\,a\,b^3\,g\right )}{9\,b^3}+\mathrm {root}\left (729\,a^2\,b^8\,z^3-729\,a^2\,b^6\,f\,z^2+54\,a\,b^5\,d\,e\,z+540\,a^3\,b^3\,g\,h\,z-216\,a^2\,b^4\,e\,g\,z-135\,a^2\,b^4\,d\,h\,z+243\,a^2\,b^4\,f^2\,z-180\,a^3\,b\,f\,g\,h-18\,a\,b^3\,d\,e\,f+72\,a^2\,b^2\,e\,f\,g+45\,a^2\,b^2\,d\,f\,h+150\,a^3\,b\,e\,h^2+12\,a\,b^3\,d^2\,g-60\,a^2\,b^2\,e^2\,h-48\,a^2\,b^2\,d\,g^2-27\,a^2\,b^2\,f^3+64\,a^3\,b\,g^3+8\,a\,b^3\,e^3-125\,a^4\,h^3-b^4\,d^3,z,k\right )\,a\,b^2\,9\right )+\frac {x\,\left (25\,a^2\,h^2-20\,a\,b\,e\,h+12\,f\,g\,a\,b+4\,b^2\,e^2-3\,d\,f\,b^2\right )}{9\,b^3}\right )\,\mathrm {root}\left (729\,a^2\,b^8\,z^3-729\,a^2\,b^6\,f\,z^2+54\,a\,b^5\,d\,e\,z+540\,a^3\,b^3\,g\,h\,z-216\,a^2\,b^4\,e\,g\,z-135\,a^2\,b^4\,d\,h\,z+243\,a^2\,b^4\,f^2\,z-180\,a^3\,b\,f\,g\,h-18\,a\,b^3\,d\,e\,f+72\,a^2\,b^2\,e\,f\,g+45\,a^2\,b^2\,d\,f\,h+150\,a^3\,b\,e\,h^2+12\,a\,b^3\,d^2\,g-60\,a^2\,b^2\,e^2\,h-48\,a^2\,b^2\,d\,g^2-27\,a^2\,b^2\,f^3+64\,a^3\,b\,g^3+8\,a\,b^3\,e^3-125\,a^4\,h^3-b^4\,d^3,z,k\right )\right )-\frac {\left (\frac {b\,e}{3}-\frac {a\,h}{3}\right )\,x^2+\left (\frac {b\,d}{3}-\frac {a\,g}{3}\right )\,x+\frac {b\,c}{3}-\frac {a\,f}{3}}{b^3\,x^3+a\,b^2}+\frac {h\,x^2}{2\,b^2}+\frac {g\,x}{b^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2*(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5))/(a + b*x^3)^2,x)

[Out]

symsum(log((9*a*b*f^2 + 2*b^2*d*e + 20*a^2*g*h - 5*a*b*d*h - 8*a*b*e*g)/(9*b^3) + root(729*a^2*b^8*z^3 - 729*a
^2*b^6*f*z^2 + 54*a*b^5*d*e*z + 540*a^3*b^3*g*h*z - 216*a^2*b^4*e*g*z - 135*a^2*b^4*d*h*z + 243*a^2*b^4*f^2*z
- 180*a^3*b*f*g*h - 18*a*b^3*d*e*f + 72*a^2*b^2*e*f*g + 45*a^2*b^2*d*f*h + 150*a^3*b*e*h^2 + 12*a*b^3*d^2*g -
60*a^2*b^2*e^2*h - 48*a^2*b^2*d*g^2 - 27*a^2*b^2*f^3 + 64*a^3*b*g^3 + 8*a*b^3*e^3 - 125*a^4*h^3 - b^4*d^3, z,
k)*((x*(9*b^4*d - 36*a*b^3*g))/(9*b^3) - 6*a*f + 9*root(729*a^2*b^8*z^3 - 729*a^2*b^6*f*z^2 + 54*a*b^5*d*e*z +
540*a^3*b^3*g*h*z - 216*a^2*b^4*e*g*z - 135*a^2*b^4*d*h*z + 243*a^2*b^4*f^2*z - 180*a^3*b*f*g*h - 18*a*b^3*d*
e*f + 72*a^2*b^2*e*f*g + 45*a^2*b^2*d*f*h + 150*a^3*b*e*h^2 + 12*a*b^3*d^2*g - 60*a^2*b^2*e^2*h - 48*a^2*b^2*d
*g^2 - 27*a^2*b^2*f^3 + 64*a^3*b*g^3 + 8*a*b^3*e^3 - 125*a^4*h^3 - b^4*d^3, z, k)*a*b^2) + (x*(4*b^2*e^2 + 25*
a^2*h^2 - 3*b^2*d*f - 20*a*b*e*h + 12*a*b*f*g))/(9*b^3))*root(729*a^2*b^8*z^3 - 729*a^2*b^6*f*z^2 + 54*a*b^5*d
*e*z + 540*a^3*b^3*g*h*z - 216*a^2*b^4*e*g*z - 135*a^2*b^4*d*h*z + 243*a^2*b^4*f^2*z - 180*a^3*b*f*g*h - 18*a*
b^3*d*e*f + 72*a^2*b^2*e*f*g + 45*a^2*b^2*d*f*h + 150*a^3*b*e*h^2 + 12*a*b^3*d^2*g - 60*a^2*b^2*e^2*h - 48*a^2
*b^2*d*g^2 - 27*a^2*b^2*f^3 + 64*a^3*b*g^3 + 8*a*b^3*e^3 - 125*a^4*h^3 - b^4*d^3, z, k), k, 1, 3) - ((b*c)/3 -
(a*f)/3 + x*((b*d)/3 - (a*g)/3) + x^2*((b*e)/3 - (a*h)/3))/(a*b^2 + b^3*x^3) + (h*x^2)/(2*b^2) + (g*x)/b^2

________________________________________________________________________________________