3.5.23 \(\int \frac {x^2 (c+d x+e x^2+f x^3+g x^4+h x^5)}{(a+b x^3)^3} \, dx\) [423]

Optimal. Leaf size=297 \[ \frac {x \left (b d-4 a g+(2 b e-5 a h) x+3 b f x^2\right )}{18 a b^2 \left (a+b x^3\right )}-\frac {c+d x+e x^2+f x^3+g x^4+h x^5}{6 b \left (a+b x^3\right )^2}-\frac {\left (b^{4/3} d+\sqrt [3]{a} b e+2 a \sqrt [3]{b} g+5 a^{4/3} h\right ) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{5/3} b^{8/3}}+\frac {\left (\sqrt [3]{b} (b d+2 a g)-\sqrt [3]{a} (b e+5 a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{5/3} b^{8/3}}-\frac {\left (\sqrt [3]{b} (b d+2 a g)-\sqrt [3]{a} (b e+5 a h)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{5/3} b^{8/3}} \]

[Out]

1/18*x*(b*d-4*a*g+(-5*a*h+2*b*e)*x+3*b*f*x^2)/a/b^2/(b*x^3+a)+1/6*(-h*x^5-g*x^4-f*x^3-e*x^2-d*x-c)/b/(b*x^3+a)
^2+1/27*(b^(1/3)*(2*a*g+b*d)-a^(1/3)*(5*a*h+b*e))*ln(a^(1/3)+b^(1/3)*x)/a^(5/3)/b^(8/3)-1/54*(b^(1/3)*(2*a*g+b
*d)-a^(1/3)*(5*a*h+b*e))*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(5/3)/b^(8/3)-1/27*(b^(4/3)*d+a^(1/3)*b*e
+2*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^(5/3)/b^(8/3)*3^(1/2)

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Rubi [A]
time = 0.29, antiderivative size = 297, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {1837, 1872, 1874, 31, 648, 631, 210, 642} \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+\sqrt [3]{a} b e+2 a \sqrt [3]{b} g+b^{4/3} d\right )}{9 \sqrt {3} a^{5/3} b^{8/3}}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right ) \left (\sqrt [3]{b} (2 a g+b d)-\sqrt [3]{a} (5 a h+b e)\right )}{54 a^{5/3} b^{8/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \left (\sqrt [3]{b} (2 a g+b d)-\sqrt [3]{a} (5 a h+b e)\right )}{27 a^{5/3} b^{8/3}}+\frac {x \left (x (2 b e-5 a h)-4 a g+b d+3 b f x^2\right )}{18 a b^2 \left (a+b x^3\right )}-\frac {c+d x+e x^2+f x^3+g x^4+h x^5}{6 b \left (a+b x^3\right )^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)^3,x]

[Out]

(x*(b*d - 4*a*g + (2*b*e - 5*a*h)*x + 3*b*f*x^2))/(18*a*b^2*(a + b*x^3)) - (c + d*x + e*x^2 + f*x^3 + g*x^4 +
h*x^5)/(6*b*(a + b*x^3)^2) - ((b^(4/3)*d + a^(1/3)*b*e + 2*a*b^(1/3)*g + 5*a^(4/3)*h)*ArcTan[(a^(1/3) - 2*b^(1
/3)*x)/(Sqrt[3]*a^(1/3))])/(9*Sqrt[3]*a^(5/3)*b^(8/3)) + ((b^(1/3)*(b*d + 2*a*g) - a^(1/3)*(b*e + 5*a*h))*Log[
a^(1/3) + b^(1/3)*x])/(27*a^(5/3)*b^(8/3)) - ((b^(1/3)*(b*d + 2*a*g) - a^(1/3)*(b*e + 5*a*h))*Log[a^(2/3) - a^
(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(54*a^(5/3)*b^(8/3))

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 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 1872

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, Module[{Q = PolynomialQuotient
[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x], R = PolynomialRemainder[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x
]}, Dist[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), Int[(a + b*x^n)^(p + 1)*ExpandToSum[a*n*(p + 1)*Q + n*(p +
1)*R + D[x*R, x], x], x], x] + Simp[(-x)*R*((a + b*x^n)^(p + 1)/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1))), x]] /
; GeQ[q, n]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 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]

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 )^3} \, dx &=-\frac {c+d x+e x^2+f x^3+g x^4+h x^5}{6 b \left (a+b x^3\right )^2}+\frac {\int \frac {d+2 e x+3 f x^2+4 g x^3+5 h x^4}{\left (a+b x^3\right )^2} \, dx}{6 b}\\ &=\frac {x \left (b d-4 a g+(2 b e-5 a h) x+3 b f x^2\right )}{18 a b^2 \left (a+b x^3\right )}-\frac {c+d x+e x^2+f x^3+g x^4+h x^5}{6 b \left (a+b x^3\right )^2}-\frac {\int \frac {-2 b (b d+2 a g)-2 b (b e+5 a h) x}{a+b x^3} \, dx}{18 a b^3}\\ &=\frac {x \left (b d-4 a g+(2 b e-5 a h) x+3 b f x^2\right )}{18 a b^2 \left (a+b x^3\right )}-\frac {c+d x+e x^2+f x^3+g x^4+h x^5}{6 b \left (a+b x^3\right )^2}-\frac {\int \frac {\sqrt [3]{a} \left (-4 b^{4/3} (b d+2 a g)-2 \sqrt [3]{a} b (b e+5 a h)\right )+\sqrt [3]{b} \left (2 b^{4/3} (b d+2 a g)-2 \sqrt [3]{a} b (b e+5 a h)\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{54 a^{5/3} b^{10/3}}+\frac {\left (\sqrt [3]{b} (b d+2 a g)-\sqrt [3]{a} (b e+5 a h)\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{5/3} b^{7/3}}\\ &=\frac {x \left (b d-4 a g+(2 b e-5 a h) x+3 b f x^2\right )}{18 a b^2 \left (a+b x^3\right )}-\frac {c+d x+e x^2+f x^3+g x^4+h x^5}{6 b \left (a+b x^3\right )^2}+\frac {\left (\sqrt [3]{b} (b d+2 a g)-\sqrt [3]{a} (b e+5 a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{5/3} b^{8/3}}+\frac {\left (b^{4/3} d+\sqrt [3]{a} b e+2 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}{18 a^{4/3} b^{7/3}}-\frac {\left (\sqrt [3]{b} (b d+2 a g)-\sqrt [3]{a} (b e+5 a h)\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}{54 a^{5/3} b^{8/3}}\\ &=\frac {x \left (b d-4 a g+(2 b e-5 a h) x+3 b f x^2\right )}{18 a b^2 \left (a+b x^3\right )}-\frac {c+d x+e x^2+f x^3+g x^4+h x^5}{6 b \left (a+b x^3\right )^2}+\frac {\left (\sqrt [3]{b} (b d+2 a g)-\sqrt [3]{a} (b e+5 a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{5/3} b^{8/3}}-\frac {\left (\sqrt [3]{b} (b d+2 a g)-\sqrt [3]{a} (b e+5 a h)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{5/3} b^{8/3}}+\frac {\left (b^{4/3} d+\sqrt [3]{a} b e+2 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 )}{9 a^{5/3} b^{8/3}}\\ &=\frac {x \left (b d-4 a g+(2 b e-5 a h) x+3 b f x^2\right )}{18 a b^2 \left (a+b x^3\right )}-\frac {c+d x+e x^2+f x^3+g x^4+h x^5}{6 b \left (a+b x^3\right )^2}-\frac {\left (b^{4/3} d+\sqrt [3]{a} b e+2 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 )}{9 \sqrt {3} a^{5/3} b^{8/3}}+\frac {\left (\sqrt [3]{b} (b d+2 a g)-\sqrt [3]{a} (b e+5 a h)\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{5/3} b^{8/3}}-\frac {\left (\sqrt [3]{b} (b d+2 a g)-\sqrt [3]{a} (b e+5 a h)\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{5/3} b^{8/3}}\\ \end {align*}

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Mathematica [A]
time = 0.16, size = 287, normalized size = 0.97 \begin {gather*} \frac {-\frac {9 b^{2/3} (b (c+x (d+e x))-a (f+x (g+h x)))}{\left (a+b x^3\right )^2}+\frac {3 b^{2/3} (b x (d+2 e x)-a (6 f+x (7 g+8 h x)))}{a \left (a+b x^3\right )}-\frac {2 \sqrt {3} \left (b^{4/3} d+\sqrt [3]{a} b e+2 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^{5/3}}+\frac {2 \left (b^{4/3} d-\sqrt [3]{a} b e+2 a \sqrt [3]{b} g-5 a^{4/3} h\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{5/3}}+\frac {\left (-b^{4/3} d+\sqrt [3]{a} b e-2 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^{5/3}}}{54 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)^3,x]

[Out]

((-9*b^(2/3)*(b*(c + x*(d + e*x)) - a*(f + x*(g + h*x))))/(a + b*x^3)^2 + (3*b^(2/3)*(b*x*(d + 2*e*x) - a*(6*f
+ x*(7*g + 8*h*x))))/(a*(a + b*x^3)) - (2*Sqrt[3]*(b^(4/3)*d + a^(1/3)*b*e + 2*a*b^(1/3)*g + 5*a^(4/3)*h)*Arc
Tan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/a^(5/3) + (2*(b^(4/3)*d - a^(1/3)*b*e + 2*a*b^(1/3)*g - 5*a^(4/3)*h)
*Log[a^(1/3) + b^(1/3)*x])/a^(5/3) + ((-(b^(4/3)*d) + a^(1/3)*b*e - 2*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^(5/3))/(54*b^(8/3))

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Maple [A]
time = 0.40, size = 311, normalized size = 1.05

method	result	size
risch	\(\frac {-\frac {\left (4 a h -b e \right ) x^{5}}{9 a b}-\frac {\left (7 a g -b d \right ) x^{4}}{18 a b}-\frac {f \,x^{3}}{3 b}-\frac {\left (5 a h +b e \right ) x^{2}}{18 b^{2}}-\frac {\left (2 a g +b d \right ) x}{9 b^{2}}-\frac {a f +b c}{6 b^{2}}}{\left (b \,x^{3}+a \right )^{2}}+\frac {\munderset {\textit {\_R} =\RootOf \left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\left (\left (5 a h +b e \right ) \textit {\_R} +2 a g +b d \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{27 a \,b^{3}}\)	\(151\)
default	\(\frac {-\frac {\left (4 a h -b e \right ) x^{5}}{9 a b}-\frac {\left (7 a g -b d \right ) x^{4}}{18 a b}-\frac {f \,x^{3}}{3 b}-\frac {\left (5 a h +b e \right ) x^{2}}{18 b^{2}}-\frac {\left (2 a g +b d \right ) x}{9 b^{2}}-\frac {a f +b c}{6 b^{2}}}{\left (b \,x^{3}+a \right )^{2}}+\frac {\left (2 a g +b d \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )+\left (5 a h +b e \right ) \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 a \,b^{2}}\)	\(311\)

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)^3,x,method=_RETURNVERBOSE)

[Out]

(-1/9*(4*a*h-b*e)/a/b*x^5-1/18*(7*a*g-b*d)/a/b*x^4-1/3*f*x^3/b-1/18*(5*a*h+b*e)/b^2*x^2-1/9*(2*a*g+b*d)/b^2*x-
1/6*(a*f+b*c)/b^2)/(b*x^3+a)^2+1/9/a/b^2*((2*a*g+b*d)*(1/3/b/(a/b)^(2/3)*ln(x+(a/b)^(1/3))-1/6/b/(a/b)^(2/3)*l
n(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)))+(5*a*h+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))))

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Maxima [A]
time = 0.50, size = 314, normalized size = 1.06 \begin {gather*} -\frac {6 \, a b f x^{3} + 2 \, {\left (4 \, a b h - b^{2} e\right )} x^{5} - {\left (b^{2} d - 7 \, a b g\right )} x^{4} + 3 \, a b c + 3 \, a^{2} f + {\left (5 \, a^{2} h + a b e\right )} x^{2} + 2 \, {\left (a b d + 2 \, a^{2} g\right )} x}{18 \, {\left (a b^{4} x^{6} + 2 \, a^{2} b^{3} x^{3} + a^{3} b^{2}\right )}} + \frac {\sqrt {3} {\left (5 \, a h \left (\frac {a}{b}\right )^{\frac {1}{3}} + b \left (\frac {a}{b}\right )^{\frac {1}{3}} e + b d + 2 \, a g\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 )}{27 \, a b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (5 \, a h \left (\frac {a}{b}\right )^{\frac {1}{3}} + b \left (\frac {a}{b}\right )^{\frac {1}{3}} e - b d - 2 \, 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 )}{54 \, a b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (5 \, a h \left (\frac {a}{b}\right )^{\frac {1}{3}} + b \left (\frac {a}{b}\right )^{\frac {1}{3}} e - b d - 2 \, a g\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a 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)^3,x, algorithm="maxima")

[Out]

-1/18*(6*a*b*f*x^3 + 2*(4*a*b*h - b^2*e)*x^5 - (b^2*d - 7*a*b*g)*x^4 + 3*a*b*c + 3*a^2*f + (5*a^2*h + a*b*e)*x
^2 + 2*(a*b*d + 2*a^2*g)*x)/(a*b^4*x^6 + 2*a^2*b^3*x^3 + a^3*b^2) + 1/27*sqrt(3)*(5*a*h*(a/b)^(1/3) + b*(a/b)^
(1/3)*e + b*d + 2*a*g)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(a*b^3*(a/b)^(2/3)) + 1/54*(5*a*h*(
a/b)^(1/3) + b*(a/b)^(1/3)*e - b*d - 2*a*g)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a*b^3*(a/b)^(2/3)) - 1/27*
(5*a*h*(a/b)^(1/3) + b*(a/b)^(1/3)*e - b*d - 2*a*g)*log(x + (a/b)^(1/3))/(a*b^3*(a/b)^(2/3))

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Fricas [C] Result contains complex when optimal does not.
time = 1.93, size = 6926, normalized size = 23.32 \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)^3,x, algorithm="fricas")

[Out]

-1/108*(36*a*b*f*x^3 - 12*(b^2*e - 4*a*b*h)*x^5 - 6*(b^2*d - 7*a*b*g)*x^4 + 18*a*b*c + 18*a^2*f + 6*(a*b*e + 5
*a^2*h)*x^2 + 2*(a*b^4*x^6 + 2*a^2*b^3*x^3 + a^3*b^2)*((1/2)^(1/3)*(I*sqrt(3) + 1)*((b^4*d^3 + a*b^3*e^3 + 6*a
*b^3*d^2*g + 12*a^2*b^2*d*g^2 + 8*a^3*b*g^3 + 15*a^2*b^2*e^2*h + 75*a^3*b*e*h^2 + 125*a^4*h^3)/(a^5*b^8) + (b^
4*d^3 - 125*a^4*h^3 + (8*g^3 - 75*e*h^2)*a^3*b + 3*(4*d*g^2 - 5*e^2*h)*a^2*b^2 - (e^3 - 6*d^2*g)*a*b^3)/(a^5*b
^8))^(1/3) - 2*(1/2)^(2/3)*(b^2*d*e + 10*a^2*g*h + (2*e*g + 5*d*h)*a*b)*(-I*sqrt(3) + 1)/(a^3*b^5*((b^4*d^3 +
a*b^3*e^3 + 6*a*b^3*d^2*g + 12*a^2*b^2*d*g^2 + 8*a^3*b*g^3 + 15*a^2*b^2*e^2*h + 75*a^3*b*e*h^2 + 125*a^4*h^3)/
(a^5*b^8) + (b^4*d^3 - 125*a^4*h^3 + (8*g^3 - 75*e*h^2)*a^3*b + 3*(4*d*g^2 - 5*e^2*h)*a^2*b^2 - (e^3 - 6*d^2*g
)*a*b^3)/(a^5*b^8))^(1/3)))*log(2*a*b^3*d*e^2 + 4*a^2*b^2*e^2*g + 1/4*(a^4*b^6*e + 5*a^5*b^5*h)*((1/2)^(1/3)*(
I*sqrt(3) + 1)*((b^4*d^3 + a*b^3*e^3 + 6*a*b^3*d^2*g + 12*a^2*b^2*d*g^2 + 8*a^3*b*g^3 + 15*a^2*b^2*e^2*h + 75*
a^3*b*e*h^2 + 125*a^4*h^3)/(a^5*b^8) + (b^4*d^3 - 125*a^4*h^3 + (8*g^3 - 75*e*h^2)*a^3*b + 3*(4*d*g^2 - 5*e^2*
h)*a^2*b^2 - (e^3 - 6*d^2*g)*a*b^3)/(a^5*b^8))^(1/3) - 2*(1/2)^(2/3)*(b^2*d*e + 10*a^2*g*h + (2*e*g + 5*d*h)*a
*b)*(-I*sqrt(3) + 1)/(a^3*b^5*((b^4*d^3 + a*b^3*e^3 + 6*a*b^3*d^2*g + 12*a^2*b^2*d*g^2 + 8*a^3*b*g^3 + 15*a^2*
b^2*e^2*h + 75*a^3*b*e*h^2 + 125*a^4*h^3)/(a^5*b^8) + (b^4*d^3 - 125*a^4*h^3 + (8*g^3 - 75*e*h^2)*a^3*b + 3*(4
*d*g^2 - 5*e^2*h)*a^2*b^2 - (e^3 - 6*d^2*g)*a*b^3)/(a^5*b^8))^(1/3)))^2 + 50*(a^3*b*d + 2*a^4*g)*h^2 - 1/2*(a^
2*b^5*d^2 + 4*a^3*b^4*d*g + 4*a^4*b^3*g^2)*((1/2)^(1/3)*(I*sqrt(3) + 1)*((b^4*d^3 + a*b^3*e^3 + 6*a*b^3*d^2*g
+ 12*a^2*b^2*d*g^2 + 8*a^3*b*g^3 + 15*a^2*b^2*e^2*h + 75*a^3*b*e*h^2 + 125*a^4*h^3)/(a^5*b^8) + (b^4*d^3 - 125
*a^4*h^3 + (8*g^3 - 75*e*h^2)*a^3*b + 3*(4*d*g^2 - 5*e^2*h)*a^2*b^2 - (e^3 - 6*d^2*g)*a*b^3)/(a^5*b^8))^(1/3)
- 2*(1/2)^(2/3)*(b^2*d*e + 10*a^2*g*h + (2*e*g + 5*d*h)*a*b)*(-I*sqrt(3) + 1)/(a^3*b^5*((b^4*d^3 + a*b^3*e^3 +
6*a*b^3*d^2*g + 12*a^2*b^2*d*g^2 + 8*a^3*b*g^3 + 15*a^2*b^2*e^2*h + 75*a^3*b*e*h^2 + 125*a^4*h^3)/(a^5*b^8) +
(b^4*d^3 - 125*a^4*h^3 + (8*g^3 - 75*e*h^2)*a^3*b + 3*(4*d*g^2 - 5*e^2*h)*a^2*b^2 - (e^3 - 6*d^2*g)*a*b^3)/(a
^5*b^8))^(1/3))) + 20*(a^2*b^2*d*e + 2*a^3*b*e*g)*h + (b^4*d^3 + a*b^3*e^3 + 6*a*b^3*d^2*g + 12*a^2*b^2*d*g^2
+ 8*a^3*b*g^3 + 15*a^2*b^2*e^2*h + 75*a^3*b*e*h^2 + 125*a^4*h^3)*x) + 12*(a*b*d + 2*a^2*g)*x - ((a*b^4*x^6 + 2
*a^2*b^3*x^3 + a^3*b^2)*((1/2)^(1/3)*(I*sqrt(3) + 1)*((b^4*d^3 + a*b^3*e^3 + 6*a*b^3*d^2*g + 12*a^2*b^2*d*g^2
+ 8*a^3*b*g^3 + 15*a^2*b^2*e^2*h + 75*a^3*b*e*h^2 + 125*a^4*h^3)/(a^5*b^8) + (b^4*d^3 - 125*a^4*h^3 + (8*g^3 -
75*e*h^2)*a^3*b + 3*(4*d*g^2 - 5*e^2*h)*a^2*b^2 - (e^3 - 6*d^2*g)*a*b^3)/(a^5*b^8))^(1/3) - 2*(1/2)^(2/3)*(b^
2*d*e + 10*a^2*g*h + (2*e*g + 5*d*h)*a*b)*(-I*sqrt(3) + 1)/(a^3*b^5*((b^4*d^3 + a*b^3*e^3 + 6*a*b^3*d^2*g + 12
*a^2*b^2*d*g^2 + 8*a^3*b*g^3 + 15*a^2*b^2*e^2*h + 75*a^3*b*e*h^2 + 125*a^4*h^3)/(a^5*b^8) + (b^4*d^3 - 125*a^4
*h^3 + (8*g^3 - 75*e*h^2)*a^3*b + 3*(4*d*g^2 - 5*e^2*h)*a^2*b^2 - (e^3 - 6*d^2*g)*a*b^3)/(a^5*b^8))^(1/3))) +
3*sqrt(1/3)*(a*b^4*x^6 + 2*a^2*b^3*x^3 + a^3*b^2)*sqrt(-(((1/2)^(1/3)*(I*sqrt(3) + 1)*((b^4*d^3 + a*b^3*e^3 +
6*a*b^3*d^2*g + 12*a^2*b^2*d*g^2 + 8*a^3*b*g^3 + 15*a^2*b^2*e^2*h + 75*a^3*b*e*h^2 + 125*a^4*h^3)/(a^5*b^8) +
(b^4*d^3 - 125*a^4*h^3 + (8*g^3 - 75*e*h^2)*a^3*b + 3*(4*d*g^2 - 5*e^2*h)*a^2*b^2 - (e^3 - 6*d^2*g)*a*b^3)/(a^
5*b^8))^(1/3) - 2*(1/2)^(2/3)*(b^2*d*e + 10*a^2*g*h + (2*e*g + 5*d*h)*a*b)*(-I*sqrt(3) + 1)/(a^3*b^5*((b^4*d^3
+ a*b^3*e^3 + 6*a*b^3*d^2*g + 12*a^2*b^2*d*g^2 + 8*a^3*b*g^3 + 15*a^2*b^2*e^2*h + 75*a^3*b*e*h^2 + 125*a^4*h^
3)/(a^5*b^8) + (b^4*d^3 - 125*a^4*h^3 + (8*g^3 - 75*e*h^2)*a^3*b + 3*(4*d*g^2 - 5*e^2*h)*a^2*b^2 - (e^3 - 6*d^
2*g)*a*b^3)/(a^5*b^8))^(1/3)))^2*a^3*b^5 + 16*b^2*d*e + 32*a*b*e*g + 80*(a*b*d + 2*a^2*g)*h)/(a^3*b^5)))*log(-
2*a*b^3*d*e^2 - 4*a^2*b^2*e^2*g - 1/4*(a^4*b^6*e + 5*a^5*b^5*h)*((1/2)^(1/3)*(I*sqrt(3) + 1)*((b^4*d^3 + a*b^3
*e^3 + 6*a*b^3*d^2*g + 12*a^2*b^2*d*g^2 + 8*a^3*b*g^3 + 15*a^2*b^2*e^2*h + 75*a^3*b*e*h^2 + 125*a^4*h^3)/(a^5*
b^8) + (b^4*d^3 - 125*a^4*h^3 + (8*g^3 - 75*e*h^2)*a^3*b + 3*(4*d*g^2 - 5*e^2*h)*a^2*b^2 - (e^3 - 6*d^2*g)*a*b
^3)/(a^5*b^8))^(1/3) - 2*(1/2)^(2/3)*(b^2*d*e + 10*a^2*g*h + (2*e*g + 5*d*h)*a*b)*(-I*sqrt(3) + 1)/(a^3*b^5*((
b^4*d^3 + a*b^3*e^3 + 6*a*b^3*d^2*g + 12*a^2*b^2*d*g^2 + 8*a^3*b*g^3 + 15*a^2*b^2*e^2*h + 75*a^3*b*e*h^2 + 125
*a^4*h^3)/(a^5*b^8) + (b^4*d^3 - 125*a^4*h^3 + (8*g^3 - 75*e*h^2)*a^3*b + 3*(4*d*g^2 - 5*e^2*h)*a^2*b^2 - (e^3
- 6*d^2*g)*a*b^3)/(a^5*b^8))^(1/3)))^2 - 50*(a^3*b*d + 2*a^4*g)*h^2 + 1/2*(a^2*b^5*d^2 + 4*a^3*b^4*d*g + 4*a^
4*b^3*g^2)*((1/2)^(1/3)*(I*sqrt(3) + 1)*((b^4*d^3 + a*b^3*e^3 + 6*a*b^3*d^2*g + 12*a^2*b^2*d*g^2 + 8*a^3*b*g^3
+ 15*a^2*b^2*e^2*h + 75*a^3*b*e*h^2 + 125*a^4*h^3)/(a^5*b^8) + (b^4*d^3 - 125*a^4*h^3 + (8*g^3 - 75*e*h^2)*a^
3*b + 3*(4*d*g^2 - 5*e^2*h)*a^2*b^2 - (e^3 - 6*d^2*g)*a*b^3)/(a^5*b^8))^(1/3) - 2*(1/2)^(2/3)*(b^2*d*e + 10*a^
2*g*h + (2*e*g + 5*d*h)*a*b)*(-I*sqrt(3) + 1)/(...

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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)**3,x)

[Out]

Timed out

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Giac [A]
time = 0.48, size = 320, normalized size = 1.08 \begin {gather*} -\frac {\sqrt {3} {\left (b^{2} d + 2 \, a b g - 5 \, \left (-a b^{2}\right )^{\frac {1}{3}} a h - \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 )}{27 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b^{2}} - \frac {{\left (b^{2} d + 2 \, a b g + 5 \, \left (-a b^{2}\right )^{\frac {1}{3}} a h + \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 )}{54 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b^{2}} - \frac {{\left (5 \, a h \left (-\frac {a}{b}\right )^{\frac {1}{3}} + b \left (-\frac {a}{b}\right )^{\frac {1}{3}} e + b d + 2 \, a g\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{2} b^{2}} - \frac {8 \, a b h x^{5} - 2 \, b^{2} x^{5} e - b^{2} d x^{4} + 7 \, a b g x^{4} + 6 \, a b f x^{3} + 5 \, a^{2} h x^{2} + a b x^{2} e + 2 \, a b d x + 4 \, a^{2} g x + 3 \, a b c + 3 \, a^{2} f}{18 \, {\left (b x^{3} + a\right )}^{2} a b^{2}} \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)^3,x, algorithm="giac")

[Out]

-1/27*sqrt(3)*(b^2*d + 2*a*b*g - 5*(-a*b^2)^(1/3)*a*h - (-a*b^2)^(1/3)*b*e)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(
1/3))/(-a/b)^(1/3))/((-a*b^2)^(2/3)*a*b^2) - 1/54*(b^2*d + 2*a*b*g + 5*(-a*b^2)^(1/3)*a*h + (-a*b^2)^(1/3)*b*e
)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/((-a*b^2)^(2/3)*a*b^2) - 1/27*(5*a*h*(-a/b)^(1/3) + b*(-a/b)^(1/3)*
e + b*d + 2*a*g)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^2*b^2) - 1/18*(8*a*b*h*x^5 - 2*b^2*x^5*e - b^2*d*x
^4 + 7*a*b*g*x^4 + 6*a*b*f*x^3 + 5*a^2*h*x^2 + a*b*x^2*e + 2*a*b*d*x + 4*a^2*g*x + 3*a*b*c + 3*a^2*f)/((b*x^3
+ a)^2*a*b^2)

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Mupad [B]
time = 5.69, size = 627, normalized size = 2.11 \begin {gather*} \left (\sum _{k=1}^3\ln \left (\mathrm {root}\left (19683\,a^5\,b^8\,z^3+810\,a^4\,b^3\,g\,h\,z+405\,a^3\,b^4\,d\,h\,z+162\,a^3\,b^4\,e\,g\,z+81\,a^2\,b^5\,d\,e\,z+75\,a^3\,b\,e\,h^2-6\,a\,b^3\,d^2\,g+15\,a^2\,b^2\,e^2\,h-12\,a^2\,b^2\,d\,g^2-8\,a^3\,b\,g^3+a\,b^3\,e^3+125\,a^4\,h^3-b^4\,d^3,z,k\right )\,\left (\mathrm {root}\left (19683\,a^5\,b^8\,z^3+810\,a^4\,b^3\,g\,h\,z+405\,a^3\,b^4\,d\,h\,z+162\,a^3\,b^4\,e\,g\,z+81\,a^2\,b^5\,d\,e\,z+75\,a^3\,b\,e\,h^2-6\,a\,b^3\,d^2\,g+15\,a^2\,b^2\,e^2\,h-12\,a^2\,b^2\,d\,g^2-8\,a^3\,b\,g^3+a\,b^3\,e^3+125\,a^4\,h^3-b^4\,d^3,z,k\right )\,a\,b^2\,9+\frac {x\,\left (54\,g\,a^2\,b^3+27\,d\,a\,b^4\right )}{81\,a^2\,b^3}\right )+\frac {b^2\,d\,e+10\,a^2\,g\,h+5\,a\,b\,d\,h+2\,a\,b\,e\,g}{81\,a^2\,b^3}+\frac {x\,\left (25\,a^2\,h^2+10\,a\,b\,e\,h+b^2\,e^2\right )}{81\,a^2\,b^3}\right )\,\mathrm {root}\left (19683\,a^5\,b^8\,z^3+810\,a^4\,b^3\,g\,h\,z+405\,a^3\,b^4\,d\,h\,z+162\,a^3\,b^4\,e\,g\,z+81\,a^2\,b^5\,d\,e\,z+75\,a^3\,b\,e\,h^2-6\,a\,b^3\,d^2\,g+15\,a^2\,b^2\,e^2\,h-12\,a^2\,b^2\,d\,g^2-8\,a^3\,b\,g^3+a\,b^3\,e^3+125\,a^4\,h^3-b^4\,d^3,z,k\right )\right )-\frac {\frac {b\,c+a\,f}{6\,b^2}+\frac {x\,\left (b\,d+2\,a\,g\right )}{9\,b^2}+\frac {f\,x^3}{3\,b}+\frac {x^2\,\left (b\,e+5\,a\,h\right )}{18\,b^2}-\frac {x^4\,\left (b\,d-7\,a\,g\right )}{18\,a\,b}-\frac {x^5\,\left (b\,e-4\,a\,h\right )}{9\,a\,b}}{a^2+2\,a\,b\,x^3+b^2\,x^6} \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)^3,x)

[Out]

symsum(log(root(19683*a^5*b^8*z^3 + 810*a^4*b^3*g*h*z + 405*a^3*b^4*d*h*z + 162*a^3*b^4*e*g*z + 81*a^2*b^5*d*e
*z + 75*a^3*b*e*h^2 - 6*a*b^3*d^2*g + 15*a^2*b^2*e^2*h - 12*a^2*b^2*d*g^2 - 8*a^3*b*g^3 + a*b^3*e^3 + 125*a^4*
h^3 - b^4*d^3, z, k)*(9*root(19683*a^5*b^8*z^3 + 810*a^4*b^3*g*h*z + 405*a^3*b^4*d*h*z + 162*a^3*b^4*e*g*z + 8
1*a^2*b^5*d*e*z + 75*a^3*b*e*h^2 - 6*a*b^3*d^2*g + 15*a^2*b^2*e^2*h - 12*a^2*b^2*d*g^2 - 8*a^3*b*g^3 + a*b^3*e
^3 + 125*a^4*h^3 - b^4*d^3, z, k)*a*b^2 + (x*(54*a^2*b^3*g + 27*a*b^4*d))/(81*a^2*b^3)) + (b^2*d*e + 10*a^2*g*
h + 5*a*b*d*h + 2*a*b*e*g)/(81*a^2*b^3) + (x*(b^2*e^2 + 25*a^2*h^2 + 10*a*b*e*h))/(81*a^2*b^3))*root(19683*a^5
*b^8*z^3 + 810*a^4*b^3*g*h*z + 405*a^3*b^4*d*h*z + 162*a^3*b^4*e*g*z + 81*a^2*b^5*d*e*z + 75*a^3*b*e*h^2 - 6*a
*b^3*d^2*g + 15*a^2*b^2*e^2*h - 12*a^2*b^2*d*g^2 - 8*a^3*b*g^3 + a*b^3*e^3 + 125*a^4*h^3 - b^4*d^3, z, k), k,
1, 3) - ((b*c + a*f)/(6*b^2) + (x*(b*d + 2*a*g))/(9*b^2) + (f*x^3)/(3*b) + (x^2*(b*e + 5*a*h))/(18*b^2) - (x^4
*(b*d - 7*a*g))/(18*a*b) - (x^5*(b*e - 4*a*h))/(9*a*b))/(a^2 + b^2*x^6 + 2*a*b*x^3)

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