3.1.7 \(\int \frac {a+b x^3+c x^6}{(d+e x^3)^2} \, dx\) [7]

Optimal. Leaf size=213 \[ \frac {c x}{e^2}+\frac {\left (c d^2-b d e+a e^2\right ) x}{3 d e^2 \left (d+e x^3\right )}+\frac {\left (4 c d^2-e (b d+2 a e)\right ) \tan ^{-1}\left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right )}{3 \sqrt {3} d^{5/3} e^{7/3}}-\frac {\left (4 c d^2-e (b d+2 a e)\right ) \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{9 d^{5/3} e^{7/3}}+\frac {\left (4 c d^2-e (b d+2 a e)\right ) \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{18 d^{5/3} e^{7/3}} \]

[Out]

c*x/e^2+1/3*(a*e^2-b*d*e+c*d^2)*x/d/e^2/(e*x^3+d)-1/9*(4*c*d^2-e*(2*a*e+b*d))*ln(d^(1/3)+e^(1/3)*x)/d^(5/3)/e^
(7/3)+1/18*(4*c*d^2-e*(2*a*e+b*d))*ln(d^(2/3)-d^(1/3)*e^(1/3)*x+e^(2/3)*x^2)/d^(5/3)/e^(7/3)+1/9*(4*c*d^2-e*(2
*a*e+b*d))*arctan(1/3*(d^(1/3)-2*e^(1/3)*x)/d^(1/3)*3^(1/2))/d^(5/3)/e^(7/3)*3^(1/2)

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Rubi [A]
time = 0.15, antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {1423, 396, 206, 31, 648, 631, 210, 642} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right ) \left (4 c d^2-e (2 a e+b d)\right )}{3 \sqrt {3} d^{5/3} e^{7/3}}+\frac {x \left (a e^2-b d e+c d^2\right )}{3 d e^2 \left (d+e x^3\right )}+\frac {\log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right ) \left (4 c d^2-e (2 a e+b d)\right )}{18 d^{5/3} e^{7/3}}-\frac {\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \left (4 c d^2-e (2 a e+b d)\right )}{9 d^{5/3} e^{7/3}}+\frac {c x}{e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 31

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

Rule 206

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 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 396

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[d*x*((a + b*x^n)^(p + 1)/(b*(n*(
p + 1) + 1))), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 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 1423

Int[((d_) + (e_.)*(x_)^(n_))^(q_)*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x_Symbol] :> Simp[(-(c*d^2 - b*
d*e + a*e^2))*x*((d + e*x^n)^(q + 1)/(d*e^2*n*(q + 1))), x] + Dist[1/(n*(q + 1)*d*e^2), Int[(d + e*x^n)^(q + 1
)*Simp[c*d^2 - b*d*e + a*e^2*(n*(q + 1) + 1) + c*d*e*n*(q + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, n}, x]
&& EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[q, -1]

Rubi steps

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

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Mathematica [A]
time = 0.14, size = 199, normalized size = 0.93 \begin {gather*} \frac {18 c \sqrt [3]{e} x+\frac {6 \sqrt [3]{e} \left (c d^2+e (-b d+a e)\right ) x}{d \left (d+e x^3\right )}+\frac {2 \sqrt {3} \left (4 c d^2-e (b d+2 a e)\right ) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{e} x}{\sqrt [3]{d}}}{\sqrt {3}}\right )}{d^{5/3}}-\frac {2 \left (4 c d^2-e (b d+2 a e)\right ) \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{d^{5/3}}+\frac {\left (4 c d^2-e (b d+2 a e)\right ) \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{d^{5/3}}}{18 e^{7/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(18*c*e^(1/3)*x + (6*e^(1/3)*(c*d^2 + e*(-(b*d) + a*e))*x)/(d*(d + e*x^3)) + (2*Sqrt[3]*(4*c*d^2 - e*(b*d + 2*
a*e))*ArcTan[(1 - (2*e^(1/3)*x)/d^(1/3))/Sqrt[3]])/d^(5/3) - (2*(4*c*d^2 - e*(b*d + 2*a*e))*Log[d^(1/3) + e^(1
/3)*x])/d^(5/3) + ((4*c*d^2 - e*(b*d + 2*a*e))*Log[d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^2])/d^(5/3))/(18*e^
(7/3))

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Maple [A]
time = 0.24, size = 156, normalized size = 0.73

method	result	size
risch	\(\frac {c x}{e^{2}}+\frac {\left (a \,e^{2}-d e b +c \,d^{2}\right ) x}{3 d \,e^{2} \left (e \,x^{3}+d \right )}+\frac {\munderset {\textit {\_R} =\RootOf \left (e \,\textit {\_Z}^{3}+d \right )}{\sum }\frac {\left (2 a \,e^{2}+d e b -4 c \,d^{2}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{9 e^{3} d}\)	\(88\)
default	\(\frac {c x}{e^{2}}+\frac {\frac {\left (a \,e^{2}-d e b +c \,d^{2}\right ) x}{3 d \left (e \,x^{3}+d \right )}+\frac {\left (2 a \,e^{2}+d e b -4 c \,d^{2}\right ) \left (\frac {\ln \left (x +\left (\frac {d}{e}\right )^{\frac {1}{3}}\right )}{3 e \left (\frac {d}{e}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {d}{e}\right )^{\frac {1}{3}} x +\left (\frac {d}{e}\right )^{\frac {2}{3}}\right )}{6 e \left (\frac {d}{e}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {d}{e}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 e \left (\frac {d}{e}\right )^{\frac {2}{3}}}\right )}{3 d}}{e^{2}}\)	\(156\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^6+b*x^3+a)/(e*x^3+d)^2,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.50, size = 168, normalized size = 0.79 \begin {gather*} c x e^{\left (-2\right )} - \frac {\sqrt {3} {\left (4 \, c d^{2} - b d e - 2 \, a e^{2}\right )} \arctan \left (-\frac {\sqrt {3} {\left (d^{\frac {1}{3}} e^{\left (-\frac {1}{3}\right )} - 2 \, x\right )} e^{\frac {1}{3}}}{3 \, d^{\frac {1}{3}}}\right ) e^{\left (-\frac {7}{3}\right )}}{9 \, d^{\frac {5}{3}}} + \frac {{\left (4 \, c d^{2} - b d e - 2 \, a e^{2}\right )} e^{\left (-\frac {7}{3}\right )} \log \left (-d^{\frac {1}{3}} x e^{\left (-\frac {1}{3}\right )} + x^{2} + d^{\frac {2}{3}} e^{\left (-\frac {2}{3}\right )}\right )}{18 \, d^{\frac {5}{3}}} - \frac {{\left (4 \, c d^{2} - b d e - 2 \, a e^{2}\right )} e^{\left (-\frac {7}{3}\right )} \log \left (d^{\frac {1}{3}} e^{\left (-\frac {1}{3}\right )} + x\right )}{9 \, d^{\frac {5}{3}}} + \frac {{\left (c d^{2} - b d e + a e^{2}\right )} x}{3 \, {\left (d x^{3} e^{3} + d^{2} e^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*x*e^(-2) - 1/9*sqrt(3)*(4*c*d^2 - b*d*e - 2*a*e^2)*arctan(-1/3*sqrt(3)*(d^(1/3)*e^(-1/3) - 2*x)*e^(1/3)/d^(1
/3))*e^(-7/3)/d^(5/3) + 1/18*(4*c*d^2 - b*d*e - 2*a*e^2)*e^(-7/3)*log(-d^(1/3)*x*e^(-1/3) + x^2 + d^(2/3)*e^(-
2/3))/d^(5/3) - 1/9*(4*c*d^2 - b*d*e - 2*a*e^2)*e^(-7/3)*log(d^(1/3)*e^(-1/3) + x)/d^(5/3) + 1/3*(c*d^2 - b*d*
e + a*e^2)*x/(d*x^3*e^3 + d^2*e^2)

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Fricas [A]
time = 0.36, size = 323, normalized size = 1.52 \begin {gather*} \frac {24 \, c d^{4} x e + 6 \, a d^{2} x e^{3} + 6 \, \sqrt {\frac {1}{3}} {\left (2 \, a d x^{3} e^{4} - 4 \, c d^{4} e + {\left (b d^{2} x^{3} + 2 \, a d^{2}\right )} e^{3} - {\left (4 \, c d^{3} x^{3} - b d^{3}\right )} e^{2}\right )} {\left (d^{2}\right )}^{\frac {1}{6}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, {\left (d^{2}\right )}^{\frac {2}{3}} x e^{\frac {2}{3}} - {\left (d^{2}\right )}^{\frac {1}{3}} d e^{\frac {1}{3}}\right )} {\left (d^{2}\right )}^{\frac {1}{6}} e^{\left (-\frac {1}{3}\right )}}{d^{2}}\right ) e^{\left (-\frac {1}{3}\right )} - {\left (2 \, a x^{3} e^{3} - 4 \, c d^{3} + {\left (b d x^{3} + 2 \, a d\right )} e^{2} - {\left (4 \, c d^{2} x^{3} - b d^{2}\right )} e\right )} {\left (d^{2}\right )}^{\frac {2}{3}} e^{\frac {2}{3}} \log \left (d x^{2} e - {\left (d^{2}\right )}^{\frac {2}{3}} x e^{\frac {2}{3}} + {\left (d^{2}\right )}^{\frac {1}{3}} d e^{\frac {1}{3}}\right ) + 2 \, {\left (2 \, a x^{3} e^{3} - 4 \, c d^{3} + {\left (b d x^{3} + 2 \, a d\right )} e^{2} - {\left (4 \, c d^{2} x^{3} - b d^{2}\right )} e\right )} {\left (d^{2}\right )}^{\frac {2}{3}} e^{\frac {2}{3}} \log \left (d x e + {\left (d^{2}\right )}^{\frac {2}{3}} e^{\frac {2}{3}}\right ) + 6 \, {\left (3 \, c d^{3} x^{4} - b d^{3} x\right )} e^{2}}{18 \, {\left (d^{3} x^{3} e^{4} + d^{4} e^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18*(24*c*d^4*x*e + 6*a*d^2*x*e^3 + 6*sqrt(1/3)*(2*a*d*x^3*e^4 - 4*c*d^4*e + (b*d^2*x^3 + 2*a*d^2)*e^3 - (4*c
*d^3*x^3 - b*d^3)*e^2)*(d^2)^(1/6)*arctan(sqrt(1/3)*(2*(d^2)^(2/3)*x*e^(2/3) - (d^2)^(1/3)*d*e^(1/3))*(d^2)^(1
/6)*e^(-1/3)/d^2)*e^(-1/3) - (2*a*x^3*e^3 - 4*c*d^3 + (b*d*x^3 + 2*a*d)*e^2 - (4*c*d^2*x^3 - b*d^2)*e)*(d^2)^(
2/3)*e^(2/3)*log(d*x^2*e - (d^2)^(2/3)*x*e^(2/3) + (d^2)^(1/3)*d*e^(1/3)) + 2*(2*a*x^3*e^3 - 4*c*d^3 + (b*d*x^
3 + 2*a*d)*e^2 - (4*c*d^2*x^3 - b*d^2)*e)*(d^2)^(2/3)*e^(2/3)*log(d*x*e + (d^2)^(2/3)*e^(2/3)) + 6*(3*c*d^3*x^
4 - b*d^3*x)*e^2)/(d^3*x^3*e^4 + d^4*e^3)

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Sympy [A]
time = 0.90, size = 206, normalized size = 0.97 \begin {gather*} \frac {c x}{e^{2}} + \frac {x \left (a e^{2} - b d e + c d^{2}\right )}{3 d^{2} e^{2} + 3 d e^{3} x^{3}} + \operatorname {RootSum} {\left (729 t^{3} d^{5} e^{7} - 8 a^{3} e^{6} - 12 a^{2} b d e^{5} + 48 a^{2} c d^{2} e^{4} - 6 a b^{2} d^{2} e^{4} + 48 a b c d^{3} e^{3} - 96 a c^{2} d^{4} e^{2} - b^{3} d^{3} e^{3} + 12 b^{2} c d^{4} e^{2} - 48 b c^{2} d^{5} e + 64 c^{3} d^{6}, \left ( t \mapsto t \log {\left (\frac {9 t d^{2} e^{2}}{2 a e^{2} + b d e - 4 c d^{2}} + x \right )} \right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*x/e**2 + x*(a*e**2 - b*d*e + c*d**2)/(3*d**2*e**2 + 3*d*e**3*x**3) + RootSum(729*_t**3*d**5*e**7 - 8*a**3*e*
*6 - 12*a**2*b*d*e**5 + 48*a**2*c*d**2*e**4 - 6*a*b**2*d**2*e**4 + 48*a*b*c*d**3*e**3 - 96*a*c**2*d**4*e**2 -
b**3*d**3*e**3 + 12*b**2*c*d**4*e**2 - 48*b*c**2*d**5*e + 64*c**3*d**6, Lambda(_t, _t*log(9*_t*d**2*e**2/(2*a*
e**2 + b*d*e - 4*c*d**2) + x)))

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Giac [A]
time = 3.95, size = 199, normalized size = 0.93 \begin {gather*} c x e^{\left (-2\right )} + \frac {\sqrt {3} {\left (4 \, c d^{2} - b d e - 2 \, a e^{2}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-d e^{\left (-1\right )}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-d e^{\left (-1\right )}\right )^{\frac {1}{3}}}\right ) e^{\left (-1\right )}}{9 \, \left (-d e^{2}\right )^{\frac {2}{3}} d} + \frac {{\left (4 \, c d^{2} - b d e - 2 \, a e^{2}\right )} e^{\left (-1\right )} \log \left (x^{2} + \left (-d e^{\left (-1\right )}\right )^{\frac {1}{3}} x + \left (-d e^{\left (-1\right )}\right )^{\frac {2}{3}}\right )}{18 \, \left (-d e^{2}\right )^{\frac {2}{3}} d} + \frac {{\left (4 \, c d^{2} - b d e - 2 \, a e^{2}\right )} \left (-d e^{\left (-1\right )}\right )^{\frac {1}{3}} e^{\left (-2\right )} \log \left ({\left | x - \left (-d e^{\left (-1\right )}\right )^{\frac {1}{3}} \right |}\right )}{9 \, d^{2}} + \frac {{\left (c d^{2} x - b d x e + a x e^{2}\right )} e^{\left (-2\right )}}{3 \, {\left (x^{3} e + d\right )} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*x*e^(-2) + 1/9*sqrt(3)*(4*c*d^2 - b*d*e - 2*a*e^2)*arctan(1/3*sqrt(3)*(2*x + (-d*e^(-1))^(1/3))/(-d*e^(-1))^
(1/3))*e^(-1)/((-d*e^2)^(2/3)*d) + 1/18*(4*c*d^2 - b*d*e - 2*a*e^2)*e^(-1)*log(x^2 + (-d*e^(-1))^(1/3)*x + (-d
*e^(-1))^(2/3))/((-d*e^2)^(2/3)*d) + 1/9*(4*c*d^2 - b*d*e - 2*a*e^2)*(-d*e^(-1))^(1/3)*e^(-2)*log(abs(x - (-d*
e^(-1))^(1/3)))/d^2 + 1/3*(c*d^2*x - b*d*x*e + a*x*e^2)*e^(-2)/((x^3*e + d)*d)

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Mupad [B]
time = 1.80, size = 187, normalized size = 0.88 \begin {gather*} \frac {c\,x}{e^2}+\frac {\ln \left (e^{1/3}\,x+d^{1/3}\right )\,\left (-4\,c\,d^2+b\,d\,e+2\,a\,e^2\right )}{9\,d^{5/3}\,e^{7/3}}+\frac {x\,\left (c\,d^2-b\,d\,e+a\,e^2\right )}{3\,d\,\left (e^3\,x^3+d\,e^2\right )}+\frac {\ln \left (2\,e^{1/3}\,x-d^{1/3}+\sqrt {3}\,d^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-4\,c\,d^2+b\,d\,e+2\,a\,e^2\right )}{9\,d^{5/3}\,e^{7/3}}-\frac {\ln \left (d^{1/3}-2\,e^{1/3}\,x+\sqrt {3}\,d^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-4\,c\,d^2+b\,d\,e+2\,a\,e^2\right )}{9\,d^{5/3}\,e^{7/3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c*x)/e^2 + (log(e^(1/3)*x + d^(1/3))*(2*a*e^2 - 4*c*d^2 + b*d*e))/(9*d^(5/3)*e^(7/3)) + (x*(a*e^2 + c*d^2 - b
*d*e))/(3*d*(d*e^2 + e^3*x^3)) + (log(3^(1/2)*d^(1/3)*1i + 2*e^(1/3)*x - d^(1/3))*((3^(1/2)*1i)/2 - 1/2)*(2*a*
e^2 - 4*c*d^2 + b*d*e))/(9*d^(5/3)*e^(7/3)) - (log(3^(1/2)*d^(1/3)*1i - 2*e^(1/3)*x + d^(1/3))*((3^(1/2)*1i)/2
+ 1/2)*(2*a*e^2 - 4*c*d^2 + b*d*e))/(9*d^(5/3)*e^(7/3))

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