\(\int \frac {c+d x+e x^2}{x (a+b x^3)^{3/2}} \, dx\) [443]

Optimal result

Integrand size = 25, antiderivative size = 579 \[ \int \frac {c+d x+e x^2}{x \left (a+b x^3\right )^{3/2}} \, dx=\frac {2 x \left (a d+a e x-b c x^2\right )}{3 a^2 \sqrt {a+b x^3}}+\frac {2 c \sqrt {a+b x^3}}{3 a^2}-\frac {2 e \sqrt {a+b x^3}}{3 a b^{2/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {2 c \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{3 a^{3/2}}+\frac {\sqrt {2-\sqrt {3}} e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{3^{3/4} a^{2/3} b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {2 \sqrt {2+\sqrt {3}} \left (\sqrt [3]{b} d+\left (1-\sqrt {3}\right ) \sqrt [3]{a} e\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} a b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \]

[Out]

-2/3*c*arctanh((b*x^3+a)^(1/2)/a^(1/2))/a^(3/2)+2/3*x*(-b*c*x^2+a*e*x+a*d)/a^2/(b*x^3+a)^(1/2)+2/3*c*(b*x^3+a)
^(1/2)/a^2-2/3*e*(b*x^3+a)^(1/2)/a/b^(2/3)/(b^(1/3)*x+a^(1/3)*(1+3^(1/2)))+1/3*e*(a^(1/3)+b^(1/3)*x)*EllipticE
((b^(1/3)*x+a^(1/3)*(1-3^(1/2)))/(b^(1/3)*x+a^(1/3)*(1+3^(1/2))),I*3^(1/2)+2*I)*(1/2*6^(1/2)-1/2*2^(1/2))*((a^
(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+3^(1/2)))^2)^(1/2)*3^(1/4)/a^(2/3)/b^(2/3)/(b*x^3+a
)^(1/2)/(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+3^(1/2)))^2)^(1/2)+2/9*(a^(1/3)+b^(1/3)*x)*Elliptic
F((b^(1/3)*x+a^(1/3)*(1-3^(1/2)))/(b^(1/3)*x+a^(1/3)*(1+3^(1/2))),I*3^(1/2)+2*I)*(b^(1/3)*d+a^(1/3)*e*(1-3^(1/
2)))*(1/2*6^(1/2)+1/2*2^(1/2))*((a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/(b^(1/3)*x+a^(1/3)*(1+3^(1/2)))^2)^(1/
2)*3^(3/4)/a/b^(2/3)/(b*x^3+a)^(1/2)/(a^(1/3)*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*x+a^(1/3)*(1+3^(1/2)))^2)^(1/2)

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 579, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {1843, 1846, 272, 65, 214, 1900, 267, 1892, 224, 1891} \[ \int \frac {c+d x+e x^2}{x \left (a+b x^3\right )^{3/2}} \, dx=\frac {2 \sqrt {2+\sqrt {3}} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a} e+\sqrt [3]{b} d\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} a b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {\sqrt {2-\sqrt {3}} e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\arcsin \left (\frac {\sqrt [3]{b} x+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} x+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{3^{3/4} a^{2/3} b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {2 c \text {arctanh}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{3 a^{3/2}}+\frac {2 x \left (a d+a e x-b c x^2\right )}{3 a^2 \sqrt {a+b x^3}}+\frac {2 c \sqrt {a+b x^3}}{3 a^2}-\frac {2 e \sqrt {a+b x^3}}{3 a b^{2/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )} \]

[In]

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

[Out]

(2*x*(a*d + a*e*x - b*c*x^2))/(3*a^2*Sqrt[a + b*x^3]) + (2*c*Sqrt[a + b*x^3])/(3*a^2) - (2*e*Sqrt[a + b*x^3])/
(3*a*b^(2/3)*((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)) - (2*c*ArcTanh[Sqrt[a + b*x^3]/Sqrt[a]])/(3*a^(3/2)) + (Sqrt
[2 - Sqrt[3]]*e*(a^(1/3) + b^(1/3)*x)*Sqrt[(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 + Sqrt[3])*a^(1/3)
+ b^(1/3)*x)^2]*EllipticE[ArcSin[((1 - Sqrt[3])*a^(1/3) + b^(1/3)*x)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)], -7
- 4*Sqrt[3]])/(3^(3/4)*a^(2/3)*b^(2/3)*Sqrt[(a^(1/3)*(a^(1/3) + b^(1/3)*x))/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x
)^2]*Sqrt[a + b*x^3]) + (2*Sqrt[2 + Sqrt[3]]*(b^(1/3)*d + (1 - Sqrt[3])*a^(1/3)*e)*(a^(1/3) + b^(1/3)*x)*Sqrt[
(a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)^2]*EllipticF[ArcSin[((1 - Sqrt
[3])*a^(1/3) + b^(1/3)*x)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)], -7 - 4*Sqrt[3]])/(3*3^(1/4)*a*b^(2/3)*Sqrt[(a^
(1/3)*(a^(1/3) + b^(1/3)*x))/((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)^2]*Sqrt[a + b*x^3])

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

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

Rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt
[2 + Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sq
rt[s*((s + r*x)/((1 + Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3])*s + r*x)
], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x] && PosQ[a]

Rule 267

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

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 1843

Int[(Pq_)*(x_)^(m_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, Module[{Q = Polynomi
alQuotient[a*b^(Floor[(q - 1)/n] + 1)*x^m*Pq, a + b*x^n, x], R = PolynomialRemainder[a*b^(Floor[(q - 1)/n] + 1
)*x^m*Pq, a + b*x^n, x], i}, Dist[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), Int[x^m*(a + b*x^n)^(p + 1)*Expand
ToSum[(n*(p + 1)*Q)/x^m + Sum[((n*(p + 1) + i + 1)/a)*Coeff[R, x, i]*x^(i - m), {i, 0, n - 1}], x], x], x] + S
imp[(-x)*R*((a + b*x^n)^(p + 1)/(a^2*n*(p + 1)*b^(Floor[(q - 1)/n] + 1))), x]]] /; FreeQ[{a, b}, x] && PolyQ[P
q, x] && IGtQ[n, 0] && LtQ[p, -1] && ILtQ[m, 0]

Rule 1846

Int[(Pq_)/((x_)*Sqrt[(a_) + (b_.)*(x_)^(n_)]), x_Symbol] :> Dist[Coeff[Pq, x, 0], Int[1/(x*Sqrt[a + b*x^n]), x
], x] + Int[ExpandToSum[(Pq - Coeff[Pq, x, 0])/x, x]/Sqrt[a + b*x^n], x] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] &
& IGtQ[n, 0] && NeQ[Coeff[Pq, x, 0], 0]

Rule 1891

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Simplify[(1 - Sqrt[3])*(d/c)]]
, s = Denom[Simplify[(1 - Sqrt[3])*(d/c)]]}, Simp[2*d*s^3*(Sqrt[a + b*x^3]/(a*r^2*((1 + Sqrt[3])*s + r*x))), x
] - Simp[3^(1/4)*Sqrt[2 - Sqrt[3]]*d*s*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]/(r^2
*Sqrt[a + b*x^3]*Sqrt[s*((s + r*x)/((1 + Sqrt[3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 - Sqrt[3])*s + r*x)/((1
+ Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && EqQ[b*c^3 - 2*(5 - 3*Sqrt[3
])*a*d^3, 0]

Rule 1892

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a,
3]]}, Dist[(c*r - (1 - Sqrt[3])*d*s)/r, Int[1/Sqrt[a + b*x^3], x], x] + Dist[d/r, Int[((1 - Sqrt[3])*s + r*x)
/Sqrt[a + b*x^3], x], x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && NeQ[b*c^3 - 2*(5 - 3*Sqrt[3])*a*d^3, 0]

Rule 1900

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[Coeff[Pq, x, n - 1], Int[x^(n - 1)*(a + b*x^n)^p, x
], x] + Int[ExpandToSum[Pq - Coeff[Pq, x, n - 1]*x^(n - 1), x]*(a + b*x^n)^p, x] /; FreeQ[{a, b, p}, x] && Pol
yQ[Pq, x] && IGtQ[n, 0] && Expon[Pq, x] == n - 1

Rubi steps \begin{align*} \text {integral}& = \frac {2 x \left (a d+a e x-b c x^2\right )}{3 a^2 \sqrt {a+b x^3}}-\frac {2 \int \frac {-\frac {3 b c}{2}-\frac {b d x}{2}+\frac {1}{2} b e x^2-\frac {3 b^2 c x^3}{2 a}}{x \sqrt {a+b x^3}} \, dx}{3 a b} \\ & = \frac {2 x \left (a d+a e x-b c x^2\right )}{3 a^2 \sqrt {a+b x^3}}-\frac {2 \int \frac {-\frac {b d}{2}+\frac {b e x}{2}-\frac {3 b^2 c x^2}{2 a}}{\sqrt {a+b x^3}} \, dx}{3 a b}+\frac {c \int \frac {1}{x \sqrt {a+b x^3}} \, dx}{a} \\ & = \frac {2 x \left (a d+a e x-b c x^2\right )}{3 a^2 \sqrt {a+b x^3}}-\frac {2 \int \frac {-\frac {b d}{2}+\frac {b e x}{2}}{\sqrt {a+b x^3}} \, dx}{3 a b}+\frac {c \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^3\right )}{3 a}+\frac {(b c) \int \frac {x^2}{\sqrt {a+b x^3}} \, dx}{a^2} \\ & = \frac {2 x \left (a d+a e x-b c x^2\right )}{3 a^2 \sqrt {a+b x^3}}+\frac {2 c \sqrt {a+b x^3}}{3 a^2}+\frac {(2 c) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x^3}\right )}{3 a b}-\frac {e \int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt {a+b x^3}} \, dx}{3 a \sqrt [3]{b}}+\frac {\left (d+\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \int \frac {1}{\sqrt {a+b x^3}} \, dx}{3 a} \\ & = \frac {2 x \left (a d+a e x-b c x^2\right )}{3 a^2 \sqrt {a+b x^3}}+\frac {2 c \sqrt {a+b x^3}}{3 a^2}-\frac {2 e \sqrt {a+b x^3}}{3 a b^{2/3} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {2 c \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{3 a^{3/2}}+\frac {\sqrt {2-\sqrt {3}} e \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{3^{3/4} a^{2/3} b^{2/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}+\frac {2 \sqrt {2+\sqrt {3}} \left (d+\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} e}{\sqrt [3]{b}}\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} a \sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.10 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.21 \[ \int \frac {c+d x+e x^2}{x \left (a+b x^3\right )^{3/2}} \, dx=\frac {4 c \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},1+\frac {b x^3}{a}\right )+x \left (4 d+2 d \sqrt {1+\frac {b x^3}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},-\frac {b x^3}{a}\right )+3 e x \sqrt {1+\frac {b x^3}{a}} \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {3}{2},\frac {5}{3},-\frac {b x^3}{a}\right )\right )}{6 a \sqrt {a+b x^3}} \]

[In]

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

[Out]

(4*c*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (b*x^3)/a] + x*(4*d + 2*d*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[1/3,
1/2, 4/3, -((b*x^3)/a)] + 3*e*x*Sqrt[1 + (b*x^3)/a]*Hypergeometric2F1[2/3, 3/2, 5/3, -((b*x^3)/a)]))/(6*a*Sqrt
[a + b*x^3])

Maple [A] (verified)

Time = 1.54 (sec) , antiderivative size = 794, normalized size of antiderivative = 1.37

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(794\)
default	\(\text {Expression too large to display}\)	\(810\)

[In]

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

[Out]

-2*b*(-1/3/a/b*x^2*e-1/3/a/b*d*x-1/3/b/a*c)/((x^3+a/b)*b)^(1/2)-2/9*I*d/a*3^(1/2)/b*(-a*b^2)^(1/3)*(I*(x+1/2/b
*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2)*((x-1/b*(-a*b^2)^(1/3))/(-3/2/
b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2)*(-I*(x+1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(
1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2)/(b*x^3+a)^(1/2)*EllipticF(1/3*3^(1/2)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3
^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2),(I*3^(1/2)/b*(-a*b^2)^(1/3)/(-3/2/b*(-a*b^2)^(1/3)+1/
2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2))+2/9*I/a*e*3^(1/2)/b*(-a*b^2)^(1/3)*(I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1
/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2)*((x-1/b*(-a*b^2)^(1/3))/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(
1/2)/b*(-a*b^2)^(1/3)))^(1/2)*(-I*(x+1/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(
1/3))^(1/2)/(b*x^3+a)^(1/2)*((-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*EllipticE(1/3*3^(1/2)*(I*(
x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2),(I*3^(1/2)/b*(-a*b^2)^(
1/3)/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2))+1/b*(-a*b^2)^(1/3)*EllipticF(1/3*3^(1/2)*(
I*(x+1/2/b*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/b*(-a*b^2)^(1/3))*3^(1/2)*b/(-a*b^2)^(1/3))^(1/2),(I*3^(1/2)/b*(-a*b^2
)^(1/3)/(-3/2/b*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/b*(-a*b^2)^(1/3)))^(1/2)))-2/3*c*arctanh((b*x^3+a)^(1/2)/a^(1/2))
/a^(3/2)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.12 (sec) , antiderivative size = 345, normalized size of antiderivative = 0.60 \[ \int \frac {c+d x+e x^2}{x \left (a+b x^3\right )^{3/2}} \, dx=\left [\frac {{\left (b^{2} c x^{3} + a b c\right )} \sqrt {a} \log \left (\frac {b^{2} x^{6} + 8 \, a b x^{3} - 4 \, {\left (b x^{3} + 2 \, a\right )} \sqrt {b x^{3} + a} \sqrt {a} + 8 \, a^{2}}{x^{6}}\right ) + 4 \, {\left (a b d x^{3} + a^{2} d\right )} \sqrt {b} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + 4 \, {\left (a b e x^{3} + a^{2} e\right )} \sqrt {b} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) + 4 \, {\left (a b e x^{2} + a b d x + a b c\right )} \sqrt {b x^{3} + a}}{6 \, {\left (a^{2} b^{2} x^{3} + a^{3} b\right )}}, \frac {{\left (b^{2} c x^{3} + a b c\right )} \sqrt {-a} \arctan \left (\frac {{\left (b x^{3} + 2 \, a\right )} \sqrt {b x^{3} + a} \sqrt {-a}}{2 \, {\left (a b x^{3} + a^{2}\right )}}\right ) + 2 \, {\left (a b d x^{3} + a^{2} d\right )} \sqrt {b} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) + 2 \, {\left (a b e x^{3} + a^{2} e\right )} \sqrt {b} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) + 2 \, {\left (a b e x^{2} + a b d x + a b c\right )} \sqrt {b x^{3} + a}}{3 \, {\left (a^{2} b^{2} x^{3} + a^{3} b\right )}}\right ] \]

[In]

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

[Out]

[1/6*((b^2*c*x^3 + a*b*c)*sqrt(a)*log((b^2*x^6 + 8*a*b*x^3 - 4*(b*x^3 + 2*a)*sqrt(b*x^3 + a)*sqrt(a) + 8*a^2)/
x^6) + 4*(a*b*d*x^3 + a^2*d)*sqrt(b)*weierstrassPInverse(0, -4*a/b, x) + 4*(a*b*e*x^3 + a^2*e)*sqrt(b)*weierst
rassZeta(0, -4*a/b, weierstrassPInverse(0, -4*a/b, x)) + 4*(a*b*e*x^2 + a*b*d*x + a*b*c)*sqrt(b*x^3 + a))/(a^2
*b^2*x^3 + a^3*b), 1/3*((b^2*c*x^3 + a*b*c)*sqrt(-a)*arctan(1/2*(b*x^3 + 2*a)*sqrt(b*x^3 + a)*sqrt(-a)/(a*b*x^
3 + a^2)) + 2*(a*b*d*x^3 + a^2*d)*sqrt(b)*weierstrassPInverse(0, -4*a/b, x) + 2*(a*b*e*x^3 + a^2*e)*sqrt(b)*we
ierstrassZeta(0, -4*a/b, weierstrassPInverse(0, -4*a/b, x)) + 2*(a*b*e*x^2 + a*b*d*x + a*b*c)*sqrt(b*x^3 + a))
/(a^2*b^2*x^3 + a^3*b)]

Sympy [A] (verification not implemented)

Time = 5.70 (sec) , antiderivative size = 265, normalized size of antiderivative = 0.46 \[ \int \frac {c+d x+e x^2}{x \left (a+b x^3\right )^{3/2}} \, dx=c \left (\frac {2 a^{3} \sqrt {1 + \frac {b x^{3}}{a}}}{3 a^{\frac {9}{2}} + 3 a^{\frac {7}{2}} b x^{3}} + \frac {a^{3} \log {\left (\frac {b x^{3}}{a} \right )}}{3 a^{\frac {9}{2}} + 3 a^{\frac {7}{2}} b x^{3}} - \frac {2 a^{3} \log {\left (\sqrt {1 + \frac {b x^{3}}{a}} + 1 \right )}}{3 a^{\frac {9}{2}} + 3 a^{\frac {7}{2}} b x^{3}} + \frac {a^{2} b x^{3} \log {\left (\frac {b x^{3}}{a} \right )}}{3 a^{\frac {9}{2}} + 3 a^{\frac {7}{2}} b x^{3}} - \frac {2 a^{2} b x^{3} \log {\left (\sqrt {1 + \frac {b x^{3}}{a}} + 1 \right )}}{3 a^{\frac {9}{2}} + 3 a^{\frac {7}{2}} b x^{3}}\right ) + \frac {d x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {3}{2} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {3}{2}} \Gamma \left (\frac {4}{3}\right )} + \frac {e x^{2} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {2}{3}, \frac {3}{2} \\ \frac {5}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {3}{2}} \Gamma \left (\frac {5}{3}\right )} \]

[In]

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

[Out]

c*(2*a**3*sqrt(1 + b*x**3/a)/(3*a**(9/2) + 3*a**(7/2)*b*x**3) + a**3*log(b*x**3/a)/(3*a**(9/2) + 3*a**(7/2)*b*
x**3) - 2*a**3*log(sqrt(1 + b*x**3/a) + 1)/(3*a**(9/2) + 3*a**(7/2)*b*x**3) + a**2*b*x**3*log(b*x**3/a)/(3*a**
(9/2) + 3*a**(7/2)*b*x**3) - 2*a**2*b*x**3*log(sqrt(1 + b*x**3/a) + 1)/(3*a**(9/2) + 3*a**(7/2)*b*x**3)) + d*x
*gamma(1/3)*hyper((1/3, 3/2), (4/3,), b*x**3*exp_polar(I*pi)/a)/(3*a**(3/2)*gamma(4/3)) + e*x**2*gamma(2/3)*hy
per((2/3, 3/2), (5/3,), b*x**3*exp_polar(I*pi)/a)/(3*a**(3/2)*gamma(5/3))

Maxima [F]

\[ \int \frac {c+d x+e x^2}{x \left (a+b x^3\right )^{3/2}} \, dx=\int { \frac {e x^{2} + d x + c}{{\left (b x^{3} + a\right )}^{\frac {3}{2}} x} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {c+d x+e x^2}{x \left (a+b x^3\right )^{3/2}} \, dx=\int { \frac {e x^{2} + d x + c}{{\left (b x^{3} + a\right )}^{\frac {3}{2}} x} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {c+d x+e x^2}{x \left (a+b x^3\right )^{3/2}} \, dx=\int \frac {e\,x^2+d\,x+c}{x\,{\left (b\,x^3+a\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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