3.5.44 \(\int \frac {c+d x+e x^2}{x^2 (a+b x^3)^{3/2}} \, dx\) [444]

Optimal. Leaf size=607 \[ \frac {2 x \left (a e-b c x-b d x^2\right )}{3 a^2 \sqrt {a+b x^3}}+\frac {2 d \sqrt {a+b x^3}}{3 a^2}-\frac {c \sqrt {a+b x^3}}{a^2 x}+\frac {5 \sqrt [3]{b} c \sqrt {a+b x^3}}{3 a^2 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {2 d \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{3 a^{3/2}}-\frac {5 \sqrt {2-\sqrt {3}} \sqrt [3]{b} c \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{2\ 3^{3/4} a^{5/3} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}}-\frac {\sqrt {2+\sqrt {3}} \left (5 \left (1-\sqrt {3}\right ) b^{2/3} c-2 a^{2/3} e\right ) \left (\sqrt [3]{a}+\sqrt [3]{b} x\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}\right )|-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} a^{5/3} \sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )^2}} \sqrt {a+b x^3}} \]

[Out]

-2/3*d*arctanh((b*x^3+a)^(1/2)/a^(1/2))/a^(3/2)+2/3*x*(-b*d*x^2-b*c*x+a*e)/a^2/(b*x^3+a)^(1/2)+2/3*d*(b*x^3+a)
^(1/2)/a^2-c*(b*x^3+a)^(1/2)/a^2/x+5/3*b^(1/3)*c*(b*x^3+a)^(1/2)/a^2/(b^(1/3)*x+a^(1/3)*(1+3^(1/2)))-5/6*b^(1/
3)*c*(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^(5/3)/(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)-1/9*(a
^(1/3)+b^(1/3)*x)*EllipticF((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)*(-2
*a^(2/3)*e+5*b^(2/3)*c*(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^(5/3)/b^(1/3)/(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)

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Rubi [A]
time = 0.37, antiderivative size = 607, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {1843, 1849, 1846, 272, 65, 214, 1900, 267, 1892, 224, 1891} \begin {gather*} -\frac {\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 (5 \left (1-\sqrt {3}\right ) b^{2/3} c-2 a^{2/3} e\right ) F\left (\text {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^{5/3} \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}}-\frac {5 \sqrt {2-\sqrt {3}} \sqrt [3]{b} c \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 (\text {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 )}{2\ 3^{3/4} a^{5/3} \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 d \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{3 a^{3/2}}+\frac {2 x \left (a e-b c x-b d x^2\right )}{3 a^2 \sqrt {a+b x^3}}-\frac {c \sqrt {a+b x^3}}{a^2 x}+\frac {5 \sqrt [3]{b} c \sqrt {a+b x^3}}{3 a^2 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {2 d \sqrt {a+b x^3}}{3 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x*(a*e - b*c*x - b*d*x^2))/(3*a^2*Sqrt[a + b*x^3]) + (2*d*Sqrt[a + b*x^3])/(3*a^2) - (c*Sqrt[a + b*x^3])/(a
^2*x) + (5*b^(1/3)*c*Sqrt[a + b*x^3])/(3*a^2*((1 + Sqrt[3])*a^(1/3) + b^(1/3)*x)) - (2*d*ArcTanh[Sqrt[a + b*x^
3]/Sqrt[a]])/(3*a^(3/2)) - (5*Sqrt[2 - Sqrt[3]]*b^(1/3)*c*(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]])/(2*3^(3/4)*a^(5/3)*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]) - (Sqrt[2 + Sqrt[3]]*(5*(1 - Sqrt[3])*b^(2/3)*c - 2*
a^(2/3)*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*S
qrt[3]])/(3*3^(1/4)*a^(5/3)*b^(1/3)*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 1849

Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{Pq0 = Coeff[Pq, x, 0]}, Simp[Pq0
*(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c*(m + 1))), x] + Dist[1/(2*a*c*(m + 1)), Int[(c*x)^(m + 1)*ExpandToSum
[2*a*(m + 1)*((Pq - Pq0)/x) - 2*b*Pq0*(m + n*(p + 1) + 1)*x^(n - 1), x]*(a + b*x^n)^p, x], x] /; NeQ[Pq0, 0]]
/; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[m, -1] && LeQ[n - 1, Expon[Pq, x]]

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*} \int \frac {c+d x+e x^2}{x^2 \left (a+b x^3\right )^{3/2}} \, dx &=\frac {2 x \left (a e-b c x-b d x^2\right )}{3 a^2 \sqrt {a+b x^3}}-\frac {2 \int \frac {-\frac {3 b c}{2}-\frac {3 b d x}{2}-\frac {1}{2} b e x^2-\frac {b^2 c x^3}{2 a}-\frac {3 b^2 d x^4}{2 a}}{x^2 \sqrt {a+b x^3}} \, dx}{3 a b}\\ &=\frac {2 x \left (a e-b c x-b d x^2\right )}{3 a^2 \sqrt {a+b x^3}}-\frac {c \sqrt {a+b x^3}}{a^2 x}+\frac {\int \frac {3 a b d+a b e x+\frac {5}{2} b^2 c x^2+3 b^2 d x^3}{x \sqrt {a+b x^3}} \, dx}{3 a^2 b}\\ &=\frac {2 x \left (a e-b c x-b d x^2\right )}{3 a^2 \sqrt {a+b x^3}}-\frac {c \sqrt {a+b x^3}}{a^2 x}+\frac {\int \frac {a b e+\frac {5}{2} b^2 c x+3 b^2 d x^2}{\sqrt {a+b x^3}} \, dx}{3 a^2 b}+\frac {d \int \frac {1}{x \sqrt {a+b x^3}} \, dx}{a}\\ &=\frac {2 x \left (a e-b c x-b d x^2\right )}{3 a^2 \sqrt {a+b x^3}}-\frac {c \sqrt {a+b x^3}}{a^2 x}+\frac {\int \frac {a b e+\frac {5}{2} b^2 c x}{\sqrt {a+b x^3}} \, dx}{3 a^2 b}+\frac {d \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,x^3\right )}{3 a}+\frac {(b d) \int \frac {x^2}{\sqrt {a+b x^3}} \, dx}{a^2}\\ &=\frac {2 x \left (a e-b c x-b d x^2\right )}{3 a^2 \sqrt {a+b x^3}}+\frac {2 d \sqrt {a+b x^3}}{3 a^2}-\frac {c \sqrt {a+b x^3}}{a^2 x}+\frac {\left (5 b^{2/3} c\right ) \int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x}{\sqrt {a+b x^3}} \, dx}{6 a^2}+\frac {(2 d) \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 {\left (5 \left (1-\sqrt {3}\right ) b^{2/3} c-2 a^{2/3} e\right ) \int \frac {1}{\sqrt {a+b x^3}} \, dx}{6 a^{5/3}}\\ &=\frac {2 x \left (a e-b c x-b d x^2\right )}{3 a^2 \sqrt {a+b x^3}}+\frac {2 d \sqrt {a+b x^3}}{3 a^2}-\frac {c \sqrt {a+b x^3}}{a^2 x}+\frac {5 \sqrt [3]{b} c \sqrt {a+b x^3}}{3 a^2 \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} x\right )}-\frac {2 d \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{3 a^{3/2}}-\frac {5 \sqrt {2-\sqrt {3}} \sqrt [3]{b} c \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 )}{2\ 3^{3/4} a^{5/3} \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}} \left (5 \left (1-\sqrt {3}\right ) b^{2/3} c-2 a^{2/3} 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}} 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^{5/3} \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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 11.61, size = 542, normalized size = 0.89 \begin {gather*} \frac {-3 a c-5 b c x^3+2 a x (d+e x)}{3 a^2 x \sqrt {a+b x^3}}-\frac {4 \sqrt {a} d \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )+\frac {4 a e \left (\sqrt [3]{-1} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {\sqrt [3]{a}+\sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt {\frac {\sqrt [3]{-1} \sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} F\left (\sin ^{-1}\left (\sqrt {\frac {\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}}\right )|\sqrt [3]{-1}\right )}{\sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt {a+b x^3}}+\frac {10 \sqrt {2} \sqrt [3]{a} \sqrt [3]{b} c \left (\sqrt [3]{-1} \sqrt [3]{a}-\sqrt [3]{b} x\right ) \sqrt {\frac {\sqrt [3]{-1} \sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt {\frac {i \left (1+\frac {\sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 i+\sqrt {3}}} \left (\left (-1+(-1)^{2/3}\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {\sqrt [6]{-1}-\frac {i \sqrt [3]{b} x}{\sqrt [3]{a}}}}{\sqrt [4]{3}}\right )|\frac {\sqrt [3]{-1}}{-1+\sqrt [3]{-1}}\right )+F\left (\sin ^{-1}\left (\frac {\sqrt {\sqrt [6]{-1}-\frac {i \sqrt [3]{b} x}{\sqrt [3]{a}}}}{\sqrt [4]{3}}\right )|\frac {\sqrt [3]{-1}}{-1+\sqrt [3]{-1}}\right )\right )}{\sqrt {\frac {\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x}{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a}}} \sqrt {a+b x^3}}}{6 a^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-3*a*c - 5*b*c*x^3 + 2*a*x*(d + e*x))/(3*a^2*x*Sqrt[a + b*x^3]) - (4*Sqrt[a]*d*ArcTanh[Sqrt[a + b*x^3]/Sqrt[a
]] + (4*a*e*((-1)^(1/3)*a^(1/3) - b^(1/3)*x)*Sqrt[(a^(1/3) + b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[((-1)
^(1/3)*a^(1/3) - (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*EllipticF[ArcSin[Sqrt[(a^(1/3) + (-1)^(2/3)
*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]], (-1)^(1/3)])/(b^(1/3)*Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-
1)^(1/3))*a^(1/3))]*Sqrt[a + b*x^3]) + (10*Sqrt[2]*a^(1/3)*b^(1/3)*c*((-1)^(1/3)*a^(1/3) - b^(1/3)*x)*Sqrt[((-
1)^(1/3)*a^(1/3) - (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[(I*(1 + (b^(1/3)*x)/a^(1/3)))/(3*I +
Sqrt[3])]*((-1 + (-1)^(2/3))*EllipticE[ArcSin[Sqrt[(-1)^(1/6) - (I*b^(1/3)*x)/a^(1/3)]/3^(1/4)], (-1)^(1/3)/(
-1 + (-1)^(1/3))] + EllipticF[ArcSin[Sqrt[(-1)^(1/6) - (I*b^(1/3)*x)/a^(1/3)]/3^(1/4)], (-1)^(1/3)/(-1 + (-1)^
(1/3))]))/(Sqrt[(a^(1/3) + (-1)^(2/3)*b^(1/3)*x)/((1 + (-1)^(1/3))*a^(1/3))]*Sqrt[a + b*x^3]))/(6*a^2)

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Maple [A]
time = 0.40, size = 825, normalized size = 1.36 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e*(2/3/a*x/((x^3+a/b)*b)^(1/2)-2/9*I/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
)))+d*(2/3/a/((x^3+a/b)*b)^(1/2)-2/3*arctanh((b*x^3+a)^(1/2)/a^(1/2))/a^(3/2))+c*(-2/3*b*x^2/a^2/((x^3+a/b)*b)
^(1/2)-(b*x^3+a)^(1/2)/a^2/x-5/9*I/a^2*3^(1/2)*(-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))))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.13, size = 377, normalized size = 0.62 \begin {gather*} \left [\frac {{\left (b^{2} d x^{4} + a b d x\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 e x^{4} + a^{2} e x\right )} \sqrt {b} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) - 10 \, {\left (b^{2} c x^{4} + a b c x\right )} \sqrt {b} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) - 2 \, {\left (5 \, b^{2} c x^{3} - 2 \, a b e x^{2} - 2 \, a b d x + 3 \, a b c\right )} \sqrt {b x^{3} + a}}{6 \, {\left (a^{2} b^{2} x^{4} + a^{3} b x\right )}}, \frac {{\left (b^{2} d x^{4} + a b d x\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 e x^{4} + a^{2} e x\right )} \sqrt {b} {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right ) - 5 \, {\left (b^{2} c x^{4} + a b c x\right )} \sqrt {b} {\rm weierstrassZeta}\left (0, -\frac {4 \, a}{b}, {\rm weierstrassPInverse}\left (0, -\frac {4 \, a}{b}, x\right )\right ) - {\left (5 \, b^{2} c x^{3} - 2 \, a b e x^{2} - 2 \, a b d x + 3 \, a b c\right )} \sqrt {b x^{3} + a}}{3 \, {\left (a^{2} b^{2} x^{4} + a^{3} b x\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/6*((b^2*d*x^4 + a*b*d*x)*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*e*x^4 + a^2*e*x)*sqrt(b)*weierstrassPInverse(0, -4*a/b, x) - 10*(b^2*c*x^4 + a*b*c*x)*sqrt(b)*
weierstrassZeta(0, -4*a/b, weierstrassPInverse(0, -4*a/b, x)) - 2*(5*b^2*c*x^3 - 2*a*b*e*x^2 - 2*a*b*d*x + 3*a
*b*c)*sqrt(b*x^3 + a))/(a^2*b^2*x^4 + a^3*b*x), 1/3*((b^2*d*x^4 + a*b*d*x)*sqrt(-a)*arctan(1/2*(b*x^3 + 2*a)*s
qrt(b*x^3 + a)*sqrt(-a)/(a*b*x^3 + a^2)) + 2*(a*b*e*x^4 + a^2*e*x)*sqrt(b)*weierstrassPInverse(0, -4*a/b, x) -
5*(b^2*c*x^4 + a*b*c*x)*sqrt(b)*weierstrassZeta(0, -4*a/b, weierstrassPInverse(0, -4*a/b, x)) - (5*b^2*c*x^3
- 2*a*b*e*x^2 - 2*a*b*d*x + 3*a*b*c)*sqrt(b*x^3 + a))/(a^2*b^2*x^4 + a^3*b*x)]

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Sympy [A]
time = 6.30, size = 267, normalized size = 0.44 \begin {gather*} d \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 {c \Gamma \left (- \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {3}{2} \\ \frac {2}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {3}{2}} x \Gamma \left (\frac {2}{3}\right )} + \frac {e 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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d*(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)) + c*g
amma(-1/3)*hyper((-1/3, 3/2), (2/3,), b*x**3*exp_polar(I*pi)/a)/(3*a**(3/2)*x*gamma(2/3)) + e*x*gamma(1/3)*hyp
er((1/3, 3/2), (4/3,), b*x**3*exp_polar(I*pi)/a)/(3*a**(3/2)*gamma(4/3))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 5.80, size = 136, normalized size = 0.22 \begin {gather*} \frac {2\,d}{3\,a\,\sqrt {b\,x^3+a}}+\frac {d\,\ln \left (\frac {{\left (\sqrt {b\,x^3+a}-\sqrt {a}\right )}^3\,\left (\sqrt {b\,x^3+a}+\sqrt {a}\right )}{x^6}\right )}{3\,a^{3/2}}-\frac {2\,c\,{\left (\frac {a}{b\,x^3}+1\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (\frac {3}{2},\frac {11}{6};\ \frac {17}{6};\ -\frac {a}{b\,x^3}\right )}{11\,x\,{\left (b\,x^3+a\right )}^{3/2}}+\frac {e\,x\,{\left (\frac {b\,x^3}{a}+1\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{3},\frac {3}{2};\ \frac {4}{3};\ -\frac {b\,x^3}{a}\right )}{{\left (b\,x^3+a\right )}^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*d)/(3*a*(a + b*x^3)^(1/2)) + (d*log((((a + b*x^3)^(1/2) - a^(1/2))^3*((a + b*x^3)^(1/2) + a^(1/2)))/x^6))/(
3*a^(3/2)) - (2*c*(a/(b*x^3) + 1)^(3/2)*hypergeom([3/2, 11/6], 17/6, -a/(b*x^3)))/(11*x*(a + b*x^3)^(3/2)) + (
e*x*((b*x^3)/a + 1)^(3/2)*hypergeom([1/3, 3/2], 4/3, -(b*x^3)/a))/(a + b*x^3)^(3/2)

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